// This file is part of Eigen, a lightweight C++ template library // for linear algebra. // // Copyright (C) 2007 Julien Pommier // Copyright (C) 2014 Pedro Gonnet (pedro.gonnet@gmail.com) // Copyright (C) 2009-2019 Gael Guennebaud <gael.guennebaud@inria.fr> // // This Source Code Form is subject to the terms of the Mozilla // Public License v. 2.0. If a copy of the MPL was not distributed // with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
// Determine exponent and mantissa from normalized_a.
exponent = pfrexp_generic_get_biased_exponent(normalized_a); // Zero, Inf and NaN return 'a' unmodified, exponent is zero // (technically the exponent is unspecified for inf/NaN, but GCC/Clang set it to zero) const Scalar scalar_non_finite_exponent = Scalar((ScalarUI(1) << int(ExponentBits)) - ScalarUI(1)); // 255 const Packet non_finite_exponent = pset1<Packet>(scalar_non_finite_exponent); const Packet is_zero_or_not_finite = por(pcmp_eq(a, zero), pcmp_eq(exponent, non_finite_exponent)); const Packet m = pselect(is_zero_or_not_finite, a, por(pand(normalized_a, sign_mantissa_mask), half));
exponent = pselect(is_zero_or_not_finite, zero, padd(exponent, exponent_offset)); return m;
}
// Safely applies ldexp, correctly handles overflows, underflows and denormals. // Assumes IEEE floating point format. template<typename Packet> EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC
Packet pldexp_generic(const Packet& a, const Packet& exponent) { // We want to return a * 2^exponent, allowing for all possible integer // exponents without overflowing or underflowing in intermediate // computations. // // Since 'a' and the output can be denormal, the maximum range of 'exponent' // to consider for a float is: // -255-23 -> 255+23 // Below -278 any finite float 'a' will become zero, and above +278 any // finite float will become inf, including when 'a' is the smallest possible // denormal. // // Unfortunately, 2^(278) cannot be represented using either one or two // finite normal floats, so we must split the scale factor into at least // three parts. It turns out to be faster to split 'exponent' into four // factors, since [exponent>>2] is much faster to compute that [exponent/3]. // // Set e = min(max(exponent, -278), 278); // b = floor(e/4); // out = ((((a * 2^(b)) * 2^(b)) * 2^(b)) * 2^(e-3*b)) // // This will avoid any intermediate overflows and correctly handle 0, inf, // NaN cases. typedeftypename unpacket_traits<Packet>::integer_packet PacketI; typedeftypename unpacket_traits<Packet>::type Scalar; typedeftypename unpacket_traits<PacketI>::type ScalarI; enum {
TotalBits = sizeof(Scalar) * CHAR_BIT,
MantissaBits = numext::numeric_limits<Scalar>::digits - 1,
ExponentBits = int(TotalBits) - int(MantissaBits) - 1
};
const Packet max_exponent = pset1<Packet>(Scalar((ScalarI(1)<<int(ExponentBits)) + ScalarI(int(MantissaBits) - 1))); // 278 const PacketI bias = pset1<PacketI>((ScalarI(1)<<(int(ExponentBits)-1)) - ScalarI(1)); // 127 const PacketI e = pcast<Packet, PacketI>(pmin(pmax(exponent, pnegate(max_exponent)), max_exponent));
PacketI b = parithmetic_shift_right<2>(e); // floor(e/4);
Packet c = preinterpret<Packet>(plogical_shift_left<int(MantissaBits)>(padd(b, bias))); // 2^b
Packet out = pmul(pmul(pmul(a, c), c), c); // a * 2^(3b)
b = psub(psub(psub(e, b), b), b); // e - 3b
c = preinterpret<Packet>(plogical_shift_left<int(MantissaBits)>(padd(b, bias))); // 2^(e-3*b)
out = pmul(out, c); return out;
}
// Explicitly multiplies // a * (2^e) // clamping e to the range // [NumTraits<Scalar>::min_exponent()-2, NumTraits<Scalar>::max_exponent()] // // This is approx 7x faster than pldexp_impl, but will prematurely over/underflow // if 2^e doesn't fit into a normal floating-point Scalar. // // Assumes IEEE floating point format template<typename Packet> struct pldexp_fast_impl { typedeftypename unpacket_traits<Packet>::integer_packet PacketI; typedeftypename unpacket_traits<Packet>::type Scalar; typedeftypename unpacket_traits<PacketI>::type ScalarI; enum {
TotalBits = sizeof(Scalar) * CHAR_BIT,
MantissaBits = numext::numeric_limits<Scalar>::digits - 1,
ExponentBits = int(TotalBits) - int(MantissaBits) - 1
};
// Natural or base 2 logarithm. // Computes log(x) as log(2^e * m) = C*e + log(m), where the constant C =log(2) // and m is in the range [sqrt(1/2),sqrt(2)). In this range, the logarithm can // be easily approximated by a polynomial centered on m=1 for stability. // TODO(gonnet): Further reduce the interval allowing for lower-degree // polynomial interpolants -> ... -> profit! template <typename Packet, bool base2>
EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS
EIGEN_UNUSED
Packet plog_impl_float(const Packet _x)
{
Packet x = _x;
// Truncate input values to the minimum positive normal.
x = pmax(x, cst_min_norm_pos);
Packet e; // extract significant in the range [0.5,1) and exponent
x = pfrexp(x,e);
// part2: Shift the inputs from the range [0.5,1) to [sqrt(1/2),sqrt(2)) // and shift by -1. The values are then centered around 0, which improves // the stability of the polynomial evaluation. // if( x < SQRTHF ) { // e -= 1; // x = x + x - 1.0; // } else { x = x - 1.0; }
Packet mask = pcmp_lt(x, cst_cephes_SQRTHF);
Packet tmp = pand(x, mask);
x = psub(x, cst_1);
e = psub(e, pand(cst_1, mask));
x = padd(x, tmp);
Packet x2 = pmul(x, x);
Packet x3 = pmul(x2, x);
// Evaluate the polynomial approximant of degree 8 in three parts, probably // to improve instruction-level parallelism.
