-/* 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 for most comments.
- *
- * Reduce x to 2^k (1+f) and calculate r = log(1+f) - f + f*f/2
- * as in log.c, then combine and scale in extra precision:
- * log2(x) = (f - f*f/2 + r)/log(2) + k
+ * Copyright (c) 2018, Arm Limited.
+ * SPDX-License-Identifier: MIT
*/
#include <math.h>
#include <stdint.h>
+#include "libm.h"
+#include "log2_data.h"
-static const double
-ivln2hi = 1.44269504072144627571e+00, /* 0x3ff71547, 0x65200000 */
-ivln2lo = 1.67517131648865118353e-10, /* 0x3de705fc, 0x2eefa200 */
-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 __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)
{
- union {double f; uint64_t i;} u = {x};
- double_t hfsq,f,s,z,R,w,t1,t2,y,hi,lo,val_hi,val_lo;
- uint32_t hx;
- int k;
-
- hx = u.i>>32;
- k = 0;
- if (hx < 0x00100000 || hx>>31) {
- if (u.i<<1 == 0)
- return -1/(x*x); /* log(+-0)=-inf */
- if (hx>>31)
- return (x-x)/0.0; /* log(-#) = NaN */
- /* subnormal number, scale x up */
- k -= 54;
- x *= 0x1p54;
- u.f = x;
- hx = u.i>>32;
- } else if (hx >= 0x7ff00000) {
- return x;
- } else if (hx == 0x3ff00000 && u.i<<32 == 0)
- return 0;
-
- /* reduce x into [sqrt(2)/2, sqrt(2)] */
- hx += 0x3ff00000 - 0x3fe6a09e;
- k += (int)(hx>>20) - 0x3ff;
- hx = (hx&0x000fffff) + 0x3fe6a09e;
- u.i = (uint64_t)hx<<32 | (u.i&0xffffffff);
- x = u.f;
+ return asuint64(x) >> 48;
+}
- f = x - 1.0;
- hfsq = 0.5*f*f;
- s = f/(2.0+f);
- z = s*s;
- w = z*z;
- t1 = w*(Lg2+w*(Lg4+w*Lg6));
- t2 = z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7)));
- R = t2 + t1;
+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;
- /*
- * 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.
- */
+ 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 (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;
+ }
- /* hi+lo = f - hfsq + s*(hfsq+R) ~ log(1+f) */
- hi = f - hfsq;
- u.f = hi;
- u.i &= (uint64_t)-1<<32;
- hi = u.f;
- lo = f - hi - hfsq + s*(hfsq+R);
+ /* 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;
- 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: */
- y = k;
- 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);
}
--- /dev/null
+/*
+ * Data for log2.
+ *
+ * Copyright (c) 2018, Arm Limited.
+ * SPDX-License-Identifier: MIT
+ */
+
+#include "log2_data.h"
+
+#define N (1 << LOG2_TABLE_BITS)
+
+const struct log2_data __log2_data = {
+// First coefficient: 0x1.71547652b82fe1777d0ffda0d24p0
+.invln2hi = 0x1.7154765200000p+0,
+.invln2lo = 0x1.705fc2eefa200p-33,
+.poly1 = {
+// relative error: 0x1.2fad8188p-63
+// in -0x1.5b51p-5 0x1.6ab2p-5
+-0x1.71547652b82fep-1,
+0x1.ec709dc3a03f7p-2,
+-0x1.71547652b7c3fp-2,
+0x1.2776c50f05be4p-2,
+-0x1.ec709dd768fe5p-3,
+0x1.a61761ec4e736p-3,
+-0x1.7153fbc64a79bp-3,
+0x1.484d154f01b4ap-3,
+-0x1.289e4a72c383cp-3,
+0x1.0b32f285aee66p-3,
+},
+.poly = {
+// relative error: 0x1.a72c2bf8p-58
+// abs error: 0x1.67a552c8p-66
+// in -0x1.f45p-8 0x1.f45p-8
+-0x1.71547652b8339p-1,
+0x1.ec709dc3a04bep-2,
+-0x1.7154764702ffbp-2,
+0x1.2776c50034c48p-2,
+-0x1.ec7b328ea92bcp-3,
+0x1.a6225e117f92ep-3,
+},
+/* Algorithm:
+
+ x = 2^k z
+ log2(x) = k + log2(c) + log2(z/c)
+ log2(z/c) = poly(z/c - 1)
+
+where z is in [1.6p-1; 1.6p0] which is split into N subintervals and z falls
+into the ith one, then table entries are computed as
+
+ tab[i].invc = 1/c
+ tab[i].logc = (double)log2(c)
+ tab2[i].chi = (double)c
+ tab2[i].clo = (double)(c - (double)c)
+
+where c is near the center of the subinterval and is chosen by trying +-2^29
+floating point invc candidates around 1/center and selecting one for which
+
+ 1) the rounding error in 0x1.8p10 + logc is 0,
+ 2) the rounding error in z - chi - clo is < 0x1p-64 and
+ 3) the rounding error in (double)log2(c) is minimized (< 0x1p-68).
