simd_granodi_math.h

/*

Functions taken from the Cephes Math Library and re-implemented to use the
SIMD_GRANODI header library, to work on 128-bit registers of 2x64 bit or 4x32
bit floating point.

The Cephes Math Library is copyright 1984, 1995, 2000 by Stephen L. Moshier
http://www.moshier.net/

The sincos() implementation's idea of calculating both sin and cos at the same
time is taken from sse_mathfun.h which is Copyright (C) 2007  Julien Pommier
http://gruntthepeon.free.fr/ssemath/sse_mathfun.h

Copyright 2022 Jon Ville.

Permission is hereby granted, free of charge, to any person obtaining a copy of
this software and associated documentation files (the "Software"), to deal in
the Software without restriction, including without limitation the rights to
use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies
of the Software, and to permit persons to whom the Software is furnished to do
so, subject to the following conditions:

The above copyright notice and this permission notices shall be included in all
copies or substantial portions of the Software.

THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
SOFTWARE.

*/

/*

!!!! WARNINGS !!!!

- These functions use faster frexp()/ldexp() implementations that do
  NOT work for denormal (subnormal) numbers.

- They do NOT return correct error values (eg log(-1) returns minus
  infinity).

- Assertions are used, so you MUST define NDEBUG as a compilier argument for
  optimized builds.

- They aim to be fast and suitable for use in audio DSP.

SINGLE PRECISION FUNCTIONS:

log2_p3():
Smooth, but very inaccurate, log2() approximation for singles or doubles
Returns -inf for x <= 0

exp2_p3():
Smooth, but very inaccurate, exp2() approximation for singles or doubles

exp_p3():
Simple wrapper that changes the base of exp2_p3() to e.

logf_cm():
Accurate logf() for singles.
Returns -inf for x <= 0

expf_cm():
Accuare expf() for singles

sincosf_cm():
Accurate sinf() and cosf() for singles where x < 8192
Scalar optimization: calculates sinf() and cosf() in parallel, whereas the
vector version calculates both sinf() and cosf() (but many of the computations
are still shared).
sinf_cm() and cosf_cm() wrap sincosf_cm() and discard unneeded results.

sqrtf_cm():
Accurate sqrtf() for singles.
Returns 0 if x <= +/-0
Scalar optimization: uses branching to select the correct polynomial, whereas
the vector version selects different polynomial coefficients via masking.

DOUBLE PRECISION FUNCTIONS:

Note: all double precision functions are overloaded such that when called with
a single precision type (ie Vec_ps or Vec_ss), they call their equivalent
single precision version.

log_cm():
Accurate log() for doubles. (However, does not include every case from the
Cephes implementation - results will be slightly different).
Returns -inf for x <= 0
Scalar optimization: computes numerator and denominator polynomials in parallel.

exp_cm():
Accurate exp() for doubles
Scalar optimization: computes numerator and denominator polynomials in parallel.

sincos_cm():
Accurate sin() and cos() for x < 1.07e9
Scalar optimization: calculates sin and cos in parallel, whereas the vector
version calculates both sin() and cos() (but many of the computations
are still shared).
sin_cm() and cos_cm() wrap sincos_cm() and discard uneeded results.

sqrt_cm():
Accurate sqrt for doubles.
Returns 0 if x <= +/-0

*/

#pragma once

#include <cassert>
#include <initializer_list>

#include "../simd_granodi/simd_granodi.h"

namespace simd_granodi {

namespace sg_math_impl {

template <typename CoeffType, int32_t N>
struct Poly {
    // Needed public because of .prepend() method
    CoeffType coeff_[N] {};

    Poly() {}
    Poly(const std::initializer_list<CoeffType>& coeff_arg) {
        static_assert(N > 1, "");
        assert(coeff_arg.size() == N);
        std::size_t i = 0;
        for (const CoeffType& c : coeff_arg) {
            //if (size_mismatch) printf("size mismatch: %.4e\n", c.data());
            if (i < static_cast<std::size_t>(N)) coeff_[i++] = c;
        }
    }

    // To evaluate two polynomials at the same time, where ArgType is
    // Vec_ps with all elements giving equal value, and the results stored in
    // 0 and 1
    Poly(const Poly<Vec_ss, N>& poly1, const Poly<Vec_ss, N>& poly0) {
        for (int32_t i = 0; i < N; ++i) {
            coeff_[i] = Vec_ps{0.0f, 0.0f, poly1.coeff_[i].data(),
                poly0.coeff_[i].data()};
        }
    }

    Poly(const Poly<Vec_sd, N>& poly1, const Poly<Vec_sd, N>& poly0) {
        for (int32_t i = 0; i < N; ++i) {
            coeff_[i] = Vec_pd{poly1.coeff_[i].data(), poly0.coeff_[i].data()};
        }
    }

    Poly<CoeffType, N+1> prepend(const CoeffType new_coeff0) const {
        Poly<CoeffType, N+1> result;
        result.coeff_[0] = new_coeff0;
        for (int32_t i = 0; i < N; ++i) result.coeff_[i+1] = coeff_[i];
        return result;
    }

    template <typename ArgType>
    ArgType sg_vectorcall(eval)(const ArgType x) const {
        ArgType result = x.mul_add(coeff_[0].template to<ArgType>(),
            coeff_[1].template to<ArgType>());
        for (int32_t i = 2; i < N; ++i) {
            result = result.mul_add(x, coeff_[i].template to<ArgType>());
        }
        return result;
    }

    template <typename ArgType>
    ArgType sg_vectorcall(eval1)(const ArgType x) const {
        ArgType result {x + coeff_[0].template to<ArgType>()};
        for (int32_t i = 1; i < N; ++i) {
            result = result.mul_add(x, coeff_[i].template to<ArgType>());
        }
        return result;
    }
};