Packet y, y1, y2;
y = pmadd(cst_cephes_log_p0, x, cst_cephes_log_p1);
y1 = pmadd(cst_cephes_log_p3, x, cst_cephes_log_p4);
y2 = pmadd(cst_cephes_log_p6, x, cst_cephes_log_p7);
y = pmadd(y, x, cst_cephes_log_p2);
y1 = pmadd(y1, x, cst_cephes_log_p5);
y2 = pmadd(y2, x, cst_cephes_log_p8);
y = pmadd(y, x3, y1);
y = pmadd(y, x3, y2);
y = pmul(y, x3);
y = pmadd(cst_neg_half, x2, y);
x = padd(x, y);
// Add the logarithm of the exponent back to the result of the interpolation. if (base2) { const Packet cst_log2e = pset1<Packet>(static_cast<float>(EIGEN_LOG2E));
x = pmadd(x, cst_log2e, e);
} else { const Packet cst_ln2 = pset1<Packet>(static_cast<float>(EIGEN_LN2));
x = pmadd(e, cst_ln2, x);
}
Packet invalid_mask = pcmp_lt_or_nan(_x, pzero(_x));
Packet iszero_mask = pcmp_eq(_x,pzero(_x));
Packet pos_inf_mask = pcmp_eq(_x,cst_pos_inf); // Filter out invalid inputs, i.e.: // - negative arg will be NAN // - 0 will be -INF // - +INF will be +INF return pselect(iszero_mask, cst_minus_inf,
por(pselect(pos_inf_mask,cst_pos_inf,x), invalid_mask));
}
/* Returns the base e (2.718...) or base 2 logarithm of x. * The argument is separated into its exponent and fractional parts. * The logarithm of the fraction in the interval [sqrt(1/2), sqrt(2)], * is approximated by * * log(1+x) = x - 0.5 x**2 + x**3 P(x)/Q(x). * * for more detail see: http://www.netlib.org/cephes/
*/ template <typename Packet, bool base2>
EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS
EIGEN_UNUSED
Packet plog_impl_double(const Packet _x)
{
Packet x = _x;
// Truncate input values to the minimum positive normal.
x = pmax(x, cst_min_norm_pos);
Packet e; // extract significant in the range [0.5,1) and exponent
x = pfrexp(x,e);
// Shift the inputs from the range [0.5,1) to [sqrt(1/2),sqrt(2)) // and shift by -1. The values are then centered around 0, which improves // the stability of the polynomial evaluation. // if( x < SQRTHF ) { // e -= 1; // x = x + x - 1.0; // } else { x = x - 1.0; }
Packet mask = pcmp_lt(x, cst_cephes_SQRTHF);
Packet tmp = pand(x, mask);
x = psub(x, cst_1);
e = psub(e, pand(cst_1, mask));
x = padd(x, tmp);
Packet x2 = pmul(x, x);
Packet x3 = pmul(x2, x);
// Evaluate the polynomial approximant , probably to improve instruction-level parallelism. // y = x - 0.5*x^2 + x^3 * polevl( x, P, 5 ) / p1evl( x, Q, 5 ) );
Packet y, y1, y_;
y = pmadd(cst_cephes_log_p0, x, cst_cephes_log_p1);
y1 = pmadd(cst_cephes_log_p3, x, cst_cephes_log_p4);
y = pmadd(y, x, cst_cephes_log_p2);
y1 = pmadd(y1, x, cst_cephes_log_p5);
y_ = pmadd(y, x3, y1);
y = pmadd(cst_cephes_log_q0, x, cst_cephes_log_q1);
y1 = pmadd(cst_cephes_log_q3, x, cst_cephes_log_q4);
y = pmadd(y, x, cst_cephes_log_q2);
y1 = pmadd(y1, x, cst_cephes_log_q5);
y = pmadd(y, x3, y1);
y_ = pmul(y_, x3);
y = pdiv(y_, y);
y = pmadd(cst_neg_half, x2, y);
x = padd(x, y);
// Add the logarithm of the exponent back to the result of the interpolation. if (base2) { const Packet cst_log2e = pset1<Packet>(static_cast<double>(EIGEN_LOG2E));
x = pmadd(x, cst_log2e, e);
} else { const Packet cst_ln2 = pset1<Packet>(static_cast<double>(EIGEN_LN2));
x = pmadd(e, cst_ln2, x);
}
Packet invalid_mask = pcmp_lt_or_nan(_x, pzero(_x));
Packet iszero_mask = pcmp_eq(_x,pzero(_x));
Packet pos_inf_mask = pcmp_eq(_x,cst_pos_inf); // Filter out invalid inputs, i.e.: // - negative arg will be NAN // - 0 will be -INF // - +INF will be +INF return pselect(iszero_mask, cst_minus_inf,
por(pselect(pos_inf_mask,cst_pos_inf,x), invalid_mask));
}
/** \internal \returns exp(x)-1 computed using W. Kahan's formula. See: http://www.plunk.org/~hatch/rightway.php
*/ template<typename Packet>
Packet generic_expm1(const Packet& x)
{ typedeftypename unpacket_traits<Packet>::type ScalarType; const Packet one = pset1<Packet>(ScalarType(1)); const Packet neg_one = pset1<Packet>(ScalarType(-1));
Packet u = pexp(x);
Packet one_mask = pcmp_eq(u, one);
Packet u_minus_one = psub(u, one);
Packet neg_one_mask = pcmp_eq(u_minus_one, neg_one);
Packet logu = plog(u); // The following comparison is to catch the case where // exp(x) = +inf. It is written in this way to avoid having // to form the constant +inf, which depends on the packet // type.
Packet pos_inf_mask = pcmp_eq(logu, u);
Packet expm1 = pmul(u_minus_one, pdiv(x, logu));
expm1 = pselect(pos_inf_mask, u, expm1); return pselect(one_mask,
x,
pselect(neg_one_mask,
neg_one,
expm1));
}
// Exponential function. Works by writing "x = m*log(2) + r" where // "m = floor(x/log(2)+1/2)" and "r" is the remainder. The result is then // "exp(x) = 2^m*exp(r)" where exp(r) is in the range [-1,1). template <typename Packet>
EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS
EIGEN_UNUSED
Packet pexp_float(const Packet _x)
{ const Packet cst_1 = pset1<Packet>(1.0f); const Packet cst_half = pset1<Packet>(0.5f); const Packet cst_exp_hi = pset1<Packet>( 88.723f); const Packet cst_exp_lo = pset1<Packet>(-88.723f);
// Clamp x.
Packet x = pmax(pmin(_x, cst_exp_hi), cst_exp_lo);
// Express exp(x) as exp(m*ln(2) + r), start by extracting // m = floor(x/ln(2) + 0.5).
