s_expm1.c revision 97413
12116Sjkh/* @(#)s_expm1.c 5.1 93/09/24 */ 22116Sjkh/* 32116Sjkh * ==================================================== 42116Sjkh * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 52116Sjkh * 62116Sjkh * Developed at SunPro, a Sun Microsystems, Inc. business. 72116Sjkh * Permission to use, copy, modify, and distribute this 88870Srgrimes * software is freely granted, provided that this notice 92116Sjkh * is preserved. 102116Sjkh * ==================================================== 112116Sjkh */ 122116Sjkh 132116Sjkh#ifndef lint 1450476Speterstatic char rcsid[] = "$FreeBSD: head/lib/msun/src/s_expm1.c 97413 2002-05-28 18:15:04Z alfred $"; 152116Sjkh#endif 162116Sjkh 172116Sjkh/* expm1(x) 182116Sjkh * Returns exp(x)-1, the exponential of x minus 1. 192116Sjkh * 202116Sjkh * Method 212116Sjkh * 1. Argument reduction: 222116Sjkh * Given x, find r and integer k such that 232116Sjkh * 248870Srgrimes * x = k*ln2 + r, |r| <= 0.5*ln2 ~ 0.34658 252116Sjkh * 268870Srgrimes * Here a correction term c will be computed to compensate 272116Sjkh * the error in r when rounded to a floating-point number. 282116Sjkh * 292116Sjkh * 2. Approximating expm1(r) by a special rational function on 302116Sjkh * the interval [0,0.34658]: 312116Sjkh * Since 322116Sjkh * r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 - r^4/360 + ... 332116Sjkh * we define R1(r*r) by 342116Sjkh * r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 * R1(r*r) 352116Sjkh * That is, 362116Sjkh * R1(r**2) = 6/r *((exp(r)+1)/(exp(r)-1) - 2/r) 372116Sjkh * = 6/r * ( 1 + 2.0*(1/(exp(r)-1) - 1/r)) 382116Sjkh * = 1 - r^2/60 + r^4/2520 - r^6/100800 + ... 398870Srgrimes * We use a special Reme algorithm on [0,0.347] to generate 408870Srgrimes * a polynomial of degree 5 in r*r to approximate R1. The 418870Srgrimes * maximum error of this polynomial approximation is bounded 422116Sjkh * by 2**-61. In other words, 432116Sjkh * R1(z) ~ 1.0 + Q1*z + Q2*z**2 + Q3*z**3 + Q4*z**4 + Q5*z**5 442116Sjkh * where Q1 = -1.6666666666666567384E-2, 452116Sjkh * Q2 = 3.9682539681370365873E-4, 462116Sjkh * Q3 = -9.9206344733435987357E-6, 472116Sjkh * Q4 = 2.5051361420808517002E-7, 482116Sjkh * Q5 = -6.2843505682382617102E-9; 492116Sjkh * (where z=r*r, and the values of Q1 to Q5 are listed below) 502116Sjkh * with error bounded by 512116Sjkh * | 5 | -61 528870Srgrimes * | 1.0+Q1*z+...+Q5*z - R1(z) | <= 2 532116Sjkh * | | 548870Srgrimes * 558870Srgrimes * expm1(r) = exp(r)-1 is then computed by the following 568870Srgrimes * specific way which minimize the accumulation rounding error: 572116Sjkh * 2 3 582116Sjkh * r r [ 3 - (R1 + R1*r/2) ] 592116Sjkh * expm1(r) = r + --- + --- * [--------------------] 602116Sjkh * 2 2 [ 6 - r*(3 - R1*r/2) ] 618870Srgrimes * 622116Sjkh * To compensate the error in the argument reduction, we use 638870Srgrimes * expm1(r+c) = expm1(r) + c + expm1(r)*c 648870Srgrimes * ~ expm1(r) + c + r*c 652116Sjkh * Thus c+r*c will be added in as the correction terms for 668870Srgrimes * expm1(r+c). Now rearrange the term to avoid optimization 