template <typename CmpType>
inline void sg_vectorcall(make_if_elseif_else)(const CmpType& cmp1, CmpType& cmp2,
    CmpType& cmp3)
{
    cmp2 = cmp2 && !cmp1;
    cmp3 = !(cmp1 || cmp2);
}

template <typename ArgType>
inline ArgType sg_vectorcall(choose3)(const typename ArgType::compare_t cmp1,
    const typename ArgType::compare_t cmp2,
    const typename ArgType::compare_t cmp3,
    const ArgType x1, const ArgType x2, const ArgType x3)
{
    assert((cmp1 != (cmp2 || cmp3)).debug_valid_eq(true));
    assert((cmp2 != (cmp1 || cmp3)).debug_valid_eq(true));
    assert((cmp3 != (cmp1 || cmp2)).debug_valid_eq(true));
    return cmp1.choose_else_zero(x1) |
        cmp2.choose_else_zero(x2) |
        cmp3.choose_else_zero(x3);
}

template <typename CoeffType, int32_t N>
struct Poly_choose3 {
    CoeffType coeff_[N*3] {};

    Poly_choose3(const Poly<CoeffType, N>& p1,
        const Poly<CoeffType, N>& p2,
        const Poly<CoeffType, N>& p3)
    {
        static_assert(N > 1, "");
        for (int32_t i = 0; i < N; ++i) {
            coeff_[3*i] = p1.coeff_[i];
            coeff_[3*i+1] = p2.coeff_[i];
            coeff_[3*i+2] = p3.coeff_[i];
        }
    }

    template <typename ArgType>
    ArgType sg_vectorcall(eval)(const typename ArgType::compare_t cmp1,
        const typename ArgType::compare_t cmp2,
        const typename ArgType::compare_t cmp3,
        const ArgType x) const
    {
        ArgType result = x.mul_add(choose3(cmp1, cmp2, cmp3,
                coeff_[0].template to<ArgType>(),
                coeff_[1].template to<ArgType>(),
                coeff_[2].template to<ArgType>()),
            choose3(cmp1, cmp2, cmp3,
                coeff_[3].template to<ArgType>(),
                coeff_[4].template to<ArgType>(),
                coeff_[5].template to<ArgType>()));
        for (int32_t i = 6; i < N*3; i += 3) {
            result = result.mul_add(x, choose3(cmp1, cmp2, cmp3,
                coeff_[i].template to<ArgType>(),
                coeff_[i+1].template to<ArgType>(),
                coeff_[i+2].template to<ArgType>()));
        }
        return result;
    }

};

//
//
// CUBIC APPROXIMATIONS: very fast, very smooth, but not accurate at all.
// Suitable for musical use in envelope generators etc

template <typename VecType> struct FloatBits {};

template <> struct FloatBits<Vec_ps> {
    static constexpr int32_t exp_shift = 23, exp_mask = 0xff, exp_bias = 127,
        mant_mask = 0x807fffff, exp1 = 0x3f800000, exph = 0x3f000000;
};
template <> struct FloatBits<Vec_ss> : public FloatBits<Vec_ps> {};
template <> struct FloatBits<Vec_f32x2> : public FloatBits<Vec_ps> {};

template <> struct FloatBits<Vec_pd> {
    static constexpr int32_t exp_shift = 52, exp_bias = 1023;
    static constexpr int64_t exp_mask = 0x7ff, mant_mask = 0x800fffffffffffff,
        exp1 = 0x3ff0000000000000, exph = 0x3fe0000000000000;
};
template <> struct FloatBits<Vec_sd> : public FloatBits<Vec_pd> {};

// These implementations of ldexp and frexp only work for finite, non-denormal
// inputs! Not comparable to standard library versions!
template <typename VecType>
inline VecType sg_vectorcall(sg_ldexp)(const VecType x,
    const typename VecType::fast_convert_int_t e)
{
    static_assert(VecType::is_float_t, "");
    using equiv_int = typename SGEquivIntType<VecType>::value;
    using fb = FloatBits<VecType>;
    return (x.template bitcast<equiv_int>() +
            e.template to<equiv_int>().template shift_l_imm<fb::exp_shift>())
        .template bitcast<VecType>();
}

template <typename VecType>
inline typename VecType::fast_convert_int_t sg_vectorcall(exponent)(
    const VecType x)
{
    static_assert(VecType::is_float_t, "");
    using equiv_int = typename SGEquivIntType<VecType>::value;
    using fast_int = typename VecType::fast_convert_int_t;
    using fb = FloatBits<VecType>;
    return ((x.template bitcast<equiv_int>()
            .template shift_rl_imm<fb::exp_shift>()
        & fb::exp_mask) - fb::exp_bias).template to<fast_int>();
}

template <typename VecType>
inline typename VecType::fast_convert_int_t sg_vectorcall(exponent_frexp)(
    const VecType x)
{
    static_assert(VecType::is_float_t, "");
    using equiv_int = typename SGEquivIntType<VecType>::value;
    using fast_int = typename VecType::fast_convert_int_t;
    using fb = FloatBits<VecType>;
    return ((x.template bitcast<equiv_int>()
            .template shift_rl_imm<fb::exp_shift>()
        & fb::exp_mask) - (fb::exp_bias-1)).template to<fast_int>();
}

template <typename VecType>
inline VecType sg_vectorcall(mantissa)(const VecType x) {
    static_assert(VecType::is_float_t, "");
    using equiv_int = typename SGEquivIntType<VecType>::value;
    using fb = FloatBits<VecType>;
    return ((x.template bitcast<equiv_int>() & fb::mant_mask) | fb::exp1)
        .template bitcast<VecType>();
}

template <typename VecType>
inline VecType sg_vectorcall(mantissa_frexp)(const VecType x) {
    static_assert(VecType::is_float_t, "");
    using equiv_int = typename SGEquivIntType<VecType>::value;
    using fb = FloatBits<VecType>;
    return ((x.template bitcast<equiv_int>() & fb::mant_mask) | fb::exph)
        .template bitcast<VecType>();
}