Packet m = pfloor(pmadd(x, cst_cephes_LOG2EF, cst_half));
// Get r = x - m*ln(2). If no FMA instructions are available, m*ln(2) is // subtracted out in two parts, m*C1+m*C2 = m*ln(2), to avoid accumulating // truncation errors. const Packet cst_cephes_exp_C1 = pset1<Packet>(-0.693359375f); const Packet cst_cephes_exp_C2 = pset1<Packet>(2.12194440e-4f);
Packet r = pmadd(m, cst_cephes_exp_C1, x);
r = pmadd(m, cst_cephes_exp_C2, r);
Packet r2 = pmul(r, r);
Packet r3 = pmul(r2, r);
// Evaluate the polynomial approximant,improved by instruction-level parallelism.
Packet y, y1, y2;
y = pmadd(cst_cephes_exp_p0, r, cst_cephes_exp_p1);
y1 = pmadd(cst_cephes_exp_p3, r, cst_cephes_exp_p4);
y2 = padd(r, cst_1);
y = pmadd(y, r, cst_cephes_exp_p2);
y1 = pmadd(y1, r, cst_cephes_exp_p5);
y = pmadd(y, r3, y1);
y = pmadd(y, r2, y2);
// Return 2^m * exp(r). // TODO: replace pldexp with faster implementation since y in [-1, 1). return pmax(pldexp(y,m), _x);
}
// clamp x
x = pmax(pmin(x, cst_exp_hi), cst_exp_lo); // Express exp(x) as exp(g + n*log(2)).
fx = pmadd(cst_cephes_LOG2EF, x, cst_half);
// Get the integer modulus of log(2), i.e. the "n" described above.
fx = pfloor(fx);
// Get the remainder modulo log(2), i.e. the "g" described above. Subtract // n*log(2) out in two steps, i.e. n*C1 + n*C2, C1+C2=log2 to get the last // digits right.
tmp = pmul(fx, cst_cephes_exp_C1);
Packet z = pmul(fx, cst_cephes_exp_C2);
x = psub(x, tmp);
x = psub(x, z);
Packet x2 = pmul(x, x);
// Evaluate the numerator polynomial of the rational interpolant.
Packet px = cst_cephes_exp_p0;
px = pmadd(px, x2, cst_cephes_exp_p1);
px = pmadd(px, x2, cst_cephes_exp_p2);
px = pmul(px, x);
// Evaluate the denominator polynomial of the rational interpolant.
Packet qx = cst_cephes_exp_q0;
qx = pmadd(qx, x2, cst_cephes_exp_q1);
qx = pmadd(qx, x2, cst_cephes_exp_q2);
qx = pmadd(qx, x2, cst_cephes_exp_q3);
// I don't really get this bit, copied from the SSE2 routines, so... // TODO(gonnet): Figure out what is going on here, perhaps find a better // rational interpolant?
x = pdiv(px, psub(qx, px));
x = pmadd(cst_2, x, cst_1);
// Construct the result 2^n * exp(g) = e * x. The max is used to catch // non-finite values in the input. // TODO: replace pldexp with faster implementation since x in [-1, 1). return pmax(pldexp(x,fx), _x);
}
// The following code is inspired by the following stack-overflow answer: // https://stackoverflow.com/questions/30463616/payne-hanek-algorithm-implementation-in-c/30465751#30465751 // It has been largely optimized: // - By-pass calls to frexp. // - Aligned loads of required 96 bits of 2/pi. This is accomplished by // (1) balancing the mantissa and exponent to the required bits of 2/pi are // aligned on 8-bits, and (2) replicating the storage of the bits of 2/pi. // - Avoid a branch in rounding and extraction of the remaining fractional part. // Overall, I measured a speed up higher than x2 on x86-64. inlinefloat trig_reduce_huge (float xf, int *quadrant)
{ using Eigen::numext::int32_t; using Eigen::numext::uint32_t; using Eigen::numext::int64_t; using Eigen::numext::uint64_t;
// 192 bits of 2/pi for Payne-Hanek reduction // Bits are introduced by packet of 8 to enable aligned reads. staticconst uint32_t two_over_pi [] =
{
0x00000028, 0x000028be, 0x0028be60, 0x28be60db,
0xbe60db93, 0x60db9391, 0xdb939105, 0x9391054a,
0x91054a7f, 0x054a7f09, 0x4a7f09d5, 0x7f09d5f4,
0x09d5f47d, 0xd5f47d4d, 0xf47d4d37, 0x7d4d3770,
0x4d377036, 0x377036d8, 0x7036d8a5, 0x36d8a566,
0xd8a5664f, 0xa5664f10, 0x664f10e4, 0x4f10e410,
0x10e41000, 0xe4100000
};
uint32_t xi = numext::bit_cast<uint32_t>(xf); // Below, -118 = -126 + 8. // -126 is to get the exponent, // +8 is to enable alignment of 2/pi's bits on 8 bits. // This is possible because the fractional part of x as only 24 meaningful bits.
uint32_t e = (xi >> 23) - 118; // Extract the mantissa and shift it to align it wrt the exponent
xi = ((xi & 0x007fffffu)| 0x00800000u) << (e & 0x7);
uint32_t i = e >> 3;
uint32_t twoopi_1 = two_over_pi[i-1];
uint32_t twoopi_2 = two_over_pi[i+3];
uint32_t twoopi_3 = two_over_pi[i+7];
// Compute x * 2/pi in 2.62-bit fixed-point format.
uint64_t p;
p = uint64_t(xi) * twoopi_3;
p = uint64_t(xi) * twoopi_2 + (p >> 32);
p = (uint64_t(xi * twoopi_1) << 32) + p;
// Round to nearest: add 0.5 and extract integral part.
uint64_t q = (p + zero_dot_five) >> 62;
*quadrant = int(q); // Now it remains to compute "r = x - q*pi/2" with high accuracy, // since we have p=x/(pi/2) with high accuracy, we can more efficiently compute r as: // r = (p-q)*pi/2, // where the product can be be carried out with sufficient accuracy using double precision.
p -= q<<62; returnfloat(double(int64_t(p)) * pio2_62);
}
// Scale x by 2/Pi to find x's octant.
Packet y = pmul(x, cst_2oPI);
// Rounding trick:
Packet y_round = padd(y, cst_rounding_magic);
EIGEN_OPTIMIZATION_BARRIER(y_round)
PacketI y_int = preinterpret<PacketI>(y_round); // last 23 digits represent integer (if abs(x)<2^24)
y = psub(y_round, cst_rounding_magic); // nearest integer to x*4/pi
// Reduce x by y octants to get: -Pi/4 <= x <= +Pi/4 // using "Extended precision modular arithmetic" #ifdefined(EIGEN_HAS_SINGLE_INSTRUCTION_MADD) // This version requires true FMA for high accuracy // It provides a max error of 1ULP up to (with absolute_error < 5.9605e-08): constfloat huge_th = ComputeSine ? 117435.992f : 71476.0625f;
x = pmadd(y, pset1<Packet>(-1.57079601287841796875f), x);
x = pmadd(y, pset1<Packet>(-3.1391647326017846353352069854736328125e-07f), x);
x = pmadd(y, pset1<Packet>(-5.390302529957764765544681040410068817436695098876953125e-15f), x); #else // Without true FMA, the previous set of coefficients maintain 1ULP accuracy // up to x<15.7 (for sin), but accuracy is immediately lost for x>15.7. // We thus use one more iteration to maintain 2ULPs up to reasonably large inputs.
