1/* origin: FreeBSD /usr/src/lib/msun/src/e_log10.c */
  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/*
 13 * Return the base 10 logarithm of x.  See log.c for most comments.
 14 *
 15 * Reduce x to 2^k (1+f) and calculate r = log(1+f) - f + f*f/2
 16 * as in log.c, then combine and scale in extra precision:
 17 *    log10(x) = (f - f*f/2 + r)/log(10) + k*log10(2)
 18 */
 19
 20#include <math.h>
 21#include <stdint.h>
 22
 23static const double
 24ivln10hi  = 4.34294481878168880939e-01, /* 0x3fdbcb7b, 0x15200000 */
 25ivln10lo  = 2.50829467116452752298e-11, /* 0x3dbb9438, 0xca9aadd5 */
 26log10_2hi = 3.01029995663611771306e-01, /* 0x3FD34413, 0x509F6000 */
 27log10_2lo = 3.69423907715893078616e-13, /* 0x3D59FEF3, 0x11F12B36 */
 28Lg1 = 6.666666666666735130e-01,  /* 3FE55555 55555593 */
 29Lg2 = 3.999999999940941908e-01,  /* 3FD99999 9997FA04 */
 30Lg3 = 2.857142874366239149e-01,  /* 3FD24924 94229359 */
 31Lg4 = 2.222219843214978396e-01,  /* 3FCC71C5 1D8E78AF */
 32Lg5 = 1.818357216161805012e-01,  /* 3FC74664 96CB03DE */
 33Lg6 = 1.531383769920937332e-01,  /* 3FC39A09 D078C69F */
 34Lg7 = 1.479819860511658591e-01;  /* 3FC2F112 DF3E5244 */
 35
 36double log10(double x)
 37{
 38	union {double f; uint64_t i;} u = {x};
 39	double_t hfsq,f,s,z,R,w,t1,t2,dk,y,hi,lo,val_hi,val_lo;
 40	uint32_t hx;
 41	int k;
 42
 43	hx = u.i>>32;
 44	k = 0;
 45	if (hx < 0x00100000 || hx>>31) {
 46		if (u.i<<1 == 0)
 47			return -1/(x*x);  /* log(+-0)=-inf */
 48		if (hx>>31)
 49			return (x-x)/0.0; /* log(-#) = NaN */
 50		/* subnormal number, scale x up */
 51		k -= 54;
 52		x *= 0x1p54;
 53		u.f = x;
 54		hx = u.i>>32;
 55	} else if (hx >= 0x7ff00000) {
 56		return x;
 57	} else if (hx == 0x3ff00000 && u.i<<32 == 0)
 58		return 0;
 59
 60	/* reduce x into [sqrt(2)/2, sqrt(2)] */
 61	hx += 0x3ff00000 - 0x3fe6a09e;
 62	k += (int)(hx>>20) - 0x3ff;
 63	hx = (hx&0x000fffff) + 0x3fe6a09e;
 64	u.i = (uint64_t)hx<<32 | (u.i&0xffffffff);
 65	x = u.f;
 66
 67	f = x - 1.0;
 68	hfsq = 0.5*f*f;
 69	s = f/(2.0+f);
 70	z = s*s;
 71	w = z*z;
 72	t1 = w*(Lg2+w*(Lg4+w*Lg6));
 73	t2 = z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7)));
 74	R = t2 + t1;
 75
 76	/* See log2.c for details. */
 77	/* hi+lo = f - hfsq + s*(hfsq+R) ~ log(1+f) */
 78	hi = f - hfsq;
 79	u.f = hi;
 80	u.i &= (uint64_t)-1<<32;
 81	hi = u.f;
 82	lo = f - hi - hfsq + s*(hfsq+R);
 83
 84	/* val_hi+val_lo ~ log10(1+f) + k*log10(2) */
 85	val_hi = hi*ivln10hi;
 86	dk = k;
 87	y = dk*log10_2hi;
 88	val_lo = dk*log10_2lo + (lo+hi)*ivln10lo + lo*ivln10hi;
 89
 90	/*
 91	 * Extra precision in for adding y is not strictly needed
 92	 * since there is no very large cancellation near x = sqrt(2) or
 93	 * x = 1/sqrt(2), but we do it anyway since it costs little on CPUs
 94	 * with some parallelism and it reduces the error for many args.
 95	 */
 96	w = y + val_hi;
 97	val_lo += (y - w) + val_hi;
 98	val_hi = w;
 99
100	return val_lo + val_hi;
101}