#ifndef COSMOPOLITAN_LIBC_TINYMATH_FREEBSD_INTERNAL_H_ #define COSMOPOLITAN_LIBC_TINYMATH_FREEBSD_INTERNAL_H_ #include "libc/assert.h" #include "libc/complex.h" #include "libc/math.h" #include "libc/runtime/fenv.h" #if !(__ASSEMBLER__ + __LINKER__ + 0) COSMOPOLITAN_C_START_ // clang-format off #define __CONCAT1(x,y) x ## y #define __CONCAT(x,y) __CONCAT1(x,y) #define __STRING(x) #x #define __XSTRING(x) __STRING(x) #ifdef __x86_64__ union IEEEl2bits { long double e; struct { unsigned int manl :32; unsigned int manh :32; unsigned int exp :15; unsigned int sign :1; unsigned int junkl :16; unsigned int junkh :32; } bits; struct { unsigned long man :64; unsigned int expsign :16; unsigned long junk :48; } xbits; }; #define LDBL_NBIT 0x80000000 #define mask_nbit_l(u) ((u).bits.manh &= ~LDBL_NBIT) #define LDBL_MANH_SIZE 32 #define LDBL_MANL_SIZE 32 #define LDBL_TO_ARRAY32(u, a) do { \ (a)[0] = (uint32_t)(u).bits.manl; \ (a)[1] = (uint32_t)(u).bits.manh; \ } while (0) #elif defined(__aarch64__) union IEEEl2bits { long double e; struct { unsigned long manl :64; unsigned long manh :48; unsigned int exp :15; unsigned int sign :1; } bits; /* TODO andrew: Check the packing here */ struct { unsigned long manl :64; unsigned long manh :48; unsigned int expsign :16; } xbits; }; #define LDBL_NBIT 0 #define LDBL_IMPLICIT_NBIT #define mask_nbit_l(u) ((void)0) #define LDBL_MANH_SIZE 48 #define LDBL_MANL_SIZE 64 #define LDBL_TO_ARRAY32(u, a) do { \ (a)[0] = (uint32_t)(u).bits.manl; \ (a)[1] = (uint32_t)((u).bits.manl >> 32); \ (a)[2] = (uint32_t)(u).bits.manh; \ (a)[3] = (uint32_t)((u).bits.manh >> 32); \ } while(0) #elif defined(__powerpc64__) union IEEEl2bits { long double e; struct { #if __BYTE_ORDER__ == __ORDER_LITTLE_ENDIAN__ unsigned int manl :32; unsigned int manh :20; unsigned int exp :11; unsigned int sign :1; #else /* _BYTE_ORDER == _LITTLE_ENDIAN */ unsigned int sign :1; unsigned int exp :11; unsigned int manh :20; unsigned int manl :32; #endif } bits; }; #define mask_nbit_l(u) ((void)0) #define LDBL_IMPLICIT_NBIT #define LDBL_NBIT 0 #define LDBL_MANH_SIZE 20 #define LDBL_MANL_SIZE 32 #define LDBL_TO_ARRAY32(u, a) do { \ (a)[0] = (uint32_t)(u).bits.manl; \ (a)[1] = (uint32_t)(u).bits.manh; \ } while(0) #endif /* __x86_64__ */ /* * The original fdlibm code used statements like: * n0 = ((*(int*)&one)>>29)^1; * index of high word * * ix0 = *(n0+(int*)&x); * high word of x * * ix1 = *((1-n0)+(int*)&x); * low word of x * * to dig two 32 bit words out of the 64 bit IEEE floating point * value. That is non-ANSI, and, moreover, the gcc instruction * scheduler gets it wrong. We instead use the following macros. * Unlike the original code, we determine the endianness at compile * time, not at run time; I don't see much benefit to selecting * endianness at run time. */ /* * A union which permits us to convert between a double and two 32 bit * ints. */ #ifdef __arm__ #if defined(__VFP_FP__) || defined(__ARM_EABI__) #define IEEE_WORD_ORDER __BYTE_ORDER__ #else #define IEEE_WORD_ORDER __ORDER_BIG_ENDIAN__ #endif #else /* __arm__ */ #define IEEE_WORD_ORDER __BYTE_ORDER__ #endif /* A union which permits us to convert between a long double and four 32 bit ints. */ #if IEEE_WORD_ORDER == __ORDER_BIG_ENDIAN__ typedef union { long double value; struct { uint32_t mswhi; uint32_t mswlo; uint32_t lswhi; uint32_t lswlo; } parts32; struct { uint64_t msw; uint64_t lsw; } parts64; } ieee_quad_shape_type; #endif #if IEEE_WORD_ORDER == __ORDER_LITTLE_ENDIAN__ typedef union { long double value; struct { uint32_t lswlo; uint32_t lswhi; uint32_t mswlo; uint32_t mswhi; } parts32; struct { uint64_t lsw; uint64_t msw; } parts64; } ieee_quad_shape_type; #endif #if IEEE_WORD_ORDER == __ORDER_BIG_ENDIAN__ typedef union { double value; struct { uint32_t msw; uint32_t lsw; } parts; struct { uint64_t w; } xparts; } ieee_double_shape_type; #endif #if IEEE_WORD_ORDER == __ORDER_LITTLE_ENDIAN__ typedef union { double value; struct { uint32_t lsw; uint32_t msw; } parts; struct { uint64_t w; } xparts; } ieee_double_shape_type; #endif /* Get two 32 bit ints from a double. */ #define EXTRACT_WORDS(ix0,ix1,d) \ do { \ ieee_double_shape_type ew_u; \ ew_u.value = (d); \ (ix0) = ew_u.parts.msw; \ (ix1) = ew_u.parts.lsw; \ } while (0) /* Get a 64-bit int from a double. */ #define EXTRACT_WORD64(ix,d) \ do { \ ieee_double_shape_type ew_u; \ ew_u.value = (d); \ (ix) = ew_u.xparts.w; \ } while (0) /* Get the more significant 32 bit int from a double. */ #define GET_HIGH_WORD(i,d) \ do { \ ieee_double_shape_type gh_u; \ gh_u.value = (d); \ (i) = gh_u.parts.msw; \ } while (0) /* Get the less significant 32 bit int from a double. */ #define