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 13176082Sbde#include <sys/cdefs.h> 14176082Sbde__FBSDID("$FreeBSD: stable/11/lib/msun/src/s_expm1.c 352835 2019-09-28 08:57:29Z dim $"); 152116Sjkh 162116Sjkh/* expm1(x) 172116Sjkh * Returns exp(x)-1, the exponential of x minus 1. 182116Sjkh * 192116Sjkh * Method 202116Sjkh * 1. Argument reduction: 212116Sjkh * Given x, find r and integer k such that 222116Sjkh * 238870Srgrimes * x = k*ln2 + r, |r| <= 0.5*ln2 ~ 0.34658 242116Sjkh * 258870Srgrimes * Here a correction term c will be computed to compensate 262116Sjkh * the error in r when rounded to a floating-point number. 272116Sjkh * 282116Sjkh * 2. Approximating expm1(r) by a special rational function on 292116Sjkh * the interval [0,0.34658]: 302116Sjkh * Since 312116Sjkh * r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 - r^4/360 + ... 322116Sjkh * we define R1(r*r) by 332116Sjkh * r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 * R1(r*r) 342116Sjkh * That is, 352116Sjkh * R1(r**2) = 6/r *((exp(r)+1)/(exp(r)-1) - 2/r) 362116Sjkh * = 6/r * ( 1 + 2.0*(1/(exp(r)-1) - 1/r)) 372116Sjkh * = 1 - r^2/60 + r^4/2520 - r^6/100800 + ... 388870Srgrimes * We use a special Reme algorithm on [0,0.347] to generate 398870Srgrimes * a polynomial of degree 5 in r*r to approximate R1. The 408870Srgrimes * maximum error of this polynomial approximation is bounded 412116Sjkh * by 2**-61. In other words, 422116Sjkh * R1(z) ~ 1.0 + Q1*z + Q2*z**2 + Q3*z**3 + Q4*z**4 + Q5*z**5 432116Sjkh * where Q1 = -1.6666666666666567384E-2, 442116Sjkh * Q2 = 3.9682539681370365873E-4, 452116Sjkh * Q3 = -9.9206344733435987357E-6, 462116Sjkh * Q4 = 2.5051361420808517002E-7, 472116Sjkh * Q5 = -6.2843505682382617102E-9; 48176128Sbde * z = r*r, 492116Sjkh * with error bounded by 502116Sjkh * | 5 | -61 518870Srgrimes * | 1.0+Q1*z+...+Q5*z - R1(z) | <= 2 522116Sjkh * | | 538870Srgrimes * 548870Srgrimes * expm1(r) = exp(r)-1 is then computed by the following 558870Srgrimes * specific way which minimize the accumulation rounding error: 562116Sjkh * 2 3 572116Sjkh * r r [ 3 - (R1 + R1*r/2) ] 582116Sjkh * expm1(r) = r + --- + --- * [--------------------] 592116Sjkh * 2 2 [ 6 - r*(3 - R1*r/2) ] 608870Srgrimes * 612116Sjkh * To compensate the error in the argument reduction, we use 628870Srgrimes * expm1(r+c) = expm1(r) + c + expm1(r)*c 638870Srgrimes * ~ expm1(r) + c + r*c 642116Sjkh * Thus c+r*c will be added in as the correction terms for 658870Srgrimes * expm1(r+c). Now rearrange the term to avoid optimization 662116Sjkh * screw up: 672116Sjkh * ( 2 2 ) 682116Sjkh * ({ ( r [ R1 - (3 - R1*r/2) ] ) } r ) 692116Sjkh * expm1(r+c)~r - ({r*(--- * [--------------------]-c)-c} - --- ) 702116Sjkh * ({ ( 2 [ 6 - r*(3 - R1*r/2) ] ) } 2 ) 712116Sjkh * ( ) 728870Srgrimes * 732116Sjkh * = r - E 742116Sjkh * 3. Scale back to obtain expm1(x): 752116Sjkh * From step 1, we have 762116Sjkh * expm1(x) = either 2^k*[expm1(r)+1] - 1 772116Sjkh * = or 2^k*[expm1(r) + (1-2^-k)] 782116Sjkh * 4. Implementation notes: 792116Sjkh * (A). To save one multiplication, we scale the coefficient Qi 802116Sjkh * to Qi*2^i, and replace z by (x^2)/2. 812116Sjkh * (B). To achieve maximum accuracy, we compute expm1(x) by 822116Sjkh * (i) if x < -56*ln2, return -1.0, (raise inexact if x!=inf) 832116Sjkh * (ii) if k=0, return r-E 842116Sjkh * (iii) if k=-1, return 0.5*(r-E)-0.5 852116Sjkh * (iv) if k=1 if r < -0.25, return 2*((r+0.5)- E) 862116Sjkh * else return 1.0+2.0*(r-E); 872116Sjkh * (v) if (k<-2||k>56) return 2^k(1-(E-r)) - 1 (or exp(x)-1) 882116Sjkh * (vi) if k <= 20, return 2^k((1-2^-k)-(E-r)), else 898870Srgrimes * (vii) return 2^k(1-((E+2^-k)-r)) 902116Sjkh * 912116Sjkh * Special cases: 922116Sjkh * expm1(INF) is INF, expm1(NaN) is NaN; 932116Sjkh * expm1(-INF) is -1, and 942116Sjkh * for finite argument, only expm1(0)=0 is exact. 952116Sjkh * 962116Sjkh * Accuracy: 972116Sjkh * according to an error analysis, the error is always less than 982116Sjkh * 1 ulp (unit in the last place). 992116Sjkh * 1002116Sjkh * Misc. info. 1018870Srgrimes * For IEEE double 1022116Sjkh * if x > 7.09782712893383973096e+02 then expm1(x) overflow 1032116Sjkh * 1042116Sjkh * Constants: 1058870Srgrimes * The hexadecimal values are the intended ones for the following 1068870Srgrimes * constants. The decimal values may be used, provided that the 1072116Sjkh * compiler will convert from decimal to binary accurately enough 1082116Sjkh * to produce the hexadecimal values shown. 1092116Sjkh */ 1102116Sjkh 111226596Sdas#include <float.h> 112226596Sdas 1132116Sjkh#include "math.h" 1142116Sjkh#include "math_private.h" 1152116Sjkh 1162116Sjkhstatic const double 1172116Sjkhone = 1.0, 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 */ 123176128Sbde/* Scaled Q's: Qn_here = 2**n * Qn_above, for R(2*z) where z = hxs = x*x/2: */ 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 130251024Sdasstatic volatile double huge = 1.0e+300; 131251024Sdas 13297413Salfreddouble 13397413Salfredexpm1(double x) 1342116Sjkh{ 135176082Sbde double y,hi,lo,c,t,e,hxs,hfx,r1,twopk; 1362116Sjkh int32_t k,xsb; 1372116Sjkh u_int32_t hx; 1382116Sjkh 1392116Sjkh GET_HIGH_WORD(hx,x); 1402116Sjkh xsb = hx&0x80000000; /* sign bit of x */ 1412116Sjkh hx &= 0x7fffffff; /* high word of |x| */ 1422116Sjkh 1432116Sjkh /* filter out huge and non-finite argument */ 1442116Sjkh if(hx >= 0x4043687A) { /* if |x|>=56*ln2 */ 1452116Sjkh if(hx >= 0x40862E42) { /* if |x|>=709.78... */ 1462116Sjkh if(hx>=0x7ff00000) { 1472116Sjkh u_int32_t low; 1482116Sjkh GET_LOW_WORD(low,x); 1498870Srgrimes if(((hx&0xfffff)|low)!