+
+Note: 1) ensures that k + logc can be computed without rounding error, 2)
+ensures that z/c - 1 can be computed as (z - chi - clo)*invc with close to a
+single rounding error when there is no fast fma for z*invc - 1, 3) ensures
+that logc + poly(z/c - 1) has small error, however near x == 1 when
+|log2(x)| < 0x1p-4, this is not enough so that is special cased. */
+.tab = {
+{0x1.724286bb1acf8p+0, -0x1.1095feecdb000p-1},
+{0x1.6e1f766d2cca1p+0, -0x1.08494bd76d000p-1},
+{0x1.6a13d0e30d48ap+0, -0x1.00143aee8f800p-1},
+{0x1.661ec32d06c85p+0, -0x1.efec5360b4000p-2},
+{0x1.623fa951198f8p+0, -0x1.dfdd91ab7e000p-2},
+{0x1.5e75ba4cf026cp+0, -0x1.cffae0cc79000p-2},
+{0x1.5ac055a214fb8p+0, -0x1.c043811fda000p-2},
+{0x1.571ed0f166e1ep+0, -0x1.b0b67323ae000p-2},
+{0x1.53909590bf835p+0, -0x1.a152f5a2db000p-2},
+{0x1.5014fed61adddp+0, -0x1.9217f5af86000p-2},
+{0x1.4cab88e487bd0p+0, -0x1.8304db0719000p-2},
+{0x1.49539b4334feep+0, -0x1.74189f9a9e000p-2},
+{0x1.460cbdfafd569p+0, -0x1.6552bb5199000p-2},
+{0x1.42d664ee4b953p+0, -0x1.56b23a29b1000p-2},
+{0x1.3fb01111dd8a6p+0, -0x1.483650f5fa000p-2},
+{0x1.3c995b70c5836p+0, -0x1.39de937f6a000p-2},
+{0x1.3991c4ab6fd4ap+0, -0x1.2baa1538d6000p-2},
+{0x1.3698e0ce099b5p+0, -0x1.1d98340ca4000p-2},
+{0x1.33ae48213e7b2p+0, -0x1.0fa853a40e000p-2},
+{0x1.30d191985bdb1p+0, -0x1.01d9c32e73000p-2},
+{0x1.2e025cab271d7p+0, -0x1.e857da2fa6000p-3},
+{0x1.2b404cf13cd82p+0, -0x1.cd3c8633d8000p-3},
+{0x1.288b02c7ccb50p+0, -0x1.b26034c14a000p-3},
+{0x1.25e2263944de5p+0, -0x1.97c1c2f4fe000p-3},
+{0x1.234563d8615b1p+0, -0x1.7d6023f800000p-3},
+{0x1.20b46e33eaf38p+0, -0x1.633a71a05e000p-3},
+{0x1.1e2eefdcda3ddp+0, -0x1.494f5e9570000p-3},
+{0x1.1bb4a580b3930p+0, -0x1.2f9e424e0a000p-3},
+{0x1.19453847f2200p+0, -0x1.162595afdc000p-3},
+{0x1.16e06c0d5d73cp+0, -0x1.f9c9a75bd8000p-4},
+{0x1.1485f47b7e4c2p+0, -0x1.c7b575bf9c000p-4},
+{0x1.12358ad0085d1p+0, -0x1.960c60ff48000p-4},
+{0x1.0fef00f532227p+0, -0x1.64ce247b60000p-4},
+{0x1.0db2077d03a8fp+0, -0x1.33f78b2014000p-4},
+{0x1.0b7e6d65980d9p+0, -0x1.0387d1a42c000p-4},
+{0x1.0953efe7b408dp+0, -0x1.a6f9208b50000p-5},
+{0x1.07325cac53b83p+0, -0x1.47a954f770000p-5},
+{0x1.05197e40d1b5cp+0, -0x1.d23a8c50c0000p-6},
