| /* 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; |
| } |