=0) 1502116Sjkh return x+x; /* NaN */ 1512116Sjkh else return (xsb==0)? x:-1.0;/* exp(+-inf)={inf,-1} */ 1522116Sjkh } 1532116Sjkh if(x > o_threshold) return huge*huge; /* overflow */ 1542116Sjkh } 1552116Sjkh if(xsb!=0) { /* x < -56*ln2, return -1.0 with inexact */ 1562116Sjkh if(x+tiny<0.0) /* raise inexact */ 1572116Sjkh return tiny-one; /* return -1 */ 1582116Sjkh } 1592116Sjkh } 1602116Sjkh 1612116Sjkh /* argument reduction */ 1628870Srgrimes if(hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */ 1632116Sjkh if(hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */ 1642116Sjkh if(xsb==0) 1652116Sjkh {hi = x - ln2_hi; lo = ln2_lo; k = 1;} 1662116Sjkh else 1672116Sjkh {hi = x + ln2_hi; lo = -ln2_lo; k = -1;} 1682116Sjkh } else { 1692116Sjkh k = invln2*x+((xsb==0)?0.5:-0.5); 1702116Sjkh t = k; 1712116Sjkh hi = x - t*ln2_hi; /* t*ln2_hi is exact here */ 1722116Sjkh lo = t*ln2_lo; 1732116Sjkh } 174226596Sdas STRICT_ASSIGN(double, x, hi - lo); 1752116Sjkh c = (hi-x)-lo; 1768870Srgrimes } 1772116Sjkh else if(hx < 0x3c900000) { /* when |x|<2**-54, return x */ 1782116Sjkh t = huge+x; /* return x with inexact flags when x!=0 */ 1798870Srgrimes return x - (t-(huge+x)); 1802116Sjkh } 1812116Sjkh else k = 0; 1822116Sjkh 1832116Sjkh /* x is now in primary range */ 1842116Sjkh hfx = 0.5*x; 1852116Sjkh hxs = x*hfx; 1862116Sjkh r1 = one+hxs*(Q1+hxs*(Q2+hxs*(Q3+hxs*(Q4+hxs*Q5)))); 1872116Sjkh t = 3.0-r1*hfx; 1882116Sjkh e = hxs*((r1-t)/(6.0 - x*t)); 1892116Sjkh if(k==0) return x - (x*e-hxs); /* c is 0 */ 1902116Sjkh else { 191352835Sdim INSERT_WORDS(twopk,((u_int32_t)(0x3ff+k))<<20,0); /* 2^k */ 1922116Sjkh e = (x*(e-c)-c); 1932116Sjkh e -= hxs; 1942116Sjkh if(k== -1) return 0.5*(x-e)-0.5; 195177711Sdas if(k==1) { 1962116Sjkh if(x < -0.25) return -2.0*(e-(x+0.5)); 1972116Sjkh else return one+2.0*(x-e); 198177711Sdas } 1992116Sjkh if (k <= -2 || k>56) { /* suffice to return exp(x)-1 */ 2002116Sjkh y = one-(e-x); 201176082Sbde if (k == 1024) y = y*2.0*0x1p1023; 202176082Sbde else y = y*twopk; 2032116Sjkh return y-one; 2042116Sjkh } 2052116Sjkh t = one; 2062116Sjkh if(k<20) { 2072116Sjkh SET_HIGH_WORD(t,0x3ff00000 - (0x200000>>k)); /* t=1-2^-k */ 2082116Sjkh y = t-(e-x); 209176082Sbde y = y*twopk; 2102116Sjkh } else { 2112116Sjkh SET_HIGH_WORD(t,((0x3ff-k)<<20)); /* 2^-k */ 2122116Sjkh y = x-(e+t); 2132116Sjkh y += one; 214176082Sbde y = y*twopk; 2152116Sjkh } 2162116Sjkh } 2172116Sjkh return y; 2182116Sjkh} 219251343Skargl 220251343Skargl#if (LDBL_MANT_DIG == 53) 221251343Skargl__weak_reference(expm1, expm1l); 222251343Skargl#endif 223