1 /*-
2  * SPDX-License-Identifier: BSD-2-Clause
4  * Copyright (c) 2009-2013 Steven G. Kargl
43 static const volatile long double
45 tiny = 0x1p-10000L;
47 static const long double
48 twom10000 = 0x1p-10000L;
50 static const long double
51 /* log(2**16384 - 0.5) rounded towards zero: */
52 /* log(2**16384 - 0.5 + 1) rounded towards zero for expm1l() is the same: */
54 /* log(2**(-16381-64-1)) rounded towards zero: */
55 u_threshold = -11433.462743336297878837243843452621503L;
57 long double
58 expl(long double x)  in expl()
61 	long double hi, lo, t, twopk;  in expl()
71 			if (hx & 0x8000)  /* x is -Inf or -NaN */  in expl()
72 				RETURNF(-1 / x);  in expl()
79 	} else if (ix < BIAS - 114) {	/* |x| < 0x1p-114 */  in expl()
113  * range for it.
116  * Setting T3 to 0 would require the |x| < 0x1p-113 condition to appear
117  * in both subintervals, so set T3 = 2**-5, which places the condition
124  * XXX these micro-optimizations are excessive.
126 static const double
127 T1 = -0.1659,				/* ~-30.625/128 * log(2) */
132  * Domain [-0.1659, 0.03125], range ~[2.9134e-44, 1.8404e-37]:
133  * |(exp(x)-1-x-x**2/2)/x - p(x)| < 2**-122.03
135  * XXX none of the long double C or D coeffs except C10 is correctly printed.
136  * If you re-print their values in %.35Le format, the result is always
138  * 67 is apparently from rounding an extra-precision value to 36 decimal
141 static const long double
142 C3  =  1.66666666666666666666666666666666667e-1L,
143 C4  =  4.16666666666666666666666666666666645e-2L,
144 C5  =  8.33333333333333333333333333333371638e-3L,
145 C6  =  1.38888888888888888888888888891188658e-3L,
146 C7  =  1.98412698412698412698412697235950394e-4L,
147 C8  =  2.48015873015873015873015112487849040e-5L,
148 C9  =  2.75573192239858906525606685484412005e-6L,
149 C10 =  2.75573192239858906612966093057020362e-7L,
150 C11 =  2.50521083854417203619031960151253944e-8L,
151 C12 =  2.08767569878679576457272282566520649e-9L,
152 C13 =  1.60590438367252471783548748824255707e-10L;
156  * XXX can start the double coeffs but not the double mults at C10.
157  * With my coeffs (C10-C17 double; s = best_s):
158  * Domain [-0.1659, 0.03125], range ~[-1.1976e-37, 1.1976e-37]:
159  * |(exp(x)-1-x-x**2/2)/x - p(x)| ~< 2**-122.65
161 static const double
162 C14 =  1.1470745580491932e-11,		/*  0x1.93974a81dae30p-37 */
163 C15 =  7.6471620181090468e-13,		/*  0x1.ae7f3820adab1p-41 */
164 C16 =  4.7793721460260450e-14,		/*  0x1.ae7cd18a18eacp-45 */
165 C17 =  2.8074757356658877e-15,		/*  0x1.949992a1937d9p-49 */
166 C18 =  1.4760610323699476e-16;		/*  0x1.545b43aabfbcdp-53 */
169  * Domain [0.03125, 0.1659], range ~[-2.7676e-37, -1.0367e-38]:
170  * |(exp(x)-1-x-x**2/2)/x - p(x)| < 2**-121.44
172 static const long double
173 D3  =  1.66666666666666666666666666666682245e-1L,
174 D4  =  4.16666666666666666666666666634228324e-2L,
175 D5  =  8.33333333333333333333333364022244481e-3L,
176 D6  =  1.38888888888888888888887138722762072e-3L,
177 D7  =  1.98412698412698412699085805424661471e-4L,
178 D8  =  2.48015873015873015687993712101479612e-5L,
179 D9  =  2.75573192239858944101036288338208042e-6L,
180 D10 =  2.75573192239853161148064676533754048e-7L,
181 D11 =  2.50521083855084570046480450935267433e-8L,
182 D12 =  2.08767569819738524488686318024854942e-9L,
183 D13 =  1.60590442297008495301927448122499313e-10L;
187  * XXX can start the double coeffs but not the double mults at D11.
188  * With my coeffs (D11-D16 double):
189  * Domain [0.03125, 0.1659], range ~[-1.1980e-37, 1.1980e-37]:
190  * |(exp(x)-1-x-x**2/2)/x - p(x)| ~< 2**-122.65
192 static const double
193 D14 =  1.1470726176204336e-11,		/*  0x1.93971dc395d9ep-37 */
194 D15 =  7.6478532249581686e-13,		/*  0x1.ae892e3D16fcep-41 */
195 D16 =  4.7628892832607741e-14,		/*  0x1.ad00Dfe41feccp-45 */
196 D17 =  3.0524857220358650e-15;		/*  0x1.D7e8d886Df921p-49 */
198 long double
199 expm1l(long double x)  in expm1l()
202 	long double hx2_hi, hx2_lo, q, r, r1, t, twomk, twopk, x_hi;  in expm1l()
203 	long double x_lo, x2;  in expm1l()
204 	double dr, dx, fn, r2;  in expm1l()
214 			if (hx & 0x8000)  /* x is -Inf or -NaN */  in expm1l()
215 				RETURNF(-1 / x - 1);  in expm1l()
227 		if (hx & 0x8000)	/* x <= -128 */  in expm1l()
228 			RETURNF(tiny - 1);	/* good for x < -114ln2 - eps */  in expm1l()
238 			if (ix < BIAS - 113) {	/* |x| < 0x1p-113 */  in expm1l()
241 				    (0x1p200 * x + fabsl(x)) * 0x1p-200);  in expm1l()
257 		x_lo = x - x_hi;  in expm1l()
260 		if (ix >= BIAS - 7)  in expm1l()
267 	fn = rnint((double)x * INV_L);  in expm1l()
271 	r1 = x - fn * L1;  in expm1l()
272 	r2 = fn * -L2;  in expm1l()
291 		t = SUM2P(tbl[n2].hi - 1, tbl[n2].lo * (r1 + 1) + t * q +  in expm1l()
295 	if (k == -1) {  in expm1l()
296 		t = SUM2P(tbl[n2].hi - 2, tbl[n2].lo * (r1 + 1) + t * q +  in expm1l()
300 	if (k < -7) {  in expm1l()
302 		RETURNI(t * twopk - 1);  in expm1l()
304 	if (k > 2 * LDBL_MANT_DIG - 1) {  in expm1l()
307 			RETURNI(t * 2 * 0x1p16383L - 1);  in expm1l()
308 		RETURNI(t * twopk - 1);  in expm1l()
311 	v.xbits.expsign = BIAS - k;  in expm1l()
314 	if (k > LDBL_MANT_DIG - 1)  in expm1l()
315 		t = SUM2P(tbl[n2].hi, tbl[n2].lo - twomk + t * (q + r1));  in expm1l()
317 		t = SUM2P(tbl[n2].hi - twomk, tbl[n2].lo + t * (q + r1));  in expm1l()