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 diff --git a/src/math/s_expm1.c b/src/math/s_expm1.cnew file mode 100644index 00000000..6f1f6675--- /dev/null+++ b/src/math/s_expm1.c@@ -0,0 +1,217 @@+/* @(#)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 +#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;+}