// Calculate log2(x) for x in [1, 2) using a cubic approximation.
// Gradient matches gradient of log2() at either end
static const Poly<Vec_sd, 4> log2_p3_poly {
 1.6404256133344508e-1,
-1.0988652862227437,
 3.1482979293341158,
-2.2134752044448169 };

// Calculate exp2(x) for x in [0, 1]. Gradient matches exp2() at both ends
static const Poly<Vec_sd, 4> exp2_p3_poly {
7.944154167983597e-2,
2.2741127776021886e-1,
6.931471805599453e-1,
1.0 };

template <typename VecType>
inline VecType sg_vectorcall(log2_p3_impl)(const VecType x) {
    VecType exponent = sg_math_impl::exponent(x).template to<VecType>(),
        mantissa = sg_math_impl::mantissa(x);
    mantissa = sg_math_impl::log2_p3_poly.eval(mantissa);
    return (x > 0.0).choose(exponent + mantissa, VecType::minus_infinity());
}

template <typename VecType>
inline VecType sg_vectorcall(exp2_p3_impl)(const VecType x) {
    const auto floor = x.template floor<typename VecType::fast_convert_int_t>();
    const VecType floor_f = floor.template to<VecType>();
    VecType frac = x - floor_f;
    frac = sg_math_impl::exp2_p3_poly.eval(frac);
    return sg_math_impl::sg_ldexp(frac, floor);
}

template <typename VecType>
inline VecType sg_vectorcall(exp_p3_impl)(const VecType x) {
    using elem = typename VecType::elem_t;
    return exp2_p3(VecType{x * elem{6.931471805599453e-1}});
}

} // namespace sg_math_impl

inline Vec_ss sg_vectorcall(log2f_p3)(const Vec_ss x) {
    return sg_math_impl::log2_p3_impl(x);
}
inline Vec_ss sg_vectorcall(log2_p3)(const Vec_ss x) {
    return sg_math_impl::log2_p3_impl(x);
}
inline Vec_f32x2 sg_vectorcall(log2f_p3)(const Vec_f32x2 x) {
    return sg_math_impl::log2_p3_impl(
        x.to<typename Vec_f32x2::fast_register_t>()).to<Vec_f32x2>();
}
inline Vec_f32x2 sg_vectorcall(log2_p3)(const Vec_f32x2 x) {
    return log2f_p3(x);
}
inline Vec_ps sg_vectorcall(log2f_p3)(const Vec_ps x) {
    return sg_math_impl::log2_p3_impl(x);
}
inline Vec_ps sg_vectorcall(log2_p3)(const Vec_ps x) {
    return sg_math_impl::log2_p3_impl(x);
}
inline Vec_sd sg_vectorcall(log2_p3)(const Vec_sd x) {
    return sg_math_impl::log2_p3_impl(x);
}
inline Vec_pd sg_vectorcall(log2_p3)(const Vec_pd x) {
    return sg_math_impl::log2_p3_impl(x);
}

inline Vec_ss sg_vectorcall(exp2f_p3)(const Vec_ss x) {
    return sg_math_impl::exp2_p3_impl(x);
}
inline Vec_ss sg_vectorcall(exp2_p3)(const Vec_ss x) {
    return sg_math_impl::exp2_p3_impl(x);
}
inline Vec_f32x2 sg_vectorcall(exp2f_p3)(const Vec_f32x2 x) {
    return sg_math_impl::exp2_p3_impl(
        x.to<typename Vec_f32x2::fast_register_t>()).to<Vec_f32x2>();
}
inline Vec_f32x2 sg_vectorcall(exp2_p3)(const Vec_f32x2 x) {
    return exp2f_p3(x);
}
inline Vec_ps sg_vectorcall(exp2f_p3)(const Vec_ps x) {
    return sg_math_impl::exp2_p3_impl(x);
}
inline Vec_ps sg_vectorcall(exp2_p3)(const Vec_ps x) {
    return sg_math_impl::exp2_p3_impl(x);
}
inline Vec_sd sg_vectorcall(exp2_p3)(const Vec_sd x) {
    return sg_math_impl::exp2_p3_impl(x);
}
inline Vec_pd sg_vectorcall(exp2_p3)(const Vec_pd x) {
    return sg_math_impl::exp2_p3_impl(x);
}

inline Vec_ss sg_vectorcall(expf_p3)(const Vec_ss x) {
    return sg_math_impl::exp_p3_impl(x);
}
inline Vec_ss sg_vectorcall(exp_p3)(const Vec_ss x) {
    return sg_math_impl::exp_p3_impl(x);
}
inline Vec_f32x2 sg_vectorcall(expf_p3)(const Vec_f32x2 x) {
    return sg_math_impl::exp_p3_impl(
        x.to<typename Vec_f32x2::fast_register_t>()).to<Vec_f32x2>();
}
inline Vec_f32x2 sg_vectorcall(exp_p3)(const Vec_f32x2 x) {
    return expf_p3(x);
}
inline Vec_ps sg_vectorcall(expf_p3)(const Vec_ps x) {
    return sg_math_impl::exp_p3_impl(x);
}
inline Vec_ps sg_vectorcall(exp_p3)(const Vec_ps x) {
    return sg_math_impl::exp_p3_impl(x);
}
inline Vec_sd sg_vectorcall(exp_p3)(const Vec_sd x) {
    return sg_math_impl::exp_p3_impl(x);
}
inline Vec_pd sg_vectorcall(exp_p3)(const Vec_pd x) {
    return sg_math_impl::exp_p3_impl(x);
}