GET_LOW_WORD(i,d) \ do { \ ieee_double_shape_type gl_u; \ gl_u.value = (d); \ (i) = gl_u.parts.lsw; \ } while (0) /* Set a double from two 32 bit ints. */ #define INSERT_WORDS(d,ix0,ix1) \ do { \ ieee_double_shape_type iw_u; \ iw_u.parts.msw = (ix0); \ iw_u.parts.lsw = (ix1); \ (d) = iw_u.value; \ } while (0) /* Set a double from a 64-bit int. */ #define INSERT_WORD64(d,ix) \ do { \ ieee_double_shape_type iw_u; \ iw_u.xparts.w = (ix); \ (d) = iw_u.value; \ } while (0) /* Set the more significant 32 bits of a double from an int. */ #define SET_HIGH_WORD(d,v) \ do { \ ieee_double_shape_type sh_u; \ sh_u.value = (d); \ sh_u.parts.msw = (v); \ (d) = sh_u.value; \ } while (0) /* Set the less significant 32 bits of a double from an int. */ #define SET_LOW_WORD(d,v) \ do { \ ieee_double_shape_type sl_u; \ sl_u.value = (d); \ sl_u.parts.lsw = (v); \ (d) = sl_u.value; \ } while (0) /* * A union which permits us to convert between a float and a 32 bit * int. */ typedef union { float value; /* FIXME: Assumes 32 bit int. */ unsigned int word; } ieee_float_shape_type; /* Get a 32 bit int from a float. */ #define GET_FLOAT_WORD(i,d) \ do { \ ieee_float_shape_type gf_u; \ gf_u.value = (d); \ (i) = gf_u.word; \ } while (0) /* Set a float from a 32 bit int. */ #define SET_FLOAT_WORD(d,i) \ do { \ ieee_float_shape_type sf_u; \ sf_u.word = (i); \ (d) = sf_u.value; \ } while (0) /* * Get expsign and mantissa as 16 bit and 64 bit ints from an 80 bit long * double. */ #define EXTRACT_LDBL80_WORDS(ix0,ix1,d) \ do { \ union IEEEl2bits ew_u; \ ew_u.e = (d); \ (ix0) = ew_u.xbits.expsign; \ (ix1) = ew_u.xbits.man; \ } while (0) /* * Get expsign and mantissa as one 16 bit and two 64 bit ints from a 128 bit * long double. */ #define EXTRACT_LDBL128_WORDS(ix0,ix1,ix2,d) \ do { \ union IEEEl2bits ew_u; \ ew_u.e = (d); \ (ix0) = ew_u.xbits.expsign; \ (ix1) = ew_u.xbits.manh; \ (ix2) = ew_u.xbits.manl; \ } while (0) /* Get expsign as a 16 bit int from a long double. */ #define GET_LDBL_EXPSIGN(i,d) \ do { \ union IEEEl2bits ge_u; \ ge_u.e = (d); \ (i) = ge_u.xbits.expsign; \ } while (0) /* * Set an 80 bit long double from a 16 bit int expsign and a 64 bit int * mantissa. */ #define INSERT_LDBL80_WORDS(d,ix0,ix1) \ do { \ union IEEEl2bits iw_u; \ iw_u.xbits.expsign = (ix0); \ iw_u.xbits.man = (ix1); \ (d) = iw_u.e; \ } while (0) /* * Set a 128 bit long double from a 16 bit int expsign and two 64 bit ints * comprising the mantissa. */ #define INSERT_LDBL128_WORDS(d,ix0,ix1,ix2) \ do { \ union IEEEl2bits iw_u; \ iw_u.xbits.expsign = (ix0); \ iw_u.xbits.manh = (ix1); \ iw_u.xbits.manl = (ix2); \ (d) = iw_u.e; \ } while (0) /* Set expsign of a long double from a 16 bit int. */ #define SET_LDBL_EXPSIGN(d,v) \ do { \ union IEEEl2bits se_u; \ se_u.e = (d); \ se_u.xbits.expsign = (v); \ (d) = se_u.e; \ } while (0) #ifdef __i386__ /* Long double constants are broken on i386. */ #define LD80C(m, ex, v) { \ .xbits.man = __CONCAT(m, ULL), \ .xbits.expsign = (0x3fff + (ex)) | ((v) < 0 ? 0x8000 : 0), \ } #else /* The above works on non-i386 too, but we use this to check v. */ #define LD80C(m, ex, v) { .e = (v), } #endif #ifdef FLT_EVAL_METHOD /* * Attempt to get strict C99 semantics for assignment with non-C99 compilers. */ #if FLT_EVAL_METHOD == 0 || __GNUC__ == 0 #define STRICT_ASSIGN(type, lval, rval) ((lval) = (rval)) #else #define STRICT_ASSIGN(type, lval, rval) do { \ volatile type __lval; \ \ if (sizeof(type) >= sizeof(long double)) \ (lval) = (rval); \ else { \ __lval = (rval); \ (lval) = __lval; \ } \ } while (0) #endif #endif /* FLT_EVAL_METHOD */ /* Support switching the mode to FP_PE if necessary. */ #if defined(__i386__) && !defined(NO_FPSETPREC) #define ENTERI() ENTERIT(long double) #define ENTERIT(returntype) \ returntype __retval; \ fp_prec_t __oprec; \ \ if ((__oprec = fpgetprec()) != FP_PE) \ fpsetprec(FP_PE) #define RETURNI(x) do { \ __retval = (x); \ if (__oprec != FP_PE) \ fpsetprec(__oprec); \ RETURNF(__retval); \ } while (0) #define ENTERV() \ fp_prec_t __oprec; \ \ if ((__oprec = fpgetprec()) != FP_PE) \ fpsetprec(FP_PE) #define RETURNV() do { \ if (__oprec != FP_PE) \ fpsetprec(__oprec); \ return; \ } while (0) #else #define ENTERI() #define ENTERIT(x) #define RETURNI(x) RETURNF(x) #define ENTERV() #define RETURNV() return #endif /* Default return statement if hack*_t() is not used. */ #define RETURNF(v) return (v) /* * 2sum gives the same result as 2sumF without requiring |a| >= |b| or * a == 0, but is slower. */ #define _2sum(a, b) do { \ __typeof(a) __s, __w; \ \ __w = (a) + (b); \ __s = __w - (a); \ (b) = ((a) - (__w - __s)) + ((b) - __s); \ (a) = __w; \ } while (0) /* * 2sumF algorithm. * * "Normalize" the