-/* origin: FreeBSD /usr/src/lib/msun/src/e_log2.c */
/*
- * ====================================================
- * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
+ * Double-precision log2(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.
- * ====================================================
- */
-/*
- * Return the base 2 logarithm of x. See log.c and __log1p.h for most
- * comments.
- *
- * This reduces x to {k, 1+f} exactly as in e_log.c, then calls the kernel,
- * then does the combining and scaling steps
- * log2(x) = (f - 0.5*f*f + k_log1p(f)) / ln2 + k
- * in not-quite-routine extra precision.
+ * Copyright (c) 2018, Arm Limited.
+ * SPDX-License-Identifier: MIT
*/
+#include <math.h>
+#include <stdint.h>
#include "libm.h"
-#include "__log1p.h"
+#include "log2_data.h"
-static const double
-two54 = 1.80143985094819840000e+16, /* 0x43500000, 0x00000000 */
-ivln2hi = 1.44269504072144627571e+00, /* 0x3ff71547, 0x65200000 */
-ivln2lo = 1.67517131648865118353e-10; /* 0x3de705fc, 0x2eefa200 */
+#define T __log2_data.tab
+#define T2 __log2_data.tab2
+#define B __log2_data.poly1
+#define A __log2_data.poly
+#define InvLn2hi __log2_data.invln2hi
+#define InvLn2lo __log2_data.invln2lo
+#define N (1 << LOG2_TABLE_BITS)
+#define OFF 0x3fe6000000000000
-double log2(double x)
+/* Top 16 bits of a double. */
+static inline uint32_t top16(double x)
{
- double f,hfsq,hi,lo,r,val_hi,val_lo,w,y;
- int32_t i,k,hx;
- uint32_t lx;
+ return asuint64(x) >> 48;
+}
- EXTRACT_WORDS(hx, lx, x);
+double log2(double x)
+{
+ double_t z, r, r2, r4, y, invc, logc, kd, hi, lo, t1, t2, t3, p;
+ 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);
+ ix = asuint64(x);
+ top = top16(x);
+#define LO asuint64(1.0 - 0x1.5b51p-5)
+#define HI asuint64(1.0 + 0x1.6ab2p-5)
+ 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;
+#if __FP_FAST_FMA
+ hi = r * InvLn2hi;
+ lo = r * InvLn2lo + __builtin_fma(r, InvLn2hi, -hi);
+#else
+ double_t rhi, rlo;
+ rhi = asdouble(asuint64(r) & -1ULL << 32);
+ rlo = r - rhi;
+ hi = rhi * InvLn2hi;
+ lo = rlo * InvLn2hi + r * InvLn2lo;
+#endif
+ r2 = r * r; /* rounding error: 0x1p-62. */
+ r4 = r2 * r2;
+ /* Worst-case error is less than 0.54 ULP (0.55 ULP without fma). */
+ p = r2 * (B[0] + r * B[1]);
+ y = hi + p;
+ lo += hi - y + p;
+ lo += r4 * (B[2] + r * B[3] + r2 * (B[4] + r * B[5]) +
+ r4 * (B[6] + r * B[7] + r2 * (B[8] + r * B[9])));
+ y += lo;
+ return eval_as_double(y);
}
- if (hx >= 0x7ff00000)
- return x+x;
- if (hx == 0x3ff00000 && lx == 0)
- return 0.0; /* log(1) = +0 */
- 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;
- y = (double)k;
- f = x - 1.0;
- hfsq = 0.5*f*f;
- r = __log1p(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 - LOG2_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);
+ kd = (double_t)k;
- /*
- * f-hfsq must (for args near 1) be evaluated in extra precision
- * to avoid a large cancellation when x is near sqrt(2) or 1/sqrt(2).
- * This is fairly efficient since f-hfsq only depends on f, so can
- * be evaluated in parallel with R. Not combining hfsq with R also
- * keeps R small (though not as small as a true `lo' term would be),
- * so that extra precision is not needed for terms involving R.
- *
- * Compiler bugs involving extra precision used to break Dekker's
- * theorem for spitting f-hfsq as hi+lo, unless double_t was used
- * or the multi-precision calculations were avoided when double_t
- * has extra precision. These problems are now automatically
- * avoided as a side effect of the optimization of combining the
- * Dekker splitting step with the clear-low-bits step.
- *
- * y must (for args near sqrt(2) and 1/sqrt(2)) be added in extra
- * precision to avoid a very large cancellation when x is very near
- * these values. Unlike the above cancellations, this problem is
- * specific to base 2. It is strange that adding +-1 is so much
- * harder than adding +-ln2 or +-log10_2.
- *
- * This uses Dekker's theorem to normalize y+val_hi, so the
- * compiler bugs are back in some configurations, sigh. And I
- * don't want to used double_t to avoid them, since that gives a
- * pessimization and the support for avoiding the pessimization
- * is not yet available.
- *
- * The multi-precision calculations for the multiplications are
- * routine.
- */
- hi = f - hfsq;
- SET_LOW_WORD(hi, 0);
- lo = (f - hi) - hfsq + r;
- val_hi = hi*ivln2hi;
- val_lo = (lo+hi)*ivln2lo + lo*ivln2hi;
+ /* log2(x) = log2(z/c) + log2(c) + k. */
+ /* r ~= z/c - 1, |r| < 1/(2*N). */
+#if __FP_FAST_FMA
+ /* rounding error: 0x1p-55/N. */
+ r = __builtin_fma(z, invc, -1.0);
+ t1 = r * InvLn2hi;
+ t2 = r * InvLn2lo + __builtin_fma(r, InvLn2hi, -t1);
+#else
+ double_t rhi, rlo;
+ /* rounding error: 0x1p-55/N + 0x1p-65. */
+ r = (z - T2[i].chi - T2[i].clo) * invc;
+ rhi = asdouble(asuint64(r) & -1ULL << 32);
+ rlo = r - rhi;
+ t1 = rhi * InvLn2hi;
+ t2 = rlo * InvLn2hi + r * InvLn2lo;
+#endif
- /* spadd(val_hi, val_lo, y), except for not using double_t: */
- w = y + val_hi;
- val_lo += (y - w) + val_hi;
- val_hi = w;
+ /* hi + lo = r/ln2 + log2(c) + k. */
+ t3 = kd + logc;
+ hi = t3 + t1;
+ lo = t3 - hi + t1 + t2;
- return val_lo + val_hi;
+ /* log2(r+1) = r/ln2 + r^2*poly(r). */
+ /* Evaluation is optimized assuming superscalar pipelined execution. */
+ r2 = r * r; /* rounding error: 0x1p-54/N^2. */
+ r4 = r2 * r2;
+ /* Worst-case error if |y| > 0x1p-4: 0.547 ULP (0.550 ULP without fma).
+ ~ 0.5 + 2/N/ln2 + abs-poly-error*0x1p56 ULP (+ 0.003 ULP without fma). */
+ p = A[0] + r * A[1] + r2 * (A[2] + r * A[3]) + r4 * (A[4] + r * A[5]);
+ y = lo + r2 * p + hi;
+ return eval_as_double(y);
}
projects / musl / blobdiff