[{"data":1,"prerenderedAt":8191},["ShallowReactive",2],{"post-\u002Fblog\u002F2023\u002F2023-12-08-fast-inverse-square-root":3},{"id":4,"title":5,"body":6,"description":452,"draft":8178,"enableComment":565,"extension":8179,"image":8180,"meta":8181,"navigation":565,"onday":8183,"path":8184,"seo":8185,"stem":8186,"summary":8187,"tags":8188,"__hash__":8190},"blog\u002Fblog\u002F2023\u002F2023-12-08-Fast-Inverse-Square-Root.md","Fast Inverse Square Root",{"type":7,"value":8,"toc":8171},"minimark",[9,14,21,412,419,422,435,442,446,742,745,763,767,1339,2038,2070,2808,3164,3167,4264,4406,5087,5430,5814,5817,6525,6695,6722,6754,6758,6761,7158,7165,7799,8099,8102,8133,8137,8149,8158,8167],[10,11,13],"h2",{"id":12},"prologue","Prologue",[15,16,17,20],"p",{},[18,19,5],"em",{}," is a rather famous algorithm, and the specific author is no longer traceable. Its prominence comes from its extensive use in the game Quake III, where it became well-known.",[15,22,23,24,28,29,411],{},"Why do we need the inverse square root instead of the square root itself? In computer graphics, many calculations involve the inverse square root rather than the square root. For example, the ",[25,26,27],"code",{},"normalize()"," operation for a vector ",[30,31,35,115],"span",{"className":32,"translate":34},[33],"katex","no",[30,36,39],{"className":37},[38],"katex-mathml",[40,41,43],"math",{"xmlns":42},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[44,45,46,110],"semantics",{},[47,48,49,91,95,97,101,103,105,107],"mrow",{},[50,51,52,56],"mfrac",{},[53,54,55],"mn",{},"1",[57,58,59],"msqrt",{},[47,60,61,71,75,82,84],{},[62,63,64,68],"msup",{},[65,66,67],"mi",{},"x",[53,69,70],{},"2",[72,73,74],"mo",{},"+",[62,76,77,80],{},[65,78,79],{},"y",[53,81,70],{},[72,83,74],{},[62,85,86,89],{},[65,87,88],{},"z",[53,90,70],{},[72,92,94],{"stretchy":93},"false","(",[65,96,67],{},[72,98,100],{"separator":99},"true",",",[65,102,79],{},[72,104,100],{"separator":99},[65,106,88],{},[72,108,109],{"stretchy":93},")",[111,112,114],"annotation",{"encoding":113},"application\u002Fx-tex","\\frac{1}{\\sqrt{x^2 + y^2 + 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0 400000 1080","xMinYMin slice",[322,323],"path",{"d":324},"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 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includes the inverse square root. In computer graphics, each frame's rendering requires at least tens of thousands of calls to normalize. If we can accelerate the calculation of the inverse square root, the rendering speed will significantly improve.",[15,413,414,415,418],{},"The algorithm originated more than 20 years ago. Today, the algorithm has research value but not practical use since modern CPUs have faster and more accurate instructions. We can simply use ",[25,416,417],{},"1\u002Fsqrt(y)",".",[15,420,421],{},"However, this doesn't hinder us from understanding the elegance of this algorithm. In a brief overview, it involves:",[423,424,425,429,432],"ol",{},[426,427,428],"li",{},"IEEE 754 Standard",[426,430,431],{},"Undefined behavior in the C language",[426,433,434],{},"Newton's iteration method.",[15,436,437,438,441],{},"The focus here is on the second step—how to derive the magic number ",[25,439,440],{},"0x5f3759df",". For more detailed information, you can refer to the source.",[10,443,445],{"id":444},"algorithm","Algorithm",[447,448,453],"pre",{"className":449,"code":450,"language":451,"meta":452,"style":452},"language-c shiki shiki-themes github-dark","float Q_rsqrt(float number)\n{\n    long i;\n    float x2, y;\n    const float threehalfs = 1.5F;\n    x2 = number * 0.5F;\n    y = number;\n\n    \u002F* Core Code *\u002F\n    \u002F\u002F 1. reinterpret_cast from float to int\n    i = *(long *)&y; \u002F\u002F evil floating-point bit-level hacking\n    \u002F\u002F 2. Estimate the inverse square root\n    i = 0x5f3759df - (i >> 1); \u002F\u002F what the heck? (....why not 0x69696969?)\n    \u002F\u002F reinterpret_cast from int to float\n    y = *(float *)&i;\n    \u002F\u002F 3. Newton's method\n    y = y * (threehalfs - (x2 * y * y)); \u002F\u002F 1st iteration\n    \u002F\u002F y = y * (threehalfs - (x2 * y * y)); \u002F\u002F 2nd iteration, this can be removed\n\n    return y;\n}\n","c","",[25,454,455,479,485,494,503,528,549,560,567,574,580,609,615,646,652,674,680,713,722,727,736],{"__ignoreMap":452},[30,456,459,463,467,470,472,476],{"class":457,"line":458},"line",1,[30,460,462],{"class":461},"snl16","float",[30,464,466],{"class":465},"svObZ"," Q_rsqrt",[30,468,94],{"class":469},"s95oV",[30,471,462],{"class":461},[30,473,475],{"class":474},"s9osk"," number",[30,477,478],{"class":469},")\n",[30,480,482],{"class":457,"line":481},2,[30,483,484],{"class":469},"{\n",[30,486,488,491],{"class":457,"line":487},3,[30,489,490],{"class":461},"    long",[30,492,493],{"class":469}," i;\n",[30,495,497,500],{"class":457,"line":496},4,[30,498,499],{"class":461},"    float",[30,501,502],{"class":469}," x2, y;\n",[30,504,506,509,512,515,518,522,525],{"class":457,"line":505},5,[30,507,508],{"class":461},"    const",[30,510,511],{"class":461}," float",[30,513,514],{"class":469}," threehalfs ",[30,516,517],{"class":461},"=",[30,519,521],{"class":520},"sDLfK"," 1.5",[30,523,524],{"class":461},"F",[30,526,527],{"class":469},";\n",[30,529,531,534,536,539,542,545,547],{"class":457,"line":530},6,[30,532,533],{"class":469},"    x2 ",[30,535,517],{"class":461},[30,537,538],{"class":469}," number ",[30,540,541],{"class":461},"*",[30,543,544],{"class":520}," 0.5",[30,546,524],{"class":461},[30,548,527],{"class":469},[30,550,552,555,557],{"class":457,"line":551},7,[30,553,554],{"class":469},"    y ",[30,556,517],{"class":461},[30,558,559],{"class":469}," number;\n",[30,561,563],{"class":457,"line":562},8,[30,564,566],{"emptyLinePlaceholder":565},true,"\n",[30,568,570],{"class":457,"line":569},9,[30,571,573],{"class":572},"sAwPA","    \u002F* Core Code *\u002F\n",[30,575,577],{"class":457,"line":576},10,[30,578,579],{"class":572},"    \u002F\u002F 1. reinterpret_cast from float to int\n",[30,581,583,586,588,591,593,596,598,600,603,606],{"class":457,"line":582},11,[30,584,585],{"class":469},"    i ",[30,587,517],{"class":461},[30,589,590],{"class":461}," *",[30,592,94],{"class":469},[30,594,595],{"class":461},"long",[30,597,590],{"class":461},[30,599,109],{"class":469},[30,601,602],{"class":461},"&",[30,604,605],{"class":469},"y;",[30,607,608],{"class":572}," \u002F\u002F evil floating-point bit-level hacking\n",[30,610,612],{"class":457,"line":611},12,[30,613,614],{"class":572},"    \u002F\u002F 2. Estimate the inverse square root\n",[30,616,618,620,622,625,628,631,634,637,640,643],{"class":457,"line":617},13,[30,619,585],{"class":469},[30,621,517],{"class":461},[30,623,624],{"class":461}," 0x",[30,626,627],{"class":520},"5f3759df",[30,629,630],{"class":461}," -",[30,632,633],{"class":469}," (i ",[30,635,636],{"class":461},">>",[30,638,639],{"class":520}," 1",[30,641,642],{"class":469},");",[30,644,645],{"class":572}," \u002F\u002F what the heck? (....why not 0x69696969?)\n",[30,647,649],{"class":457,"line":648},14,[30,650,651],{"class":572},"    \u002F\u002F reinterpret_cast from int to float\n",[30,653,655,657,659,661,663,665,667,669,671],{"class":457,"line":654},15,[30,656,554],{"class":469},[30,658,517],{"class":461},[30,660,590],{"class":461},[30,662,94],{"class":469},[30,664,462],{"class":461},[30,666,590],{"class":461},[30,668,109],{"class":469},[30,670,602],{"class":461},[30,672,673],{"class":469},"i;\n",[30,675,677],{"class":457,"line":676},16,[30,678,679],{"class":572},"    \u002F\u002F 3. Newton's method\n",[30,681,683,685,687,690,692,695,698,701,703,705,707,710],{"class":457,"line":682},17,[30,684,554],{"class":469},[30,686,517],{"class":461},[30,688,689],{"class":469}," y ",[30,691,541],{"class":461},[30,693,694],{"class":469}," (threehalfs ",[30,696,697],{"class":461},"-",[30,699,700],{"class":469}," (x2 ",[30,702,541],{"class":461},[30,704,689],{"class":469},[30,706,541],{"class":461},[30,708,709],{"class":469}," y));",[30,711,712],{"class":572}," \u002F\u002F 1st iteration\n",[30,714,716,719],{"class":457,"line":715},18,[30,717,718],{"class":572},"    \u002F\u002F y = y * (threehalfs - (x2 * y * y));",[30,720,721],{"class":572}," \u002F\u002F 2nd iteration, this can be removed\n",[30,723,725],{"class":457,"line":724},19,[30,726,566],{"emptyLinePlaceholder":565},[30,728,730,733],{"class":457,"line":729},20,[30,731,732],{"class":461},"    return",[30,734,735],{"class":469}," y;\n",[30,737,739],{"class":457,"line":738},21,[30,740,741],{"class":469},"}\n",[15,743,744],{},"The algorithm roughly consists of three parts:",[746,747,748,754,760],"ul",{},[426,749,750,753],{},[25,751,752],{},"reinterpret_cast"," used for conversion between float and int because any compiler would otherwise prohibit bitwise operations on float types.",[426,755,756,757,759],{},"Estimation using the magic number ",[25,758,440],{}," for the inverse square root (explained later).",[426,761,762],{},"One iteration of Newton's method (explained later).",[10,764,766],{"id":765},"estimating-the-inverse-square-root","Estimating the Inverse Square Root",[15,768,769,770,801,802,832,833,418],{},"In the IEEE 754 standard, a 32-bit floating-point number contains 1 sign bit, 8 exponent bits ",[30,771,773,787],{"className":772,"translate":34},[33],[30,774,776],{"className":775},[38],[40,777,778],{"xmlns":42},[44,779,780,785],{},[47,781,782],{},[65,783,784],{},"E",[111,786,784],{"encoding":113},[30,788,790],{"className":789,"ariaHidden":99},[118],[30,791,793,797],{"className":792},[122],[30,794],{"className":795,"style":796},[126],"height:0.6833em;",[30,798,784],{"className":799,"style":800},[131,205],"margin-right:0.05764em;",", and 23 mantissa bits ",[30,803,805,819],{"className":804,"translate":34},[33],[30,806,808],{"className":807},[38],[40,809,810],{"xmlns":42},[44,811,812,817],{},[47,813,814],{},[65,815,816],{},"M",[111,818,816],{"encoding":113},[30,820,822],{"className":821,"ariaHidden":99},[118],[30,823,825,828],{"className":824},[122],[30,826],{"className":827,"style":796},[126],[30,829,816],{"className":830,"style":831},[131,205],"margin-right:0.10903em;",". Any floating-point number can be represented as: 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\\times 2^{E - 127} = (1 + \\frac{M}{2^{23}}) \\times 2^{E - 127}, M \\in[0, 2^{23} ), E \\in 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calculate ",[30,1343,1345,1358],{"className":1344,"translate":34},[33],[30,1346,1348],{"className":1347},[38],[40,1349,1350],{"xmlns":42},[44,1351,1352,1356],{},[47,1353,1354],{},[65,1355,79],{},[111,1357,79],{"encoding":113},[30,1359,1361],{"className":1360,"ariaHidden":99},[118],[30,1362,1364,1368],{"className":1363},[122],[30,1365],{"className":1366,"style":1367},[126],"height:0.625em;vertical-align:-0.1944em;",[30,1369,79],{"className":1370,"style":245},[131,205],"'s inverse square root ",[30,1373,1375,1411],{"className":1374,"translate":34},[33],[30,1376,1378],{"className":1377},[38],[40,1379,1380],{"xmlns":42},[44,1381,1382,1408],{},[47,1383,1384,1392,1394],{},[50,1385,1386,1388],{},[53,1387,55],{},[57,1389,1390],{},[65,1391,79],{},[72,1393,517],{},[62,1395,1396,1398],{},[65,1397,79],{},[47,1399,1400,1402],{},[72,1401,867],{},[50,1403,1404,1406],{},[53,1405,55],{},[53,1407,70],{},[111,1409,1410],{"encoding":113},"\\frac{1}{\\sqrt{y}} = y^{-\\frac{1}{2}}",[30,1412,1414,1550],{"className":1413,"ariaHidden":99},[118],[30,1415,1417,1421,1541,1544,1547],{"className":1416},[122],[30,1418],{"className":1419,"style":1420},[126],"height:1.422em;vertical-align:-0.5769em;",[30,1422,1424,1427,1538],{"className":1423},[131],[30,1425],{"className":1426},[135,136],[30,1428,1430],{"className":1429},[50],[30,1431,1433,1529],{"className":1432},[143,144],[30,1434,1436,1526],{"className":1435},[148],[30,1437,1439,1504,1512],{"className":1438,"style":153},[152],[30,1440,1441,1444],{"style":1088},[30,1442],{"className":1443,"style":161},[160],[30,1445,1447],{"className":1446},[165,166,167,168],[30,1448,1450],{"className":1449},[131,168],[30,1451,1453],{"className":1452},[131,175,168],[30,1454,1456,1495],{"className":1455},[143,144],[30,1457,1459,1492],{"className":1458},[148],[30,1460,1463,1477],{"className":1461,"style":1462},[152],"height:0.7087em;",[30,1464,1467,1470],{"className":1465,"style":1466},[189],"top:-3em;",[30,1468],{"className":1469,"style":161},[160],[30,1471,1474],{"className":1472,"style":1473},[131,168],"padding-left:0.833em;",[30,1475,79],{"className":1476,"style":245},[131,205,168],[30,1478,1480,1483],{"style":1479},"top:-2.6687em;",[30,1481],{"className":1482,"style":161},[160],[30,1484,1487],{"className":1485,"style":1486},[311,168],"min-width:0.853em;height:1.08em;",[314,1488,1490],{"xmlns":316,"width":317,"height":1489,"viewBox":319,"preserveAspectRatio":320},"1.08em",[322,1491],{"d":324},[30,1493,329],{"className":1494},[328],[30,1496,1498],{"className":1497},[148],[30,1499,1502],{"className":1500,"style":1501},[152],"height:0.3313em;",[30,1503],{},[30,1505,1506,1509],{"style":341},[30,1507],{"className":1508,"style":161},[160],[30,1510],{"className":1511,"style":349},[348],[30,1513,1514,1517],{"style":352},[30,1515],{"className":1516,"style":161},[160],[30,1518,1520],{"className":1519},[165,166,167,168],[30,1521,1523],{"className":1522},[131,168],[30,1524,55],{"className":1525},[131,168],[30,1527,329],{"className":1528},[328],[30,1530,1532],{"className":1531},[148],[30,1533,1536],{"className":1534,"style":1535},[152],"height:0.5769em;",[30,1537],{},[30,1539],{"className":1540},[380,136],[30,1542],{"className":1543,"style":1031},[394],[30,1545,517],{"className":1546},[1035],[30,1548],{"className":1549,"style":1031},[394],[30,1551,1553,1557],{"className":1552},[122],[30,1554],{"className":1555,"style":1556},[126],"height:1.1485em;vertical-align:-0.1944em;",[30,1558,1560,1563],{"className":1559},[131],[30,1561,79],{"className":1562,"style":245},[131,205],[30,1564,1566],{"className":1565},[209],[30,1567,1569],{"className":1568},[143],[30,1570,1572],{"className":1571},[148],[30,1573,1576],{"className":1574,"style":1575},[152],"height:0.954em;",[30,1577,1579,1582],{"style":1578},"top:-3.363em;margin-right:0.05em;",[30,1580],{"className":1581,"style":161},[160],[30,1583,1585],{"className":1584},[165,166,167,168],[30,1586,1588,1591],{"className":1587},[131,168],[30,1589,867],{"className":1590},[131,168],[30,1592,1594,1598,1663],{"className":1593},[131,168],[30,1595],{"className":1596},[135,136,165,230,1597],"size6",[30,1599,1601],{"className":1600},[50],[30,1602,1604,1654],{"className":1603},[143,144],[30,1605,1607,1651],{"className":1606},[148],[30,1608,1611,1626,1636],{"className":1609,"style":1610},[152],"height:0.8443em;",[30,1612,1614,1617],{"style":1613},"top:-2.656em;",[30,1615],{"className":1616,"style":161},[160],[30,1618,1620],{"className":1619},[165,230,231,168],[30,1621,1623],{"className":1622},[131,168],[30,1624,70],{"className":1625},[131,168],[30,1627,1629,1632],{"style":1628},"top:-3.2255em;",[30,1630],{"className":1631,"style":161},[160],[30,1633],{"className":1634,"style":1635},[348,168],"border-bottom-width:0.049em;",[30,1637,1639,1642],{"style":1638},"top:-3.384em;",[30,1640],{"className":1641,"style":161},[160],[30,1643,1645],{"className":1644},[165,230,231,168],[30,1646,1648],{"className":1647},[131,168],[30,1649,55],{"className":1650},[131,168],[30,1652,329],{"className":1653},[328],[30,1655,1657],{"className":1656},[148],[30,1658,1661],{"className":1659,"style":1660},[152],"height:0.344em;",[30,1662],{},[30,1664],{"className":1665},[380,136,165,230,1597],", the IEEE 754 form of ",[30,1668,1670,1683],{"className":1669,"translate":34},[33],[30,1671,1673],{"className":1672},[38],[40,1674,1675],{"xmlns":42},[44,1676,1677,1681],{},[47,1678,1679],{},[65,1680,79],{},[111,1682,79],{"encoding":113},[30,1684,1686],{"className":1685,"ariaHidden":99},[118],[30,1687,1689,1692],{"className":1688},[122],[30,1690],{"className":1691,"style":1367},[126],[30,1693,79],{"className":1694,"style":245},[131,205]," is ",[30,1697,1699,1756],{"className":1698,"translate":34},[33],[30,1700,1702],{"className":1701},[38],[40,1703,1704],{"xmlns":42},[44,1705,1706,1753],{},[47,1707,1708,1710,1712,1714,1716,1718,1733,1735,1737],{},[65,1709,79],{},[72,1711,517],{},[72,1713,94],{"stretchy":93},[53,1715,55],{},[72,1717,74],{},[50,1719,1720,1727],{},[1721,1722,1723,1725],"msub",{},[65,1724,816],{},[65,1726,79],{},[62,1728,1729,1731],{},[53,1730,70],{},[53,1732,889],{},[72,1734,109],{"stretchy":93},[72,1736,856],{},[62,1738,1739,1741],{},[53,1740,70],{},[47,1742,1743,1749,1751],{},[1721,1744,1745,1747],{},[65,1746,784],{},[65,1748,79],{},[72,1750,867],{},[53,1752,870],{},[111,1754,1755],{"encoding":113},"y = (1 + \\frac{M_y}{2^{23}}) \\times 2^{E_y - 127}",[30,1757,1759,1777,1798,1956],{"className":1758,"ariaHidden":99},[118],[30,1760,1762,1765,1768,1771,1774],{"className":1761},[122],[30,1763],{"className":1764,"style":1367},[126],[30,1766,79],{"className":1767,"style":245},[131,205],[30,1769],{"className":1770,"style":1031},[394],[30,1772,517],{"className":1773},[1035],[30,1775],{"className":1776,"style":1031},[394],[30,1778,1780,1783,1786,1789,1792,1795],{"className":1779},[122],[30,1781],{"className":1782,"style":958},[126],[30,1784,94],{"className":1785},[135],[30,1787,55],{"className":1788},[131],[30,1790],{"className":1791,"style":974},[394],[30,1793,74],{"className":1794},[238],[30,1796],{"className":1797,"style":974},[394],[30,1799,1801,1805,1944,1947,1950,1953],{"className":1800},[122],[30,1802],{"className":1803,"style":1804},[126],"height:1.3307em;vertical-align:-0.345em;",[30,1806,1808,1811,1941],{"className":1807},[131],[30,1809],{"className":1810},[135,136],[30,1812,1814],{"className":1813},[50],[30,1815,1817,1933],{"className":1816},[143,144],[30,1818,1820,1930],{"className":1819},[148],[30,1821,1824,1867,1875],{"className":1822,"style":1823},[152],"height:0.9857em;",[30,1825,1826,1829],{"style":1088},[30,1827],{"className":1828,"style":161},[160],[30,1830,1832],{"className":1831},[165,166,167,168],[30,1833,1835],{"className":1834},[131,168],[30,1836,1838,1841],{"className":1837},[131,168],[30,1839,70],{"className":1840},[131,168],[30,1842,1844],{"className":1843},[209],[30,1845,1847],{"className":1846},[143],[30,1848,1850],{"className":1849},[148],[30,1851,1853],{"className":1852,"style":219},[152],[30,1854,1855,1858],{"style":222},[30,1856],{"className":1857,"style":226},[160],[30,1859,1861],{"className":1860},[165,230,231,168],[30,1862,1864],{"className":1863},[131,168],[30,1865,889],{"className":1866},[131,168],[30,1868,1869,1872],{"style":341},[30,1870],{"className":1871,"style":161},[160],[30,1873],{"className":1874,"style":349},[348],[30,1876,1878,1881],{"style":1877},"top:-3.5073em;",[30,1879],{"className":1880,"style":161},[160],[30,1882,1884],{"className":1883},[165,166,167,168],[30,1885,1887],{"className":1886},[131,168],[30,1888,1890,1893],{"className":1889},[131,168],[30,1891,816],{"className":1892,"style":831},[131,205,168],[30,1894,1896],{"className":1895},[209],[30,1897,1899,1921],{"className":1898},[143,144],[30,1900,1902,1918],{"className":1901},[148],[30,1903,1906],{"className":1904,"style":1905},[152],"height:0.1645em;",[30,1907,1909,1912],{"style":1908},"top:-2.357em;margin-left:-0.109em;margin-right:0.0714em;",[30,1910],{"className":1911,"style":226},[160],[30,1913,1915],{"className":1914},[165,230,231,168],[30,1916,79],{"className":1917,"style":245},[131,205,168],[30,1919,329],{"className":1920},[328],[30,1922,1924],{"className":1923},[148],[30,1925,1928],{"className":1926,"style":1927},[152],"height:0.2819em;",[30,1929],{},[30,1931,329],{"className":1932},[328],[30,1934,1936],{"className":1935},[148],[30,1937,1939],{"className":1938,"style":1161},[152],[30,1940],{},[30,1942],{"className":1943},[380,136],[30,1945,109],{"className":1946},[380],[30,1948],{"className":1949,"style":974},[394],[30,1951,856],{"className":1952},[238],[30,1954],{"className":1955,"style":974},[394],[30,1957,1959,1962],{"className":1958},[122],[30,1960],{"className":1961,"style":987},[126],[30,1963,1965,1968],{"className":1964},[131],[30,1966,70],{"className":1967},[131],[30,1969,1971],{"className":1970},[209],[30,1972,1974],{"className":1973},[143],[30,1975,1977],{"className":1976},[148],[30,1978,1980],{"className":1979,"style":987},[152],[30,1981,1982,1985],{"style":1008},[30,1983],{"className":1984,"style":1012},[160],[30,1986,1988],{"className":1987},[165,166,167,168],[30,1989,1991,2032,2035],{"className":1990},[131,168],[30,1992,1994,1997],{"className":1993},[131,168],[30,1995,784],{"className":1996,"style":800},[131,205,168],[30,1998,2000],{"className":1999},[209],[30,2001,2003,2024],{"className":2002},[143,144],[30,2004,2006,2021],{"className":2005},[148],[30,2007,2009],{"className":2008,"style":1905},[152],[30,2010,2012,2015],{"style":2011},"top:-2.357em;margin-left:-0.0576em;margin-right:0.0714em;",[30,2013],{"className":2014,"style":226},[160],[30,2016,2018],{"className":2017},[165,230,231,168],[30,2019,79],{"className":2020,"style":245},[131,205,168],[30,2022,329],{"className":2023},[328],[30,2025,2027],{"className":2026},[148],[30,2028,2030],{"className":2029,"style":1927},[152],[30,2031],{},[30,2033,867],{"className":2034},[238,168],[30,2036,870],{"className":2037},[131,168],[15,2039,2040,2041,2069],{},"For simplicity, let's take the logarithm of ",[30,2042,2044,2057],{"className":2043,"translate":34},[33],[30,2045,2047],{"className":2046},[38],[40,2048,2049],{"xmlns":42},[44,2050,2051,2055],{},[47,2052,2053],{},[65,2054,79],{},[111,2056,79],{"encoding":113},[30,2058,2060],{"className":2059,"ariaHidden":99},[118],[30,2061,2063,2066],{"className":2062},[122],[30,2064],{"className":2065,"style":1367},[126],[30,2067,79],{"className":2068,"style":245},[131,205]," to the base 2:",[30,2071,2074],{"className":2072,"translate":34},[2073],"katex-display",[30,2075,2077,2243],{"className":2076,"translate":34},[33],[30,2078,2080],{"className":2079},[38],[40,2081,2083],{"xmlns":42,"display":2082},"block",[44,2084,2085,2240],{},[2086,2087,2091,2173],"mtable",{"rowspacing":2088,"columnalign":2089,"columnspacing":2090},"0.25em","right left","0em",[2092,2093,2094,2099,2114,2166,2169],"mtr",{},[2095,2096],"mtd",{"className":2097},[2098],"mtr-glue",[2095,2100,2101],{},[2102,2103,2104],"mstyle",{"scriptlevel":918,"displaystyle":99},[47,2105,2106,2109,2112],{},[65,2107,2108],{},"log",[72,2110,2111],{},"⁡",[65,2113,79],{},[2095,2115,2116],{},[2102,2117,2118],{"scriptlevel":918,"displaystyle":99},[47,2119,2120,2122,2124,2126,2128],{},[47,2121],{},[72,2123,517],{},[65,2125,2108],{},[72,2127,2111],{},[47,2129,2130,2132,2134,2136,2138,2144,2146,2148,2164],{},[72,2131,94],{"stretchy":93},[72,2133,94],{"stretchy":93},[53,2135,55],{},[72,2137,74],{},[1721,2139,2140,2142],{},[65,2141,816],{},[65,2143,79],{},[72,2145,109],{"stretchy":93},[72,2147,856],{},[62,2149,2150,2152],{},[53,2151,70],{},[47,2153,2154,2160,2162],{},[1721,2155,2156,2158],{},[65,2157,784],{},[65,2159,79],{},[72,2161,867],{},[53,2163,870],{},[72,2165,109],{"stretchy":93},[2095,2167],{"className":2168},[2098],[2095,2170],{"className":2171},[2172],"mml-eqn-num",[2092,2174,2175,2178,2184,2234,2237],{},[2095,2176],{"className":2177},[2098],[2095,2179,2180],{},[2102,2181,2182],{"scriptlevel":918,"displaystyle":99},[47,2183],{},[2095,2185,2186],{},[2102,2187,2188],{"scriptlevel":918,"displaystyle":99},[47,2189,2190,2192,2194,2196,2198,2222,2224,2230,2232],{},[47,2191],{},[72,2193,517],{},[65,2195,2108],{},[72,2197,2111],{},[47,2199,2200,2202,2204,2206,2220],{},[72,2201,94],{"stretchy":93},[53,2203,55],{},[72,2205,74],{},[50,2207,2208,2214],{},[1721,2209,2210,2212],{},[65,2211,816],{},[65,2213,79],{},[62,2215,2216,2218],{},[53,2217,70],{},[53,2219,889],{},[72,2221,109],{"stretchy":93},[72,2223,74],{},[1721,2225,2226,2228],{},[65,2227,784],{},[65,2229,79],{},[72,2231,867],{},[53,2233,870],{},[2095,2235],{"className":2236},[2098],[2095,2238],{"className":2239},[2172],[111,2241,2242],{"encoding":113},"\\begin{align}\n\\log y &= \\log{((1 + M_y) \\times 2^{E_y - 127})} \\\\\n&= \\log{(1+\\frac{M_y}{2^{23}})} + E_y - 