//
//
// CEPHES MATH LIBRARY 32-BIT FLOAT IMPLEMENTATIONS

namespace sg_math_impl {

static constexpr double sqrt_half = 7.07106781186547524401e-1;

// logf and expf constants
static constexpr double log_q1 = -2.12194440e-4,
    log_q2 = 6.93359375e-1,
    log2e = 1.44269504088896341; // log2(e)

static const Poly<Vec_ss, 9> logf_poly {
 7.0376836292e-2f,
-1.1514610310e-1f,
 1.1676998740e-1f,
-1.2420140846e-1f,
 1.4249322787e-1f,
-1.6668057665e-1f,
 2.0000714765e-1f,
-2.4999993993e-1f,
 3.3333331174e-1f };
static const Poly<Vec_ss, 6> expf_poly {
1.9875691500e-4f,
1.3981999507e-3f,
8.3334519073e-3f,
4.1665795894e-2f,
1.6666665459e-1f,
5.0000001201e-1f };

template <typename VecType>
inline VecType sg_vectorcall(logf_impl)(const VecType a) {
    VecType x = mantissa_frexp(a);
    VecType e = exponent_frexp(a).template to<VecType>();
    auto x_lt_sqrth = x < static_cast<float>(sqrt_half);
    e -= x_lt_sqrth.choose_else_zero(1.0f);
    x += x_lt_sqrth.choose_else_zero(x);
    x -= 1.0f;

    VecType z = x * x;

    VecType y = logf_poly.eval(x) * x * z;

    y += e*static_cast<float>(log_q1);
    y += -0.5f * z;
    z = x + y;
    z += e*static_cast<float>(log_q2);

    return (a > 0.0f).choose(z, VecType::minus_infinity());
}

} // namespace sg_math_impl

inline Vec_ss sg_vectorcall(logf_cm)(const Vec_ss a) {
    return sg_math_impl::logf_impl(a);
}
inline Vec_f32x2 sg_vectorcall(logf_cm)(const Vec_f32x2 a) {
    return sg_math_impl::logf_impl(a.to<Vec_f32x2::fast_register_t>())
        .to<Vec_f32x2>();
}
inline Vec_ps sg_vectorcall(logf_cm)(const Vec_ps a) {
    return sg_math_impl::logf_impl(a);
}
inline Vec_ss sg_vectorcall(log_cm)(const Vec_ss a) {
    return sg_math_impl::logf_impl(a);
}
inline Vec_f32x2 sg_vectorcall(log_cm)(const Vec_f32x2 a) {
    return logf_cm(a);
}
inline Vec_ps sg_vectorcall(log_cm)(const Vec_ps a) {
    return sg_math_impl::logf_impl(a);
}

namespace sg_math_impl {

template <typename VecType>
inline VecType sg_vectorcall(expf_impl)(const VecType a) {
    VecType x = a;
    VecType z = x * static_cast<float>(log2e);
    typedef typename SGType<int32_t, VecType::elem_count>::value
        equiv_int_type;
    auto n = z.template nearest<equiv_int_type>();
    z = n.template to<VecType>();

    x -= z*static_cast<float>(log_q2) + z*static_cast<float>(log_q1);
    z = x * x;
    z *= expf_poly.eval(x);
    z += x + 1.0;
    return sg_ldexp(z, n);
}

} // namespace sg_math_impl

inline Vec_ss sg_vectorcall(expf_cm)(const Vec_ss a) {
    return sg_math_impl::expf_impl(a);
}
inline Vec_f32x2 sg_vectorcall(expf_cm)(const Vec_f32x2 a) {
    return sg_math_impl::expf_impl(a.to<Vec_f32x2::fast_register_t>())
        .to<Vec_f32x2>();
}
inline Vec_ps sg_vectorcall(expf_cm)(const Vec_ps a) {
    return sg_math_impl::expf_impl(a);
}
inline Vec_ss sg_vectorcall(exp_cm)(const Vec_ss a) {
    return sg_math_impl::expf_impl(a);
}
inline Vec_f32x2 sg_vectorcall(exp_cm)(const Vec_f32x2 a) {
    return expf_cm(a);
}
inline Vec_ps sg_vectorcall(exp_cm)(const Vec_ps a) {
    return sg_math_impl::expf_impl(a);
}

namespace sg_math_impl {

// sincos constants
static constexpr double four_over_pi = 1.2732395447351628;
static constexpr float dp1_f = 7.8515625e-1f;
static constexpr float dp2_f = 2.4187564849853515625e-4f;
static constexpr float dp3_f = 3.77489497744594108e-8f;

static const Poly<Vec_ss, 3> sinf_poly {
-1.9515295891e-4f,
 8.3321608736e-3f,
-1.6666654611e-1f };
static const Poly<Vec_ss, 3> cosf_poly {
 2.443315711809948e-5f,
-1.388731625493765e-3f,
 4.166664568298827e-2f };

// {cos, sin}
static const Poly<Vec_ps, 3> sincosf_poly { cosf_poly, sinf_poly };

} // namespace sg_math_impl

template <typename VecType>
struct sincos_result { VecType sin_result, cos_result; };