terms in the infinite-precision expression a + b for * the sum of 2 floating point values so that b is as small as possible * relative to 'a'. (The resulting 'a' is the value of the expression in * the same precision as 'a' and the resulting b is the rounding error.) * |a| must be >= |b| or 0, b's type must be no larger than 'a's type, and * exponent overflow or underflow must not occur. This uses a Theorem of * Dekker (1971). See Knuth (1981) 4.2.2 Theorem C. The name "TwoSum" * is apparently due to Skewchuk (1997). * * For this to always work, assignment of a + b to 'a' must not retain any * extra precision in a + b. This is required by C standards but broken * in many compilers. The brokenness cannot be worked around using * STRICT_ASSIGN() like we do elsewhere, since the efficiency of this * algorithm would be destroyed by non-null strict assignments. (The * compilers are correct to be broken -- the efficiency of all floating * point code calculations would be destroyed similarly if they forced the * conversions.) * * Fortunately, a case that works well can usually be arranged by building * any extra precision into the type of 'a' -- 'a' should have type float_t, * double_t or long double. b's type should be no larger than 'a's type. * Callers should use these types with scopes as large as possible, to * reduce their own extra-precision and efficiciency problems. In * particular, they shouldn't convert back and forth just to call here. */ #ifdef DEBUG #define _2sumF(a, b) do { \ __typeof(a) __w; \ volatile __typeof(a) __ia, __ib, __r, __vw; \ \ __ia = (a); \ __ib = (b); \ assert(__ia == 0 || fabsl(__ia) >= fabsl(__ib)); \ \ __w = (a) + (b); \ (b) = ((a) - __w) + (b); \ (a) = __w; \ \ /* The next 2 assertions are weak if (a) is already long double. */ \ assert((long double)__ia + __ib == (long double)(a) + (b)); \ __vw = __ia + __ib; \ __r = __ia - __vw; \ __r += __ib; \ assert(__vw == (a) && __r == (b)); \ } while (0) #else /* !DEBUG */ #define _2sumF(a, b) do { \ __typeof(a) __w; \ \ __w = (a) + (b); \ (b) = ((a) - __w) + (b); \ (a) = __w; \ } while (0) #endif /* DEBUG */ /* * Set x += c, where x is represented in extra precision as a + b. * x must be sufficiently normalized and sufficiently larger than c, * and the result is then sufficiently normalized. * * The details of ordering are that |a| must be >= |c| (so that (a, c) * can be normalized without extra work to swap 'a' with c). The details of * the normalization are that b must be small relative to the normalized 'a'. * Normalization of (a, c) makes the normalized c tiny relative to the * normalized a, so b remains small relative to 'a' in the result. However, * b need not ever be tiny relative to 'a'. For example, b might be about * 2**20 times smaller than 'a' to give about 20 extra bits of precision. * That is usually enough, and adding c (which by normalization is about * 2**53 times smaller than a) cannot change b significantly. However, * cancellation of 'a' with c in normalization of (a, c) may reduce 'a' * significantly relative to b. The caller must ensure that significant * cancellation doesn't occur, either by having c of the same sign as 'a', * or by having |c| a few percent smaller than |a|. Pre-normalization of * (a, b) may help. * * This is a variant of an algorithm of Kahan (see Knuth (1981) 4.2.2 * exercise 19). We gain considerable efficiency by requiring the terms to * be sufficiently normalized and sufficiently increasing. */ #define _3sumF(a, b, c) do { \ __typeof(a) __tmp; \ \ __tmp = (c); \ _2sumF(__tmp, (a)); \ (b) += (a); \ (a) = __tmp; \ } while (0) /* * Common routine to process the arguments to nan(), nanf(), and nanl(). */ void _scan_nan(uint32_t *__words, int __num_words, const char *__s); /* * Mix 0, 1 or 2 NaNs. First add 0 to each arg. This normally just turns * signaling NaNs into quiet NaNs by setting a quiet bit. We do this * because we want to never return a signaling NaN, and also because we * don't want the quiet bit to affect the result. Then mix the converted * args using the specified operation. * * When one arg is NaN, the result is typically that arg quieted. When both * args are NaNs, the result is typically the quietening of the arg whose * mantissa is largest after quietening. When neither arg is NaN, the * result may be NaN because it is indeterminate, or finite for subsequent * construction of a NaN as the indeterminate 0.0L/0.0L. * * Technical complications: the result in bits after rounding to the final * precision might depend on the runtime precision and/or on compiler * optimizations, especially when different register sets are used for * different