127\n\\end{align}",[30,2244,2246,2767],{"className":2245,"ariaHidden":99},[118],[30,2247,2249,2253],{"className":2248},[122],[30,2250],{"className":2251,"style":2252},[126],"height:3.8977em;vertical-align:-1.6988em;",[30,2254,2256,2316],{"className":2255},[2086],[30,2257,2260],{"className":2258},[2259],"col-align-r",[30,2261,2263,2307],{"className":2262},[143,144],[30,2264,2266,2304],{"className":2265},[148],[30,2267,2270,2295],{"className":2268,"style":2269},[152],"height:2.1988em;",[30,2271,2273,2277],{"style":2272},"top:-4.6678em;",[30,2274],{"className":2275,"style":2276},[160],"height:3.3603em;",[30,2278,2280,2289,2292],{"className":2279},[131],[30,2281,2284,2285],{"className":2282},[2283],"mop","lo",[30,2286,2288],{"style":2287},"margin-right:0.01389em;","g",[30,2290],{"className":2291,"style":395},[394],[30,2293,79],{"className":2294,"style":245},[131,205],[30,2296,2298,2301],{"style":2297},"top:-2.6475em;",[30,2299],{"className":2300,"style":2276},[160],[30,2302],{"className":2303},[131],[30,2305,329],{"className":2306},[328],[30,2308,2310],{"className":2309},[148],[30,2311,2314],{"className":2312,"style":2313},[152],"height:1.6988em;",[30,2315],{},[30,2317,2320],{"className":2318},[2319],"col-align-l",[30,2321,2323,2759],{"className":2322},[143,144],[30,2324,2326,2756],{"className":2325},[148],[30,2327,2329,2511],{"className":2328,"style":2269},[152],[30,2330,2331,2334],{"style":2272},[30,2332],{"className":2333,"style":2276},[160],[30,2335,2337,2340,2343,2346,2349,2354,2357],{"className":2336},[131],[30,2338],{"className":2339},[131],[30,2341],{"className":2342,"style":1031},[394],[30,2344,517],{"className":2345},[1035],[30,2347],{"className":2348,"style":1031},[394],[30,2350,2284,2352],{"className":2351},[2283],[30,2353,2288],{"style":2287},[30,2355],{"className":2356,"style":395},[394],[30,2358,2360,2364,2367,2370,2373,2376,2419,2422,2425,2428,2431,2508],{"className":2359},[131],[30,2361,2363],{"className":2362},[135],"((",[30,2365,55],{"className":2366},[131],[30,2368],{"className":2369,"style":974},[394],[30,2371,74],{"className":2372},[238],[30,2374],{"className":2375,"style":974},[394],[30,2377,2379,2382],{"className":2378},[131],[30,2380,816],{"className":2381,"style":831},[131,205],[30,2383,2385],{"className":2384},[209],[30,2386,2388,2410],{"className":2387},[143,144],[30,2389,2391,2407],{"className":2390},[148],[30,2392,2395],{"className":2393,"style":2394},[152],"height:0.1514em;",[30,2396,2398,2401],{"style":2397},"top:-2.55em;margin-left:-0.109em;margin-right:0.05em;",[30,2399],{"className":2400,"style":1012},[160],[30,2402,2404],{"className":2403},[165,166,167,168],[30,2405,79],{"className":2406,"style":245},[131,205,168],[30,2408,329],{"className":2409},[328],[30,2411,2413],{"className":2412},[148],[30,2414,2417],{"className":2415,"style":2416},[152],"height:0.2861em;",[30,2418],{},[30,2420,109],{"className":2421},[380],[30,2423],{"className":2424,"style":974},[394],[30,2426,856],{"className":2427},[238],[30,2429],{"className":2430,"style":974},[394],[30,2432,2434,2437],{"className":2433},[131],[30,2435,70],{"className":2436},[131],[30,2438,2440],{"className":2439},[209],[30,2441,2443],{"className":2442},[143],[30,2444,2446],{"className":2445},[148],[30,2447,2450],{"className":2448,"style":2449},[152],"height:0.8913em;",[30,2451,2453,2456],{"style":2452},"top:-3.113em;margin-right:0.05em;",[30,2454],{"className":2455,"style":1012},[160],[30,2457,2459],{"className":2458},[165,166,167,168],[30,2460,2462,2502,2505],{"className":2461},[131,168],[30,2463,2465,2468],{"className":2464},[131,168],[30,2466,784],{"className":2467,"style":800},[131,205,168],[30,2469,2471],{"className":2470},[209],[30,2472,2474,2494],{"className":2473},[143,144],[30,2475,2477,2491],{"className":2476},[148],[30,2478,2480],{"className":2479,"style":1905},[152],[30,2481,2482,2485],{"style":2011},[30,2483],{"className":2484,"style":226},[160],[30,2486,2488],{"className":2487},[165,230,231,168],[30,2489,79],{"className":2490,"style":245},[131,205,168],[30,2492,329],{"className":2493},[328],[30,2495,2497],{"className":2496},[148],[30,2498,2500],{"className":2499,"style":1927},[152],[30,2501],{},[30,2503,867],{"className":2504},[238,168],[30,2506,870],{"className":2507},[131,168],[30,2509,109],{"className":2510},[380],[30,2512,2513,2516],{"style":2297},[30,2514],{"className":2515,"style":2276},[160],[30,2517,2519,2522,2525,2528,2531,2536,2539,2694,2697,2700,2703,2744,2747,2750,2753],{"className":2518},[131],[30,2520],{"className":2521},[131],[30,2523],{"className":2524,"style":1031},[394],[30,2526,517],{"className":2527},[1035],[30,2529],{"className":2530,"style":1031},[394],[30,2532,2284,2534],{"className":2533},[2283],[30,2535,2288],{"style":2287},[30,2537],{"className":2538,"style":395},[394],[30,2540,2542,2545,2548,2551,2554,2557,2691],{"className":2541},[131],[30,2543,94],{"className":2544},[135],[30,2546,55],{"className":2547},[131],[30,2549],{"className":2550,"style":974},[394],[30,2552,74],{"className":2553},[238],[30,2555],{"className":2556,"style":974},[394],[30,2558,2560,2563,2688],{"className":2559},[131],[30,2561],{"className":2562},[135,136],[30,2564,2566],{"className":2565},[50],[30,2567,2569,2679],{"className":2568},[143,144],[30,2570,2572,2676],{"className":2571},[148],[30,2573,2576,2619,2627],{"className":2574,"style":2575},[152],"height:1.3603em;",[30,2577,2579,2582],{"style":2578},"top:-2.314em;",[30,2580],{"className":2581,"style":161},[160],[30,2583,2585],{"className":2584},[131],[30,2586,2588,2591],{"className":2587},[131],[30,2589,70],{"className":2590},[131],[30,2592,2594],{"className":2593},[209],[30,2595,2597],{"className":2596},[143],[30,2598,2600],{"className":2599},[148],[30,2601,2604],{"className":2602,"style":2603},[152],"height:0.7401em;",[30,2605,2607,2610],{"style":2606},"top:-2.989em;margin-right:0.05em;",[30,2608],{"className":2609,"style":1012},[160],[30,2611,2613],{"className":2612},[165,166,167,168],[30,2614,2616],{"className":2615},[131,168],[30,2617,889],{"className":2618},[131,168],[30,2620,2621,2624],{"style":341},[30,2622],{"className":2623,"style":161},[160],[30,2625],{"className":2626,"style":349},[348],[30,2628,2630,2633],{"style":2629},"top:-3.677em;",[30,2631],{"className":2632,"style":161},[160],[30,2634,2636],{"className":2635},[131],[30,2637,2639,2642],{"className":2638},[131],[30,2640,816],{"className":2641,"style":831},[131,205],[30,2643,2645],{"className":2644},[209],[30,2646,2648,2668],{"className":2647},[143,144],[30,2649,2651,2665],{"className":2650},[148],[30,2652,2654],{"className":2653,"style":2394},[152],[30,2655,2656,2659],{"style":2397},[30,2657],{"className":2658,"style":1012},[160],[30,2660,2662],{"className":2661},[165,166,167,168],[30,2663,79],{"className":2664,"style":245},[131,205,168],[30,2666,329],{"className":2667},[328],[30,2669,2671],{"className":2670},[148],[30,2672,2674],{"className":2673,"style":2416},[152],[30,2675],{},[30,2677,329],{"className":2678},[328],[30,2680,2682],{"className":2681},[148],[30,2683,2686],{"className":2684,"style":2685},[152],"height:0.686em;",[30,2687],{},[30,2689],{"className":2690},[380,136],[30,2692,109],{"className":2693},[380],[30,2695],{"className":2696,"style":974},[394],[30,2698,74],{"className":2699},[238],[30,2701],{"className":2702,"style":974},[394],[30,2704,2706,2709],{"className":2705},[131],[30,2707,784],{"className":2708,"style":800},[131,205],[30,2710,2712],{"className":2711},[209],[30,2713,2715,2736],{"className":2714},[143,144],[30,2716,2718,2733],{"className":2717},[148],[30,2719,2721],{"className":2720,"style":2394},[152],[30,2722,2724,2727],{"style":2723},"top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;",[30,2725],{"className":2726,"style":1012},[160],[30,2728,2730],{"className":2729},[165,166,167,168],[30,2731,79],{"className":2732,"style":245},[131,205,168],[30,2734,329],{"className":2735},[328],[30,2737,2739],{"className":2738},[148],[30,2740,2742],{"className":2741,"style":2416},[152],[30,2743],{},[30,2745],{"className":2746,"style":974},[394],[30,2748,867],{"className":2749},[238],[30,2751],{"className":2752,"style":974},[394],[30,2754,870],{"className":2755},[131],[30,2757,329],{"className":2758},[328],[30,2760,2762],{"className":2761},[148],[30,2763,2765],{"className":2764,"style":2313},[152],[30,2766],{},[30,2768,2771],{"className":2769},[2770],"tag",[30,2772,2774,2800],{"className":2773},[143,144],[30,2775,2777,2797],{"className":2776},[148],[30,2778,2780,2789],{"className":2779,"style":2269},[152],[30,2781,2782,2785],{"style":2272},[30,2783],{"className":2784,"style":2276},[160],[30,2786],{"className":2787},[2788],"eqn-num",[30,2790,2791,2794],{"style":2297},[30,2792],{"className":2793,"style":2276},[160],[30,2795],{"className":2796},[2788],[30,2798,329],{"className":2799},[328],[30,2801,2803],{"className":2802},[148],[30,2804,2806],{"className":2805,"style":2313},[152],[30,2807],{},[15,2809,2810,2811,2983,2984,3037,3038,3132,3133,3163],{},"It's clear that ",[30,2812,2814,2840],{"className":2813,"translate":34},[33],[30,2815,2817],{"className":2816},[38],[40,2818,2819],{"xmlns":42},[44,2820,2821,2837],{},[47,2822,2823],{},[50,2824,2825,2831],{},[1721,2826,2827,2829],{},[65,2828,816],{},[65,2830,79],{},[62,2832,2833,2835],{},[53,2834,70],{},[53,2836,889],{},[111,2838,2839],{"encoding":113},"\\frac{M_y}{2^{23}}",[30,2841,2843],{"className":2842,"ariaHidden":99},[118],[30,2844,2846,2849],{"className":2845},[122],[30,2847],{"className":2848,"style":1804},[126],[30,2850,2852,2855,2980],{"className":2851},[131],[30,2853],{"className":2854},[135,136],[30,2856,2858],{"className":2857},[50],[30,2859,2861,2972],{"className":2860},[143,144],[30,2862,2864,2969],{"className":2863},[148],[30,2865,2867,2910,2918],{"className":2866,"style":1823},[152],[30,2868,2869,2872],{"style":1088},[30,2870],{"className":2871,"style":161},[160],[30,2873,2875],{"className":2874},[165,166,167,168],[30,2876,2878],{"className":2877},[131,168],[30,2879,2881,2884],{"className":2880},[131,168],[30,2882,70],{"className":2883},[131,168],[30,2885,2887],{"className":2886},[209],[30,2888,2890],{"className":2889},[143],[30,2891,2893],{"className":2892},[148],[30,2894,2896],{"className":2895,"style":219},[152],[30,2897,2898,2901],{"style":222},[30,2899],{"className":2900,"style":226},[160],[30,2902,2904],{"className":2903},[165,230,231,168],[30,2905,2907],{"className":2906},[131,168],[30,2908,889],{"className":2909},[131,168],[30,2911,2912,2915],{"style":341},[30,2913],{"className":2914,"style":161},[160],[30,2916],{"className":2917,"style":349},[348],[30,2919,2920,2923],{"style":1877},[30,2921],{"className":2922,"style":161},[160],[30,2924,2926],{"className":2925},[165,166,167,168],[30,2927,2929],{"className":2928},[131,168],[30,2930,2932,2935],{"className":2931},[131,168],[30,2933,816],{"className":2934,"style":831},[131,205,168],[30,2936,2938],{"className":2937},[209],[30,2939,2941,2961],{"className":2940},[143,144],[30,2942,2944,2958],{"className":2943},[148],[30,2945,2947],{"className":2946,"style":1905},[152],[30,2948,2949,2952],{"style":1908},[30,2950],{"className":2951,"style":226},[160],[30,2953,2955],{"className":2954},[165,230,231,168],[30,2956,79],{"className":2957,"style":245},[131,205,168],[30,2959,329],{"className":2960},[328],[30,2962,2964],{"className":2963},[148],[30,2965,2967],{"className":2966,"style":1927},[152],[30,2968],{},[30,2970,329],{"className":2971},[328],[30,2973,2975],{"className":2974},[148],[30,2976,2978],{"className":2977,"style":1161},[152],[30,2979],{},[30,2981],{"className":2982},[380,136]," is in the interval ",[30,2985,2987,3010],{"className":2986,"translate":34},[33],[30,2988,2990],{"className":2989},[38],[40,2991,2992],{"xmlns":42},[44,2993,2994,3007],{},[47,2995,2996,2998,3000,3002,3004],{},[72,2997,915],{"stretchy":93},[53,2999,918],{},[72,3001,100],{"separator":99},[53,3003,55],{},[72,3005,3006],{"stretchy":93},"]",[111,3008,3009],{"encoding":113},"[0,1]",[30,3011,3013],{"className":3012,"ariaHidden":99},[118],[30,3014,3016,3019,3022,3025,3028,3031,3034],{"className":3015},[122],[30,3017],{"className":3018,"style":958},[126],[30,3020,915],{"className":3021},[135],[30,3023,918],{"className":3024},[131],[30,3026,100],{"className":3027},[390],[30,3029],{"className":3030,"style":395},[394],[30,3032,55],{"className":3033},[131],[30,3035,3006],{"className":3036},[380],". In this interval, ",[30,3039,3041,3072],{"className":3040,"translate":34},[33],[30,3042,3044],{"className":3043},[38],[40,3045,3046],{"xmlns":42},[44,3047,3048,3069],{},[47,3049,3050,3052,3054,3056,3058,3060,3062,3064,3067],{},[65,3051,2108],{},[72,3053,2111],{},[72,3055,94],{"stretchy":93},[53,3057,55],{},[72,3059,74],{},[65,3061,67],{},[72,3063,109],{"stretchy":93},[72,3065,3066],{},"≈",[65,3068,67],{},[111,3070,3071],{"encoding":113},"\\log(1+x) \\approx x",[30,3073,3075,3101,3122],{"className":3074,"ariaHidden":99},[118],[30,3076,3078,3081,3086,3089,3092,3095,3098],{"className":3077},[122],[30,3079],{"className":3080,"style":958},[126],[30,3082,2284,3084],{"className":3083},[2283],[30,3085,2288],{"style":2287},[30,3087,94],{"className":3088},[135],[30,3090,55],{"className":3091},[131],[30,3093],{"className":3094,"style":974},[394],[30,3096,74],{"className":3097},[238],[30,3099],{"className":3100,"style":974},[394],[30,3102,3104,3107,3110,3113,3116,3119],{"className":3103},[122],[30,3105],{"className":3106,"style":958},[126],[30,3108,67],{"className":3109},[131,205],[30,3111,109],{"className":3112},[380],[30,3114],{"className":3115,"style":1031},[394],[30,3117,3066],{"className":3118},[1035],[30,3120],{"className":3121,"style":1031},[394],[30,3123,3125,3129],{"className":3124},[122],[30,3126],{"className":3127,"style":3128},[126],"height:0.4306em;",[30,3130,67],{"className":3131},[131,205],". If we add a constant ",[30,3134,3136,3151],{"className":3135,"translate":34},[33],[30,3137,3139],{"className":3138},[38],[40,3140,3141],{"xmlns":42},[44,3142,3143,3148],{},[47,3144,3145],{},[65,3146,3147],{},"μ",[111,3149,3150],{"encoding":113},"\\mu",[30,3152,3154],{"className":3153,"ariaHidden":99},[118],[30,3155,3157,3160],{"className":3156},[122],[30,3158],{"className":3159,"style":1367},[126],[30,3161,3147],{"className":3162},[131,205],", the approximation improves:",[15,3165,3166],{},"Using this approximation, we can further simplify:",[30,3168,3170],{"className":3169,"translate":34},[2073],[30,3171,3173,3386],{"className":3172,"translate":34},[33],[30,3174,3176],{"className":3175},[38],[40,3177,3178],{"xmlns":42,"display":2082},[44,3179,3180,3383],{},[2086,3181,3182,3255,3312],{"rowspacing":2088,"columnalign":2089,"columnspacing":2090},[2092,3183,3184,3187,3199,3249,3252],{},[2095,3185],{"className":3186},[2098],[2095,3188,3189],{},[2102,3190,3191],{"scriptlevel":918,"displaystyle":99},[47,3192,3193,3195,3197],{},[65,3194,2108],{},[72,3196,2111],{},[65,3198,79],{},[2095,3200,3201],{},[2102,3202,3203],{"scriptlevel":918,"displaystyle":99},[47,3204,3205,3207,3209,3211,3213,3237,3239,3245,3247],{},[47,3206],{},[72,3208,517],{},[65,3210,2108],{},[72,3212,2111],{},[47,3214,3215,3217,3219,3221,3235],{},[72,3216,94],{"stretchy":93},[53,3218,55],{},[72,3220,74],{},[50,3222,3223,3229],{},[1721,3224,3225,3227],{},[65,3226,816],{},[65,3228,79],{},[62,3230,3231,3233],{},[53,3232,70],{},[53,3234,889],{},[72,3236,109],{"stretchy":93},[72,3238,74],{},[1721,3240,3241,3243],{},[65,3242,784],{},[65,3244,79],{},[72,3246,867],{},[53,3248,870],{},[2095,3250],{"className":3251},[2098],[2095,3253],{"className":3254},[2172],[2092,3256,3257,3260,3266,3306,3309],{},[2095,3258],{"className":3259},[2098],[2095,3261,3262],{},[2102,3263,3264],{"scriptlevel":918,"displaystyle":99},[47,3265],{},[2095,3267,3268],{},[2102,3269,3270],{"scriptlevel":918,"displaystyle":99},[47,3271,3272,3274,3276,3290,3292,3294,3296,3302,3304],{},[47,3273],{},[72,3275,3066],{},[50,3277,3278,3284],{},[1721,3279,3280,3282],{},[65,3281,816],{},[65,3283,79],{},[62,3285,3286,3288],{},[53,3287,70],{},[53,3289,889],{},[72,3291,74],{},[65,3293,3147],{},[72,3295,74],{},[1721,3297,3298,3300],{},[65,3299,784],{},[65,3301,79],{},[72,3303,867],{},[53,3305,870],{},[2095,3307],{"className":3308},[2098],[2095,3310],{"className":3311},[2172],[2092,3313,3314,3317,3323,3377,3380],{},[2095,3315],{"className":3316},[2098],[2095,3318,3319],{},[2102,3320,3321],{"scriptlevel":918,"displaystyle":99},[47,3322],{},[2095,3324,3325],{},[2102,3326,3327],{"scriptlevel":918,"displaystyle":99},[47,3328,3329,3331,3333,3343,3345,3351,3353,3359,3361,3367,3369,3371,3373,3375],{},[47,3330],{},[72,3332,517],{},[50,3334,3335,3337],{},[53,3336,55],{},[62,3338,3339,3341],{},[53,3340,70],{},[53,3342,889],{},[72,3344,94],{"stretchy":93},[1721,3346,3347,3349],{},[65,3348,816],{},[65,3350,79],{},[72,3352,74],{},[1721,3354,3355,3357],{},[65,3356,784],{},[65,3358,79],{},[72,3360,856],{},[62,3362,3363,3365],{},[53,3364,70],{},[53,3366,889],{},[72,3368,109],{"stretchy":93},[72,3370,74],{},[65,3372,3147],{},[72,3374,867],{},[53,3376,870],{},[2095,3378],{"className":3379},[2098],[2095,3381],{"className":3382},[2172],[111,3384,3385],{"encoding":113},"\\begin{align}\n\\log y\n&= \\log{(1+\\frac{M_y}{2^{23}})} + E_y - 127 \\\\\n&\\approx \\frac{M_y}{2^{23}} + \\mu + E_y - 127 \\\\\n&= \\frac{1}{2^{23}}(M_y + E_y \\times 2^{23}) + \\mu - 127\n\\end{align}",[30,3387,3389,4217],{"className":3388,"ariaHidden":99},[118],[30,3390,3392,3396],{"className":3391},[122],[30,3393],{"className":3394,"style":3395},[126],"height:7.0001em;vertical-align:-3.2501em;",[30,3397,3399,3462],{"className":3398},[2086],[30,3400,3402],{"className":3401},[2259],[30,3403,3405,3453],{"className":3404},[143,144],[30,3406,3408,3450],{"className":3407},[148],[30,3409,3412,3432,3441],{"className":3410,"style":3411},[152],"height:3.75em;",[30,3413,3415,3418],{"style":3414},"top:-5.75em;",[30,3416],{"className":3417,"style":2276},[160],[30,3419,3421,3426,3429],{"className":3420},[131],[30,3422,2284,3424],{"className":3423},[2283],[30,3425,2288],{"style":2287},[30,3427],{"className":3428,"style":395},[394],[30,3430,79],{"className":3431,"style":245},[131,205],[30,3433,3435,3438],{"style":3434},"top:-3.4037em;",[30,3436],{"className":3437,"style":2276},[160],[30,3439],{"className":3440},[131],[30,3442,3444,3447],{"style":3443},"top:-1.0963em;",[30,3445],{"className":3446,"style":2276},[160],[30,3448],{"className":3449},[131],[30,3451,329],{"className":3452},[328],[30,3454,3456],{"className":3455},[148],[30,3457,3460],{"className":3458,"style":3459},[152],"height:3.2501em;",[30,3461],{},[30,3463,3465],{"className":3464},[2319],[30,3466,3468,4209],{"className":3467},[143,144],[30,3469,3471,4206],{"className":3470},[148],[30,3472,3474,3712,3933],{"className":3473,"style":3411},[152],[30,3475,3476,3479],{"style":3414},[30,3477],{"className":3478,"style":2276},[160],[30,3480,3482,3485,3488,3491,3494,3499,3502,3651,3654,3657,3660,3700,3703,3706,3709],{"className":3481},[131],[30,3483],{"className":3484},[131],[30,3486],{"className":3487,"style":1031},[394],[30,3489,517],{"className":3490},[1035],[30,3492],{"className":3493,"style":1031},[394],[30,3495,2284,3497],{"className":3496},[2283],[30,3498,2288],{"style":2287},[30,3500],{"className":3501,"style":395},[394],[30,3503,3505,3508,3511,3514,3517,3520,3648],{"className":3504},[131],[30,3506,94],{"className":3507},[135],[30,3509,55],{"className":3510},[131],[30,3512],{"className":3513,"style":974},[394],[30,3515,74],{"className":3516},[238],[30,3518],{"className":3519,"style":974},[394],[30,3521,3523,3526,3645],{"className":3522},[131],[30,3524],{"className":3525},[135,136],[30,3527,3529],{"className":3528},[50],[30,3530,3532,3637],{"className":3531},[143,144],[30,3533,3535,3634],{"className":3534},[148],[30,3536,3538,3578,3586],{"className":3537,"style":2575},[152],[30,3539,3540,3543],{"style":2578},[30,3541],{"className":3542,"style":161},[160],[30,3544,3546],{"className":3545},[131],[30,3547,3549,3552],{"className":3548},[131],[30,3550,70],{"className":3551},[131],[30,3553,3555],{"className":3554},[209],[30,3556,3558],{"className":3557},[143],[30,3559,3561],{"className":3560},[148],[30,3562,3564],{"className":3563,"style":2603},[152],[30,3565,3566,3569],{"style":2606},[30,3567],{"className":3568,"style":1012},[160],[30,3570,3572],{"className":3571},[165,166,167,168],[30,3573,3575],{"className":3574},[131,168],[30,3576,889],{"className":3577},[131,168],[30,3579,3580,3583],{"style":341},[30,3581],{"className":3582,"style":161},[160],[30,3584],{"className":3585,"style":349},[348],[30,3587,3588,3591],{"style":2629},[30,3589],{"className":3590,"style":161},[160],[30,3592,3594],{"className":3593},[131],[30,3595,3597,3600],{"className":3596},[131],[30,3598,816],{"className":3599,"style":831},[131,205],[30,3601,3603],{"className":3602},[209],[30,3604,3606,3626],{"className":3605},[143,144],[30,3607,3609,3623],{"className":3608},[148],[30,3610,3612],{"className":3611,"style":2394},[152],[30,3613,3614,3617],{"style":2397},[30,3615],{"className":3616,"style":1012},[160],[30,3618,3620],{"className":3619},[165,166,167,168],[30,3621,79],{"className":3622,"style":245},[131,205,168],[30,3624,329],{"className":3625},[328],[30,3627,3629],{"className":3628},[148],[30,3630,3632],{"className":3631,"style":2416},[152],[30,3633],{},[30,3635,329],{"className":3636},[328],[30,3638,3640],{"className":3639},[148],[30,3641,3643],{"className":3642,"style":2685},[152],[30,3644],{},[30,3646],{"className":3647},[380,136],[30,3649,109],{"className":3650},[380],[30,3652],{"className":3653,"style":974},[394],[30,3655,74],{"className":3656},[238],[30,3658],{"className":3659,"style":974},[394],[30,3661,3663,3666],{"className":3662},[131],[30,3664,784],{"className":3665,"style":800},[131,205],[30,3667,3669],{"className":3668},[209],[30,3670,3672,3692],{"className":3671},[143,144],[30,3673,3675,3689],{"className":3674},[148],[30,3676,3678],{"className":3677,"style":2394},[152],[30,3679,3680,3683],{"style":2723},[30,3681],{"className":3682,"style":1012},[160],[30,3684,3686],{"className":3685},[165,166,167,168],[30,3687,79],{"className":3688,"style":245},[131,205,168],[30,3690,329],{"className":3691},[328],[30,3693,3695],{"className":3694},[148],[30,3696,3698],{"className":3697,"style":2416},[152],[30,3699],{},[30,3701],{"className":3702,"style":974},[394],[30,3704,867],{"className":3705},[238],[30,3707],{"className":3708,"style":974},[394],[30,3710,870],{"className":3711},[131],[30,3713,3714,3717],{"style":3434},[30,3715],{"className":3716,"style":2276},[160],[30,3718,3720,3723,3726,3729,3732,3860,3863,3866,3869,3872,3875,3878,3881,3921,3924,3927,3930],{"className":3719},[131],[30,3721],{"className":3722},[131],[30,3724],{"className":3725,"style":1031},[394],[30,3727,3066],{"className":3728},[1035],[30,3730],{"className":3731,"style":1031},[394],[30,3733,3735,3738,3857],{"className":3734},[131],[30,3736],{"className":3737},[135,136],[30,3739,3741],{"className":3740},[50],[30,3742,3744,3849],{"className":3743},[143,144],[30,3745,3747,3846],{"className":3746},[148],[30,3748,3750,3790,3798],{"className":3749,"style":2575},[152],[30,3751,3752,3755],{"style":2578},[30,3753],{"className":3754,"style":161},[160],[30,3756,3758],{"className":3757},[131],[30,3759,3761,3764],{"className":3760},[131],[30,3762,70],{"className":3763},[131],[30,3765,3767],{"className":3766},[209],[30,3768,3770],{"className":3769},[143],[30,3771,3773],{"className":3772},[148],[30,3774,3776],{"className":3775,"style":2603},[152],[30,3777,3778,3781],{"style":2606},[30,3779],{"className":3780,"style":1012},[160],[30,3782,3784],{"className":3783},[165,166,167,168],[30,3785,3787],{"className":3786},[131,168],[30,3788,889],{"className":3789},[131,168],[30,3791,3792,3795],{"style":341},[30,3793],{"className":3794,"style":161},[160],[30,3796],{"className":3797,"style":349},[348],[30,3799,3800,3803],{"style":2629},[30,3801],{"className":3802,"style":161},[160],[30,3804,3806],{"className":3805},[131],[30,3807,3809,3812],{"className":3808},[131],[30,3810,816],{"className":3811,"style":831},[131,205],[30,3813,3815],{"className":3814},[209],[30,3816,3818,3838],{"className":3817},[143,144],[30,3819,3821,3835],{"className":3820},[148],[30,3822,3824],{"className":3823,"style":2394},[152],[30,3825,3826,3829],{"style":2397},[30,3827],{"className":3828,"style":1012},[160],[30,3830,3832],{"className":3831},[165,166,167,168],[30,3833,79],{"className":3834,"style":245},[131,205,168],[30,3836,329],{"className":3837},[328],[30,3839,3841],{"className":3840},[148],[30,3842,3844],{"className":3843,"style":2416},[152],[30,3845],{},[30,3847,329],{"className":3848},[328],[30,3850,3852],{"className":3851},[148],[30,3853,3855],{"className":3854,"style":2685},[152],[30,3856],{},[30,3858],{"className":3859},[380,136],[30,3861],{"className":3862,"style":974},[394],[30,3864,74],{"className":3865},[238],[30,3867],{"className":3868,"style":974},[394],[30,3870,3147],{"className":3871},[131,205],[30,3873],{"className":3874,"style":974},[394],[30,3876,74],{"className":3877},[238],[30,3879],{"className":3880,"style":974},[394],[30,3882,3884,3887],{"className":3883},[131],[30,3885,784],{"className":3886,"style":800},[131,205],[30,3888,3890],{"className":3889},[209],[30,3891,3893,3913],{"className":3892},[143,144],[30,3894,3896,3910],{"className":3895},[148],[30,3897,3899],{"className":3898,"style":2394},[152],[30,3900,3901,3904],{"style":2723},[30,3902],{"className":3903,"style":1012},[160],[30,3905,3907],{"className":3906},[165,166,167,168],[30,3908,79],{"className":3909,"style":245},[131,205,168],[30,3911,329],{"className":3912},[328],[30,3914,3916],{"className":3915},[148],[30,3917,3919],{"className":3918,"style":2416},[152],[30,3920],{},[30,3922],{"className":3923,"style":974},[394],[30,3925,867],{"className":3926},[238],[30,3928],{"className":3929,"style":974},[394],[30,3931,870],{"className":3932},[131],[30,3934,3935,3938],{"style":3443},[30,3936],{"className":3937,"style":2276},[160],[30,3939,3941,3944,3947,3950,3953,4045,4048,4088,4091,4094,4097,4137,4140,4143,4146,4179,4182,4185,4188,4191,4194,4197,4200,4203],{"className":3940},[131],[30,3942],{"className":3943},[131],[30,3945],{"className":3946,"style":1031},[394],[30,3948,517],{"className":3949},[1035],[30,3951],{"className":3952,"style":1031},[394],[30,3954,3956,3959,4042],{"className":3955},[131],[30,3957],{"className":3958},[135,136],[30,3960,3962],{"className":3961},[50],[30,3963,3965,4034],{"className":3964},[143,144],[30,3966,3968,4031],{"className":3967},[148],[30,3969,3972,4012,4020],{"className":3970,"style":3971},[152],"height:1.3214em;",[30,3973,3974,3977],{"style":2578},[30,3975],{"className":3976,"style":161},[160],[30,3978,3980],{"className":3979},[131],[30,3981,3983,3986],{"className":3982},[131],[30,3984,70],{"className":3985},[131],[30,3987,3989],{"className":3988},[209],[30,3990,3992],{"className":3991},[143],[30,3993,3995],{"className":3994},[148],[30,3996,3998],{"className":3997,"style":2603},[152],[30,3999,4000,4003],{"style":2606},[30,4001],{"className":4002,"style":1012},[160],[30,4004,4006],{"className":4005},[165,166,167,168],[30,4007,4009],{"className":4008},[131,168],[30,4010,889],{"className":4011},[131,168],[30,4013,4014,4017],{"style":341},[30,4015],{"className":4016,"style":161},[160],[30,4018],{"className":4019,"style":349},[348],[30,4021,4022,4025],{"s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let's