// The following set of coefficients maintain 1ULP up to 9.43 and 14.16 for sin and cos respectively. // and 2 ULP up to: constfloat huge_th = ComputeSine ? 25966.f : 18838.f;
x = pmadd(y, pset1<Packet>(-1.5703125), x); // = 0xbfc90000
EIGEN_OPTIMIZATION_BARRIER(x)
x = pmadd(y, pset1<Packet>(-0.000483989715576171875), x); // = 0xb9fdc000
EIGEN_OPTIMIZATION_BARRIER(x)
x = pmadd(y, pset1<Packet>(1.62865035235881805419921875e-07), x); // = 0x342ee000
x = pmadd(y, pset1<Packet>(5.5644315544167710640977020375430583953857421875e-11), x); // = 0x2e74b9ee
// For the record, the following set of coefficients maintain 2ULP up // to a slightly larger range: // const float huge_th = ComputeSine ? 51981.f : 39086.125f; // but it slightly fails to maintain 1ULP for two values of sin below pi. // x = pmadd(y, pset1<Packet>(-3.140625/2.), x); // x = pmadd(y, pset1<Packet>(-0.00048351287841796875), x); // x = pmadd(y, pset1<Packet>(-3.13855707645416259765625e-07), x); // x = pmadd(y, pset1<Packet>(-6.0771006282767103812147979624569416046142578125e-11), x);
// For the record, with only 3 iterations it is possible to maintain // 1 ULP up to 3PI (maybe more) and 2ULP up to 255. // The coefficients are: 0xbfc90f80, 0xb7354480, 0x2e74b9ee #endif
// Compute the sign to apply to the polynomial. // sin: sign = second_bit(y_int) xor signbit(_x) // cos: sign = second_bit(y_int+1)
Packet sign_bit = ComputeSine ? pxor(_x, preinterpret<Packet>(plogical_shift_left<30>(y_int)))
: preinterpret<Packet>(plogical_shift_left<30>(padd(y_int,csti_1)));
sign_bit = pand(sign_bit, cst_sign_mask); // clear all but left most bit
// Get the polynomial selection mask from the second bit of y_int // We'll calculate both (sin and cos) polynomials and then select from the two.
Packet poly_mask = preinterpret<Packet>(pcmp_eq(pand(y_int, csti_1), pzero(y_int)));
// Computes the principal sqrt of the complex numbers in the input. // // For example, for packets containing 2 complex numbers stored in interleaved format // a = [a0, a1] = [x0, y0, x1, y1], // where x0 = real(a0), y0 = imag(a0) etc., this function returns // b = [b0, b1] = [u0, v0, u1, v1], // such that b0^2 = a0, b1^2 = a1. // // To derive the formula for the complex square roots, let's consider the equation for // a single complex square root of the number x + i*y. We want to find real numbers // u and v such that // (u + i*v)^2 = x + i*y <=> // u^2 - v^2 + i*2*u*v = x + i*v. // By equating the real and imaginary parts we get: // u^2 - v^2 = x // 2*u*v = y. // // For x >= 0, this has the numerically stable solution // u = sqrt(0.5 * (x + sqrt(x^2 + y^2))) // v = 0.5 * (y / u) // and for x < 0, // v = sign(y) * sqrt(0.5 * (-x + sqrt(x^2 + y^2))) // u = 0.5 * (y / v) // // To avoid unnecessary over- and underflow, we compute sqrt(x^2 + y^2) as // l = max(|x|, |y|) * sqrt(1 + (min(|x|, |y|) / max(|x|, |y|))^2) ,
// In the following, without lack of generality, we have annotated the code, assuming // that the input is a packet of 2 complex numbers. // // Step 1. Compute l = [l0, l0, l1, l1], where // l0 = sqrt(x0^2 + y0^2), l1 = sqrt(x1^2 + y1^2) // To avoid over- and underflow, we use the stable formula for each hypotenuse // l0 = (min0 == 0 ? max0 : max0 * sqrt(1 + (min0/max0)**2)), // where max0 = max(|x0|, |y0|), min0 = min(|x0|, |y0|), and similarly for l1.
RealPacket a_abs = pabs(a.v); // [|x0|, |y0|, |x1|, |y1|]
RealPacket a_abs_flip = pcplxflip(Packet(a_abs)).v; // [|y0|, |x0|, |y1|, |x1|]
RealPacket a_max = pmax(a_abs, a_abs_flip);
RealPacket a_min = pmin(a_abs, a_abs_flip);
RealPacket a_min_zero_mask = pcmp_eq(a_min, pzero(a_min));
RealPacket a_max_zero_mask = pcmp_eq(a_max, pzero(a_max));
RealPacket r = pdiv(a_min, a_max); const RealPacket cst_one = pset1<RealPacket>(RealScalar(1));
RealPacket l = pmul(a_max, psqrt(padd(cst_one, pmul(r, r)))); // [l0, l0, l1, l1] // Set l to a_max if a_min is zero.
l = pselect(a_min_zero_mask, a_max, l);
// Step 2. Compute [rho0, *, rho1, *], where // rho0 = sqrt(0.5 * (l0 + |x0|)), rho1 = sqrt(0.5 * (l1 + |x1|)) // We don't care about the imaginary parts computed here. They will be overwritten later. const RealPacket cst_half = pset1<RealPacket>(RealScalar(0.5));
Packet rho;
rho.v = psqrt(pmul(cst_half, padd(a_abs, l)));
// Step 3. Compute [rho0, eta0, rho1, eta1], where // eta0 = (y0 / l0) / 2, and eta1 = (y1 / l1) / 2. // set eta = 0 of input is 0 + i0.