precisions. Try to make the result not depend on at least the * runtime precision by always doing the main mixing step in long double * precision. Try to reduce dependencies on optimizations by adding the * the 0's in different precisions (unless everything is in long double * precision). */ #define nan_mix(x, y) (nan_mix_op((x), (y), +)) #define nan_mix_op(x, y, op) (((x) + 0.0L) op ((y) + 0)) #ifdef _COMPLEX_H /* * C99 specifies that complex numbers have the same representation as * an array of two elements, where the first element is the real part * and the second element is the imaginary part. */ typedef union { float complex f; float a[2]; } float_complex; typedef union { double complex f; double a[2]; } double_complex; typedef union { long double complex f; long double a[2]; } long_double_complex; #define REALPART(z) ((z).a[0]) #define IMAGPART(z) ((z).a[1]) /* * Inline functions that can be used to construct complex values. * * The C99 standard intends x+I*y to be used for this, but x+I*y is * currently unusable in general since gcc introduces many overflow, * underflow, sign and efficiency bugs by rewriting I*y as * (0.0+I)*(y+0.0*I) and laboriously computing the full complex product. * In particular, I*Inf is corrupted to NaN+I*Inf, and I*-0 is corrupted * to -0.0+I*0.0. * * The C11 standard introduced the macros CMPLX(), CMPLXF() and CMPLXL() * to construct complex values. Compilers that conform to the C99 * standard require the following functions to avoid the above issues. */ #ifndef CMPLXF static __inline float complex CMPLXF(float x, float y) { float_complex z; REALPART(z) = x; IMAGPART(z) = y; return (z.f); } #endif #ifndef CMPLX static __inline double complex CMPLX(double x, double y) { double_complex z; REALPART(z) = x; IMAGPART(z) = y; return (z.f); } #endif #ifndef CMPLXL static __inline long double complex CMPLXL(long double x, long double y) { long_double_complex z; REALPART(z) = x; IMAGPART(z) = y; return (z.f); } #endif #endif /* _COMPLEX_H */ /* * The rnint() family rounds to the nearest integer for a restricted range * range of args (up to about 2**MANT_DIG). We assume that the current * rounding mode is FE_TONEAREST so that this can be done efficiently. * Extra precision causes more problems in practice, and we only centralize * this here to reduce those problems, and have not solved the efficiency * problems. The exp2() family uses a more delicate version of this that * requires extracting bits from the intermediate value, so it is not * centralized here and should copy any solution of the efficiency problems. */ static inline double rnint(double_t x) { /* * This casts to double to kill any extra precision. This depends * on the cast being applied to a double_t to avoid compiler bugs * (this is a cleaner version of STRICT_ASSIGN()). This is * inefficient if there actually is extra precision, but is hard * to improve on. We use double_t in the API to minimise conversions * for just calling here. Note that we cannot easily change the * magic number to the one that works directly with double_t, since * the rounding precision is variable at runtime on x86 so the * magic number would need to be variable. Assuming that the * rounding precision is always the default is too fragile. This * and many other complications will move when the default is * changed to FP_PE. */ return ((double)(x + 0x1.8p52) - 0x1.8p52); } static inline float rnintf(float_t x) { /* * As for rnint(), except we could just call that to handle the * extra precision case, usually without losing efficiency. */ return ((float)(x + 0x1.8p23F) - 0x1.8p23F); } #ifdef LDBL_MANT_DIG /* * The complications for extra precision are smaller for rnintl() since it * can safely assume that the rounding precision has been increased from * its default to FP_PE on x86. We don't exploit that here to get small * optimizations from limiting the rangle to double. We just need it for * the magic number to work with long doubles. ld128 callers should use * rnint() instead of this if possible. ld80 callers should prefer * rnintl() since for amd64 this avoids swapping the register set, while * for i386 it makes no difference (assuming FP_PE), and for other arches * it makes little difference. */ static inline long double rnintl(long double x) { return (x + __CONCAT(0x1.8p, LDBL_MANT_DIG) / 2 - __CONCAT(0x1.8p, LDBL_MANT_DIG) / 2); } #endif /* LDBL_MANT_DIG */ /* * irint() and i64rint() give the same result as casting to their integer * return type provided their arg is a floating point integer. They can * sometimes be more efficient because no rounding is required. */ #if defined(amd64) || defined(__i386__) #define