672116Sjkh * screw up: 682116Sjkh * ( 2 2 ) 692116Sjkh * ({ ( r [ R1 - (3 - R1*r/2) ] ) } r ) 702116Sjkh * expm1(r+c)~r - ({r*(--- * [--------------------]-c)-c} - --- ) 712116Sjkh * ({ ( 2 [ 6 - r*(3 - R1*r/2) ] ) } 2 ) 722116Sjkh * ( ) 738870Srgrimes * 742116Sjkh * = r - E 752116Sjkh * 3. Scale back to obtain expm1(x): 762116Sjkh * From step 1, we have 772116Sjkh * expm1(x) = either 2^k*[expm1(r)+1] - 1 782116Sjkh * = or 2^k*[expm1(r) + (1-2^-k)] 792116Sjkh * 4. Implementation notes: 802116Sjkh * (A). To save one multiplication, we scale the coefficient Qi 812116Sjkh * to Qi*2^i, and replace z by (x^2)/2. 822116Sjkh * (B). To achieve maximum accuracy, we compute expm1(x) by 832116Sjkh * (i) if x < -56*ln2, return -1.0, (raise inexact if x!=inf) 842116Sjkh * (ii) if k=0, return r-E 852116Sjkh * (iii) if k=-1, return 0.5*(r-E)-0.5 862116Sjkh * (iv) if k=1 if r < -0.25, return 2*((r+0.5)- E) 872116Sjkh * else return 1.0+2.0*(r-E); 882116Sjkh * (v) if (k<-2||k>56) return 2^k(1-(E-r)) - 1 (or exp(x)-1) 892116Sjkh * (vi) if k <= 20, return 2^k((1-2^-k)-(E-r)), else 908870Srgrimes * (vii) return 2^k(1-((E+2^-k)-r)) 912116Sjkh * 922116Sjkh * Special cases: 932116Sjkh * expm1(INF) is INF, expm1(NaN) is NaN; 942116Sjkh * expm1(-INF) is -1, and 952116Sjkh * for finite argument, only expm1(0)=0 is exact. 962116Sjkh * 972116Sjkh * Accuracy: 982116Sjkh * according to an error analysis, the error is always less than 992116Sjkh * 1 ulp (unit in the last place). 1002116Sjkh * 1012116Sjkh * Misc. info. 1028870Srgrimes * For IEEE double 1032116Sjkh * if x > 7.09782712893383973096e+02 then expm1(x) overflow 1042116Sjkh * 1052116Sjkh * Constants: 1068870Srgrimes * The hexadecimal values are the intended ones for the following 1078870Srgrimes * constants. The decimal values may be used, provided that the 1082116Sjkh * compiler will convert from decimal to binary accurately enough 1092116Sjkh * to produce the hexadecimal values shown. 1102116Sjkh */ 1112116Sjkh 1122116Sjkh#include "math.h" 1132116Sjkh#include "math_private.h" 1142116Sjkh 1152116Sjkhstatic const double 1162116Sjkhone = 1.0, 1172116Sjkhhuge = 1.0e+300, 1182116Sjkhtiny = 1.0e-300, 1192116Sjkho_threshold = 7.09782712893383973096e+02,/* 0x40862E42, 0xFEFA39EF */ 1202116Sjkhln2_hi = 6.93147180369123816490e-01,/* 0x3fe62e42, 0xfee00000 */ 1212116Sjkhln2_lo = 1.90821492927058770002e-10,/* 0x3dea39ef, 0x35793c76 */ 1222116Sjkhinvln2 = 1.44269504088896338700e+00,/* 0x3ff71547, 0x652b82fe */ 1232116Sjkh /* scaled coefficients related to expm1 */ 1242116SjkhQ1 = -3.33333333333331316428e-02, /* BFA11111 111110F4 */ 1252116SjkhQ2 = 1.58730158725481460165e-03, /* 3F5A01A0 19FE5585 */ 1262116SjkhQ3 = -7.93650757867487942473e-05, /* BF14CE19 9EAADBB7 */ 1272116SjkhQ4 = 4.00821782732936239552e-06, /* 3ED0CFCA 86E65239 */ 1282116SjkhQ5 = -2.01099218183624371326e-07; /* BE8AFDB7 6E09C32D */ 1292116Sjkh 13097413Salfreddouble 13197413Salfredexpm1(double x) 1322116Sjkh{ 1332116Sjkh double y,hi,lo,c,t,e,hxs,hfx,r1; 1342116Sjkh int32_t k,xsb; 1352116Sjkh u_int32_t hx; 1362116Sjkh 1372116Sjkh GET_HIGH_WORD(hx,x); 