+{0x1.03091c1208ea2p+0, -0x1.16a2629780000p-6},
+{0x1.0101025b37e21p+0, -0x1.720f8d8e80000p-8},
+{0x1.fc07ef9caa76bp-1, 0x1.6fe53b1500000p-7},
+{0x1.f4465d3f6f184p-1, 0x1.11ccce10f8000p-5},
+{0x1.ecc079f84107fp-1, 0x1.c4dfc8c8b8000p-5},
+{0x1.e573a99975ae8p-1, 0x1.3aa321e574000p-4},
+{0x1.de5d6f0bd3de6p-1, 0x1.918a0d08b8000p-4},
+{0x1.d77b681ff38b3p-1, 0x1.e72e9da044000p-4},
+{0x1.d0cb5724de943p-1, 0x1.1dcd2507f6000p-3},
+{0x1.ca4b2dc0e7563p-1, 0x1.476ab03dea000p-3},
+{0x1.c3f8ee8d6cb51p-1, 0x1.7074377e22000p-3},
+{0x1.bdd2b4f020c4cp-1, 0x1.98ede8ba94000p-3},
+{0x1.b7d6c006015cap-1, 0x1.c0db86ad2e000p-3},
+{0x1.b20366e2e338fp-1, 0x1.e840aafcee000p-3},
+{0x1.ac57026295039p-1, 0x1.0790ab4678000p-2},
+{0x1.a6d01bc2731ddp-1, 0x1.1ac056801c000p-2},
+{0x1.a16d3bc3ff18bp-1, 0x1.2db11d4fee000p-2},
+{0x1.9c2d14967feadp-1, 0x1.406464ec58000p-2},
+{0x1.970e4f47c9902p-1, 0x1.52dbe093af000p-2},
+{0x1.920fb3982bcf2p-1, 0x1.651902050d000p-2},
+{0x1.8d30187f759f1p-1, 0x1.771d2cdeaf000p-2},
+{0x1.886e5ebb9f66dp-1, 0x1.88e9c857d9000p-2},
+{0x1.83c97b658b994p-1, 0x1.9a80155e16000p-2},
+{0x1.7f405ffc61022p-1, 0x1.abe186ed3d000p-2},
+{0x1.7ad22181415cap-1, 0x1.bd0f2aea0e000p-2},
+{0x1.767dcf99eff8cp-1, 0x1.ce0a43dbf4000p-2},
+},
+#if !__FP_FAST_FMA
+.tab2 = {
+{0x1.6200012b90a8ep-1, 0x1.904ab0644b605p-55},
+{0x1.66000045734a6p-1, 0x1.1ff9bea62f7a9p-57},
+{0x1.69fffc325f2c5p-1, 0x1.27ecfcb3c90bap-55},
+{0x1.6e00038b95a04p-1, 0x1.8ff8856739326p-55},
+{0x1.71fffe09994e3p-1, 0x1.afd40275f82b1p-55},
+{0x1.7600015590e1p-1, -0x1.2fd75b4238341p-56},
+{0x1.7a00012655bd5p-1, 0x1.808e67c242b76p-56},
+{0x1.7e0003259e9a6p-1, -0x1.208e426f622b7p-57},
+{0x1.81fffedb4b2d2p-1, -0x1.402461ea5c92fp-55},
+{0x1.860002dfafcc3p-1, 0x1.df7f4a2f29a1fp-57},
+{0x1.89ffff78c6b5p-1, -0x1.e0453094995fdp-55},
+{0x1.8e00039671566p-1, -0x1.a04f3bec77b45p-55},
+{0x1.91fffe2bf1745p-1, -0x1.7fa34400e203cp-56},
+{0x1.95fffcc5c9fd1p-1, -0x1.6ff8005a0695dp-56},
+{0x1.9a0003bba4767p-1, 0x1.0f8c4c4ec7e03p-56},
+{0x1.9dfffe7b92da5p-1, 0x1.e7fd9478c4602p-55},
+{0x1.a1fffd72efdafp-1, -0x1.a0c554dcdae7ep-57},
+{0x1.a5fffde04ff95p-1, 0x1.67da98ce9b26bp-55},
+{0x1.a9fffca5e8d2bp-1, -0x1.284c9b54c13dep-55},
+{0x1.adfffddad03eap-1, 0x1.812c8ea602e3cp-58},