irint(x) \ (sizeof(x) == sizeof(float) && \ sizeof(float_t) == sizeof(long double) ? irintf(x) : \ sizeof(x) == sizeof(double) && \ sizeof(double_t) == sizeof(long double) ? irintd(x) : \ sizeof(x) == sizeof(long double) ? irintl(x) : (int)(x)) #else #define irint(x) ((int)(x)) #endif #define i64rint(x) ((int64_t)(x)) /* only needed for ld128 so not opt. */ #if defined(__i386__) static __inline int irintf(float x) { int n; __asm("fistl %0" : "=m" (n) : "t" (x)); return (n); } static __inline int irintd(double x) { int n; __asm("fistl %0" : "=m" (n) : "t" (x)); return (n); } #endif #if defined(__amd64__) || defined(__i386__) static __inline int irintl(long double x) { int n; __asm("fistl %0" : "=m" (n) : "t" (x)); return (n); } #endif #ifdef DEBUG #if defined(__amd64__) || defined(__i386__) #define breakpoint() asm("int $3") #else #define breakpoint() raise(SIGTRAP) #endif #endif /* Write a pari script to test things externally. */ #ifdef DOPRINT #ifndef DOPRINT_SWIZZLE #define DOPRINT_SWIZZLE 0 #endif #ifdef DOPRINT_LD80 #define DOPRINT_START(xp) do { \ uint64_t __lx; \ uint16_t __hx; \ \ /* Hack to give more-problematic args. */ \ EXTRACT_LDBL80_WORDS(__hx, __lx, *xp); \ __lx ^= DOPRINT_SWIZZLE; \ INSERT_LDBL80_WORDS(*xp, __hx, __lx); \ printf("x = %.21Lg; ", (long double)*xp); \ } while (0) #define DOPRINT_END1(v) \ printf("y = %.21Lg; z = 0; show(x, y, z);\n", (long double)(v)) #define DOPRINT_END2(hi, lo) \ printf("y = %.21Lg; z = %.21Lg; show(x, y, z);\n", \ (long double)(hi), (long double)(lo)) #elif defined(DOPRINT_D64) #define DOPRINT_START(xp) do { \ uint32_t __hx, __lx; \ \ EXTRACT_WORDS(__hx, __lx, *xp); \ __lx ^= DOPRINT_SWIZZLE; \ INSERT_WORDS(*xp, __hx, __lx); \ printf("x = %.21Lg; ", (long double)*xp); \ } while (0) #define DOPRINT_END1(v) \ printf("y = %.21Lg; z = 0; show(x, y, z);\n", (long double)(v)) #define DOPRINT_END2(hi, lo) \ printf("y = %.21Lg; z = %.21Lg; show(x, y, z);\n", \ (long double)(hi), (long double)(lo)) #elif defined(DOPRINT_F32) #define DOPRINT_START(xp) do { \ uint32_t __hx; \ \ GET_FLOAT_WORD(__hx, *xp); \ __hx ^= DOPRINT_SWIZZLE; \ SET_FLOAT_WORD(*xp, __hx); \ printf("x = %.21Lg; ", (long double)*xp); \ } while (0) #define DOPRINT_END1(v) \ printf("y = %.21Lg; z = 0; show(x, y, z);\n", (long double)(v)) #define DOPRINT_END2(hi, lo) \ printf("y = %.21Lg; z = %.21Lg; show(x, y, z);\n", \ (long double)(hi), (long double)(lo)) #else /* !DOPRINT_LD80 && !DOPRINT_D64 (LD128 only) */ #ifndef DOPRINT_SWIZZLE_HIGH #define DOPRINT_SWIZZLE_HIGH 0 #endif #define DOPRINT_START(xp) do { \ uint64_t __lx, __llx; \ uint16_t __hx; \ \ EXTRACT_LDBL128_WORDS(__hx, __lx, __llx, *xp); \ __llx ^= DOPRINT_SWIZZLE; \ __lx ^= DOPRINT_SWIZZLE_HIGH; \ INSERT_LDBL128_WORDS(*xp, __hx, __lx, __llx); \ printf("x = %.36Lg; ", (long double)*xp); \ } while (0) #define DOPRINT_END1(v) \ printf("y = %.36Lg; z = 0; show(x, y, z);\n", (long double)(v)) #define DOPRINT_END2(hi, lo) \ printf("y = %.36Lg; z = %.36Lg; show(x, y, z);\n", \ (long double)(hi), (long double)(lo)) #endif /* DOPRINT_LD80 */ #else /* !DOPRINT */ #define DOPRINT_START(xp) #define DOPRINT_END1(v) #define DOPRINT_END2(hi, lo) #endif /* DOPRINT */ #define RETURNP(x) do { \ DOPRINT_END1(x); \ RETURNF(x); \ } while (0) #define RETURNPI(x) do { \ DOPRINT_END1(x); \ RETURNI(x); \ } while (0) #define RETURN2P(x, y) do { \ DOPRINT_END2((x), (y)); \ RETURNF((x) + (y)); \ } while (0) #define RETURN2PI(x, y) do { \ DOPRINT_END2((x), (y)); \ RETURNI((x) + (y)); \ } while (0) #define RETURNSP(rp) do { \ if (!(rp)->lo_set) \ RETURNP((rp)->hi); \ RETURN2P((rp)->hi, (rp)->lo); \ } while (0) #define RETURNSPI(rp) do { \ if (!(rp)->lo_set) \ RETURNPI((rp)->hi); \ RETURN2PI((rp)->hi, (rp)->lo); \ } while (0) #define SUM2P(x, y) ({ \ const __typeof (x) __x = (x); \ const __typeof (y) __y = (y); \ \ DOPRINT_END2(__x, __y); \ __x + __y; \ }) /* fdlibm kernel function */ int __kernel_rem_pio2(double*,double*,int,int,int); /* double precision kernel functions */ #ifndef INLINE_REM_PIO2 int __ieee754_rem_pio2(double,double*); #endif double __kernel_sin(double,double,int); double __kernel_cos(double,double); double __kernel_tan(double,double,int); double __ldexp_exp(double,int); #ifdef _COMPLEX_H double complex __ldexp_cexp(double complex,int); #endif /* float precision kernel functions */ #ifndef INLINE_REM_PIO2F int __ieee754_rem_pio2f(float,double*); #endif #ifndef INLINE_KERNEL_SINDF float __kernel_sindf(double); #endif #ifndef INLINE_KERNEL_COSDF float __kernel_cosdf(double); #endif #ifndef INLINE_KERNEL_TANDF float __kernel_tandf(double,int); #endif float __ldexp_expf(float,int); #ifdef _COMPLEX_H float complex __ldexp_cexpf(float