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Diffstat (limited to 'src/math/s_expm1.c')
| -rw-r--r-- | src/math/s_expm1.c | 217 |
1 files changed, 0 insertions, 217 deletions
diff --git a/src/math/s_expm1.c b/src/math/s_expm1.c deleted file mode 100644 index 6f1f6675..00000000 --- a/src/math/s_expm1.c +++ /dev/null @@ -1,217 +0,0 @@ -/* @(#)s_expm1.c 5.1 93/09/24 */ -/* - * ==================================================== - * 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. - * ==================================================== - */ - -/* expm1(x) - * Returns exp(x)-1, the exponential of x minus 1. - * - * Method - * 1. Argument reduction: - * Given x, find r and integer k such that - * - * x = k*ln2 + r, |r| <= 0.5*ln2 ~ 0.34658 - * - * Here a correction term c will be computed to compensate - * the error in r when rounded to a floating-point number. - * - * 2. Approximating expm1(r) by a special rational function on - * the interval [0,0.34658]: - * Since - * r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 - r^4/360 + ... - * we define R1(r*r) by - * r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 * R1(r*r) - * That is, - * R1(r**2) = 6/r *((exp(r)+1)/(exp(r)-1) - 2/r) - * = 6/r * ( 1 + 2.0*(1/(exp(r)-1) - 1/r)) - * = 1 - r^2/60 + r^4/2520 - r^6/100800 + ... - * We use a special Reme algorithm on [0,0.347] to generate - * a polynomial of degree 5 in r*r to approximate R1. The - * maximum error of this polynomial approximation is bounded - * by 2**-61. In other words, - * R1(z) ~ 1.0 + Q1*z + Q2*z**2 + Q3*z**3 + Q4*z**4 + Q5*z**5 - * where Q1 = -1.6666666666666567384E-2, - * Q2 = 3.9682539681370365873E-4, - * Q3 = -9.9206344733435987357E-6, - * Q4 = 2.5051361420808517002E-7, - * Q5 = -6.2843505682382617102E-9; - * (where z=r*r, and the values of Q1 to Q5 are listed below) - * with error bounded by - * | 5 | -61 - * | 1.0+Q1*z+...+Q5*z - R1(z) | <= 2 - * | | - * - * expm1(r) = exp(r)-1 is then computed by the following - * specific way which minimize the accumulation rounding error: - * 2 3 - * r r [ 3 - (R1 + R1*r/2) ] - * expm1(r) = r + --- + --- * [--------------------] - * 2 2 [ 6 - r*(3 - R1*r/2) ] - * - * To compensate the error in the argument reduction, we use - * expm1(r+c) = expm1(r) + c + expm1(r)*c - * ~ expm1(r) + c + r*c - * Thus c+r*c will be added in as the correction terms for - * expm1(r+c). Now rearrange the term to avoid optimization - * screw up: - * ( 2 2 ) - * ({ ( r [ R1 - (3 - R1*r/2) ] ) } r ) - * expm1(r+c)~r - ({r*(--- * [--------------------]-c)-c} - --- ) - * ({ ( 2 [ 6 - r*(3 - R1*r/2) ] ) } 2 ) - * ( ) - * - * = r - E - * 3. Scale back to obtain expm1(x): - * From step 1, we have - * expm1(x) = either 2^k*[expm1(r)+1] - 1 - * = or 2^k*[expm1(r) + (1-2^-k)] - * 4. Implementation notes: - * (A). To save one multiplication, we scale the coefficient Qi - * to Qi*2^i, and replace z by (x^2)/2. - * (B). To achieve maximum accuracy, we compute expm1(x) by - * (i) if x < -56*ln2, return -1.0, (raise inexact if x!