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| Original file line number | Diff line number | Diff line change |
|---|---|---|
| @@ -0,0 +1,182 @@ | ||
| /* | ||
| * ==================================================== | ||
| * 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_exp(x) | ||
| * Returns the exponential of x. | ||
| * | ||
| * Method | ||
| * 1. Argument reduction: | ||
| * Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658. | ||
| * Given x, find r and integer k such that | ||
| * | ||
| * x = k*ln2 + r, |r| <= 0.5*ln2. | ||
| * | ||
| * Here r will be represented as r = hi-lo for better | ||
| * accuracy. | ||
| * | ||
| * 2. Approximation of exp(r) by a special rational function on | ||
| * the interval [0,0.34658]: | ||
| * Write | ||
| * R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ... | ||
| * We use a special Reme algorithm on [0,0.34658] to generate | ||
| * a polynomial of degree 5 to approximate R. The maximum error | ||
| * of this polynomial approximation is bounded by 2**-59. In | ||
| * other words, | ||
| * R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5 | ||
| * (where z=r*r, and the values of P1 to P5 are listed below) | ||
| * and | ||
| * | 5 | -59 | ||
| * | 2.0+P1*z+...+P5*z - R(z) | <= 2 | ||
| * | | | ||
| * The computation of exp(r) thus becomes | ||
| * 2*r | ||
| * exp(r) = 1 + ------- | ||
| * R - r | ||
| * r*R1(r) | ||
| * = 1 + r + ----------- (for better accuracy) | ||
| * 2 - R1(r) | ||
| * where | ||
| * 2 4 10 | ||
| * R1(r) = r - (P1*r + P2*r + ... + P5*r ). | ||
| * | ||
| * 3. Scale back to obtain exp(x): | ||
| * From step 1, we have | ||
| * exp(x) = 2^k * exp(r) | ||
| * | ||
| * Special cases: | ||
| * exp(INF) is INF, exp(NaN) is NaN; | ||
| * exp(-INF) is 0, and | ||
| * for finite argument, only exp(0)=1 is exact. | ||
| * | ||
| * Accuracy: | ||
| * according to an error analysis, the error is always less than | ||
| * 1 ulp (unit in the last place). | ||
| * | ||
| * Misc. info. | ||
| * For IEEE double | ||
| * if x > 7.09782712893383973096e+02 then exp(x) overflow | ||
| * if x < -7.45133219101941108420e+02 then exp(x) underflow | ||
| * | ||
| * 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_libm.h" | ||
| #include "math_private.h" | ||
|
|
||
| static const double | ||
| one = 1.0, | ||
| halF[2] = {0.5,-0.5,}, | ||
| huge = 1.0e+300, | ||
| twom1000= 9.33263618503218878990e-302, /* 2**-1000=0x01700000,0*/ | ||
| o_threshold= 7.09782712893383973096e+02, /* 0x40862E42, 0xFEFA39EF */ | ||
| u_threshold= -7.45133219101941108420e+02, /* 0xc0874910, 0xD52D3051 */ | ||
| ln2HI[2] ={ 6.93147180369123816490e-01, /* 0x3fe62e42, 0xfee00000 */ | ||
| -6.93147180369123816490e-01,},/* 0xbfe62e42, 0xfee00000 */ | ||
| ln2LO[2] ={ 1.90821492927058770002e-10, /* 0x3dea39ef, 0x35793c76 */ | ||
| -1.90821492927058770002e-10,},/* 0xbdea39ef, 0x35793c76 */ | ||
| invln2 = 1.44269504088896338700e+00, /* 0x3ff71547, 0x652b82fe */ | ||
| P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */ | ||
| P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */ | ||
| P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */ | ||
| P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */ | ||
| P5 = 4.13813679705723846039e-08; /* 0x3E663769, 0x72BEA4D0 */ | ||
|
|
||
| double __ieee754_exp(double x) /* default IEEE double exp */ | ||
| { | ||
| double y; | ||
| double hi = 0.0; | ||
| double lo = 0.0; | ||
| double c; | ||
| double t; | ||
| int32_t k=0; | ||
| int32_t xsb; | ||
| u_int32_t hx; | ||
|
|
||
| GET_HIGH_WORD(hx,x); | ||
| xsb = (hx>>31)&1; /* sign bit of x */ | ||
| hx &= 0x7fffffff; /* high word of |x| */ | ||
|
|
||
| /* filter out non-finite argument */ | ||
| if(hx >= 0x40862E42) { /* if |x|>=709.78... */ | ||
| if(hx>=0x7ff00000) { | ||
| u_int32_t lx; | ||
| GET_LOW_WORD(lx,x); | ||
| if(((hx&0xfffff)|lx)!=0) | ||
| return x+x; /* NaN */ | ||
| else return (xsb==0)? x:0.0; /* exp(+-inf)={inf,0} */ | ||
| } | ||
| if(x > o_threshold) return huge*huge; /* overflow */ | ||
| if(x < u_threshold) return twom1000*twom1000; /* underflow */ | ||
| } | ||
|
|
||
| /* argument reduction */ | ||
| if(hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */ | ||
| if(hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */ | ||
| hi = x-ln2HI[xsb]; lo=ln2LO[xsb]; k = 1-xsb-xsb; | ||
| } else { | ||
| k = invln2*x+halF[xsb]; | ||
| t = k; | ||
| hi = x - t*ln2HI[0]; /* t*ln2HI is exact here */ | ||
| lo = t*ln2LO[0]; | ||
| } | ||
| x = hi - lo; | ||
| } | ||
| else if(hx < 0x3e300000) { /* when |x|<2**-28 */ | ||
| if(huge+x>one) return one+x;/* trigger inexact */ | ||
| } | ||
| else k = 0; | ||
|
|
||
| /* x is now in primary range */ | ||
| t = x*x; | ||
| c = x - t*(P1+t*(P2+t*(P3+t*(P4+t*P5)))); | ||
| if(k==0) return one-((x*c)/(c-2.0)-x); | ||
| else y = one-((lo-(x*c)/(2.0-c))-hi); | ||
| if(k >= -1021) { | ||
| u_int32_t hy; | ||
| GET_HIGH_WORD(hy,y); | ||
| SET_HIGH_WORD(y,hy+(k<<20)); /* add k to y's exponent */ | ||
| return y; | ||
| } else { | ||
| u_int32_t hy; | ||
| GET_HIGH_WORD(hy,y); | ||
| SET_HIGH_WORD(y,hy+((k+1000)<<20)); /* add k to y's exponent */ | ||
| return y*twom1000; | ||
| } | ||
| } | ||
|
|
||
| /* | ||
| * wrapper exp(x) | ||
| */ | ||
| #ifndef _IEEE_LIBM | ||
| double exp(double x) | ||
| { | ||
| static const double o_threshold = 7.09782712893383973096e+02; /* 0x40862E42, 0xFEFA39EF */ | ||
| static const double u_threshold = -7.45133219101941108420e+02; /* 0xc0874910, 0xD52D3051 */ | ||
|
|
||
| double z = __ieee754_exp(x); | ||
| if (_LIB_VERSION == _IEEE_) | ||
| return z; | ||
| if (isfinite(x)) { | ||
| if (x > o_threshold) | ||
| return __kernel_standard(x, x, 6); /* exp overflow */ | ||
| if (x < u_threshold) | ||
| return __kernel_standard(x, x, 7); /* exp underflow */ | ||
| } | ||
| return z; | ||
| } | ||
| #else | ||
| strong_alias(__ieee754_exp, exp) | ||
| #endif | ||
| libm_hidden_def(exp) |
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