complex,int); #endif /* long double precision kernel functions */ long double __kernel_sinl(long double, long double, int); long double __kernel_cosl(long double, long double); long double __kernel_tanl(long double, long double, int); /* * ld128 version of k_expl.h. See ../ld80/s_expl.c for most comments. * * See ../src/e_exp.c and ../src/k_exp.h for precision-independent comments * about the secondary kernels. */ #define INTERVALS 128 #define LOG2_INTERVALS 7 #define BIAS (LDBL_MAX_EXP - 1) static const double /* * ln2/INTERVALS = L1+L2 (hi+lo decomposition for multiplication). L1 must * have at least 22 (= log2(|LDBL_MIN_EXP-extras|) + log2(INTERVALS)) lowest * bits zero so that multiplication of it by n is exact. */ INV_L = 1.8466496523378731e+2, /* 0x171547652b82fe.0p-45 */ L2 = -1.0253670638894731e-29; /* -0x1.9ff0342542fc3p-97 */ static const long double /* 0x1.62e42fefa39ef35793c768000000p-8 */ L1 = 5.41521234812457272982212595914567508e-3L; /* * XXX values in hex in comments have been lost (or were never present) * from here. */ static const long double /* * Domain [-0.002708, 0.002708], range ~[-2.4021e-38, 2.4234e-38]: * |exp(x) - p(x)| < 2**-124.9 * (0.002708 is ln2/(2*INTERVALS) rounded up a little). * * XXX the coeffs aren't very carefully rounded, and I get 3.6 more bits. */ A2 = 0.5, A3 = 1.66666666666666666666666666651085500e-1L, A4 = 4.16666666666666666666666666425885320e-2L, A5 = 8.33333333333333333334522877160175842e-3L, A6 = 1.38888888888888888889971139751596836e-3L; static const double A7 = 1.9841269841269470e-4, /* 0x1.a01a01a019f91p-13 */ A8 = 2.4801587301585286e-5, /* 0x1.71de3ec75a967p-19 */ A9 = 2.7557324277411235e-6, /* 0x1.71de3ec75a967p-19 */ A10 = 2.7557333722375069e-7; /* 0x1.27e505ab56259p-22 */ static const struct { /* * hi must be rounded to at most 106 bits so that multiplication * by r1 in expm1l() is exact, but it is rounded to 88 bits due to * historical accidents. * * XXX it is wasteful to use long double for both hi and lo. ld128 * exp2l() uses only float for lo (in a very differently organized * table; ld80 exp2l() is different again. It uses 2 doubles in a * table organized like this one. 1 double and 1 float would * suffice). There are different packing/locality/alignment/caching * problems with these methods. * * XXX C's bad %a format makes the bits unreadable. They happen * to all line up for the hi values 1 before the point and 88 * in 22 nybbles, but for the low values the nybbles are shifted * randomly. */ long double hi; long double lo; } tbl[INTERVALS] = { {0x1p0L, 0x0p0L}, {0x1.0163da9fb33356d84a66aep0L, 0x3.36dcdfa4003ec04c360be2404078p-92L}, {0x1.02c9a3e778060ee6f7cacap0L, 0x4.f7a29bde93d70a2cabc5cb89ba10p-92L}, {0x1.04315e86e7f84bd738f9a2p0L, 0xd.a47e6ed040bb4bfc05af6455e9b8p-96L}, {0x1.059b0d31585743ae7c548ep0L, 0xb.68ca417fe53e3495f7df4baf84a0p-92L}, {0x1.0706b29ddf6ddc6dc403a8p0L, 0x1.d87b27ed07cb8b092ac75e311753p-88L}, {0x1.0874518759bc808c35f25cp0L, 0x1.9427fa2b041b2d6829d8993a0d01p-88L}, {0x1.09e3ecac6f3834521e060cp0L, 0x5.84d6b74ba2e023da730e7fccb758p-92L}, {0x1.0b5586cf9890f6298b92b6p0L, 0x1.1842a98364291408b3ceb0a2a2bbp-88L}, {0x1.0cc922b7247f7407b705b8p0L, 0x9.3dc5e8aac564e6fe2ef1d431fd98p-92L}, {0x1.0e3ec32d3d1a2020742e4ep0L, 0x1.8af6a552ac4b358b1129e9f966a4p-88L}, {0x1.0fb66affed31af232091dcp0L, 0x1.8a1426514e0b627bda694a400a27p-88L}, {0x1.11301d0125b50a4ebbf1aep0L, 0xd.9318ceac5cc47ab166ee57427178p-92L}, {0x1.12abdc06c31cbfb92bad32p0L, 0x4.d68e2f7270bdf7cedf94eb1cb818p-92L}, {0x1.1429aaea92ddfb34101942p0L, 0x1.b2586d01844b389bea7aedd221d4p-88L}, {0x1.15a98c8a58e512480d573cp0L, 0x1.d5613bf92a2b618ee31b376c2689p-88L}, {0x1.172b83c7d517adcdf7c8c4p0L, 0x1.0eb14a792035509ff7d758693f24p-88L}, {0x1.18af9388c8de9bbbf70b9ap0L, 0x3.c2505c97c0102e5f1211941d2840p-92L}, {0x1.1a35beb6fcb753cb698f68p0L, 0x1.2d1c835a6c30724d5cfae31b84e5p-88L}, {0x1.1bbe084045cd39ab1e72b4p0L, 0x4.27e35f9acb57e473915519a1b448p-92L}, {0x1.1d4873168b9aa7805b8028p0L, 0x9.90f07a98b42206e46166cf051d70p-92L}, {0x1.1ed5022fcd91cb8819ff60p0L, 0x1.121d1e504d36c47474c9b7de6067p-88L}, {0x1.2063b88628cd63b8eeb028p0L, 0x1.50929d0fc487d21c2b84004264dep-88L}, {0x1.21f49917ddc962552fd292p0L, 0x9.4bdb4b61ea62477caa1dce823ba0p-92L}, {0x1.2387a6e75623866c1fadb0p0L, 0x1.c15cb593b0328566902df69e4de2p-88L}, {0x1.251ce4fb2a63f3582ab7dep0L, 0x9.e94811a9c8afdcf796934bc652d0p-92L}, {0x1.26b4565e27cdd257a67328p0L, 0x1.d3b249dce4e9186ddd5ff44e6b08p-92L}, {0x1.284dfe1f5638096cf15cf0p0L, 0x3.ca0967fdaa2e52d7c8106f2e262cp-92L}, {0x1.29e9df51fdee12c25d15f4p0L, 0x1.a24aa3bca890ac08d203fed80a07p-88L}, {0x1.2b87fd0dad98ffddea4652p0L, 0x1.8fcab88442fdc3cb6de4519165edp-88L}, {0x1.2d285a6e4030b40091d536p0L, 