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 /* @(#)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. * ==================================================== */ #ifndef lint static char rcsid[] = "\$FreeBSD: src/lib/msun/src/s_expm1.c,v 1.7 2002/05/28 18:15:04 alfred Exp \$"; #endif /* 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,t,e,hxs,hfx,r1; int32_t k,xsb; u_int32_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) { u_int32_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 */ u_int32_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) { u_int32_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 { u_int32_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; }