X-Git-Url: http://nsz.repo.hu/git/?a=blobdiff_plain;f=src%2Fmath%2Flog.c;h=cc52585a949886be2788a80f46af23f0784fde7f;hb=8274aaaaa1948c50c661aa32e21b3db27a5c0eab;hp=98051205f80ec607d416a645a1d77d63cb0eebc8;hpb=634c3a63027aa4a693b64fae0e2f6e1635558e93;p=musl diff --git a/src/math/log.c b/src/math/log.c index 98051205..cc52585a 100644 --- a/src/math/log.c +++ b/src/math/log.c @@ -1,138 +1,112 @@ -/* origin: FreeBSD /usr/src/lib/msun/src/e_log.c */ /* - * ==================================================== - * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * Double-precision log(x) function. * - * Developed at SunSoft, a Sun Microsystems, Inc. business. - * Permission to use, copy, modify, and distribute this - * software is freely granted, provided that this notice - * is preserved. - * ==================================================== - */ -/* log(x) - * Return the logrithm of x - * - * Method : - * 1. Argument Reduction: find k and f such that - * x = 2^k * (1+f), - * where sqrt(2)/2 < 1+f < sqrt(2) . - * - * 2. Approximation of log(1+f). - * Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s) - * = 2s + 2/3 s**3 + 2/5 s**5 + ....., - * = 2s + s*R - * We use a special Remez algorithm on [0,0.1716] to generate - * a polynomial of degree 14 to approximate R The maximum error - * of this polynomial approximation is bounded by 2**-58.45. In - * other words, - * 2 4 6 8 10 12 14 - * R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s +Lg6*s +Lg7*s - * (the values of Lg1 to Lg7 are listed in the program) - * and - * | 2 14 | -58.45 - * | Lg1*s +...+Lg7*s - R(z) | <= 2 - * | | - * Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2. - * In order to guarantee error in log below 1ulp, we compute log - * by - * log(1+f) = f - s*(f - R) (if f is not too large) - * log(1+f) = f - (hfsq - s*(hfsq+R)). (better accuracy) - * - * 3. Finally, log(x) = k*ln2 + log(1+f). - * = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo))) - * Here ln2 is split into two floating point number: - * ln2_hi + ln2_lo, - * where n*ln2_hi is always exact for |n| < 2000. - * - * Special cases: - * log(x) is NaN with signal if x < 0 (including -INF) ; - * log(+INF) is +INF; log(0) is -INF with signal; - * log(NaN) is that NaN with no signal. - * - * Accuracy: - * according to an error analysis, the error is always less than - * 1 ulp (unit in the last place). - * - * Constants: - * The hexadecimal values are the intended ones for the following - * constants. The decimal values may be used, provided that the - * compiler will convert from decimal to binary accurately enough - * to produce the hexadecimal values shown. + * Copyright (c) 2018, Arm Limited. + * SPDX-License-Identifier: MIT */ +#include +#include #include "libm.h" +#include "log_data.h" -static const double -ln2_hi = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */ -ln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */ -two54 = 1.80143985094819840000e+16, /* 43500000 00000000 */ -Lg1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */ -Lg2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */ -Lg3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */ -Lg4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */ -Lg5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */ -Lg6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */ -Lg7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */ +#define T __log_data.tab +#define T2 __log_data.tab2 +#define B __log_data.poly1 +#define A __log_data.poly +#define Ln2hi __log_data.ln2hi +#define Ln2lo __log_data.ln2lo +#define N (1 << LOG_TABLE_BITS) +#define OFF 0x3fe6000000000000 -double log(double x) +/* Top 16 bits of a double. */ +static inline uint32_t top16(double x) { - double hfsq,f,s,z,R,w,t1,t2,dk; - int32_t