-/* 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 <math.h>
+#include <stdint.h>
#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);
}
projects / musl / blobdiff