take the logarithm of 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y^{-\\frac{1}{2}} = -\\frac{1}{2} \\log y \\approx -\\frac{1}{2}\\left( \\frac{1}{2^{23}}(M_y + E_y \\times 2^{23}) + \\mu - 127 \\right)",[30,4519,4521,4647,4742],{"className":4520,"ariaHidden":99},[118],[30,4522,4524,4528,4533,4536,4638,4641,4644],{"className":4523},[122],[30,4525],{"className":4526,"style":4527},[126],"height:1.1985em;vertical-align:-0.1944em;",[30,4529,2284,4531],{"className":4530},[2283],[30,4532,2288],{"style":2287},[30,4534],{"className":4535,"style":395},[394],[30,4537,4539,4542],{"className":4538},[131],[30,4540,79],{"className":4541,"style":245},[131,205],[30,4543,4545],{"className":4544},[209],[30,4546,4548],{"className":4547},[143],[30,4549,4551],{"className":4550},[148],[30,4552,4555],{"className":4553,"style":4554},[152],"height:1.004em;",[30,4556,4558,4561],{"style":4557},"top:-3.413em;margin-right:0.05em;",[30,4559],{"className":4560,"style":161},[160],[30,4562,4564],{"className":4563},[165,166,167,168],[30,4565,4567,4570],{"className":4566},[131,168],[30,4568,867],{"className":4569},[131,168],[30,4571,4573,4576,4635],{"className":4572},[131,168],[30,4574],{"className":4575},[135,136,165,230,1597],[30,4577,4579],{"className":4578},[50],[30,4580,4582,4627],{"className":4581},[143,144],[30,4583,4585,4624],{"className":4584},[148],[30,4586,4588,4602,4610],{"className":4587,"style":1610},[152],[30,4589,4590,4593],{"style":1613},[30,4591],{"className":4592,"style":161},[160],[30,4594,4596],{"className":4595},[165,230,231,168],[30,4597,4599],{"className":4598},[131,168],[30,4600,70],{"className":4601},[131,168],[30,4603,4604,4607],{"style":1628},[30,4605],{"className":4606,"style":161},[160],[30,4608],{"className":4609,"style":1635},[348,168],[30,4611,4612,4615],{"style":1638},[30,4613],{"className":4614,"style":161},[160],[30,4616,4618],{"className":4617},[165,230,231,168],[30,4619,4621],{"className":4620},[131,168],[30,4622,55],{"className":4623},[131,168],[30,4625,329],{"className":4626},[328],[30,4628,4630],{"className":4629},[148],[30,4631,4633],{"className":4632,"style":1660},[152],[30,4634],{},[30,4636],{"className":4637},[380,136,165,230,1597],[30,4639],{"className":4640,"style":1031},[394],[30,4642,517],{"className":4643},[1035],[30,4645],{"className":4646,"style":1031},[394],[30,4648,4650,4654,4657,4719,4722,4727,4730,4733,4736,4739],{"className":4649},[122],[30,4651],{"className":4652,"style":4653},[126],"height:2.0074em;vertical-align:-0.686em;",[30,4655,867],{"className":4656},[131],[30,4658,4660,4663,4716],{"className":4659},[131],[30,4661],{"className":4662},[135,136],[30,4664,4666],{"className":4665},[50],[30,4667,4669,4708],{"className":4668},[143,144],[30,4670,4672,4705],{"className":4671},[148],[30,4673,4675,4686,4694],{"className":4674,"style":3971},[152],[30,4676,4677,4680],{"style":2578},[30,4678],{"className":4679,"style":161},[160],[30,4681,4683],{"className":4682},[131],[30,4684,70],{"className":4685},[131],[30,4687,4688,4691],{"style":341},[30,4689],{"className":4690,"style":161},[160],[30,4692],{"className":4693,"style":349},[348],[30,4695,4696,4699],{"style":2629},[30,4697],{"className":4698,"style":161},[160],[30,4700,4702],{"className":4701},[131],[30,4703,55],{"className":4704},[131],[30,4706,329],{"className":4707},[328],[30,4709,4711],{"className":4710},[148],[30,4712,4714],{"className":4713,"style":2685},[152],[30,4715],{},[30,4717],{"className":4718},[380,136],[30,4720],{"className":4721,"style":395},[394],[30,4723,2284,4725],{"className":4724},[2283],[30,4726,2288],{"style":2287},[30,4728],{"className":4729,"style":395},[394],[30,4731,79],{"className":4732,"style":245},[131,205],[30,4734],{"className":4735,"style":1031},[394],[30,4737,3066],{"className":4738},[1035],[30,4740],{"className":4741,"style":1031},[394],[30,4743,4745,4749,4752,4814,4817],{"className":4744},[122],[30,4746],{"className":4747,"style":4748},[126],"height:2.4em;vertical-align:-0.95em;",[30,4750,867],{"className":4751},[131],[30,4753,4755,4758,4811],{"className":4754},[131],[30,4756],{"className":4757},[135,136],[30,4759,4761],{"className":4760},[50],[30,4762,4764,4803],{"className":4763},[143,144],[30,4765,4767,4800],{"className":4766},[148],[30,4768,4770,4781,4789],{"className":4769,"style":3971},[152],[30,4771,4772,4775],{"style":2578},[30,4773],{"className":4774,"style":161},[160],[30,4776,4778],{"className":4777},[131],[30,4779,70],{"className":4780},[131],[30,4782,4783,4786],{"style":341},[30,4784],{"className":4785,"style":161},[160],[30,4787],{"className":4788,"style":349},[348],[30,4790,4791,4794],{"style":2629},[30,4792],{"className":4793,"style":161},[160],[30,4795,4797],{"className":4796},[131],[30,4798,55],{"className":4799},[131],[30,4801,329],{"className":4802},[328],[30,4804,4806],{"className":4805},[148],[30,4807,4809],{"className":4808,"style":2685},[152],[30,4810],{},[30,4812],{"className":4813},[380,136],[30,4815],{"className":4816,"style":395},[394],[30,4818,4821,4830,4921,4924,4964,4967,4970,4973,5013,5016,5019,5022,5054,5057,5060,5063,5066,5069,5072,5075,5078,5081],{"className":4819},[4820],"minner",[30,4822,4826],{"className":4823,"style":4825},[135,4824],"delimcenter","top:0em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the accurate value of ",[30,5091,5093,5118],{"className":5092,"translate":34},[33],[30,5094,5096],{"className":5095},[38],[40,5097,5098],{"xmlns":42},[44,5099,5100,5116],{},[47,5101,5102],{},[62,5103,5104,5106],{},[65,5105,79],{},[47,5107,5108,5110],{},[72,5109,867],{},[50,5111,5112,5114],{},[53,5113,55],{},[53,5115,70],{},[111,5117,4295],{"encoding":113},[30,5119,5121],{"className":5120,"ariaHidden":99},[118],[30,5122,5124,5127],{"className":5123},[122],[30,5125],{"className":5126,"style":1556},[126],[30,5128,5130,5133],{"className":5129},[131],[30,5131,79],{"className":5132,"style":245},[131,205],[30,5134,5136],{"className":5135},[209],[30,5137,5139],{"className":5138},[143],[30,5140,5142],{"className":5141},[148],[30,5143,5145],{"className":5144,"style":1575},[152],[30,5146,5147,5150],{"style":1578},[30,5148],{"className":5149,"style":161},[160],[30,5151,5153],{"className":5152},[165,166,167,168],[30,5154,5156,5159],{"className":5155},[131,168],[30,5157,867],{"className":5158},[131,168],[30,5160,5162,5165,5224],{"className":5161},[131,168],[30,5163],{"className":5164},[135,136,165,230,1597],[30,5166,5168],{"className":5167},[50],[30,5169,5171,5216],{"className":5170},[143,144],[30,5172,5174,5213],{"className":5173},[148],[30,5175,5177,5191,5199],{"className":5176,"style":1610},[152],[30,5178,5179,5182],{"style":1613},[30,5180],{"className":5181,"style":161},[160],[30,5183,5185],{"className":5184},[165,230,231,168],[30,5186,5188],{"className":5187},[131,168],[30,5189,70],{"className":5190},[131,168],[30,5192,5193,5196],{"style":1628},[30,5194],{"className":5195,"style":161},[160],[30,5197],{"className":5198,"style":1635},[348,168],[30,5200,5201,5204],{"style":1638},[30,5202],{"className":5203,"style":161},[160],[30,5205,5207],{"className":5206},[165,230,231,168],[30,5208,5210],{"className":5209},[131,168],[30,5211,55],{"className":5212},[131,168],[30,5214,329],{"className":5215},[328],[30,5217,5219],{"className":5218},[148],[30,5220,5222],{"className":5221,"style":1660},[152],[30,5223],{},[30,5225],{"className":5226},[380,136,165,230,1597],[30,5228,5230,5244],{"className":5229,"translate":34},[33],[30,5231,5233],{"className":5232},[38],[40,5234,5235],{"xmlns":42},[44,5236,5237,5242],{},[47,5238,5239],{},[65,5240,5241],{},"A",[111,5243,5241],{"encoding":113},[30,5245,5247],{"className":5246,"ariaHidden":99},[118],[30,5248,5250,5253],{"className":5249},[122],[30,5251],{"className":5252,"style":796},[126],[30,5254,5241],{"className":5255},[131,205],", with exponent and mantissa ",[30,5258,5260,5278],{"className":5259,"translate":34},[33],[30,5261,5263],{"className":5262},[38],[40,5264,5265],{"xmlns":42},[44,5266,5267,5275],{},[47,5268,5269],{},[1721,5270,5271,5273],{},[65,5272,784],{},[65,5274,5241],{},[111,5276,5277],{"encoding":113},"E_A",[30,5279,5281],{"className":5280,"ariaHidden":99},[118],[30,5282,5284,5288],{"className":5283},[122],[30,5285],{"className":5286,"style":5287},[126],"height:0.8333em;vertical-align:-0.15em;",[30,5289,5291,5294],{"className":5290},[131],[30,5292,784],{"className":5293,"style":800},[131,205],[30,5295,5297],{"className":5296},[209],[30,5298,5300,5321],{"className":5299},[143,144],[30,5301,5303,5318],{"className":5302},[148],[30,5304,5307],{"className":5305,"style":5306},[152],"height:0.3283em;",[30,5308,5309,5312],{"style":2723},[30,5310],{"className":5311,"style":1012},[160],[30,5313,5315],{"className":5314},[165,166,167,168],[30,5316,5241],{"className":5317},[131,205,168],[30,5319,329],{"className":5320},[328],[30,5322,5324],{"className":5323},[148],[30,5325,5328],{"className":5326,"style":5327},[152],"height:0.15em;",[30,5329],{}," and ",[30,5332,5334,5352],{"className":5333,"translate":34},[33],[30,5335,5337],{"className":5336},[38],[40,5338,5339],{"xmlns":42},[44,5340,5341,5349],{},[47,5342,5343],{},[1721,5344,5345,5347],{},[65,5346,816],{},[65,5348,5241],{},[111,5350,5351],{"encoding":113},"M_A",[30,5353,5355],{"className":5354,"ariaHidden":99},[118],[30,5356,5358,5361],{"className":5357},[122],[30,5359],{"className":5360,"style":5287},[126],[30,5362,5364,5367],{"className":5363},[131],[30,5365,816],{"className":5366,"style":831},[131,205],[30,5368,5370],{"className":5369},[209],[30,5371,5373,5393],{"className":5372},[143,144],[30,5374,5376,5390],{"className":5375},[148],[30,5377,5379],{"className":5378,"style":5306},[152],[30,5380,5381,5384],{"style":2397},[30,5382],{"className":5383,"style":1012},[160],[30,5385,5387],{"className":5386},[165,166,167,168],[30,5388,5241],{"className":5389},[131,205,168],[30,5391,329],{"className":5392},[328],[30,5394,5396],{"className":5395},[148],[30,5397,5399],{"className":5398,"style":5327},[152],[30,5400],{}," respectively, we can take the logarithm of ",[30,5403,5405,5418],{"className":5404,"translate":34},[33],[30,5406,5408],{"className":5407},[38],[40,5409,5410],{"xmlns":42},[44,5411,5412,5416],{},[47,5413,5414],{},[65,5415,5241],{},[111,5417,5241],{"encoding":113},[30,5419,5421],{"className":5420,"ariaHidden":99},[118],[30,5422,5424,5427],{"className":5423},[122],[30,5425],{"className":5426,"style":796},[126],[30,5428,5241],{"className":5429},[131,205],[30,5431,5433],{"className":5432,"translate":34},[2073],[30,5434,5436,5500],{"className":5435,"translate":34},[33],[30,5437,5439],{"className":5438},[38],[40,5440,5441],{"xmlns":42,"display":2082},[44,5442,5443,5497],{},[47,5444,5445,5447,5449,5451,5453,5463,5465,5471,5473,5479,5481,5487,5489,5491,5493,5495],{},[65,5446,2108],{},[72,5448,2111],{},[65,5450,5241],{},[72,5452,3066],{},[50,5454,5455,5457],{},[53,5456,55],{},[62,5458,5459,5461],{},[53,5460,70],{},[53,5462,889],{},[72,5464,94],{"stretchy":93},[1721,5466,5467,5469],{},[65,5468,816],{},[65,5470,5241],{},[72,5472,74],{},[1721,5474,5475,5477],{},[65,5476,784],{},[65,5478,5241],{},[72,5480,856],{},[62,5482,5483,5485],{},[53,5484,70],{},[53,5486,889],{},[72,5488,109],{"stretchy":93},[72,5490,74],{},[65,5492,3147],{},[72,5494,867],{},[53,5496,870],{},[111,5498,5499],{"encoding":113},"\\log A \\approx \\frac{1}{2^{23}}(M_A + E_A \\times 2^{23}) + \\mu - 