// sin and cos for f32 break when x >= 8192
inline sincos_result<Vec_ss> sg_vectorcall(sincosf_cm)(const Vec_ss xx) {
    // {cos sign bit, sin sign bit}
    Vec_pi32 signbits {0, 0, 0, xx.bitcast<Vec_s32x1>().data() & (~sg_fp_signmask_s32)};
    float x = xx.abs().data();
    int32_t j = static_cast<int32_t>(x *
        static_cast<float>(sg_math_impl::four_over_pi));
    float y = static_cast<float>(j);
    if (j & 1) {
        ++j;
        y += 1.0f;
    }
    j &= 7;
    if (j > 3) {
        signbits ^= Vec_pi32{0, 0, ~sg_fp_signmask_s32, ~sg_fp_signmask_s32};
        j -= 4;
    }
    if (j > 1) signbits ^= Vec_pi32{0, 0, ~sg_fp_signmask_s32, 0};
    x = ((x - y * sg_math_impl::dp1_f) - y * sg_math_impl::dp2_f) -
        y * sg_math_impl::dp3_f;
    const float z = x * x;
    // From here, calculate both {cos, sin} results in parallel
    Vec_ps result = sg_math_impl::sincosf_poly.eval(Vec_ps{0.0f, 0.0f, z, z})
        * z;
    result *= Vec_ps{0.0f, 0.0f, z, x};
    result += Vec_ps{0.0f, 0.0f, 1.0f - 0.5f*z, x};
    if ((j == 1) || (j == 2)) result = result.shuffle<3, 2, 0, 1>();
    result = (result.bitcast<Vec_pi32>() ^ signbits).bitcast<Vec_ps>();
    sincos_result<Vec_ss> r;
    r.sin_result = result.f0();
    r.cos_result = result.f1();
    return r;
}

inline Vec_ss sg_vectorcall(sinf_cm)(const Vec_ss x) {
    return sincosf_cm(x).sin_result;
}
inline Vec_ss sg_vectorcall(cosf_cm)(const Vec_ss x) {
    return sincosf_cm(x).cos_result;
}

inline sincos_result<Vec_ps> sg_vectorcall(sincosf_cm)(const Vec_ps xx) {
    Vec_pi32 cos_signbit = 0, sin_signbit = xx.bitcast<Vec_pi32>() & (~sg_fp_signmask_s32);
    Vec_ps x = xx.abs();
    Vec_pi32 j = (x * static_cast<float>(sg_math_impl::four_over_pi))
        .truncate<Vec_pi32>();
    Vec_ps y = j.to<Vec_ps>();
    const Compare_pi32 j_odd {(j & 1) != 0};
    j += j_odd.choose_else_zero(1);
    y += j_odd.to<Compare_ps>().choose_else_zero(1.0f);
    j &= 7;
    const Compare_pi32 j_gt_3 { j > 3 };
    j -= j_gt_3.choose_else_zero(4);
    cos_signbit ^= j_gt_3.choose_else_zero(~sg_fp_signmask_s32);
    sin_signbit ^= j_gt_3.choose_else_zero(~sg_fp_signmask_s32);
    const Compare_pi32 j_gt_1 = (j > 1);
    cos_signbit ^= j_gt_1.choose_else_zero(~sg_fp_signmask_s32);
    x = ((x - y * sg_math_impl::dp1_f) - y * sg_math_impl::dp2_f) -
        y * sg_math_impl::dp3_f;
    const Vec_ps z = x * x;
    // Brackets on following line needed for identical scalar / vec behaviour
    const Vec_ps cos_result = sg_math_impl::cosf_poly.eval(z) * z * z +
            (1.0f - 0.5f*z),
        sin_result = sg_math_impl::sinf_poly.eval(z) * z * x + x;
    const Compare_ps swap = ((j == 1) || (j == 2)).to<Compare_ps>();
    sincos_result<Vec_ps> result;
    result.cos_result = swap.choose(sin_result, cos_result);
    result.cos_result = (result.cos_result.bitcast<Vec_pi32>() ^ cos_signbit).bitcast<Vec_ps>();
    result.sin_result = swap.choose(cos_result, sin_result);
    result.sin_result = (result.sin_result.bitcast<Vec_pi32>() ^ sin_signbit).bitcast<Vec_ps>();
    return result;
}

inline sincos_result<Vec_f32x2> sg_vectorcall(sincosf_cm)(const Vec_f32x2 xx) {
    const auto result_ps = sincosf_cm(xx.to<Vec_ps>());
    sincos_result<Vec_f32x2> result;
    result.sin_result = result_ps.sin_result.to<Vec_f32x2>();
    result.cos_result = result_ps.cos_result.to<Vec_f32x2>();
    return result;
}
inline Vec_f32x2 sg_vectorcall(sinf_cm)(const Vec_f32x2 x) {
    return sincosf_cm(x).sin_result;
}
inline Vec_f32x2 sg_vectorcall(cosf_cm)(const Vec_f32x2 x) {
    return sincosf_cm(x).cos_result;
}

inline Vec_ps sg_vectorcall(sinf_cm)(const Vec_ps x) {
    return sincosf_cm(x).sin_result;
}
inline Vec_ps sg_vectorcall(cosf_cm)(const Vec_ps x) {
    return sincosf_cm(x).cos_result;
}