RealPacket eta = pandnot(pmul(cst_half, pdiv(a.v, pcplxflip(rho).v)), a_max_zero_mask);
RealPacket real_mask = peven_mask(a.v);
Packet positive_real_result; // Compute result for inputs with positive real part.
positive_real_result.v = pselect(real_mask, rho.v, eta);
// Step 4. Compute solution for inputs with negative real part: // [|eta0|, sign(y0)*rho0, |eta1|, sign(y1)*rho1] const RealScalar neg_zero = RealScalar(numext::bit_cast<float>(0x80000000u)); const RealPacket cst_imag_sign_mask = pset1<Packet>(Scalar(RealScalar(0.0), neg_zero)).v;
RealPacket imag_signs = pand(a.v, cst_imag_sign_mask);
Packet negative_real_result; // Notice that rho is positive, so taking it's absolute value is a noop.
negative_real_result.v = por(pabs(pcplxflip(positive_real_result).v), imag_signs);
// Step 5. Select solution branch based on the sign of the real parts.
Packet negative_real_mask;
negative_real_mask.v = pcmp_lt(pand(real_mask, a.v), pzero(a.v));
negative_real_mask.v = por(negative_real_mask.v, pcplxflip(negative_real_mask).v);
Packet result = pselect(negative_real_mask, negative_real_result, positive_real_result);
// Step 6. Handle special cases for infinities: // * If z is (x,+∞), the result is (+∞,+∞) even if x is NaN // * If z is (x,-∞), the result is (+∞,-∞) even if x is NaN // * If z is (-∞,y), the result is (0*|y|,+∞) for finite or NaN y // * If z is (+∞,y), the result is (+∞,0*|y|) for finite or NaN y const RealPacket cst_pos_inf = pset1<RealPacket>(NumTraits<RealScalar>::infinity());
Packet is_inf;
is_inf.v = pcmp_eq(a_abs, cst_pos_inf);
Packet is_real_inf;
is_real_inf.v = pand(is_inf.v, real_mask);
is_real_inf = por(is_real_inf, pcplxflip(is_real_inf)); // prepare packet of (+∞,0*|y|) or (0*|y|,+∞), depending on the sign of the infinite real part.
Packet real_inf_result;
real_inf_result.v = pmul(a_abs, pset1<Packet>(Scalar(RealScalar(1.0), RealScalar(0.0))).v);
real_inf_result.v = pselect(negative_real_mask.v, pcplxflip(real_inf_result).v, real_inf_result.v); // prepare packet of (+∞,+∞) or (+∞,-∞), depending on the sign of the infinite imaginary part.
Packet is_imag_inf;
is_imag_inf.v = pandnot(is_inf.v, real_mask);
is_imag_inf = por(is_imag_inf, pcplxflip(is_imag_inf));
Packet imag_inf_result;
imag_inf_result.v = por(pand(cst_pos_inf, real_mask), pandnot(a.v, real_mask));
// TODO(rmlarsen): The following set of utilities for double word arithmetic // should perhaps be refactored as a separate file, since it would be generally // useful for special function implementation etc. Writing the algorithms in // terms if a double word type would also make the code more readable.
// This function splits x into the nearest integer n and fractional part r, // such that x = n + r holds exactly. template<typename Packet>
EIGEN_STRONG_INLINE void absolute_split(const Packet& x, Packet& n, Packet& r) {
n = pround(x);
r = psub(x, n);
}
// This function computes the sum {s, r}, such that x + y = s_hi + s_lo // holds exactly, and s_hi = fl(x+y), if |x| >= |y|. template<typename Packet>
EIGEN_STRONG_INLINE void fast_twosum(const Packet& x, const Packet& y, Packet& s_hi, Packet& s_lo) {
s_hi = padd(x, y); const Packet t = psub(s_hi, x);
s_lo = psub(y, t);
}
#ifdef EIGEN_HAS_SINGLE_INSTRUCTION_MADD // This function implements the extended precision product of // a pair of floating point numbers. Given {x, y}, it computes the pair // {p_hi, p_lo} such that x * y = p_hi + p_lo holds exactly and // p_hi = fl(x * y). template<typename Packet>
EIGEN_STRONG_INLINE void twoprod(const Packet& x, const Packet& y,
Packet& p_hi, Packet& p_lo) {
p_hi = pmul(x, y);
p_lo = pmadd(x, y, pnegate(p_hi));
}
#else
// This function implements the Veltkamp splitting. Given a floating point // number x it returns the pair {x_hi, x_lo} such that x_hi + x_lo = x holds // exactly and that half of the significant of x fits in x_hi. // This is Algorithm 3 from Jean-Michel Muller, "Elementary Functions", // 3rd edition, Birkh\"auser, 2016. template<typename Packet>
EIGEN_STRONG_INLINE void veltkamp_splitting(const Packet& x, Packet& x_hi, Packet& x_lo) { typedeftypename unpacket_traits<Packet>::type Scalar;
EIGEN_CONSTEXPR int shift = (NumTraits<Scalar>::digits() + 1) / 2; const Scalar shift_scale = Scalar(uint64_t(1) << shift); // Scalar constructor not necessarily constexpr. const Packet gamma = pmul(pset1<Packet>(shift_scale + Scalar(1)), x);
Packet rho = psub(x, gamma);
x_hi = padd(rho, gamma);
x_lo = psub(x, x_hi);
}
// This function implements Dekker's algorithm for products x * y. // Given floating point numbers {x, y} computes the pair // {p_hi, p_lo} such that x * y = p_hi + p_lo holds exactly and // p_hi = fl(x * y). template<typename Packet>
EIGEN_STRONG_INLINE void twoprod(const Packet& x, const Packet& y,
Packet& p_hi, Packet& p_lo) {
Packet x_hi, x_lo, y_hi, y_lo;
veltkamp_splitting(x, x_hi, x_lo);
veltkamp_splitting(y, y_hi, y_lo);
// This is a version of twosum for double word numbers, // which assumes that |x_hi| >= |y_hi|. template<typename Packet>
EIGEN_STRONG_INLINE void fast_twosum(const Packet& x_hi, const Packet& x_lo, const Packet& y_hi, const Packet& y_lo,
Packet& s_hi, Packet& s_lo) {
Packet r_hi, r_lo;
fast_twosum(x_hi, y_hi, r_hi, r_lo); const Packet s = padd(padd(y_lo, r_lo), x_lo);
fast_twosum(r_hi, s, s_hi, s_lo);
}
// This is a version of twosum for adding a floating point number x to // double word number {y_hi, y_lo} number, with the assumption // that |x| >= |y_hi|. template<typename Packet>