0xd.075384589c1cd1b3e4018a6b1348p-92L}, {0x1.2ecafa93e2f5611ca0f45cp0L, 0x1.523833af611bdcda253c554cf278p-88L}, {0x1.306fe0a31b7152de8d5a46p0L, 0x3.05c85edecbc27343629f502f1af2p-92L}, {0x1.32170fc4cd8313539cf1c2p0L, 0x1.008f86dde3220ae17a005b6412bep-88L}, {0x1.33c08b26416ff4c9c8610cp0L, 0x1.96696bf95d1593039539d94d662bp-88L}, {0x1.356c55f929ff0c94623476p0L, 0x3.73af38d6d8d6f9506c9bbc93cbc0p-92L}, {0x1.371a7373aa9caa7145502ep0L, 0x1.4547987e3e12516bf9c699be432fp-88L}, {0x1.38cae6d05d86585a9cb0d8p0L, 0x1.bed0c853bd30a02790931eb2e8f0p-88L}, {0x1.3a7db34e59ff6ea1bc9298p0L, 0x1.e0a1d336163fe2f852ceeb134067p-88L}, {0x1.3c32dc313a8e484001f228p0L, 0xb.58f3775e06ab66353001fae9fca0p-92L}, {0x1.3dea64c12342235b41223ep0L, 0x1.3d773fba2cb82b8244267c54443fp-92L}, {0x1.3fa4504ac801ba0bf701aap0L, 0x4.1832fb8c1c8dbdff2c49909e6c60p-92L}, {0x1.4160a21f72e29f84325b8ep0L, 0x1.3db61fb352f0540e6ba05634413ep-88L}, {0x1.431f5d950a896dc7044394p0L, 0x1.0ccec81e24b0caff7581ef4127f7p-92L}, {0x1.44e086061892d03136f408p0L, 0x1.df019fbd4f3b48709b78591d5cb5p-88L}, {0x1.46a41ed1d005772512f458p0L, 0x1.229d97df404ff21f39c1b594d3a8p-88L}, {0x1.486a2b5c13cd013c1a3b68p0L, 0x1.062f03c3dd75ce8757f780e6ec99p-88L}, {0x1.4a32af0d7d3de672d8bcf4p0L, 0x6.f9586461db1d878b1d148bd3ccb8p-92L}, {0x1.4bfdad5362a271d4397afep0L, 0xc.42e20e0363ba2e159c579f82e4b0p-92L}, {0x1.4dcb299fddd0d63b36ef1ap0L, 0x9.e0cc484b25a5566d0bd5f58ad238p-92L}, {0x1.4f9b2769d2ca6ad33d8b68p0L, 0x1.aa073ee55e028497a329a7333dbap-88L}, {0x1.516daa2cf6641c112f52c8p0L, 0x4.d822190e718226177d7608d20038p-92L}, {0x1.5342b569d4f81df0a83c48p0L, 0x1.d86a63f4e672a3e429805b049465p-88L}, {0x1.551a4ca5d920ec52ec6202p0L, 0x4.34ca672645dc6c124d6619a87574p-92L}, {0x1.56f4736b527da66ecb0046p0L, 0x1.64eb3c00f2f5ab3d801d7cc7272dp-88L}, {0x1.58d12d497c7fd252bc2b72p0L, 0x1.43bcf2ec936a970d9cc266f0072fp-88L}, {0x1.5ab07dd48542958c930150p0L, 0x1.91eb345d88d7c81280e069fbdb63p-88L}, {0x1.5c9268a5946b701c4b1b80p0L, 0x1.6986a203d84e6a4a92f179e71889p-88L}, {0x1.5e76f15ad21486e9be4c20p0L, 0x3.99766a06548a05829e853bdb2b52p-92L}, {0x1.605e1b976dc08b076f592ap0L, 0x4.86e3b34ead1b4769df867b9c89ccp-92L}, {0x1.6247eb03a5584b1f0fa06ep0L, 0x1.d2da42bb1ceaf9f732275b8aef30p-88L}, {0x1.6434634ccc31fc76f8714cp0L, 0x4.ed9a4e41000307103a18cf7a6e08p-92L}, {0x1.66238825522249127d9e28p0L, 0x1.b8f314a337f4dc0a3adf1787ff74p-88L}, {0x1.68155d44ca973081c57226p0L, 0x1.b9f32706bfe4e627d809a85dcc66p-88L}, {0x1.6a09e667f3bcc908b2fb12p0L, 0x1.66ea957d3e3adec17512775099dap-88L}, {0x1.6c012750bdabeed76a9980p0L, 0xf.4f33fdeb8b0ecd831106f57b3d00p-96L}, {0x1.6dfb23c651a2ef220e2cbep0L, 0x1.bbaa834b3f11577ceefbe6c1c411p-92L}, {0x1.6ff7df9519483cf87e1b4ep0L, 0x1.3e213bff9b702d5aa477c12523cep-88L}, {0x1.71f75e8ec5f73dd2370f2ep0L, 0xf.0acd6cb434b562d9e8a20adda648p-92L}, {0x1.73f9a48a58173bd5c9a4e6p0L, 0x8.ab1182ae217f3a7681759553e840p-92L}, {0x1.75feb564267c8bf6e9aa32p0L, 0x1.a48b27071805e61a17b954a2dad8p-88L}, {0x1.780694fde5d3f619ae0280p0L, 0x8.58b2bb2bdcf86cd08e35fb04c0f0p-92L}, {0x1.7a11473eb0186d7d51023ep0L, 0x1.6cda1f5ef42b66977960531e821bp-88L}, {0x1.7c1ed0130c1327c4933444p0L, 0x1.937562b2dc933d44fc828efd4c9cp-88L}, {0x1.7e2f336cf4e62105d02ba0p0L, 0x1.5797e170a1427f8fcdf5f3906108p-88L}, {0x1.80427543e1a11b60de6764p0L, 0x9.a354ea706b8e4d8b718a672bf7c8p-92L}, {0x1.82589994cce128acf88afap0L, 0xb.34a010f6ad65cbbac0f532d39be0p-92L}, {0x1.8471a4623c7acce52f6b96p0L, 0x1.c64095370f51f48817914dd78665p-88L}, {0x1.868d99b4492ec80e41d90ap0L, 0xc.251707484d73f136fb5779656b70p-92L}, {0x1.88ac7d98a669966530bcdep0L, 0x1.2d4e9d61283ef385de170ab20f96p-88L}, {0x1.8ace5422aa0db5ba7c55a0p0L, 0x1.92c9bb3e6ed61f2733304a346d8fp-88L}, {0x1.8cf3216b5448bef2aa1cd0p0L, 0x1.61c55d84a9848f8c453b3ca8c946p-88L}, {0x1.8f1ae991577362b982745cp0L, 0x7.2ed804efc9b4ae1458ae946099d4p-92L}, {0x1.9145b0b91ffc588a61b468p0L, 0x1.f6b70e01c2a90229a4c4309ea719p-88L}, {0x1.93737b0cdc5e4f4501c3f2p0L, 0x5.40a22d2fc4af581b63e8326efe9cp-92L}, {0x1.95a44cbc8520ee9b483694p0L, 0x1.a0fc6f7c7d61b2b3a22a0eab2cadp-88L}, {0x1.97d829fde4e4f8b9e920f8p0L, 0x1.1e8bd7edb9d7144b6f6818084cc7p-88L}, {0x1.9a0f170ca07b9ba3109b8cp0L, 0x4.6737beb19e1eada6825d3c557428p-92L}, {0x1.9c49182a3f0901c7c46b06p0L, 0x1.1f2be58ddade50c217186c90b457p-88L}, {0x1.9e86319e323231824ca78ep0L, 0x6.4c6e010f92c082bbadfaf605cfd4p-92L}, {0x1.a0c667b5de564b29ada8b8p0L, 0xc.ab349aa0422a8da7d4512edac548p-92L}, {0x1.a309bec4a2d3358c171f76p0L, 0x1.0daad547fa22c26d168ea762d854p-88L}, {0x1.a5503b23e255c8b424491cp0L, 0xa.f87bc8050a405381703ef7caff50p-92L}, {0x1.a799e1330b3586f2dfb2b0p0L, 0x1.58f1a98796ce8908ae852236ca94p-88L}, {0x1.a9e6b5579fdbf43eb243bcp0L, 