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 ```/* origin: FreeBSD /usr/src/lib/msun/src/s_expm1.c */ ``` ```/* ``` ``` * ==================================================== ``` ``` * 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 Remez 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; ``` ``` * z = r*r, ``` ``` * 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 "libm.h" ``` ``` ``` ```static const double ``` ```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 Q's: Qn_here = 2**n * Qn_above, for R(2*z) where z = hxs = x*x/2: */ ``` ```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_t y,hi,lo,c,t,e,hxs,hfx,r1,twopk; ``` ``` union {double f; uint64_t i;} u = {x}; ``` ``` uint32_t hx = u.i>>32 & 0x7fffffff; ``` ``` int k, sign = u.i>>63; ``` ``` ``` ``` /* filter out huge and non-finite argument */ ``` ``` if (hx >= 0x4043687A) { /* if |x|>=56*ln2 */ ``` ``` if (isnan(x)) ``` ``` return x; ``` ``` if (sign) ``` ``` return -1; ``` ``` if (x > o_threshold) { ``` ``` x *= 0x1p1023; ``` ``` return x; ``` ``` } ``` ``` } ``` ``` ``` ``` /* argument reduction */ ``` ``` if (hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */ ``` ``` if (hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */ ``` ``` if (!sign) { ``` ``` hi = x - ln2_hi; ``` ``` lo = ln2_lo; ``` ``` k = 1; ``` ``` } else { ``` ``` hi = x + ln2_hi; ``` ``` lo = -ln2_lo; ``` ``` k = -1; ``` ``` } ``` ``` } else { ``` ``` k = invln2*x + (sign ? -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) { /* |x| < 2**-54, return x */ ``` ``` if (hx < 0x00100000) ``` ``` FORCE_EVAL((float)x); ``` ``` return x; ``` ``` } else ``` ``` k = 0; ``` ``` ``` ``` /* x is now in primary range */ ``` ``` hfx = 0.5*x; ``` ``` hxs = x*hfx; ``` ``` r1 = 1.0+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) /* c is 0 */ ``` ``` return x - (x*e-hxs); ``` ``` e = x*(e-c) - c; ``` ``` e -= hxs; ``` ``` /* exp(x) ~ 2^k (x_reduced - e + 1) */ ``` ``` if (k == -1) ``` ``` return 0.5*(x-e) - 0.5; ``` ``` if (k == 1) { ``` ``` if (x < -0.25) ``` ``` return -2.0*(e-(x+0.5)); ``` ``` return 1.0+2.0*(x-e); ``` ``` } ``` ``` u.i = (uint64_t)(0x3ff + k)<<52; /* 2^k */ ``` ``` twopk = u.f; ``` ``` if (k < 0 || k > 56) { /* suffice to return exp(x)-1 */ ``` ``` y = x - e + 1.0; ``` ``` if (k == 1024) ``` ``` y = y*2.0*0x1p1023; ``` ``` else ``` ``` y = y*twopk; ``` ``` return y - 1.0; ``` ``` } ``` ``` u.i = (uint64_t)(0x3ff - k)<<52; /* 2^-k */ ``` ``` if (k < 20) ``` ``` y = (x-e+(1-u.f))*twopk; ``` ``` else ``` ``` y = (x-(e+u.f)+1)*twopk; ``` ``` return y; ``` ```} ``` ``` ```