inline sincos_result<Vec_ss> sg_vectorcall(sincos_cm)(const Vec_ss xx) {
    return sincosf_cm(xx);
}
inline sincos_result<Vec_f32x2> sg_vectorcall(sincos_cm)(const Vec_f32x2 xx) {
    return sincosf_cm(xx);
}
inline sincos_result<Vec_ps> sg_vectorcall(sincos_cm)(const Vec_ps xx) {
    return sincosf_cm(xx);
}
inline Vec_ss sg_vectorcall(sin_cm)(const Vec_ss x) {
    return sincosf_cm(x).sin_result;
}
inline Vec_f32x2 sg_vectorcall(sin_cm)(const Vec_f32x2 x) {
    return sincosf_cm(x).sin_result;
}
inline Vec_ps sg_vectorcall(sin_cm)(const Vec_ps x) {
    return sincosf_cm(x).sin_result;
}
inline Vec_ss sg_vectorcall(cos_cm)(const Vec_ss x) {
    return sincosf_cm(x).cos_result;
}
inline Vec_f32x2 sg_vectorcall(cos_cm)(const Vec_f32x2 x) {
    return sincosf_cm(x).cos_result;
}
inline Vec_ps sg_vectorcall(cos_cm)(const Vec_ps x) {
    return sincosf_cm(x).cos_result;
}

namespace sg_math_impl {

static constexpr double sqrt_2 = 1.4142135623730951;

static const Poly<Vec_ss, 7> sqrtf_poly1 {
-9.8843065718e-4f,
 7.9479950957e-4f,
-3.5890535377e-3f,
 1.1028809744e-2f,
-4.4195203560e-2f,
 3.5355338194e-1f,
 1.41421356237f };
static const Poly<Vec_ss, 6> sqrtf_poly2 {
 1.35199291026e-2f,
-2.26657767832e-2f,
 2.78720776889e-2f,
-3.89582788321e-2f,
 6.24811144548e-2f,
-1.25001503933e-1f };
static const Poly<Vec_ss, 7> sqrtf_poly3 {
-3.9495006054e-1f,
 5.1743034569e-1f,
-4.3214437330e-1f,
 3.5310730460e-1f,
-3.5354581892e-1f,
 7.0710676017e-1f,
 7.07106781187e-1f };
static const Poly_choose3<Vec_ss, 7> sqrtf_choose3{ sqrtf_poly1,
    sqrtf_poly2.prepend(0.0f), sqrtf_poly3 };

} // namespace sg_math_impl

inline Vec_ss sg_vectorcall(sqrtf_cm)(const Vec_ss a) {
    if (a.data() <= 0.0f) return 0.0f;
    auto e = sg_math_impl::exponent_frexp(a);
    Vec_ss x = sg_math_impl::mantissa_frexp(a);
    if (e.data() & 1) {
        --e;
        x += x;
    }
    e = e.shift_ra_imm<1>();
    Vec_ss y;
    if (x.data() > static_cast<float>(sg_math_impl::sqrt_2)) {
        x -= 2.0f;
        y = sg_math_impl::sqrtf_poly1.eval(x);
    } else if (x.data() > static_cast<float>(sg_math_impl::sqrt_half)) {
        x -= 1.0f;
        // Brackets needed to match output of vector version
        y = sg_math_impl::sqrtf_poly2.eval(x) * (x * x) + (0.5f*x + 1.0f);
    } else {
        x -= 0.5f;
        y = sg_math_impl::sqrtf_poly3.eval(x);
    }
    return sg_math_impl::sg_ldexp(y, e);
}

inline Vec_ps sg_vectorcall(sqrtf_cm)(const Vec_ps a) {
    Vec_pi32 e = sg_math_impl::exponent_frexp(a);
    Vec_ps x = sg_math_impl::mantissa_frexp(a);
    const Compare_pi32 e_odd { (e & 1) != 0 };
    e -= e_odd.choose_else_zero(1);
    x += e_odd.to<Compare_ps>().choose_else_zero(x);
    e = e.shift_ra_imm<1>();
    Vec_ps y;
    const Compare_ps x_gt_sqrt2 {x > static_cast<float>(sg_math_impl::sqrt_2)};
    Compare_ps x_gt_sqrth { x > static_cast<float>(sg_math_impl::sqrt_half) },
        x_else;
    sg_math_impl::make_if_elseif_else(x_gt_sqrt2, x_gt_sqrth, x_else);
    x -= sg_math_impl::choose3(x_gt_sqrt2, x_gt_sqrth, x_else,
        Vec_ps{2.0f}, Vec_ps{1.0f}, Vec_ps{0.5f});
    y = sg_math_impl::sqrtf_choose3.eval(x_gt_sqrt2, x_gt_sqrth, x_else, x);
    y *= x_gt_sqrth.choose(x * x, 1.0f);
    y += x_gt_sqrth.choose_else_zero(0.5f*x + 1.0f);
    return (a > 0.0f).choose_else_zero(sg_math_impl::sg_ldexp(y, e));
}

inline Vec_f32x2 sg_vectorcall(sqrtf_cm)(const Vec_f32x2 a) {
    return sqrtf_cm(a.to<Vec_ps>()).to<Vec_f32x2>();
}

inline Vec_ss sg_vectorcall(sqrt_cm)(const Vec_ss a) {
    return sqrtf_cm(a);
}
inline Vec_f32x2 sg_vectorcall(sqrt_cm)(const Vec_f32x2 a) {
    return sqrtf_cm(a);
}
inline Vec_ps sg_vectorcall(sqrt_cm)(const Vec_ps a) {
    return sqrtf_cm(a);
}

//
//
// CEPHES 64-BIT IMPLEMENTATIONS

namespace sg_math_impl {

static constexpr double log_c1 = 2.121944400546905827679e-4,
    log_c2 = 6.93359375e-1;

static const Poly<Vec_sd, 3> log_poly_R {
-7.89580278884799154124e-1,
 1.63866645699558079767e1,
 -6.41409952958715622951e1 };
static const Poly<Vec_sd, 3> log_poly_S {
// 1.0,
-3.56722798256324312549e1,
 3.12093766372244180303e2,
-7.69691943550460008604e2 };
static const Poly<Vec_pd, 4> log_poly_R_S {
    log_poly_R.prepend(0.0), log_poly_S.prepend(1.0)
};