127",[30,5501,5503,5530,5679,5734,5785,5804],{"className":5502,"ariaHidden":99},[118],[30,5504,5506,5510,5515,5518,5521,5524,5527],{"className":5505},[122],[30,5507],{"className":5508,"style":5509},[126],"height:0.8889em;vertical-align:-0.1944em;",[30,5511,2284,5513],{"className":5512},[2283],[30,5514,2288],{"style":2287},[30,5516],{"className":5517,"style":395},[394],[30,5519,5241],{"className":5520},[131,205],[30,5522],{"className":5523,"style":1031},[394],[30,5525,3066],{"className":5526},[1035],[30,5528],{"className":5529,"style":1031},[394],[30,5531,5533,5536,5627,5630,5670,5673,5676],{"className":5532},[122],[30,5534],{"className":5535,"style":4653},[126],[30,5537,5539,5542,5624],{"className":5538},[131],[30,5540],{"className":5541},[135,136],[30,5543,5545],{"className":5544},[50],[30,5546,5548,5616],{"className":5547},[143,144],[30,5549,5551,5613],{"className":5550},[148],[30,5552,5554,5594,5602],{"className":5553,"style":3971},[152],[30,5555,5556,5559],{"style":2578},[30,5557],{"className":5558,"style":161},[160],[30,5560,5562],{"className":5561},[131],[30,5563,5565,5568],{"className":5564},[131],[30,5566,70],{"className":5567},[131],[30,5569,5571],{"className":5570},[209],[30,5572,5574],{"className":5573},[143],[30,5575,5577],{"className":5576},[148],[30,5578,5580],{"className":5579,"style":2603},[152],[30,5581,5582,5585],{"style":2606},[30,5583],{"className":5584,"style":1012},[160],[30,5586,5588],{"className":5587},[165,166,167,168],[30,5589,5591],{"className":5590},[131,168],[30,5592,889],{"className":5593},[131,168],[30,5595,5596,5599],{"style":341},[30,5597],{"className":5598,"style":161},[160],[30,5600],{"className":5601,"style":349},[348],[30,5603,5604,5607],{"style":2629},[30,5605],{"className":5606,"style":161},[160],[30,5608,5610],{"className":5609},[131],[30,5611,55],{"className":5612},[131],[30,5614,329],{"className":5615},[328],[30,5617,5619],{"className":5618},[148],[30,5620,5622],{"className":5621,"style":2685},[152],[30,5623],{},[30,5625],{"className":5626},[380,136],[30,5628,94],{"className":5629},[135],[30,5631,5633,5636],{"className":5632},[131],[30,5634,816],{"className":5635,"style":831},[131,205],[30,5637,5639],{"className":5638},[209],[30,5640,5642,5662],{"className":5641},[143,144],[30,5643,5645,5659],{"className":5644},[148],[30,5646,5648],{"className":5647,"style":5306},[152],[30,5649,5650,5653],{"style":2397},[30,5651],{"className":5652,"style":1012},[160],[30,5654,5656],{"className":5655},[165,166,167,168],[30,5657,5241],{"className":5658},[131,205,168],[30,5660,329],{"className":5661},[328],[30,5663,5665],{"className":5664},[148],[30,5666,5668],{"className":5667,"style":5327},[152],[30,5669],{},[30,5671],{"className":5672,"style":974},[394],[30,5674,74],{"className":5675},[238],[30,5677],{"className":5678,"style":974},[394],[30,5680,5682,5685,5725,5728,5731],{"className":5681},[122],[30,5683],{"className":5684,"style":5287},[126],[30,5686,5688,5691],{"className":5687},[131],[30,5689,784],{"className":5690,"style":800},[131,205],[30,5692,5694],{"className":5693},[209],[30,5695,5697,5717],{"className":5696},[143,144],[30,5698,5700,5714],{"className":5699},[148],[30,5701,5703],{"className":5702,"style":5306},[152],[30,5704,5705,5708],{"style":2723},[30,5706],{"className":5707,"style":1012},[160],[30,5709,5711],{"className":5710},[165,166,167,168],[30,5712,5241],{"className":5713},[131,205,168],[30,5715,329],{"className":5716},[328],[30,5718,5720],{"className":5719},[148],[30,5721,5723],{"className":5722,"style":5327},[152],[30,5724],{},[30,5726],{"className":5727,"style":974},[394],[30,5729,856],{"className":5730},[238],[30,5732],{"className":5733,"style":974},[394],[30,5735,5737,5741,5773,5776,5779,5782],{"className":5736},[122],[30,5738],{"className":5739,"style":5740},[126],"height:1.1141em;vertical-align:-0.25em;",[30,5742,5744,5747],{"className":5743},[131],[30,5745,70],{"className":5746},[131],[30,5748,5750],{"className":5749},[209],[30,5751,5753],{"className":5752},[143],[30,5754,5756],{"className":5755},[148],[30,5757,5759],{"className":5758,"style":4164},[152],[30,5760,5761,5764],{"style":2452},[30,5762],{"className":5763,"style":1012},[160],[30,5765,5767],{"className":5766},[165,166,167,168],[30,5768,5770],{"className":5769},[131,168],[30,5771,889],{"className":5772},[131,168],[30,5774,109],{"className":5775},[380],[30,5777],{"className":5778,"style":974},[394],[30,5780,74],{"className":5781},[238],[30,5783],{"className":5784,"style":974},[394],[30,5786,5788,5792,5795,5798,5801],{"className":5787},[122],[30,5789],{"className":5790,"style":5791},[126],"height:0.7778em;vertical-align:-0.1944em;",[30,5793,3147],{"className":5794},[131,205],[30,5796],{"className":5797,"style":974},[394],[30,5799,867],{"className":5800},[238],[30,5802],{"className":5803,"style":974},[394],[30,5805,5807,5811],{"className":5806},[122],[30,5808],{"className":5809,"style":5810},[126],"height:0.6444em;",[30,5812,870],{"className":5813},[131],[15,5815,5816],{},"By equating the two logarithmic expressions, we get:",[30,5818,5820],{"className":5819,"translate":34},[2073],[30,5821,5823,5943],{"className":5822,"translate":34},[33],[30,5824,5826],{"className":5825},[38],[40,5827,5828],{"xmlns":42,"display":2082},[44,5829,5830,5940],{},[2086,5831,5832],{"rowspacing":2088,"columnalign":2089,"columnspacing":2090},[2092,5833,5834,5837,5865,5934,5937],{},[2095,5835],{"className":5836},[2098],[2095,5838,5839],{},[2102,5840,5841],{"scriptlevel":918,"displaystyle":99},[47,5842,5843,5849,5851,5857,5859],{},[1721,5844,5845,5847],{},[65,5846,816],{},[65,5848,5241],{},[72,5850,74],{},[1721,5852,5853,5855],{},[65,5854,784],{},[65,5856,5241],{},[72,5858,856],{},[62,5860,5861,5863],{},[53,5862,70],{},[53,5864,889],{},[2095,5866,5867],{},[2102,5868,5869],{"scriptlevel":918,"displaystyle":99},[47,5870,5871,5873,5875,5881,5883,5890,5892,5894,5896,5898,5900,5902,5908,5910,5916,5918,5924,5926,5932],{},[47,5872],{},[72,5874,517],{},[62,5876,5877,5879],{},[53,5878,70],{},[53,5880,889],{},[72,5882,856],{},[50,5884,5885,5888],{},[53,5886,5887],{},"3",[53,5889,70],{},[72,5891,94],{"stretchy":93},[53,5893,870],{},[72,5895,867],{},[65,5897,3147],{},[72,5899,109],{"stretchy":93},[72,5901,867],{},[50,5903,5904,5906],{},[53,5905,55],{},[53,5907,70],{},[72,5909,94],{"stretchy":93},[1721,5911,5912,5914],{},[65,5913,816],{},[65,5915,79],{},[72,5917,74],{},[1721,5919,5920,5922],{},[65,5921,784],{},[65,5923,79],{},[72,5925,856],{},[62,5927,5928,5930],{},[53,5929,70],{},[53,5931,889],{},[72,5933,109],{"stretchy":93},[2095,5935],{"className":5936},[2098],[2095,5938],{"className":5939},[2172],[111,5941,5942],{"encoding":113},"\\begin{align}\nM_A + E_A \\times 2^{23} &= 2^{23} \\times \\frac{3}{2}(127 - \\mu) - \\frac{1}{2}(M_y + E_y \\times 2^{23})\n\\end{align}",[30,5944,5946,6494],{"className":5945,"ariaHidden":99},[118],[30,5947,5949,5953],{"className":5948},[122],[30,5950],{"className":5951,"style":5952},[126],"height:2.3074em;vertical-align:-0.9037em;",[30,5954,5956,6120],{"className":5955},[2086],[30,5957,5959],{"className":5958},[2259],[30,5960,5962,6111],{"className":5961},[143,144],[30,5963,5965,6108],{"className":5964},[148],[30,5966,5969],{"className":5967,"style":5968},[152],"height:1.4037em;",[30,5970,5971,5975],{"style":3434},[30,5972],{"className":5973,"style":5974},[160],"height:3.3214em;",[30,5976,5978,6018,6021,6024,6027,6067,6070,6073,6076],{"className":5977},[131],[30,5979,5981,5984],{"className":5980},[131],[30,5982,816],{"className":5983,"style":831},[131,205],[30,5985,5987],{"className":5986},[209],[30,5988,5990,6010],{"className":5989},[143,144],[30,5991,5993,6007],{"className":5992},[148],[30,5994,5996],{"className":5995,"style":5306},[152],[30,5997,5998,6001],{"style":2397},[30,5999],{"className":6000,"style":1012},[160],[30,6002,6004],{"className":6003},[165,166,167,168],[30,6005,5241],{"className":6006},[131,205,168],[30,6008,329],{"className":6009},[328],[30,6011,6013],{"className":6012},[148],[30,6014,6016],{"className":6015,"style":5327},[152],[30,6017],{},[30,6019],{"className":6020,"style":974},[394],[30,6022,74],{"className":6023},[238],[30,6025],{"className":6026,"style":974},[394],[30,6028,6030,6033],{"className":6029},[131],[30,6031,784],{"className":6032,"style":800},[131,205],[30,6034,6036],{"className":6035},[209],[30,6037,6039,6059],{"className":6038},[143,144],[30,6040,6042,6056],{"className":6041},[148],[30,6043,6045],{"className":6044,"style":5306},[152],[30,6046,6047,6050],{"style":2723},[30,6048],{"className":6049,"style":1012},[160],[30,6051,6053],{"className":6052},[165,166,167,168],[30,6054,5241],{"className":6055},[131,205,168],[30,6057,329],{"className":6058},[328],[30,6060,6062],{"className":6061},[148],[30,6063,6065],{"className":6064,"style":5327},[152],[30,6066],{},[30,6068],{"className":6069,"style":974},[394],[30,6071,856],{"className":6072},[238],[30,6074],{"className":6075,"style":974},[394],[30,6077,6079,6082],{"className":6078},[131],[30,6080,70],{"className":6081},[131],[30,6083,6085],{"className":6084},[209],[30,6086,6088],{"className":6087},[143],[30,6089,6091],{"className":6090},[148],[30,6092,6094],{"className":6093,"style":4164},[152],[30,6095,6096,6099],{"style":2452},[30,6097],{"className":6098,"style":1012},[160],[30,6100,6102],{"className":6101},[165,166,167,168],[30,6103,6105],{"className":6104},[131,168],[30,6106,889],{"className":6107},[131,168],[30,6109,329],{"className":6110},[328],[30,6112,6114],{"className":6113},[148],[30,6115,6118],{"className":6116,"style":6117},[152],"height:0.9037em;",[30,6119],{},[30,6121,6123],{"className":6122},[2319],[30,6124,6126,6486],{"className":6125},[143,144],[30,6127,6129,6483],{"className":6128},[148],[30,6130,6132],{"className":6131,"style":5968},[152],[30,6133,6134,6137],{"style":3434},[30,6135],{"className":6136,"style":5974},[160],[30,6138,6140,6143,6146,6149,6152,6184,6187,6190,6193,6255,6258,6261,6264,6267,6270,6273,6276,6279,6282,6285,6347,6350,6390,6393,6396,6399,6439,6442,6445,6448,6480],{"className":6139},[131],[30,6141],{"className":6142},[131],[30,6144],{"className":6145,"style":1031},[394],[30,6147,517],{"className":6148},[1035],[30,6150],{"className":6151,"style":1031},[394],[30,6153,6155,6158],{"className":6154},[131],[30,6156,70],{"className":6157},[131],[30,6159,6161],{"className":6160},[209],[30,6162,6164],{"className":6163},[143],[30,6165,6167],{"className":6166},[148],[30,6168,6170],{"className":6169,"style":4164},[152],[30,6171,6172,6175],{"style":2452},[30,6173],{"className":6174,"style":1012},[160],[30,6176,6178],{"className":6177},[165,166,167,168],[30,6179,6181],{"className":6180},[131,168],[30,6182,889],{"className":6183},[131,168],[30,6185],{"className":6186,"style":974},[394],[30,6188,856],{"className":6189},[238],[30,6191],{"className":6192,"style":974},[394],[30,6194,6196,6199,6252],{"className":6195},[131],[30,6197],{"className":6198},[135,136],[30,6200,6202],{"className":6201},[50],[30,6203,6205,6244],{"className":6204},[143,144],[30,6206,6208,6241],{"className":6207},[148],[30,6209,6211,6222,6230],{"className":6210,"style":3971},[152],[30,6212,6213,6216],{"style":2578},[30,6214],{"className":6215,"style":161},[160],[30,6217,6219],{"className":6218},[131],[30,6220,70],{"className":6221},[131],[30,6223,6224,6227],{"style":341},[30,6225],{"className":6226,"style":161},[160],[30,6228],{"className":6229,"style":349},[348],[30,6231,6232,6235],{"style":2629},[30,6233],{"className":6234,"style":161},[160],[30,6236,6238],{"className":6237},[131],[30,6239,5887],{"className":6240},[131],[30,6242,329],{"className":6243},[328],[30,6245,6247],{"className":6246},[148],[30,6248,6250],{"className":6249,"style":2685},[152],[30,6251],{},[30,6253],{"className":6254},[380,136],[30,6256,94],{"className":6257},[135],[30,625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",[30,6529,6531,6544],{"className":6530,"translate":34},[33],[30,6532,6534],{"className":6533},[38],[40,6535,6536],{"xmlns":42},[44,6537,6538,6542],{},[47,6539,6540],{},[65,6541,3147],{},[111,6543,3150],{"encoding":113},[30,6545,6547],{"className":6546,"ariaHidden":99},[118],[30,6548,6550,6553],{"className":6549},[122],[30,6551],{"className":6552,"style":1367},[126],[30,6554,3147],{"className":6555},[131,205]," is set to ",[30,6558,6560,6574],{"className":6559,"translate":34},[33],[30,6561,6563],{"className":6562},[38],[40,6564,6565],{"xmlns":42},[44,6566,6567,6572],{},[47,6568,6569],{},[53,6570,6571],{},"0.0450461875791687011756",[111,6573,6571],{"encoding":113},[30,6575,6577],{"className":6576,"ariaHidden":99},[118],[30,6578,6580,6583],{"className":6579},[122],[30,6581],{"className":6582,"style":5810},[126],[30,6584,6571],{"className":6585},[131],", the first term becomes ",[25,6588,6589],{},"1597463011"," in hexadecimal (",[25,6592,6593],{},"0x5F3759e3","). If we replace the second term's ",[30,6596,6598,6616],{"className":6597,"translate":34},[33],[30,6599,6601],{"className":6600},[38],[40,6602,6603],{"xmlns":42},[44,6604,6605,6613],{},[47,6606,6607],{},[50,6608,6609,6611],{},[53,6610,55],{},[53,6612,70],{},[111,6614,6615],{"encoding":113},"\\frac{1}{2}",[30,6617,6619],{"className":6618,"ariaHidden":99},[118],[30,6620,6622,6626],{"className":6621},[122],[30,6623],{"className":6624,"style":6625},[126],"height:1.1901em;vertical-align:-0.345em;",[30,6627,6629,6632,6691],{"className":6628},[131],[30,6630],{"className":6631},[135,136],[30,6633,6635],{"className":6634},[50],[30,6636,6638,6683],{"className":6637},[143,144],[30,6639,6641,6680],{"className":6640},[148],[30,6642,6644,6658,6666],{"className":6643,"style":153},[152],[30,6645,6646,6649],{"style":1088},[30,6647],{"className":6648,"style":161},[160],[30,6650,6652],{"className":6651},[165,166,167,168],[30,6653,6655],{"className":6654},[131,168],[30,6656,70],{"className":6657},[131,168],[30,6659,6660,6663],{"style":341},[30,6661],{"className":6662,"style":161},[160],[30,6664],{"className":6665,"style":349},[348],[30,6667,6668,6671],{"style":352},[30,6669],{"className":6670,"style":161},[160],[30,6672,6674],{"className":6673},[165,166,167,168],[30,6675,6677],{"className":6676},[131,168],[30,6678,55],{"className":6679},[131,168],[30,6681,329],{"className":6682},[328],[30,6684,6686],{"className":6685},[148],[30,6687,6689],{"className":6688,"style":1161},[152],[30,6690],{},[30,6692],{"className":6693},[380,136]," with a right shift operation, the result is remarkably close to the line in the code:",[447,6696,6698],{"className":449,"code":6697,"language":451,"meta":452,"style":452},"i = 0x5f3759df - (i >> 1);\n",[25,6699,6700],{"__ignoreMap":452},[30,6701,6702,6705,6707,6709,6711,6713,6715,6717,6719],{"class":457,"line":458},[30,6703,6704],{"class":469},"i ",[30,6706,517],{"class":461},[30,6708,624],{"class":461},[30,6710,627],{"class":520},[30,6712,630],{"class":461},[30,6714,633],{"class":469},[30,6716,636],{"class":461},[30,6718,639],{"class":520},[30,6720,6721],{"class":469},");\n",[15,6723,6724,6725,6753],{},"The magic number is slightly different, but with an appropriate ",[30,6726,6728,6741],{"className":6727,"translate":34},[33],[30,6729,6731],{"className":6730},[38],[40,6732,6733],{"xmlns":42},[44,6734,6735,6739],{},[47,6736,6737],{},[65,6738,3147],{},[111,6740,3150],{"encoding":113},[30,6742,6744],{"className":6743,"ariaHidden":99},[118],[30,6745,6747,6750],{"className":6746},[122],[30,6748],{"className":6749,"style":1367},[126],[30,6751,3147],{"className":6752},[131,205]," value, it should yield exactly the same number.",[10,6755,6757],{"id":6756},"newtons-iteration-method","Newton's Iteration Method",[15,6759,6760],{},"The update formula for Newton's iteration method is:",[30,6762,6764],{"className":6763,"translate":34},[2073],[30,6765,6767,6839],{"className":6766,"translate":34},[33],[30,6768,6770],{"className":6769},[38],[40,6771,6772],{"xmlns":42,"display":2082},[44,6773,6774,6836],{},[47,6775,6776,6789,6791,6797,6799],{},[1721,6777,6778,6780],{},[65,6779,67],{},[47,6781,6782,6785,6787],{},[65,6783,6784],{},"t",[72,6786,74],{},[53,6788,55],{},[72,6790,517],{},[1721,6792,6793,6795],{},[65,6794,67],{},[65,6796,6784],{},[72,6798,867],{},[50,6800,6801,6816],{},[47,6802,6803,6806,6808,6814],{},[65,6804,6805],{},"f",[72,6807,94],{"stretchy":93},[1721,6809,6810,6812],{},[65,6811,67],{},[65,6813,6784],{},[72,6815,109],{"stretchy":93},[47,6817,6818,6826,6828,6834],{},[62,6819,6820,6822],{},[65,6821,6805],{},[72,6823,6825],{"mathvariant":6824},"normal","′",[72,6827,94],{"stretchy":93},[1721,6829,6830,6832],{},[65,6831,67],{},[65,6833,6784],{},[72,6835,109],{"stretchy":93},[111,6837,6838],{"encoding":113},"x_{t+1} = x_t - \\frac{f(x_t)}{f^\\prime(x_t)}",[30,6840,6842,6910,6967],{"className":6841,"ariaHidden":99},[118],[30,6843,6845,6849,6901,6904,6907],{"className":6844},[122],[30,6846],{"className":6847,"style":6848},[126],"height:0.6389em;vertical-align:-0.2083em;",[30,6850,6852,6855],{"className":6851},[131],[30,6853,67],{"className":6854},[131,205],[30,6856,6858],{"className":6857},[209],[30,6859,6861,6892],{"className":6860},[143,144],[30,6862,6864,6889],{"className":6863},[148],[30,6865,6868],{"className":6866,"style":6867},[152],"height:0.3011em;",[30,6869,6871,6874],{"style":6870},"top:-2.55em;margin-left:0em;margin-right:0.05em;",[30,6872],{"className":6873,"style":1012},[160],[30,6875,6877],{"className":6876},[165,166,167,168],[30,6878,6880,6883,6886],{"className":6879},[131,168],[30,6881,6784],{"className":6882},[131,205,168],[30,6884,74],{"className":6885},[238,168],[30,6887,55],{"className":6888},[131,168],[30,6890,329],{"className":6891},[328],[30,6893,6895],{"className":6894},[148],[30,6896,6899],{"className":6897,"style":6898},[152],"height:0.2083em;",[30,6900],{},[30,6902],{"className":6903,"style":1031},[394],[30,6905,517],{"className":6906},[1035],[30,6908],{"className":6909,"style":1031},[394],[30,6911,6913,6917,6958,6961,6964],{"className":6912},[122],[30,6914],{"className":6915,"style":6916},[126],"height:0.7333em;vertical-align:-0.15em;",[30,6918,6920,6923],{"className":6919},[131],[30,6921,67],{"className":6922},[131,205],[30,6924,6926],{"className":6925},[209],[30,6927,6929,6950],{"className":6928},[143,144],[30,6930,6932,6947],{"className":6931},[148],[30,6933,6936],{"className":6934,"style":6935},[152],"height:0.2806em;",[30,6937,6938,6941],{"style":6870},[30,6939],{"className":6940,"style":1012},[160],[30,6942,6944],{"className":6943},[165,166,167,168],[30,6945,6784],{"className":6946},[131,205,168],[30,6948,329],{"className":6949},[328],[30,6951,6953],{"className":6952},[148],[30,6954,6956],{"className":6955,"style":5327},[152],[30,6957],{},[30,6959],{"className":6960,"style":974},[394],[30,6962,867],{"className":6963},[238],[30,6965],{"className":6966,"style":974},[394],[30,6968,6970,6974],{"className":6969},[122],[30,6971],{"className":6972,"style":6973},[126],"height:2.363em;vertical-align:-0.936em;",[30,6975,6977,6980,7155],{"className":6976},[131],[30,6978],{"className":6979},[135,136],[30,6981,6983],{"className":6982},[50],[30,6984,6986,7146],{"className":6985},[143,144],[30,6987,6989,7143],{"className":6988},[148],[30,6990,6993,7078,7086],{"className":6991,"style":6992},[152],"height:1.427em;",[30,6994,6995,6998],{"style":2578},[30,6996],{"className":6997,"style":161},[160],[30,6999,7001,7032,7035,7075],{"className":7000},[131],[30,7002,7004,7008],{"className":7003},[131],[30,7005,6805],{"className":7006,"style":7007},[131,205],"margin-right:0.10764em;",[30,7009,7011],{"className":7010},[209],[30,7012,7014],{"className":7013},[143],[30,7015,7017],{"className":7016},[148],[30,7018,7021],{"className":7019,"style":7020},[152],"height:0.6779em;",[30,7022,7023,7026],{"style":2606},[30,7024],{"className":7025,"style":1012},[160],[30,7027,7029],{"className":7028},[165,166,167,168],[30,7030,6825],{"className":7031},[131,168],[30,7033,94],{"className":7034},[135],[30,7036,7038,7041],{"className":7037},[131],[30,7039,67],{"className":7040},[131,205],[30,7042,7044],{"className":7043},[209],[30,7045,7047,7067],{"className":7046},[143,144],[30,7048,7050,7064],{"className":7049},[148],[30,7051,7053],{"className":7052,"style":6935},[152],[30,7054,7055,7058],{"style":6870},[30,7056],{"className":7057,"style":1012},[160],[30,7059,7061],{"className":7060},[165,166,167,168],[30,7062,6784],{"className":7063},[131,205,168],[30,7065,329],{"className":7066},[328],[30,7068,7070],{"className":7069},[148],[30,7071,7073],{"className":7072,"style":5327},[152],[30,7074],{},[30,7076,109],{"className":7077},[380],[30,7079,7080,7083],{"style":341},[30,7081],{"className":7082,"style":161},[160],[30,7084],{"className":7085,"style":349},[348],[30,7087,7088,7091],{"style":2629},[30,7089],{"className":7090,"style":161},[160],[30,7092,7094,7097,7100,7140],{"className":7093},[131],[30,7095,6805],{"className":7096,"style":7007},[131,205],[30,7098,94],{"className":7099},[135],[30,7101,7103,7106],{"className":7102},[131],[30,7104,67],{"className":7105},[131,205],[30,7107,7109],{"className":7108},[209],[30,7110,7112,7132],{"className":7111},[143,144],[30,7113,7115,7129],{"className":7114},[148],[30,7116,7118],{"className":7117,"style":6935},[152],[30,7119,7120,7123],{"style":6870},[30,7121],{"className":7122,"style":1012},[160],[30,7124,7126],{"className":7125},[165,166,167,168],[30,7127,6784],{"className":7128},[131,205,168],[30,7130,329],{"className":7131},[328],[30,7133,7135],{"className":7134},[148],[30,7136,7138],{"className":7137,"style":5327},[152],[30,7139],{},[30,7141,109],{"className":7142},[380],[30,7144,329],{"className":7145},[328],[30,7147,7149],{"className":7148},[148],[30,7150,7153],{"className":7151,"style":7152},[152],"height:0.936em;",[30,7154],{},[30,7156],{"className":7157},[380,136],[15,7159,7160,7161,7164],{},"The precondition for Newton's iteration method is that the initial point should be near the function's zero. Since the estimation from ",[25,7162,7163],{},"i = 0x5f3759df - (i >> 1);"," should have a small error, we can assume the precondition is satisfied.",[15,7166,7167,7168,7196,7197,7225,7226,7392,7393,7543,7544,7753,7754,7798],{},"Now, our problem is, given ",[30,7169,7171,7184],{"className":7170,"translate":34},[33],[30,7172,7174],{"className":7173},[38],[40,7175,7176],{"xmlns":42},[44,7177,7178,7182],{},[47,7179,7180],{},[65,7181,79],{},[111,7183,79],{"encoding":113},[30,7185,7187],{"className":7186,"ariaHidden":99},[118],[30,7188,7190,7193],{"className":7189},[122],[30,7191],{"className":7192,"style":1367},[126],[30,7194,79],{"className":7195,"style":245},[131,205],", we need to find ",[30,7198,7200,7213],{"className":7199,"translate":34},[33],[30,7201,7203],{"className":7202},[38],[40,7204,7205],{"xmlns":42},[44,7206,7207,7211],{},[47,7208,7209],{},[65,7210,67],{},[111,7212,67],{"encoding":113},[30,7214,7216],{"className":7215,"ariaHidden":99},[118],[30,7217,7219,7222],{"className":7218},[122],[30,7220],{"className":7221,"style":3128},[126],[30,7223,67],{"className":7224},[131,205]," such that ",[30,7227,7229,7253],{"className":7228,"translate":34},[33],[30,7230,7232],{"className":7231},[38],[40,7233,7234],{"xmlns":42},[44,7235,7236,7250],{},[47,7237,7238,7246,7248],{},[50,7239,7240,7242],{},[53,7241,55],{},[57,7243,7244],{},[65,7245,79],{},[72,7247,517],{},[65,7249,67],{},[111,7251,7252],{"encoding":113},"\\frac{1}{\\sqrt{y}} = x",[30,7254,7256,7383],{"className":7255,"ariaHidden":99},[118],[30,7257,7259,7262,7374,7377,7380],{"className":7258},[122],[30,7260],{"className":7261,"style":1420},[126],[30,7263,7265,7268,7371],{"className":7264},[131],[30,7266],{"className":7267},[135,136],[30,7269,7271],{"className":7270},[50],[30,7272,7274,7363],{"className":7273},[143,144],[30,7275,7277,7360],{"className":7276},[148],[30,7278,7280,7338,7346],{"className":7279,"style":153},[152],[30,7281,7282,7285],{"style":1088},[30,7283],{"className":7284,"style":161},[160],[30,7286,7288],{"className":7287},[165,166,167,168],[30,7289,7291],{"className":7290},[131,168],[30,7292,7294],{"className":7293},[131,175,168],[30,7295,7297,7330],{"className":7296},[143,144],[30,7298,7300,7327],{"className":7299},[148],[30,7301,7303,7315],{"className":7302,"style":1462},[152],[30,7304,7306,7309],{"className":7305,"style":1466},[189],[30,7307],{"className":7308,"style":161},[160],[30,7310,7312],{"className":7311,"style":1473},[131,168],[30,7313,79],{"className":7314,"style":245},[131,205,168],[30,7316,7317,7320],{"style":1479},[30,7318],{"className":7319,"style":161},[160],[30,7321,7323],{"className":7322,"style":1486},[311,168],[314,7324,7325],{"xmlns":316,"width":317,"height":1489,"viewBox":319,"preserveAspectRatio":320},[322,7326],{"d":324},[30,7328,329],{"className":7329},[328],[30,7331,7333],{"className":7332},[148],[30,7334,7336],{"className":7335,"style":1501},[152],[30,7337],{},[30,7339,7340,7343],{"style":341},[30,7341],{"className":7342,"style":161},[160],[30,7344],{"className":7345,"style":349},[348],[30,7347,7348,7351],{"style":352},[30,7349],{"className":7350,"style":161},[160],[30,7352,7354],{"className":7353},[165,166,167,168],[30,7355,7357],{"className":7356},[131,168],[30,7358,55],{"className":7359},[131,168],[30,7361,329],{"className":7362},[328],[30,7364,7366],{"className":7365},[148],[30,7367,7369],{"className":7368,"style":1535},[152],[30,7370],{},[30,7372],{"className":7373},[380,136],[30,7375],{"className":7376,"style":1031},[394],[30,7378,517],{"className":7379},[1035],[30,7381],{"className":7382,"style":1031},[394],[30,7384,7386,7389],{"className":7385},[122],[30,7387],{"className":7388,"style":3128},[126],[30,7390,67],{"className":7391},[131,205],", i.e., ",[30,7394,7396,7422],{"className":7395,"translate":34},[33],[30,7397,7399],{"className":7398},[38],[40,7400,7401],{"xmlns":42},[44,7402,7403,7419],{},[47,7404,7405,7415,7417],{},[50,7406,7407,7409],{},[53,7408,55],{},[62,7410,7411,7413],{},[65,7412,67],{},[53,7414,70],{},[72,7416,517],{},[65,7418,79],{},[111,7420,7421],{"encoding":113},"\\frac{1}{x^2} = y",[30,7423,7425,7534],{"className":7424,"ariaHidden":99},[118],[30,7426,7428,7431,7525,7528,7531],{"className":7427},[122],[30,7429],{"className":7430,"style":6625},[126],[30,7432,7434,7437,7522],{"className":7433},[131],[30,7435],{"className":7436},[135,136],[30,7438,7440],{"className":7439},[50],[30,7441,7443,7514],{"className":7442},[143,144],[30,7444,7446,7511],{"className":7445},[148],[30,7447,7449,7489,7497],{"className":7448,"style":153},[152],[30,7450,7451,7454],{"style":1088},[30,7452],{"className":7453,"style":161},[160],[30,7455,7457],{"className":7456},[165,166,167,168],[30,7458,7460],{"className":7459},[131,168],[30,7461,7463,7466],{"className":7462},[131,168],[30,7464,67],{"className":7465},[131,205,168],[30,7467,7469],{"className":7468},[209],[30,7470,7472],{"className":7471},[143],[30,7473,7475],{"className":7474},[148],[30,7476,7478],{"className":7477,"style":219},[152],[30,7479,7480,7483],{"style":222},[30,7481],{"className":7482,"style":226},[160],[30,7484,7486],{"className":7485},[165,230,231,168],[30,7487,70],{"className":7488},[131,168],[30,7490,7491,7494],{"style":341},[30,7492],{"className":7493,"style":161},[160],[30,7495],{"className":7496,"style":349},[348],[30,7498,7499,7502],{"style":352},[30,7500],{"className":7501,"style":161},[160],[30,7503,7505],{"className":7504},[165,166,167,168],[30,7506,7508],{"className":7507},[131,168],[30,7509,55],{"className":7510},[131,168],[30,7512,329],{"className":7513},[328],[30,7515,7517],{"className":7516},[148],[30,7518,7520],{"className":7519,"style":1161},[152],[30,7521],{},[30,7523],{"className":7524},[380,136],[30,7526],{"className":7527,"style":1031},[394],[30,7529,517],{"className":7530},[1035],[30,7532],{"className":7533,"style":1031},[394],[30,7535,7537,7540],{"className":7536},[122],[30,7538],{"className":7539,"style":1367},[126],[30,7541,79],{"className":7542,"style":245},[131,205],". Rearranging, we