1382116Sjkh xsb = hx&0x80000000; /* sign bit of x */ 1392116Sjkh if(xsb==0) y=x; else y= -x; /* y = |x| */ 1402116Sjkh hx &= 0x7fffffff; /* high word of |x| */ 1412116Sjkh 1422116Sjkh /* filter out huge and non-finite argument */ 1432116Sjkh if(hx >= 0x4043687A) { /* if |x|>=56*ln2 */ 1442116Sjkh if(hx >= 0x40862E42) { /* if |x|>=709.78... */ 1452116Sjkh if(hx>=0x7ff00000) { 1462116Sjkh u_int32_t low; 1472116Sjkh GET_LOW_WORD(low,x); 1488870Srgrimes if(((hx&0xfffff)|low)!=0) 1492116Sjkh return x+x; /* NaN */ 1502116Sjkh else return (xsb==0)? x:-1.0;/* exp(+-inf)={inf,-1} */ 1512116Sjkh } 1522116Sjkh if(x > o_threshold) return huge*huge; /* overflow */ 1532116Sjkh } 1542116Sjkh if(xsb!=0) { /* x < -56*ln2, return -1.0 with inexact */ 1552116Sjkh if(x+tiny<0.0) /* raise inexact */ 1562116Sjkh return tiny-one; /* return -1 */ 1572116Sjkh } 1582116Sjkh } 1592116Sjkh 1602116Sjkh /* argument reduction */ 1618870Srgrimes if(hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */ 1622116Sjkh if(hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */ 1632116Sjkh if(xsb==0) 1642116Sjkh {hi = x - ln2_hi; lo = ln2_lo; k = 1;} 1652116Sjkh else 1662116Sjkh {hi = x + ln2_hi; lo = -ln2_lo; k = -1;} 1672116Sjkh } else { 1682116Sjkh k = invln2*x+((xsb==0)?0.5:-0.5); 1692116Sjkh t = k; 1702116Sjkh hi = x - t*ln2_hi; /* t*ln2_hi is exact here */ 1712116Sjkh lo = t*ln2_lo; 1722116Sjkh } 1732116Sjkh x = hi - lo; 1742116Sjkh c = (hi-x)-lo; 1758870Srgrimes } 1762116Sjkh else if(hx < 0x3c900000) { /* when |x|<2**-54, return x */ 1772116Sjkh t = huge+x; /* return x with inexact flags when x!=0 */ 1788870Srgrimes return x - (t-(huge+x)); 1792116Sjkh } 1802116Sjkh else k = 0; 1812116Sjkh 1822116Sjkh /* x is now in primary range */ 1832116Sjkh hfx = 0.5*x; 1842116Sjkh hxs = x*hfx; 1852116Sjkh r1 = one+hxs*(Q1+hxs*(Q2+hxs*(Q3+hxs*(Q4+hxs*Q5)))); 1862116Sjkh t = 3.0-r1*hfx; 1872116Sjkh e = hxs*((r1-t)/(6.0 - x*t)); 1882116Sjkh if(k==0) return x - (x*e-hxs); /* c is 0 */ 1892116Sjkh else { 1902116Sjkh e = (x*(e-c)-c); 1912116Sjkh e -= hxs; 1922116Sjkh if(k== -1) return 0.5*(x-e)-0.5; 1938870Srgrimes if(k==1) 1942116Sjkh if(x < -0.25) return -2.0*(e-(x+0.5)); 1952116Sjkh else return one+2.0*(x-e); 1962116Sjkh if (k <= -2 || k>56) { /* suffice to return exp(x)-1 */ 1972116Sjkh u_int32_t high; 1982116Sjkh y = one-(e-x); 1992116Sjkh GET_HIGH_WORD(high,y); 2002116Sjkh SET_HIGH_WORD(y,high+(k<<20)); /* add k to y's exponent */ 2012116Sjkh return y-one; 2022116Sjkh } 2032116Sjkh t = one; 2042116Sjkh if(k<20) { 2052116Sjkh u_int32_t high; 2062116Sjkh SET_HIGH_WORD(t,0x3ff00000 - (0x200000>>k)); /* t=1-2^-k */ 2072116Sjkh y = t-(e-x); 2082116Sjkh GET_HIGH_WORD(high,y); 2092116Sjkh SET_HIGH_WORD(y,high+(k<<20)); /* add k to y's exponent */ 2102116Sjkh } else { 2112116Sjkh u_int32_t high; 2122116Sjkh SET_HIGH_WORD(t,((0x3ff-k)<<20)); /* 2^-k */ 2132116Sjkh y = x-(e+t); 2142116Sjkh y += one; 2152116Sjkh GET_HIGH_WORD(high,y); 2162116Sjkh SET_HIGH_WORD(y,high+(k<<20)); /* add k to y's exponent */ 2172116Sjkh } 2182116Sjkh } 2192116Sjkh return y; 2202116Sjkh} 221