} // namespace sg_math_impl

inline Vec_sd sg_vectorcall(log_cm)(const Vec_sd a) {
    if (a.data() <= 0.0) return sg_minus_infinity_f64x1;
    double x = sg_math_impl::mantissa_frexp(a).data();
    auto e = sg_math_impl::exponent_frexp(a).data();
    double y, z;
    if (x < sg_math_impl::sqrt_half) {
        e -= 1;
        z = x - 0.5;
        y = 0.5*z + 0.5;
    } else {
        z = x - 1.0;
        y = 0.5*x + 0.5;
    }
    x = z / y;
    z = x * x;
    // {R, S}
    Vec_pd  poly_eval = sg_math_impl::log_poly_R_S.eval(Vec_pd{z});
    z = x * ((z * poly_eval.d1()) / poly_eval.d0());
    const double e_double = static_cast<double>(e);
    y = e_double;
    z -= y * sg_math_impl::log_c1;
    z += x;
    z += e_double * sg_math_impl::log_c2;
    return z;
}

inline Vec_pd sg_vectorcall(log_cm)(const Vec_pd a) {
    Vec_pd x = sg_math_impl::mantissa_frexp(a);
    auto e = sg_math_impl::exponent_frexp(a);
    Vec_pd y, z;
    Compare_pd x_lt_sqrth {x < sg_math_impl::sqrt_half};
    e -= x_lt_sqrth.to<Vec_pd::fast_convert_int_t::compare_t>()
        .choose_else_zero(1);
    z = x - x_lt_sqrth.choose(0.5, 1.0);
    y = 0.5 * x_lt_sqrth.choose(z, x) + 0.5;
    x = z / y;
    z = x * x;
    Vec_pd R = sg_math_impl::log_poly_R.eval(Vec_pd{z}),
        S = sg_math_impl::log_poly_S.eval1(Vec_pd{z});
    z = x * ((z * R) / S);
    const Vec_pd e_pd {e.to<Vec_pd>()};
    y = e_pd;
    z -= y * sg_math_impl::log_c1;
    z += x;
    z += e_pd * sg_math_impl::log_c2;
    return (a > 0.0).choose(z, Vec_pd::minus_infinity());
}

namespace sg_math_impl {

static const Poly<Vec_sd, 3>exp_poly_P {
1.26177193074810590878e-4,
3.02994407707441961300e-2,
9.99999999999999999910e-1 };
static const Poly<Vec_sd, 4>exp_poly_Q {
3.00198505138664455042e-6,
2.52448340349684104192e-3,
2.27265548208155028766e-1,
2.00000000000000000009 };
static const Poly<Vec_pd, 4>exp_poly_P_Q {
    exp_poly_P.prepend(0.0), exp_poly_Q
};

static constexpr double exp_c1 = 6.93145751953125e-1,
    exp_c2 = 1.42860682030941723212e-6;

} // namespace sg_math_impl

inline Vec_sd sg_vectorcall(exp_cm)(const Vec_sd a) {
    Vec_sd x = a.data();
    auto n = (x * sg_math_impl::log2e).nearest<Vec_sd::fast_convert_int_t>();
    Vec_sd px = n.to<Vec_f64x1>();
    x -= px*sg_math_impl::exp_c1 + px*sg_math_impl::exp_c2;
    Vec_sd xx = x * x;
    // {P, Q}
    const Vec_pd poly = Vec_pd{x.data(), 1.0} *
        sg_math_impl::exp_poly_P_Q.eval(xx.to<Vec_pd>());
    const Vec_sd P = poly.d1(), Q = poly.d0();
    x = P / (Q - P);
    x = 2.0*x + 1.0;
    return sg_math_impl::sg_ldexp(x, n);
}

inline Vec_pd sg_vectorcall(exp_cm)(const Vec_pd a) {
    Vec_pd x = a;
    auto n = (x * sg_math_impl::log2e).nearest<Vec_pd::fast_convert_int_t>();
    Vec_pd px = n.to<Vec_pd>();
    x -= px*sg_math_impl::exp_c1 + px*sg_math_impl::exp_c2;
    Vec_pd xx = x * x;
    Vec_pd P = x * sg_math_impl::exp_poly_P.eval(xx),
        Q = sg_math_impl::exp_poly_Q.eval(xx);
    x = P / (Q - P);
    x = 2.0*x + 1.0;
    return sg_math_impl::sg_ldexp(x, n);
}

namespace sg_math_impl {

static const Poly<Vec_sd, 6> sin_poly {
 1.58962301576546568060e-10,
-2.50507477628578072866e-8,
 2.75573136213857245213e-6,
-1.98412698295895385996e-4,
 8.33333333332211858878e-3,
-1.66666666666666307295e-1 };
static const Poly<Vec_sd, 6> cos_poly {
-1.13585365213876817300e-11,
 2.08757008419747316778e-9,
-2.75573141792967388112e-7,
 2.48015872888517045348e-5,
-1.38888888888730564116e-3,
 4.16666666666665929218e-2 };
static const Poly<Vec_pd, 6> cossin_poly { cos_poly, sin_poly };

static constexpr double dp1 = 7.85398125648498535156e-1;
static constexpr double dp2 = 3.77489470793079817668e-8;
static constexpr double dp3 = 2.69515142907905952645e-15;