k,hx,i,j; - uint32_t lx; + return asuint64(x) >> 48; +} - EXTRACT_WORDS(hx, lx, x); +double log(double x) +{ + double_t w, z, r, r2, r3, y, invc, logc, kd, hi, lo; + uint64_t ix, iz, tmp; + uint32_t top; + int k, i; - k = 0; - if (hx < 0x00100000) { /* x < 2**-1022 */ - if (((hx&0x7fffffff)|lx) == 0) - return -two54/0.0; /* log(+-0)=-inf */ - if (hx < 0) - return (x-x)/0.0; /* log(-#) = NaN */ - /* subnormal number, scale up x */ - k -= 54; - x *= two54; - GET_HIGH_WORD(hx,x); - } - if (hx >= 0x7ff00000) - return x+x; - k += (hx>>20) - 1023; - hx &= 0x000fffff; - i = (hx+0x95f64)&0x100000; - SET_HIGH_WORD(x, hx|(i^0x3ff00000)); /* normalize x or x/2 */ - k += i>>20; - f = x - 1.0; - if ((0x000fffff&(2+hx)) < 3) { /* -2**-20 <= f < 2**-20 */ - if (f == 0.0) { - if (k == 0) { - return 0.0; - } - dk = (double)k; - return dk*ln2_hi + dk*ln2_lo; - } - R = f*f*(0.5-0.33333333333333333*f); - if (k == 0) - return f - R; - dk = (double)k; - return dk*ln2_hi - ((R-dk*ln2_lo)-f); + ix = asuint64(x); + top = top16(x); +#define LO asuint64(1.0 - 0x1p-4) +#define HI asuint64(1.0 + 0x1.09p-4) + if (predict_false(ix - LO < HI - LO)) { + /* Handle close to 1.0 inputs separately. */ + /* Fix sign of zero with downward rounding when x==1. */ + if (WANT_ROUNDING && predict_false(ix == asuint64(1.0))) + return 0; + r = x - 1.0; + r2 = r * r; + r3 = r * r2; + y = r3 * + (B[1] + r * B[2] + r2 * B[3] + + r3 * (B[4] + r * B[5] + r2 * B[6] + + r3 * (B[7] + r * B[8] + r2 * B[9] + r3 * B[10]))); + /* Worst-case error is around 0.507 ULP. */ + w = r * 0x1p27; + double_t rhi = r + w - w; + double_t rlo = r - rhi; + w = rhi * rhi * B[0]; /* B[0] == -0.5. */ + hi = r + w; + lo = r - hi + w; + lo += B[0] * rlo * (rhi + r); + y += lo; + y += hi; + return eval_as_double(y); } - s = f/(2.0+f); - dk = (double)k; - z = s*s; - i = hx - 0x6147a; - w = z*z; - j = 0x6b851 - hx; - t1 = w*(Lg2+w*(Lg4+w*Lg6)); - t2 = z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7))); - i |= j; - R = t2 + t1; - if (i > 0) { - hfsq = 0.5*f*f; - if (k == 0) - return f - (hfsq-s*(hfsq+R)); - return dk*ln2_hi - ((hfsq-(s*(hfsq+R)+dk*ln2_lo))-f); - } else { - if (k == 0) - return f - s*(f-R); - return dk*ln2_hi - ((s*(f-R)-dk*ln2_lo)-f); + if (predict_false(top - 0x0010 >= 0x7ff0 - 0x0010)) { + /* x < 0x1p-1022 or inf or nan. */ + if (ix * 2 == 0) + return __math_divzero(1); + if (ix == asuint64(INFINITY)) /* log(inf) == inf. */ + return x; + if ((top & 0x8000) || (top & 0x7ff0) == 0x7ff0) + return __math_invalid(x); + /* x is subnormal, normalize it. */ + ix = asuint64(x * 0x1p52); + ix -= 52ULL << 52; } + + /* x = 2^k z; where z is in range [OFF,2*OFF) and exact. + The range is split into N subintervals. + The ith subinterval contains z and c is near its center. */ + tmp = ix - OFF; + i = (tmp >> (52 - LOG_TABLE_BITS)) % N; + k = (int64_t)tmp >> 52; /* arithmetic shift */ + iz = ix - (tmp & 0xfffULL << 52); + invc = T[i].invc; + logc = T[i].logc; + z = asdouble(iz); + + /* log(x) = log1p(z/c-1) + log(c) + k*Ln2. */ + /* r ~= z/c - 1, |r| < 1/(2*N). */ +#if __FP_FAST_FMA + /* rounding error: 0x1p-55/N. */ + r = __builtin_fma(z, invc, -1.0); +#else + /* rounding error: 0x1p-55/N + 0x1p-66. */ + r = (z - T2[i].chi - T2[i].clo) * invc; +#endif + kd = (double_t)k; + + /* hi + lo = r + log(c) + k*Ln2. */ + w = kd * Ln2hi + logc; + hi = w + r; + lo = w - hi + r + kd * Ln2lo; + + /* log(x) = lo + (log1p(r) - r) + hi. */ + r2 = r * r; /* rounding error: 0x1p-54/N^2. */ + /* Worst case error if |y| > 0x1p-5: + 0.5 + 4.13/N + abs-poly-error*2^57 ULP (+ 0.002 ULP without fma) + Worst case error if |y| > 0x1p-4: + 0.5 + 2.06/N + abs-poly-error*2^56 ULP (+ 0.001 ULP without fma). */ + y = lo + r2 * A[0] + + r * r2 * (A[1] + r * A[2] + r2 * (A[3] + r * A[4])) + hi; + return eval_as_double(y); }