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| 1 // The following is adapted from fdlibm (http://www.netlib.org/fdlibm). |
| 2 // |
| 3 // ==================================================== |
| 4 // Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. |
| 5 // |
| 6 // Developed at SunSoft, a Sun Microsystems, Inc. business. |
| 7 // Permission to use, copy, modify, and distribute this |
| 8 // software is freely granted, provided that this notice |
| 9 // is preserved. |
| 10 // ==================================================== |
| 11 // |
| 12 // The original source code covered by the above license above has been |
| 13 // modified significantly by Google Inc. |
| 14 // Copyright 2016 the V8 project authors. All rights reserved. |
| 15 |
| 16 #include "src/base/ieee754.h" |
| 17 |
| 18 #include <limits> |
| 19 |
| 20 #include "src/base/build_config.h" |
| 21 #include "src/base/macros.h" |
| 22 |
| 23 namespace v8 { |
| 24 namespace base { |
| 25 namespace ieee754 { |
| 26 |
| 27 namespace { |
| 28 |
| 29 union Float64 { |
| 30 double v; |
| 31 uint64_t w; |
| 32 struct { |
| 33 #if V8_TARGET_LITTLE_ENDIAN |
| 34 uint32_t lw; |
| 35 uint32_t hw; |
| 36 #else |
| 37 uint32_t hw; |
| 38 uint32_t lw; |
| 39 #endif |
| 40 } words; |
| 41 }; |
| 42 |
| 43 // Extract the less significant 32-bit word from a double. |
| 44 V8_INLINE uint32_t extractLowWord32(double v) { |
| 45 Float64 f; |
| 46 f.v = v; |
| 47 return f.words.lw; |
| 48 } |
| 49 |
| 50 // Extract the most significant 32-bit word from a double. |
| 51 V8_INLINE uint32_t extractHighWord32(double v) { |
| 52 Float64 f; |
| 53 f.v = v; |
| 54 return f.words.hw; |
| 55 } |
| 56 |
| 57 // Insert the most significant 32-bit word into a double. |
| 58 V8_INLINE double insertHighWord32(double v, uint32_t hw) { |
| 59 Float64 f; |
| 60 f.v = v; |
| 61 f.words.hw = hw; |
| 62 return f.v; |
| 63 } |
| 64 |
| 65 double const kLn2Hi = 6.93147180369123816490e-01; // 3fe62e42 fee00000 |
| 66 double const kLn2Lo = 1.90821492927058770002e-10; // 3dea39ef 35793c76 |
| 67 double const kTwo54 = 1.80143985094819840000e+16; // 43500000 00000000 |
| 68 double const kLg1 = 6.666666666666735130e-01; // 3FE55555 55555593 |
| 69 double const kLg2 = 3.999999999940941908e-01; // 3FD99999 9997FA04 |
| 70 double const kLg3 = 2.857142874366239149e-01; // 3FD24924 94229359 |
| 71 double const kLg4 = 2.222219843214978396e-01; // 3FCC71C5 1D8E78AF |
| 72 double const kLg5 = 1.818357216161805012e-01; // 3FC74664 96CB03DE |
| 73 double const kLg6 = 1.531383769920937332e-01; // 3FC39A09 D078C69F |
| 74 double const kLg7 = 1.479819860511658591e-01; // 3FC2F112 DF3E5244 |
| 75 |
| 76 } // namespace |
| 77 |
| 78 /* log(x) |
| 79 * Return the logrithm of x |
| 80 * |
| 81 * Method : |
| 82 * 1. Argument Reduction: find k and f such that |
| 83 * x = 2^k * (1+f), |
| 84 * where sqrt(2)/2 < 1+f < sqrt(2) . |
| 85 * |
| 86 * 2. Approximation of log(1+f). |
| 87 * Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s) |
| 88 * = 2s + 2/3 s**3 + 2/5 s**5 + ....., |
| 89 * = 2s + s*R |
| 90 * We use a special Reme algorithm on [0,0.1716] to generate |
| 91 * a polynomial of degree 14 to approximate R The maximum error |
| 92 * of this polynomial approximation is bounded by 2**-58.45. In |
| 93 * other words, |
| 94 * 2 4 6 8 10 12 14 |
| 95 * R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s +Lg6*s +Lg7*s |
| 96 * (the values of Lg1 to Lg7 are listed in the program) |