} // namespace sg_math_impl

inline sincos_result<Vec_sd> sg_vectorcall(sincos_cm)(const Vec_sd a) {
    // { cos sign bit, sin sign bit }
    Vec_pi64 signbits { 0, (a.bitcast<Vec_s64x1>() & (~sg_fp_signmask_s64)).data() };
    double x = a.abs().data();
    double y = Vec_sd{x * sg_math_impl::four_over_pi}
        .floor<Vec_sd::fast_convert_int_t>().to<Vec_f64x1>().data();
    double z = Vec_sd{y * 0.0625}.floor<Vec_sd::fast_convert_int_t>()
        .to<Vec_f64x1>().data();
    z = y - 16.0*z;
    int32_t j = static_cast<int32_t>(z);
    if (j & 1) {
        ++j;
        y += 1.0;
    }
    j &= 7;
    if (j > 3) {
        signbits ^= Vec_pi64{~sg_fp_signmask_s64};
        j -= 4;
    }
    if (j > 1) signbits ^= Vec_pi64{~sg_fp_signmask_s64, 0};
    z = ((x - sg_math_impl::dp1*y) - sg_math_impl::dp2*y)
        - sg_math_impl::dp3*y;
    const double zz = z * z;

    Vec_pd cossin_poly = Vec_pd{1.0 - 0.5*zz, z} + Vec_pd{zz, z} *
        zz * sg_math_impl::cossin_poly.eval(Vec_pd{zz});
    if ((j == 1) || (j == 2)) cossin_poly = cossin_poly.shuffle<0, 1>();
    cossin_poly = (cossin_poly.bitcast<Vec_pi64>() ^ signbits).bitcast<Vec_pd>();

    sincos_result<Vec_sd> result;
    result.sin_result = cossin_poly.d0();
    result.cos_result = cossin_poly.d1();
    return result;
}

inline Vec_sd sg_vectorcall(sin_cm)(const Vec_sd a) {
    return sincos_cm(a).sin_result;
}
inline Vec_sd sg_vectorcall(cos_cm)(const Vec_sd a) {
    return sincos_cm(a).cos_result;
}

inline sincos_result<Vec_pd> sg_vectorcall(sincos_cm)(const Vec_pd a) {
    Vec_pi64 cos_signbits {0}, sin_signbits{(a.bitcast<Vec_pi64>() & (~sg_fp_signmask_s64)).data()};
    Vec_pd x = a.abs();
    Vec_pd y = (x * sg_math_impl::four_over_pi)
        .floor<Vec_pd::fast_convert_int_t>().to<Vec_pd>();
    Vec_pd z = (y * 0.0625).floor<Vec_pd::fast_convert_int_t>().to<Vec_pd>();
    z = y - 16.0*z;
    auto j = z.truncate<Vec_pd::fast_convert_int_t>();
    const auto j_odd {(j & 1) != 0};
    j += j_odd.choose_else_zero(1);
    y += j_odd.to<Compare_pd>().choose_else_zero(1.0);
    j &= 7;
    const auto j_gt_3 {j > 3};
    j -= j_gt_3.choose_else_zero(4);
    const Compare_pi64 j_gt_3_pi64 = j_gt_3.to<Compare_pi64>();
    cos_signbits ^= j_gt_3_pi64.choose_else_zero(~sg_fp_signmask_s64);
    sin_signbits ^= j_gt_3_pi64.choose_else_zero(~sg_fp_signmask_s64);
    const auto j_gt_1 {j > 1};
    cos_signbits ^= j_gt_1.to<Compare_pi64>().choose_else_zero(~sg_fp_signmask_s64);
    z = ((x - sg_math_impl::dp1*y) - sg_math_impl::dp2*y)
        - sg_math_impl::dp3*y;
    const Vec_pd zz = z * z;
    const Vec_pd sin_result = z + z*zz*sg_math_impl::sin_poly.eval(zz);
    const Vec_pd cos_result = 1.0 - 0.5*zz +
        zz*zz*sg_math_impl::cos_poly.eval(zz);
    const Compare_pd swap_results = ((j == 1) || (j == 2)).to<Compare_pd>();
    sincos_result<Vec_pd> result;
    result.sin_result = swap_results.choose(cos_result, sin_result);
    result.sin_result = (result.sin_result.bitcast<Vec_pi64>() ^ sin_signbits).bitcast<Vec_pd>();
    result.cos_result = swap_results.choose(sin_result, cos_result);
    result.cos_result = (result.cos_result.bitcast<Vec_pi64>() ^ cos_signbits).bitcast<Vec_pd>();

    return result;
}

inline Vec_pd sg_vectorcall(sin_cm)(const Vec_pd a) {
    return sincos_cm(a).sin_result;
}
inline Vec_pd sg_vectorcall(cos_cm)(const Vec_pd a) {
    return sincos_cm(a).cos_result;
}

namespace sg_math_impl {

template <typename VecType>
VecType sg_vectorcall(sqrt_impl)(const VecType a) {
    VecType x = a;
    const VecType w = x;
    const VecType z = mantissa_frexp(a);
    const auto e = exponent_frexp(a);
    x = 4.173075996388649989089e-1 + 5.9016206709064458299663e-1*z;
    x *= ((e & 1) != 0)
        .template to<typename VecType::compare_t>()
        .choose(sqrt_2, 1.0);
    x = sg_ldexp(x, e.template shift_ra_imm<1>());
    x = 0.5*(x + w.safe_divide_by(x));
    x = 0.5*(x + w.safe_divide_by(x));
    x = 0.5*(x + w.safe_divide_by(x));
    return (a > 0.0).choose_else_zero(x);
}

} // namespace sg_math_impl

inline Vec_pd sg_vectorcall(sqrt_cm)(const Vec_pd a) {
    return sg_math_impl::sqrt_impl(a);
}
inline Vec_sd sg_vectorcall(sqrt_cm)(const Vec_sd a) {
    return sg_math_impl::sqrt_impl(a);
}

} // namespace simd_granodi