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- /* Adapted for log2 by Ulrich Drepper <drepper@cygnus.com>. */
- /*
- * ====================================================
- * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
- *
- * Developed at SunPro, a Sun Microsystems, Inc. business.
- * Permission to use, copy, modify, and distribute this
- * software is freely granted, provided that this notice
- * is preserved.
- * ====================================================
- */
- /* __ieee754_log2(x)
- * Return the logarithm to base 2 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 Reme 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 + log(1+f).
- * = k+(f-(hfsq-(s*(hfsq+R))))
- *
- * Special cases:
- * log2(x) is NaN with signal if x < 0 (including -INF) ;
- * log2(+INF) is +INF; log(0) is -INF with signal;
- * log2(NaN) is that NaN with no signal.
- *
- * 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.
- */
- #include "math.h"
- #include "math_private.h"
- static const double
- ln2 = 0.69314718055994530942,
- 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 */
- static const double zero = 0.0;
- double __ieee754_log2(double x)
- {
- double hfsq,f,s,z,R,w,t1,t2,dk;
- int32_t k,hx,i,j;
- u_int32_t lx;
- EXTRACT_WORDS(hx,lx,x);
- k=0;
- if (hx < 0x00100000) { /* x < 2**-1022 */
- if (((hx&0x7fffffff)|lx)==0)
- return -two54/(x-x); /* log(+-0)=-inf */
- if (hx<0) return (x-x)/(x-x); /* log(-#) = NaN */
- k -= 54; x *= two54; /* subnormal number, scale up x */
- 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);
- dk = (double) k;
- f = x-1.0;
- if((0x000fffff&(2+hx))<3) { /* |f| < 2**-20 */
- if(f==zero) return dk;
- R = f*f*(0.5-0.33333333333333333*f);
- return dk-(R-f)/ln2;
- }
- s = f/(2.0+f);
- 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;
- return dk-((hfsq-(s*(hfsq+R)))-f)/ln2;
- } else {
- return dk-((s*(f-R))-f)/ln2;
- }
- }
- strong_alias(__ieee754_log2,log2)
- libm_hidden_def(log2)
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