EIGEN_STRONG_INLINE void fast_twosum(const Packet& x, const Packet& y_hi, const Packet& y_lo,
Packet& s_hi, Packet& s_lo) {
Packet r_hi, r_lo;
fast_twosum(x, y_hi, r_hi, r_lo); const Packet s = padd(y_lo, r_lo);
fast_twosum(r_hi, s, s_hi, s_lo);
}
// This function implements the multiplication of a double word // number represented by {x_hi, x_lo} by a floating point number y. // It returns the result as a pair {p_hi, p_lo} such that // (x_hi + x_lo) * y = p_hi + p_lo hold with a relative error // of less than 2*2^{-2p}, where p is the number of significand bit // in the floating point type. // This is Algorithm 7 from Jean-Michel Muller, "Elementary Functions", // 3rd edition, Birkh\"auser, 2016. template<typename Packet>
EIGEN_STRONG_INLINE void twoprod(const Packet& x_hi, const Packet& x_lo, const Packet& y,
Packet& p_hi, Packet& p_lo) {
Packet c_hi, c_lo1;
twoprod(x_hi, y, c_hi, c_lo1); const Packet c_lo2 = pmul(x_lo, y);
Packet t_hi, t_lo1;
fast_twosum(c_hi, c_lo2, t_hi, t_lo1); const Packet t_lo2 = padd(t_lo1, c_lo1);
fast_twosum(t_hi, t_lo2, p_hi, p_lo);
}
// This function implements the multiplication of two double word // numbers represented by {x_hi, x_lo} and {y_hi, y_lo}. // It returns the result as a pair {p_hi, p_lo} such that // (x_hi + x_lo) * (y_hi + y_lo) = p_hi + p_lo holds with a relative error // of less than 2*2^{-2p}, where p is the number of significand bit // in the floating point type. template<typename Packet>
EIGEN_STRONG_INLINE void twoprod(const Packet& x_hi, const Packet& x_lo, const Packet& y_hi, const Packet& y_lo,
Packet& p_hi, Packet& p_lo) {
Packet p_hi_hi, p_hi_lo;
twoprod(x_hi, x_lo, y_hi, p_hi_hi, p_hi_lo);
Packet p_lo_hi, p_lo_lo;
twoprod(x_hi, x_lo, y_lo, p_lo_hi, p_lo_lo);
fast_twosum(p_hi_hi, p_hi_lo, p_lo_hi, p_lo_lo, p_hi, p_lo);
}
// This function computes the reciprocal of a floating point number // with extra precision and returns the result as a double word. template <typename Packet> void doubleword_reciprocal(const Packet& x, Packet& recip_hi, Packet& recip_lo) { typedeftypename unpacket_traits<Packet>::type Scalar; // 1. Approximate the reciprocal as the reciprocal of the high order element.
Packet approx_recip = prsqrt(x);
approx_recip = pmul(approx_recip, approx_recip);
// 2. Run one step of Newton-Raphson iteration in double word arithmetic // to get the bottom half. The NR iteration for reciprocal of 'a' is // x_{i+1} = x_i * (2 - a * x_i)
// This function computes log2(x) and returns the result as a double word. template <typename Scalar> struct accurate_log2 { template <typename Packet>
EIGEN_STRONG_INLINE voidoperator()(const Packet& x, Packet& log2_x_hi, Packet& log2_x_lo) {
log2_x_hi = plog2(x);
log2_x_lo = pzero(x);
}
};
// This specialization uses a more accurate algorithm to compute log2(x) for // floats in [1/sqrt(2);sqrt(2)] with a relative accuracy of ~6.42e-10. // This additional accuracy is needed to counter the error-magnification // inherent in multiplying by a potentially large exponent in pow(x,y). // The minimax polynomial used was calculated using the Sollya tool. // See sollya.org. template <> struct accurate_log2<float> { template <typename Packet>
EIGEN_STRONG_INLINE voidoperator()(const Packet& z, Packet& log2_x_hi, Packet& log2_x_lo) { // The function log(1+x)/x is approximated in the interval // [1/sqrt(2)-1;sqrt(2)-1] by a degree 10 polynomial of the form // Q(x) = (C0 + x * (C1 + x * (C2 + x * (C3 + x * P(x))))), // where the degree 6 polynomial P(x) is evaluated in single precision, // while the remaining 4 terms of Q(x), as well as the final multiplication by x // to reconstruct log(1+x) are evaluated in extra precision using // double word arithmetic. C0 through C3 are extra precise constants // stored as double words. // // The polynomial coefficients were calculated using Sollya commands: // > n = 10; // > f = log2(1+x)/x; // > interval = [sqrt(0.5)-1;sqrt(2)-1]; // > p = fpminimax(f,n,[|double,double,double,double,single...|],interval,relative,floating);
const Packet x = psub(z, one); // Evaluate P(x) in working precision. // We evaluate it in multiple parts to improve instruction level // parallelism.
Packet x2 = pmul(x,x);
Packet p_even = pmadd(p6, x2, p4);
p_even = pmadd(p_even, x2, p2);
p_even = pmadd(p_even, x2, p0);
Packet p_odd = pmadd(p5, x2, p3);
p_odd = pmadd(p_odd, x2, p1);
Packet p = pmadd(p_odd, x, p_even);
// Now evaluate the low-order tems of Q(x) in double word precision. // In the following, due to the alternating signs and the fact that // |x| < sqrt(2)-1, we can assume that |C*_hi| >= q_i, and use // fast_twosum instead of the slower twosum.
Packet q_hi, q_lo;
Packet t_hi, t_lo; // C3 + x * p(x)
twoprod(p, x, t_hi, t_lo);
fast_twosum(C3_hi, C3_lo, t_hi, t_lo, q_hi, q_lo); // C2 + x * p(x)
twoprod(q_hi, q_lo, x, t_hi, t_lo);
fast_twosum(C2_hi, C2_lo, t_hi, t_lo, q_hi, q_lo); // C1 + x * p(x)
twoprod(q_hi, q_lo, x, t_hi, t_lo);
fast_twosum(C1_hi, C1_lo, t_hi, t_lo, q_hi, q_lo); // C0 + x * p(x)
twoprod(q_hi, q_lo, x, t_hi, t_lo);
fast_twosum(C0_hi, C0_lo, t_hi, t_lo, q_hi, q_lo);
// This specialization uses a more accurate algorithm to compute log2(x) for // floats in [1/sqrt(2);sqrt(2)] with a relative accuracy of ~1.27e-18. // This additional accuracy is needed to counter the error-magnification // inherent in multiplying by a potentially large exponent in pow(x,y). // The minimax polynomial used was calculated using the Sollya tool. // See sollya.org.