get ",[30,7545,7547,7587],{"className":7546,"translate":34},[33],[30,7548,7550],{"className":7549},[38],[40,7551,7552],{"xmlns":42},[44,7553,7554,7584],{},[47,7555,7556,7558,7560,7562,7564,7566,7576,7578,7580,7582],{},[65,7557,6805],{},[72,7559,94],{"stretchy":93},[65,7561,67],{},[72,7563,109],{"stretchy":93},[72,7565,517],{},[50,7567,7568,7570],{},[53,7569,55],{},[62,7571,7572,7574],{},[65,7573,67],{},[53,7575,70],{},[72,7577,867],{},[65,7579,79],{},[72,7581,517],{},[53,7583,918],{},[111,7585,7586],{"encoding":113},"f(x) = \\frac{1}{x^2} - y = 0",[30,7588,7590,7617,7726,7744],{"className":7589,"ariaHidden":99},[118],[30,7591,7593,7596,7599,7602,7605,7608,7611,7614],{"className":7592},[122],[30,7594],{"className":7595,"style":958},[126],[30,7597,6805],{"className":7598,"style":7007},[131,205],[30,7600,94],{"className":7601},[135],[30,7603,67],{"className":7604},[131,205],[30,7606,109],{"className":7607},[380],[30,7609],{"className":7610,"style":1031},[394],[30,7612,517],{"className":7613},[1035],[30,7615],{"className":7616,"style":1031},[394],[30,7618,7620,7623,7717,7720,7723],{"className":7619},[122],[30,7621],{"className":7622,"style":6625},[126],[30,7624,7626,7629,7714],{"className":7625},[131],[30,7627],{"className":7628},[135,136],[30,7630,7632],{"className":7631},[50],[30,7633,7635,7706],{"className":7634},[143,144],[30,7636,7638,7703],{"className":7637},[148],[30,7639,7641,7681,7689],{"className":7640,"style":153},[152],[30,7642,7643,7646],{"style":1088},[30,7644],{"className":7645,"style":161},[160],[30,7647,7649],{"className":7648},[165,166,167,168],[30,7650,7652],{"className":7651},[131,168],[30,7653,7655,7658],{"className":7654},[131,168],[30,7656,67],{"className":7657},[131,205,168],[30,7659,7661],{"className":7660},[209],[30,7662,7664],{"className":7663},[143],[30,7665,7667],{"className":7666},[148],[30,7668,7670],{"className":7669,"style":219},[152],[30,7671,7672,7675],{"style":222},[30,7673],{"className":7674,"style":226},[160],[30,7676,7678],{"className":7677},[165,230,231,168],[30,7679,70],{"className":7680},[131,168],[30,7682,7683,7686],{"style":341},[30,7684],{"className":7685,"style":161},[160],[30,7687],{"className":7688,"style":349},[348],[30,7690,7691,7694],{"style":352},[30,7692],{"className":7693,"style":161},[160],[30,7695,7697],{"className":7696},[165,166,167,168],[30,7698,7700],{"className":7699},[131,168],[30,7701,55],{"className":7702},[131,168],[30,7704,329],{"className":7705},[328],[30,7707,7709],{"className":7708},[148],[30,7710,7712],{"className":7711,"style":1161},[152],[30,7713],{},[30,7715],{"className":7716},[380,136],[30,7718],{"className":7719,"style":974},[394],[30,7721,867],{"className":7722},[238],[30,7724],{"className":7725,"style":974},[394],[30,7727,7729,7732,7735,7738,7741],{"className":7728},[122],[30,7730],{"className":7731,"style":1367},[126],[30,7733,79],{"className":7734,"style":245},[131,205],[30,7736],{"className":7737,"style":1031},[394],[30,7739,517],{"className":7740},[1035],[30,7742],{"className":7743,"style":1031},[394],[30,7745,7747,7750],{"className":7746},[122],[30,7748],{"className":7749,"style":5810},[126],[30,7751,918],{"className":7752},[131],". Applying Newton's iteration to ",[30,7755,7757,7777],{"className":7756,"translate":34},[33],[30,7758,7760],{"className":7759},[38],[40,7761,7762],{"xmlns":42},[44,7763,7764,7774],{},[47,7765,7766,7768,7770,7772],{},[65,7767,6805],{},[72,7769,94],{"stretchy":93},[65,7771,67],{},[72,7773,109],{"stretchy":93},[111,7775,7776],{"encoding":113},"f(x)",[30,7778,7780],{"className":7779,"ariaHidden":99},[118],[30,7781,7783,7786,7789,7792,7795],{"className":7782},[122],[30,7784],{"className":7785,"style":958},[126],[30,7787,6805],{"className":7788,"style":7007},[131,205],[30,7790,94],{"className":7791},[135],[30,7793,67],{"className":7794},[131,205],[30,7796,109],{"className":7797},[380]," gives the iteration formula:",[30,7800,7802],{"className":7801,"translate":34},[2073],[30,7803,7805,7854],{"className":7804,"translate":34},[33],[30,7806,7808],{"className":7807},[38],[40,7809,7810],{"xmlns":42,"display":2082},[44,7811,7812,7851],{},[47,7813,7814,7826,7828,7834,7836,7842],{},[1721,7815,7816,7818],{},[65,7817,67],{},[47,7819,7820,7822,7824],{},[65,7821,6784],{},[72,7823,74],{},[53,7825,55],{},[72,7827,517],{},[1721,7829,7830,7832],{},[65,7831,67],{},[65,7833,6784],{},[72,7835,867],{},[50,7837,7838,7840],{},[65,7839,79],{},[53,7841,70],{},[7843,7844,7845,7847,7849],"msubsup",{},[65,7846,67],{},[65,7848,6784],{},[53,7850,5887],{},[111,7852,7853],{"encoding":113},"x_{t+1} = x_t - \\frac{y}{2}x_t^3",[30,7855,7857,7921,7976],{"className":7856,"ariaHidden":99},[118],[30,7858,7860,7863,7912,7915,7918],{"className":7859},[122],[30,7861],{"className":7862,"style":6848},[126],[30,7864,7866,7869],{"className":7865},[131],[30,7867,67],{"className":7868},[131,205],[30,7870,7872],{"className":7871},[209],[30,7873,7875,7904],{"className":7874},[143,144],[30,7876,7878,7901],{"className":7877},[148],[30,7879,7881],{"className":7880,"style":6867},[152],[30,7882,7883,7886],{"style":6870},[30,7884],{"className":7885,"style":1012},[160],[30,7887,7889],{"className":7888},[165,166,167,168],[30,7890,7892,7895,7898],{"className":7891},[131,168],[30,7893,6784],{"className":7894},[131,205,168],[30,7896,74],{"className":7897},[238,168],[30,7899,55],{"className":7900},[131,168],[30,7902,329],{"className":7903},[328],[30,7905,7907],{"className":7906},[148],[30,7908,7910],{"className":7909,"style":6898},[152],[30,7911],{},[30,7913],{"className":7914,"style":1031},[394],[30,7916,517],{"className":7917},[1035],[30,7919],{"className":7920,"style":1031},[394],[30,7922,7924,7927,7967,7970,7973],{"className":7923},[122],[30,7925],{"className":7926,"style":6916},[126],[30,7928,7930,7933],{"className":7929},[131],[30,7931,67],{"className":7932},[131,205],[30,7934,7936],{"className":7935},[209],[30,7937,7939,7959],{"className":7938},[143,144],[30,7940,7942,7956],{"className":7941},[148],[30,7943,7945],{"className":7944,"style":6935},[152],[30,7946,7947,7950],{"style":6870},[30,7948],{"className":7949,"style":1012},[160],[30,7951,7953],{"className":7952},[165,166,167,168],[30,7954,6784],{"className":7955},[131,205,168],[30,7957,329],{"className":7958},[328],[30,7960,7962],{"className":7961},[148],[30,7963,7965],{"className":7964,"style":5327},[152],[30,7966],{},[30,7968],{"className":7969,"style":974},[394],[30,7971,867],{"className":7972},[238],[30,7974],{"className":7975,"style":974},[394],[30,7977,7979,7983,8046],{"className":7978},[122],[30,7980],{"className":7981,"style":7982},[126],"height:1.7936em;vertical-align:-0.686em;",[30,7984,7986,7989,8043],{"className":7985},[131],[30,7987],{"className":7988},[135,136],[30,7990,7992],{"className":7991},[50],[30,7993,7995,8035],{"className":7994},[143,144],[30,7996,7998,8032],{"className":7997},[148],[30,7999,8002,8013,8021],{"className":8000,"style":8001},[152],"height:1.1076em;",[30,8003,8004,8007],{"style":2578},[30,8005],{"className":8006,"style":161},[160],[30,8008,8010],{"className":8009},[131],[30,8011,70],{"className":8012},[131],[30,8014,8015,8018],{"style":341},[30,8016],{"className":8017,"style":161},[160],[30,8019],{"className":8020,"style":349},[348],[30,8022,8023,8026],{"style":2629},[30,8024],{"className":8025,"style":161},[160],[30,8027,8029],{"className":8028},[131],[30,8030,79],{"className":8031,"style":245},[131,205],[30,8033,329],{"className":8034},[328],[30,8036,8038],{"className":8037},[148],[30,8039,8041],{"className":8040,"style":2685},[152],[30,8042],{},[30,8044],{"className":8045},[380,136],[30,8047,8049,8052],{"className":8048},[131],[30,8050,67],{"className":8051},[131,205],[30,8053,8055],{"className":8054},[209],[30,8056,8058,8090],{"className":8057},[143,144],[30,8059,8061,8087],{"className":8060},[148],[30,8062,8064,8076],{"className":8063,"style":4164},[152],[30,8065,8067,8070],{"style":8066},"top:-2.453em;margin-left:0em;margin-right:0.05em;",[30,8068],{"className":8069,"style":1012},[160],[30,8071,8073],{"className":8072},[165,166,167,168],[30,8074,6784],{"className":8075},[131,205,168],[30,8077,8078,8081],{"style":2452},[30,8079],{"className":8080,"style":1012},[160],[30,8082,8084],{"className":8083},[165,166,167,168],[30,8085,5887],{"className":8086},[131,168],[30,8088,329],{"className":8089},[328],[30,8091,8093],{"className":8092},[148],[30,8094,8097],{"className":8095,"style":8096},[152],"height:0.247em;",[30,8098],{},[15,8100,8101],{},"This is translated into code as:",[447,8103,8105],{"className":449,"code":8104,"language":451,"meta":452,"style":452},"y = y * (threehalfs - (x2 * y * y));\n",[25,8106,8107],{"__ignoreMap":452},[30,8108,8109,8112,8114,8116,8118,8120,8122,8124,8126,8128,8130],{"class":457,"line":458},[30,8110,8111],{"class":469},"y ",[30,8113,517],{"class":461},[30,8115,689],{"class":469},[30,8117,541],{"class":461},[30,8119,694],{"class":469},[30,8121,697],{"class":461},[30,8123,700],{"class":469},[30,8125,541],{"class":461},[30,8127,689],{"class":469},[30,8129,541],{"class":461},[30,8131,8132],{"class":469}," y));\n",[10,8134,8136],{"id":8135},"references","References",[15,8138,8139,8141,8142],{},[30,8140,55],{}," ",[8143,8144,8148],"a",{"href":8145,"rel":8146},"https:\u002F\u002Fwww.quora.com\u002FIs-fast-inverse-square-root-still-used",[8147],"nofollow","Is fast inverse square root still used?",[15,8150,8151,8141,8153],{},[30,8152,70],{},[8143,8154,8157],{"href":8155,"rel":8156},"https:\u002F\u002Fwww.youtube.com\u002Fwatch?v=p8u_k2LIZyo",[8147],"Fast Inverse Square Root — A Quake III Algorithm",[15,8159,8160,8141,8162],{},[30,8161,5887],{},[8143,8163,8166],{"href":8164,"rel":8165},"https:\u002F\u002Fwww.lomont.org\u002Fpapers\u002F2003\u002FInvSqrt.pdf",[8147],"Relevant Thesis",[8168,8169,8170],"style",{},"html pre.shiki code .snl16, html code.shiki .snl16{--shiki-default:#F97583}html pre.shiki code .svObZ, html code.shiki .svObZ{--shiki-default:#B392F0}html pre.shiki code .s95oV, html code.shiki .s95oV{--shiki-default:#E1E4E8}html pre.shiki code .s9osk, html code.shiki .s9osk{--shiki-default:#FFAB70}html pre.shiki code .sDLfK, html code.shiki .sDLfK{--shiki-default:#79B8FF}html pre.shiki code .sAwPA, html code.shiki .sAwPA{--shiki-default:#6A737D}html .default .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}",{"title":452,"searchDepth":481,"depth":481,"links":8172},[8173,8174,8175,8176,8177],{"id":12,"depth":481,"text":13},{"id":444,"depth":481,"text":445},{"id":765,"depth":481,"text":766},{"id":6756,"depth":481,"text":6757},{"id":8135,"depth":481,"text":8136},false,"md",null,{"author":8182},"Ethan Zhang","2023-12-08","\u002Fblog\u002F2023\u002F2023-12-08-fast-inverse-square-root",{"title":5,"description":452},"blog\u002F2023\u002F2023-12-08-Fast-Inverse-Square-Root","The article introduces an algorithm called Fast Inverse Square Root, which is used to quickly calculate the inverse square root of a floating point number.",[8189,444],"mathematics","EZpOtfgOhPikaPDVcyuktjWZ4aThW1rZcuP1fQ9SOw4",1777090295696]