0x1.ff4c4c58b571cf465caf07b4b9f5p-88L}, {0x1.ac36bbfd3f379c0db966a2p0L, 0x1.1265fc73e480712d20f8597a8e7bp-88L}, {0x1.ae89f995ad3ad5e8734d16p0L, 0x1.73205a7fbc3ae675ea440b162d6cp-88L}, {0x1.b0e07298db66590842acdep0L, 0x1.c6f6ca0e5dcae2aafffa7a0554cbp-88L}, {0x1.b33a2b84f15faf6bfd0e7ap0L, 0x1.d947c2575781dbb49b1237c87b6ep-88L}, {0x1.b59728de559398e3881110p0L, 0x1.64873c7171fefc410416be0a6525p-88L}, {0x1.b7f76f2fb5e46eaa7b081ap0L, 0xb.53c5354c8903c356e4b625aacc28p-92L}, {0x1.ba5b030a10649840cb3c6ap0L, 0xf.5b47f297203757e1cc6eadc8bad0p-92L}, {0x1.bcc1e904bc1d2247ba0f44p0L, 0x1.b3d08cd0b20287092bd59be4ad98p-88L}, {0x1.bf2c25bd71e088408d7024p0L, 0x1.18e3449fa073b356766dfb568ff4p-88L}, {0x1.c199bdd85529c2220cb12ap0L, 0x9.1ba6679444964a36661240043970p-96L}, {0x1.c40ab5fffd07a6d14df820p0L, 0xf.1828a5366fd387a7bdd54cdf7300p-92L}, {0x1.c67f12e57d14b4a2137fd2p0L, 0xf.2b301dd9e6b151a6d1f9d5d5f520p-96L}, {0x1.c8f6d9406e7b511acbc488p0L, 0x5.c442ddb55820171f319d9e5076a8p-96L}, {0x1.cb720dcef90691503cbd1ep0L, 0x9.49db761d9559ac0cb6dd3ed599e0p-92L}, {0x1.cdf0b555dc3f9c44f8958ep0L, 0x1.ac51be515f8c58bdfb6f5740a3a4p-88L}, {0x1.d072d4a07897b8d0f22f20p0L, 0x1.a158e18fbbfc625f09f4cca40874p-88L}, {0x1.d2f87080d89f18ade12398p0L, 0x9.ea2025b4c56553f5cdee4c924728p-92L}, {0x1.d5818dcfba48725da05aeap0L, 0x1.66e0dca9f589f559c0876ff23830p-88L}, {0x1.d80e316c98397bb84f9d04p0L, 0x8.805f84bec614de269900ddf98d28p-92L}, {0x1.da9e603db3285708c01a5ap0L, 0x1.6d4c97f6246f0ec614ec95c99392p-88L}, {0x1.dd321f301b4604b695de3cp0L, 0x6.30a393215299e30d4fb73503c348p-96L}, {0x1.dfc97337b9b5eb968cac38p0L, 0x1.ed291b7225a944efd5bb5524b927p-88L}, {0x1.e264614f5a128a12761fa0p0L, 0x1.7ada6467e77f73bf65e04c95e29dp-88L}, {0x1.e502ee78b3ff6273d13014p0L, 0x1.3991e8f49659e1693be17ae1d2f9p-88L}, {0x1.e7a51fbc74c834b548b282p0L, 0x1.23786758a84f4956354634a416cep-88L}, {0x1.ea4afa2a490d9858f73a18p0L, 0xf.5db301f86dea20610ceee13eb7b8p-92L}, {0x1.ecf482d8e67f08db0312fap0L, 0x1.949cef462010bb4bc4ce72a900dfp-88L}, {0x1.efa1bee615a27771fd21a8p0L, 0x1.2dac1f6dd5d229ff68e46f27e3dfp-88L}, {0x1.f252b376bba974e8696fc2p0L, 0x1.6390d4c6ad5476b5162f40e1d9a9p-88L}, {0x1.f50765b6e4540674f84b76p0L, 0x2.862baff99000dfc4352ba29b8908p-92L}, {0x1.f7bfdad9cbe138913b4bfep0L, 0x7.2bd95c5ce7280fa4d2344a3f5618p-92L}, {0x1.fa7c1819e90d82e90a7e74p0L, 0xb.263c1dc060c36f7650b4c0f233a8p-92L}, {0x1.fd3c22b8f71f10975ba4b2p0L, 0x1.2bcf3a5e12d269d8ad7c1a4a8875p-88L} }; /* * Kernel for expl(x). x must be finite and not tiny or huge. * "tiny" is anything that would make us underflow (|A6*x^6| < ~LDBL_MIN). * "huge" is anything that would make fn*L1 inexact (|x| > ~2**17*ln2). */ static inline void __k_expl(long double x, long double *hip, long double *lop, int *kp) { long double q, r, r1, t; double dr, fn, r2; int n, n2; /* Reduce x to (k*ln2 + endpoint[n2] + r1 + r2). */ fn = rnint((double)x * INV_L); n = irint(fn); n2 = (unsigned)n % INTERVALS; /* Depend on the sign bit being propagated: */ *kp = n >> LOG2_INTERVALS; r1 = x - fn * L1; r2 = fn * -L2; r = r1 + r2; /* Evaluate expl(endpoint[n2] + r1 + r2) = tbl[n2] * expl(r1 + r2). */ dr = r; q = r2 + r * r * (A2 + r * (A3 + r * (A4 + r * (A5 + r * (A6 + dr * (A7 + dr * (A8 + dr * (A9 + dr * A10)))))))); t = tbl[n2].lo + tbl[n2].hi; *hip = tbl[n2].hi; *lop = tbl[n2].lo + t * (q + r1); } /* * XXX: the rest of the functions are identical for ld80 and ld128. * However, we should use scalbnl() for ld128, since long double * multiplication was very slow on sparc64 and no new evaluation has * been made for aarch64 and/or riscv. */ static inline void k_hexpl(long double x, long double *hip, long double *lop) { float twopkm1; int k; __k_expl(x, hip, lop, &k); SET_FLOAT_WORD(twopkm1, 0x3f800000 + ((k - 1) << 23)); *hip *= twopkm1; *lop *= twopkm1; } static inline long double hexpl(long double x) { long double hi, lo, twopkm2; int k; twopkm2 = 1; __k_expl(x, &hi, &lo, &k); SET_LDBL_EXPSIGN(twopkm2, BIAS + k - 2); return (lo + hi) * 2 * twopkm2; } #ifdef _COMPLEX_H /* * See ../src/k_exp.c for details. */ static inline long double complex __ldexp_cexpl(long double complex z, int expt) { long double c, exp_x, hi, lo, s; long double x, y, scale1, scale2; int half_expt, k; x = creall(z); y = cimagl(z); __k_expl(x, &hi, &lo, &k); exp_x = (lo + hi) * 0x1p16382L; expt += k - 16382; scale1 = 1; half_expt = expt / 2; SET_LDBL_EXPSIGN(scale1, BIAS + half_expt); scale2 = 1; SET_LDBL_EXPSIGN(scale2, BIAS + expt - half_expt); sincosl(y, &s, &c); return (CMPLXL(c * exp_x * scale1 * scale2, s * exp_x * scale1 * scale2)); } #endif /* _COMPLEX_H */ COSMOPOLITAN_C_END_ #endif /* !(__ASSEMBLER__ + __LINKER__ + 0) */ #endif /* COSMOPOLITAN_LIBC_TINYMATH_FREEBSD_INTERNAL_H_ */