template <> struct accurate_log2<double> { template <typename Packet>
EIGEN_STRONG_INLINE voidoperator()(const Packet& x, Packet& log2_x_hi, Packet& log2_x_lo) { // We use a transformation of variables: // r = c * (x-1) / (x+1), // such that // log2(x) = log2((1 + r/c) / (1 - r/c)) = f(r). // The function f(r) can be approximated well using an odd polynomial // of the form // P(r) = ((Q(r^2) * r^2 + C) * r^2 + 1) * r, // For the implementation of log2<double> here, Q is of degree 6 with // coefficient represented in working precision (double), while C is a // constant represented in extra precision as a double word to achieve // full accuracy. // // The polynomial coefficients were computed by the Sollya script: // // c = 2 / log(2); // trans = c * (x-1)/(x+1); // itrans = (1+x/c)/(1-x/c); // interval=[trans(sqrt(0.5)); trans(sqrt(2))]; // print(interval); // f = log2(itrans(x)); // p=fpminimax(f,[|1,3,5,7,9,11,13,15,17|],[|1,DD,double...|],interval,relative,floating); const Packet q12 = pset1<Packet>(2.87074255468000586e-9); const Packet q10 = pset1<Packet>(2.38957980901884082e-8); const Packet q8 = pset1<Packet>(2.31032094540014656e-7); const Packet q6 = pset1<Packet>(2.27279857398537278e-6); const Packet q4 = pset1<Packet>(2.31271023278625638e-5); const Packet q2 = pset1<Packet>(2.47556738444535513e-4); const Packet q0 = pset1<Packet>(2.88543873228900172e-3); const Packet C_hi = pset1<Packet>(0.0400377511598501157); const Packet C_lo = pset1<Packet>(-4.77726582251425391e-19); const Packet one = pset1<Packet>(1.0);
const Packet cst_2_log2e_hi = pset1<Packet>(2.88539008177792677); const Packet cst_2_log2e_lo = pset1<Packet>(4.07660016854549667e-17); // c * (x - 1)
Packet num_hi, num_lo;
twoprod(cst_2_log2e_hi, cst_2_log2e_lo, psub(x, one), num_hi, num_lo); // TODO(rmlarsen): Investigate if using the division algorithm by // Muller et al. is faster/more accurate. // 1 / (x + 1)
Packet denom_hi, denom_lo;
doubleword_reciprocal(padd(x, one), denom_hi, denom_lo); // r = c * (x-1) / (x+1),
Packet r_hi, r_lo;
twoprod(num_hi, num_lo, denom_hi, denom_lo, r_hi, r_lo); // r2 = r * r
Packet r2_hi, r2_lo;
twoprod(r_hi, r_lo, r_hi, r_lo, r2_hi, r2_lo); // r4 = r2 * r2
Packet r4_hi, r4_lo;
twoprod(r2_hi, r2_lo, r2_hi, r2_lo, r4_hi, r4_lo);
// Evaluate Q(r^2) in working precision. We evaluate it in two parts // (even and odd in r^2) to improve instruction level parallelism.
Packet q_even = pmadd(q12, r4_hi, q8);
Packet q_odd = pmadd(q10, r4_hi, q6);
q_even = pmadd(q_even, r4_hi, q4);
q_odd = pmadd(q_odd, r4_hi, q2);
q_even = pmadd(q_even, r4_hi, q0);
Packet q = pmadd(q_odd, r2_hi, q_even);
// Now evaluate the low order terms of P(x) in double word precision. // In the following, due to the increasing magnitude of the coefficients // and r being constrained to [-0.5, 0.5] we can use fast_twosum instead // of the slower twosum. // Q(r^2) * r^2
Packet p_hi, p_lo;
twoprod(r2_hi, r2_lo, q, p_hi, p_lo); // Q(r^2) * r^2 + C
Packet p1_hi, p1_lo;
fast_twosum(C_hi, C_lo, p_hi, p_lo, p1_hi, p1_lo); // (Q(r^2) * r^2 + C) * r^2
Packet p2_hi, p2_lo;
twoprod(r2_hi, r2_lo, p1_hi, p1_lo, p2_hi, p2_lo); // ((Q(r^2) * r^2 + C) * r^2 + 1)
Packet p3_hi, p3_lo;
fast_twosum(one, p2_hi, p2_lo, p3_hi, p3_lo);
// This specialization uses a faster algorithm to compute exp2(x) for floats // in [-0.5;0.5] with a relative accuracy of 1 ulp. // The minimax polynomial used was calculated using the Sollya tool. // See sollya.org. template <> struct fast_accurate_exp2<float> { template <typename Packet>
EIGEN_STRONG_INLINE
Packet operator()(const Packet& x) { // This function approximates exp2(x) by a degree 6 polynomial of the form // Q(x) = 1 + x * (C + x * P(x)), where the degree 4 polynomial P(x) is evaluated in // single precision, and the remaining steps are evaluated with extra precision using // double word arithmetic. C is an extra precise constant stored as a double word. // // The polynomial coefficients were calculated using Sollya commands: // > n = 6; // > f = 2^x; // > interval = [-0.5;0.5]; // > p = fpminimax(f,n,[|1,double,single...|],interval,relative,floating);
// Evaluate P(x) in working precision. // We evaluate even and odd parts of the polynomial separately // to gain some instruction level parallelism.
Packet x2 = pmul(x,x);
Packet p_even = pmadd(p4, x2, p2);
Packet p_odd = pmadd(p3, x2, p1);
p_even = pmadd(p_even, x2, p0);
Packet p = pmadd(p_odd, x, p_even);
// Evaluate the remaining terms of Q(x) with extra precision using // double word arithmetic.
Packet p_hi, p_lo; // x * p(x)
twoprod(p, x, p_hi, p_lo); // C + x * p(x)
Packet q1_hi, q1_lo;
twosum(p_hi, p_lo, C_hi, C_lo, q1_hi, q1_lo); // x * (C + x * p(x))
Packet q2_hi, q2_lo;
twoprod(q1_hi, q1_lo, x, q2_hi, q2_lo); // 1 + x * (C + x * p(x))
Packet q3_hi, q3_lo; // Since |q2_hi| <= sqrt(2)-1 < 1, we can use fast_twosum // for adding it to unity here.
fast_twosum(one, q2_hi, q3_hi, q3_lo); return padd(q3_hi, padd(q2_lo, q3_lo));
}
};
// in [-0.5;0.5] with a relative accuracy of 1 ulp. // The minimax polynomial used was calculated using the Sollya tool. // See sollya.org. template <> struct fast_accurate_exp2<double> { template <typename Packet>
EIGEN_STRONG_INLINE
Packet operator()(const Packet& x) { // This function approximates exp2(x) by a degree 10 polynomial of the form // Q(x) = 1 + x * (C + x * P(x)), where the degree 8 polynomial P(x) is evaluated in // single precision, and the remaining steps are evaluated with extra precision using // double word arithmetic. C is an extra precise constant stored as a double word. // // The polynomial coefficients were calculated using Sollya commands: // > n = 11; // > f = 2^x; // > interval = [-0.5;0.5]; // > p = fpminimax(f,n,[|1,DD,double...|],interval,relative,floating);
// Evaluate P(x) in working precision. // We evaluate even and odd parts of the polynomial separately // to gain some instruction level parallelism.