=inf) - * (ii) if k=0, return r-E - * (iii) if k=-1, return 0.5*(r-E)-0.5 - * (iv) if k=1 if r < -0.25, return 2*((r+0.5)- E) - * else return 1.0+2.0*(r-E); - * (v) if (k<-2||k>56) return 2^k(1-(E-r)) - 1 (or exp(x)-1) - * (vi) if k <= 20, return 2^k((1-2^-k)-(E-r)), else - * (vii) return 2^k(1-((E+2^-k)-r)) - * - * Special cases: - * expm1(INF) is INF, expm1(NaN) is NaN; - * expm1(-INF) is -1, and - * for finite argument, only expm1(0)=0 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 expm1(x) overflow - * - * 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 -one = 1.0, -huge = 1.0e+300, -tiny = 1.0e-300, -o_threshold = 7.09782712893383973096e+02,/* 0x40862E42, 0xFEFA39EF */ -ln2_hi = 6.93147180369123816490e-01,/* 0x3fe62e42, 0xfee00000 */ -ln2_lo = 1.90821492927058770002e-10,/* 0x3dea39ef, 0x35793c76 */ -invln2 = 1.44269504088896338700e+00,/* 0x3ff71547, 0x652b82fe */ - /* scaled coefficients related to expm1 */ -Q1 = -3.33333333333331316428e-02, /* BFA11111 111110F4 */ -Q2 = 1.58730158725481460165e-03, /* 3F5A01A0 19FE5585 */ -Q3 = -7.93650757867487942473e-05, /* BF14CE19 9EAADBB7 */ -Q4 = 4.00821782732936239552e-06, /* 3ED0CFCA 86E65239 */ -Q5 = -2.01099218183624371326e-07; /* BE8AFDB7 6E09C32D */ - -double -expm1(double x) -{ - double y,hi,lo,c=0.0,t,e,hxs,hfx,r1; - int32_t k,xsb; - uint32_t hx; - - GET_HIGH_WORD(hx,x); - xsb = hx&0x80000000; /* sign bit of x */ - if(xsb==0) y=x; else y= -x; /* y = |x| */ - hx &= 0x7fffffff; /* high word of |x| */ - - /* filter out huge and non-finite argument */ - if(hx >= 0x4043687A) { /* if |x|>=56*ln2 */ - if(hx >= 0x40862E42) { /* if |x|>=709.78... */ - if(hx>=0x7ff00000) { - uint32_t low; - GET_LOW_WORD(low,x); - if(((hx&0xfffff)|low)!=0) - return x+x; /* NaN */ - else return (xsb==0)? x:-1.0;/* exp(+-inf)={inf,-1} */ - } - if(x > o_threshold) return huge*huge; /* overflow */ - } - if(xsb!=0) { /* x < -56*ln2, return -1.0 with inexact */ - if(x+tiny<0.0) /* raise inexact */ - return tiny-one; /* return -1 */ - } - } - - /* argument reduction */ - if(hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */ - if(hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */ - if(xsb==0) - {hi = x - ln2_hi; lo = ln2_lo; k = 1;} - else - {hi = x + ln2_hi; lo = -ln2_lo; k = -1;} - } else { - k = invln2*x+((xsb==0)?0.5:-0.5); - t = k; - hi = x - t*ln2_hi; /* t*ln2_hi is exact here */ - lo = t*ln2_lo; - } - x = hi - lo; - c = (hi-x)-lo; - } - else if(hx < 0x3c900000) { /* when |x|<2**-54, return x */ - t = huge+x; /* return x with inexact flags when x!=0 */ - return x - (t-(huge+x)); - } - else k = 0; - - /* x is now in primary range */ - hfx = 0.5*x; - hxs = x*hfx; - r1 = one+hxs*(Q1+hxs*(Q2+hxs*(Q3+hxs*(Q4+hxs*Q5)))); - t = 3.0-r1*hfx; - e = hxs*((r1-t)/(6.0 - x*t)); - if(k==0) return x - (x*e-hxs); /* c is 0 */ - else { - e = (x*(e-c)-c); - e -= hxs; - if(k== -1) return 0.5*(x-e)-0.5; - if(k==1) { - if(x < -0.25) return -2.0*(e-(x+0.5)); - else return one+2.0*(x-e); - } - if (k <= -2 || k>56) { /* suffice to return exp(x)-1 */ - uint32_t high; - y = one-(e-x); - GET_HIGH_WORD(high,y); - SET_HIGH_WORD(y,high+(k<<20)); /* add k to y's exponent */ - return y-one; - } - t = one; - if(k<20) { - uint32_t high; - SET_HIGH_WORD(t,0x3ff00000 - (0x200000>>k)); /* t=1-2^-k */ - y = t-(e-x); - GET_HIGH_WORD(high,y); - SET_HIGH_WORD(y,high+(k<<20)); /* add k to y's exponent */ - } else { - uint32_t high; - SET_HIGH_WORD(t,((0x3ff-k)<<20)); /* 2^-k */ - y = x-(e+t); - y += one; - GET_HIGH_WORD(high,y); - SET_HIGH_WORD(y,high+(k<<20)); /* add k to y's exponent */ - } - } - return y; -} |