+{0x1.b1ffff10d3d4dp-1, -0x1.efaddad27789cp-55},
+{0x1.b5fffce21165ap-1, 0x1.3cb1719c61237p-58},
+{0x1.b9fffd950e674p-1, 0x1.3f7d94194cep-56},
+{0x1.be000139ca8afp-1, 0x1.50ac4215d9bcp-56},
+{0x1.c20005b46df99p-1, 0x1.beea653e9c1c9p-57},
+{0x1.c600040b9f7aep-1, -0x1.c079f274a70d6p-56},
+{0x1.ca0006255fd8ap-1, -0x1.a0b4076e84c1fp-56},
+{0x1.cdfffd94c095dp-1, 0x1.8f933f99ab5d7p-55},
+{0x1.d1ffff975d6cfp-1, -0x1.82c08665fe1bep-58},
+{0x1.d5fffa2561c93p-1, -0x1.b04289bd295f3p-56},
+{0x1.d9fff9d228b0cp-1, 0x1.70251340fa236p-55},
+{0x1.de00065bc7e16p-1, -0x1.5011e16a4d80cp-56},
+{0x1.e200002f64791p-1, 0x1.9802f09ef62ep-55},
+{0x1.e600057d7a6d8p-1, -0x1.e0b75580cf7fap-56},
+{0x1.ea00027edc00cp-1, -0x1.c848309459811p-55},
+{0x1.ee0006cf5cb7cp-1, -0x1.f8027951576f4p-55},
+{0x1.f2000782b7dccp-1, -0x1.f81d97274538fp-55},
+{0x1.f6000260c450ap-1, -0x1.071002727ffdcp-59},
+{0x1.f9fffe88cd533p-1, -0x1.81bdce1fda8bp-58},
+{0x1.fdfffd50f8689p-1, 0x1.7f91acb918e6ep-55},
+{0x1.0200004292367p+0, 0x1.b7ff365324681p-54},
+{0x1.05fffe3e3d668p+0, 0x1.6fa08ddae957bp-55},
+{0x1.0a0000a85a757p+0, -0x1.7e2de80d3fb91p-58},
+{0x1.0e0001a5f3fccp+0, -0x1.1823305c5f014p-54},
+{0x1.11ffff8afbaf5p+0, -0x1.bfabb6680bac2p-55},
+{0x1.15fffe54d91adp+0, -0x1.d7f121737e7efp-54},
+{0x1.1a00011ac36e1p+0, 0x1.c000a0516f5ffp-54},
+{0x1.1e00019c84248p+0, -0x1.082fbe4da5dap-54},
+{0x1.220000ffe5e6ep+0, -0x1.8fdd04c9cfb43p-55},
+{0x1.26000269fd891p+0, 0x1.cfe2a7994d182p-55},
+{0x1.2a00029a6e6dap+0, -0x1.00273715e8bc5p-56},
+{0x1.2dfffe0293e39p+0, 0x1.b7c39dab2a6f9p-54},
+{0x1.31ffff7dcf082p+0, 0x1.df1336edc5254p-56},
+{0x1.35ffff05a8b6p+0, -0x1.e03564ccd31ebp-54},
+{0x1.3a0002e0eaeccp+0, 0x1.5f0e74bd3a477p-56},
+{0x1.3e000043bb236p+0, 0x1.c7dcb149d8833p-54},
+{0x1.4200002d187ffp+0, 0x1.e08afcf2d3d28p-56},
+{0x1.460000d387cb1p+0, 0x1.20837856599a6p-55},
+{0x1.4a00004569f89p+0, -0x1.9fa5c904fbcd2p-55},
+{0x1.4e000043543f3p+0, -0x1.81125ed175329p-56},
+{0x1.51fffcc027f0fp+0, 0x1.883d8847754dcp-54},
+{0x1.55ffffd87b36fp+0, -0x1.709e731d02807p-55},
+{0x1.59ffff21df7bap+0, 0x1.7f79f68727b02p-55},
+{0x1.5dfffebfc3481p+0, -0x1.180902e30e93ep-54},
+},
+#endif
+};
projects / oweals / musl.git / commitdiff