Packet x2 = pmul(x,x);
Packet p_even = pmadd(p8, x2, p6);
Packet p_odd = pmadd(p9, x2, p7);
p_even = pmadd(p_even, x2, p4);
p_odd = pmadd(p_odd, x2, p5);
p_even = pmadd(p_even, x2, p2);
p_odd = pmadd(p_odd, x2, p3);
p_even = pmadd(p_even, x2, p0);
p_odd = pmadd(p_odd, x2, p1);
Packet p = pmadd(p_odd, x, p_even);
// Evaluate the remaining terms of Q(x) with extra precision using // double word arithmetic.
Packet p_hi, p_lo; // x * p(x)
twoprod(p, x, p_hi, p_lo); // C + x * p(x)
Packet q1_hi, q1_lo;
twosum(p_hi, p_lo, C_hi, C_lo, q1_hi, q1_lo); // x * (C + x * p(x))
Packet q2_hi, q2_lo;
twoprod(q1_hi, q1_lo, x, q2_hi, q2_lo); // 1 + x * (C + x * p(x))
Packet q3_hi, q3_lo; // Since |q2_hi| <= sqrt(2)-1 < 1, we can use fast_twosum // for adding it to unity here.
fast_twosum(one, q2_hi, q3_hi, q3_lo); return padd(q3_hi, padd(q2_lo, q3_lo));
}
};
// This function implements the non-trivial case of pow(x,y) where x is // positive and y is (possibly) non-integer. // Formally, pow(x,y) = exp2(y * log2(x)), where exp2(x) is shorthand for 2^x. // TODO(rmlarsen): We should probably add this as a packet up 'ppow', to make it // easier to specialize or turn off for specific types and/or backends.x template <typename Packet>
EIGEN_STRONG_INLINE Packet generic_pow_impl(const Packet& x, const Packet& y) { typedeftypename unpacket_traits<Packet>::type Scalar; // Split x into exponent e_x and mantissa m_x.
Packet e_x;
Packet m_x = pfrexp(x, e_x);
// Adjust m_x to lie in [1/sqrt(2):sqrt(2)] to minimize absolute error in log2(m_x).
EIGEN_CONSTEXPR Scalar sqrt_half = Scalar(0.70710678118654752440); const Packet m_x_scale_mask = pcmp_lt(m_x, pset1<Packet>(sqrt_half));
m_x = pselect(m_x_scale_mask, pmul(pset1<Packet>(Scalar(2)), m_x), m_x);
e_x = pselect(m_x_scale_mask, psub(e_x, pset1<Packet>(Scalar(1))), e_x);
// Compute log2(m_x) with 6 extra bits of accuracy.
Packet rx_hi, rx_lo;
accurate_log2<Scalar>()(m_x, rx_hi, rx_lo);
// Compute the two terms {y * e_x, y * r_x} in f = y * log2(x) with doubled // precision using double word arithmetic.
Packet f1_hi, f1_lo, f2_hi, f2_lo;
twoprod(e_x, y, f1_hi, f1_lo);
twoprod(rx_hi, rx_lo, y, f2_hi, f2_lo); // Sum the two terms in f using double word arithmetic. We know // that |e_x| > |log2(m_x)|, except for the case where e_x==0. // This means that we can use fast_twosum(f1,f2). // In the case e_x == 0, e_x * y = f1 = 0, so we don't lose any // accuracy by violating the assumption of fast_twosum, because // it's a no-op.
Packet f_hi, f_lo;
fast_twosum(f1_hi, f1_lo, f2_hi, f2_lo, f_hi, f_lo);
// Split f into integer and fractional parts.
Packet n_z, r_z;
absolute_split(f_hi, n_z, r_z);
r_z = padd(r_z, f_lo);
Packet n_r;
absolute_split(r_z, n_r, r_z);
n_z = padd(n_z, n_r);
// We now have an accurate split of f = n_z + r_z and can compute // x^y = 2**{n_z + r_z) = exp2(r_z) * 2**{n_z}. // Since r_z is in [-0.5;0.5], we compute the first factor to high accuracy // using a specialized algorithm. Multiplication by the second factor can // be done exactly using pldexp(), since it is an integer power of 2. const Packet e_r = fast_accurate_exp2<Scalar>()(r_z); return pldexp(e_r, n_z);
}
// General computation of pow(x,y) for positive x or negative x and integer y. const Packet negate_pow_abs = pandnot(x_is_neg, y_is_even); const Packet pow_abs = generic_pow_impl(abs_x, y); return pselect(y_is_one, x,
pselect(pow_is_one, cst_one,
pselect(pow_is_nan, cst_nan,
pselect(pow_is_inf, cst_pos_inf,
pselect(pow_is_zero, cst_zero,
pselect(negate_pow_abs, pnegate(pow_abs), pow_abs))))));
}
/* polevl (modified for Eigen) * * Evaluate polynomial * * * * SYNOPSIS: * * int N; * Scalar x, y, coef[N+1]; * * y = polevl<decltype(x), N>( x, coef); * * * * DESCRIPTION: * * Evaluates polynomial of degree N: * * 2 N * y = C + C x + C x +...+ C x * 0 1 2 N * * Coefficients are stored in reverse order: * * coef[0] = C , ..., coef[N] = C . * N 0 * * The function p1evl() assumes that coef[N] = 1.0 and is * omitted from the array. Its calling arguments are * otherwise the same as polevl(). * * * The Eigen implementation is templatized. For best speed, store * coef as a const array (constexpr), e.g. * * const double coef[] = {1.0, 2.0, 3.0, ...}; *
*/ template <typename Packet, int N> struct ppolevl { static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Packet run(const Packet& x, consttypename unpacket_traits<Packet>::type coeff[]) {
EIGEN_STATIC_ASSERT((N > 0), YOU_MADE_A_PROGRAMMING_MISTAKE); return pmadd(ppolevl<Packet, N-1>::run(x, coeff), x, pset1<Packet>(coeff[N]));
}
--> --------------------
--> maximum size reached
--> --------------------
Messung V0.5
¤ Dauer der Verarbeitung: 0.22 Sekunden
(vorverarbeitet)
¤
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