| 97 * and |
| 98 * | 2 14 | -58.45 |
| 99 * | Lg1*s +...+Lg7*s - R(z) | <= 2 |
| 100 * | | |
| 101 * Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2. |
| 102 * In order to guarantee error in log below 1ulp, we compute log |
| 103 * by |
| 104 * log(1+f) = f - s*(f - R) (if f is not too large) |
| 105 * log(1+f) = f - (hfsq - s*(hfsq+R)). (better accuracy) |
| 106 * |
| 107 * 3. Finally, log(x) = k*ln2 + log(1+f). |
| 108 * = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo))) |
| 109 * Here ln2 is split into two floating point number: |
| 110 * ln2_hi + ln2_lo, |
| 111 * where n*ln2_hi is always exact for |n| < 2000. |
| 112 * |
| 113 * Special cases: |
| 114 * log(x) is NaN with signal if x < 0 (including -INF) ; |
| 115 * log(+INF) is +INF; log(0) is -INF with signal; |
| 116 * log(NaN) is that NaN with no signal. |
| 117 * |
| 118 * Accuracy: |
| 119 * according to an error analysis, the error is always less than |
| 120 * 1 ulp (unit in the last place). |
| 121 * |
| 122 * Constants: |
| 123 * The hexadecimal values are the intended ones for the following |
| 124 * constants. The decimal values may be used, provided that the |
| 125 * compiler will convert from decimal to binary accurately enough |
| 126 * to produce the hexadecimal values shown. |
| 127 */ |
| 128 double log(double x) { |
| 129 double hfsq, f, s, z, r, w, t1, t2, dk; |
| 130 int32_t k = 0, i, j; |
| 131 int32_t hx = extractHighWord32(x); |
| 132 uint32_t lx = extractLowWord32(x); |
| 133 |
| 134 if (hx < 0x00100000) { /* x < 2**-1022 */ |
| 135 if (((hx & 0x7fffffff) | lx) == 0) { |
| 136 return -std::numeric_limits<double>::infinity(); |
| 137 } |
| 138 if (hx < 0) { |
| 139 return std::numeric_limits<double>::quiet_NaN(); |
| 140 } |
| 141 k -= 54; |
| 142 x *= kTwo54; /* subnormal number, scale up x */ |
| 143 hx = extractHighWord32(x); |
| 144 } |
| 145 if (hx >= 0x7ff00000) return x + x; |
| 146 k += (hx >> 20) - 1023; |
| 147 hx &= 0x000fffff; |
| 148 i = (hx + 0x95f64) & 0x100000; |
| 149 x = insertHighWord32(x, hx | (i ^ 0x3ff00000)); /* normalize x or x/2 */ |
| 150 k += (i >> 20); |
| 151 f = x - 1.0; |
| 152 if ((0x000fffff & (2 + hx)) < 3) { /* -2**-20 <= f < 2**-20 */ |
| 153 if (f == 0.0) { |
| 154 if (k == 0) { |
| 155 return 0.0; |
| 156 } else { |
| 157 dk = static_cast<double>(k); |
| 158 return dk * kLn2Hi + dk * kLn2Lo; |
| 159 } |
| 160 } |
| 161 r = f * f * (0.5 - 0.33333333333333333 * f); |
| 162 if (k == 0) { |
| 163 return f - r; |
| 164 } else { |
| 165 dk = static_cast<double>(k); |
| 166 return dk * kLn2Hi - ((r - dk * kLn2Lo) - f); |
| 167 } |
| 168 } |
| 169 s = f / (2.0 + f); |
| 170 dk = static_cast<double>(k); |
| 171 z = s * s; |
| 172 i = hx - 0x6147a; |
| 173 w = z * z; |
| 174 j = 0x6b851 - hx; |
| 175 t1 = w * (kLg2 + w * (kLg4 + w * kLg6)); |
| 176 t2 = z * (kLg1 + w * (kLg3 + w * (kLg5 + w * kLg7))); |
| 177 i |= j; |
| 178 r = t2 + t1; |
| 179 if (i > 0) { |
| 180 hfsq = 0.5 * f * f; |
| 181 if (k == 0) { |
| 182 return f - (hfsq - s * (hfsq + r)); |
| 183 } else { |
| 184 return dk * kLn2Hi - ((hfsq - (s * (hfsq + r) + dk * kLn2Lo)) - f); |
| 185 } |
| 186 } else { |
| 187 if (k == 0) { |
| 188 return f - s * (f - r); |
| 189 } else { |
| 190 return dk * kLn2Hi - ((s * (f - r) - dk * kLn2Lo) - f); |
| 191 } |
| 192 } |
| 193 } |
| 194 |
| 195 } // namespace ieee754 |
| 196 } // namespace base |
| 197 } // namespace v8 |
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