Jungfraujoch/image_analysis/bragg_integration/BraggExp.h

// SPDX-FileCopyrightText: 2026 Filip Leonarski, Paul Scherrer Institute <filip.leonarski@psi.ch>
// SPDX-License-Identifier: GPL-3.0-only

#pragma once

// exp in single precision, the same bits wherever it is compiled: the host compiler (no FMA contraction,
// see CMakeLists.txt) and the Metal compiler (fast math off) both see only IEEE +, -, * rounded to
// nearest and exact scaling by a power of two, so the profile fit's Gaussian is the same on the CPU and
// on an Apple GPU, whose exp no host function reproduces. It is closer to correctly rounded than macOS's
// expf: at most 0.53 ulp for a normal result, and away from a correctly rounded float for 0.2% of
// arguments below -1 and 0.004% in [-1, 0].
//
// exp(x) = 2^k exp(r), k = rint(x / ln2), r = x - k ln2 with ln2 split three ways, the first two parts
// of 16 bits so that k times them is exact (Cody & Waite). r is carried as rr + re, rr^2 is formed exactly
// as zh + zl (Dekker's product with Veltkamp's split) and the sums into 1 + rr + rr^2/2 are error-free
// (2Sum, Fast2Sum), so the only rounding that matters is the last one. The rest of the series is a
// degree-4 minimax fit, relative error 1.3e-10.

#ifdef __METAL_VERSION__
inline float BraggBitsToFloat(int i) { return as_type<float>(i); }
inline int BraggFloatToBits(float f) { return as_type<int>(f); }
#else
#include <bit>
#include <cstdint>
inline float BraggBitsToFloat(int32_t i) { return std::bit_cast<float>(i); }
inline int32_t BraggFloatToBits(float f) { return std::bit_cast<int32_t>(f); }
#endif

// p * 2^k, rounded once where the result is subnormal: below k = -125 the scaling is split in two. The
// two cases differ only in their operands - the normal one scales by 1 the second time, which is exact -
// so a loop over it has no branch and vectorises.
inline float BraggScale2(float p, int k) {
    const bool subnormal = k < -125;
    const float first = BraggBitsToFloat(((subnormal ? k + 100 : k) + 127) << 23);
    const float second = subnormal ? BraggBitsToFloat((-100 + 127) << 23) : 1.0f;
    return (p * first) * second;
}

// Following Cody & Waite (1980) Software Manual for the Elementary Functions; Dekker (1971) Numer. Math.
// 18, 224-242; Knuth, The Art of Computer Programming Vol. 2, 4.2.2 (2Sum).
// The two ends are told apart on the bits of x and applied to the bits of the result, with integer masks
// rather than branches or selects: the compiler moves arithmetic behind a branch or a select that only
// one side needs, and may not run float arithmetic on a path that does not need it, which either way
// keeps a loop over BraggExp from vectorising. Out of range the arithmetic runs on 0 and its result is
// masked off; in range it is the same arithmetic on the same x. (A negative float's bits are a negative
// integer that grows as the float falls; a positive one's grow with it.)
inline float BraggExp(float x_in) {
    const int bits = BraggFloatToBits(x_in);
    const bool under = (bits < 0) & (bits > BraggFloatToBits(-104.0f));   // below half the smallest subnormal
    const bool over = bits > BraggFloatToBits(88.72283f);                  // overflows
    const int in_range = -static_cast<int>(!(under | over));              // every bit set, or none
    const float x = BraggBitsToFloat(bits & in_range);
    const float shifter = 12582912.0f;                             // 1.5 * 2^23: adding it rounds to an integer
    const float kf = (x * 1.44269504f + shifter) - shifter;
    const int k = static_cast<int>(kf);
    const float r1 = x - kf * 0.693145751953125f;                  // exact
    const float t2 = kf * 1.4286197256296873e-06f;                 // exact
    const float rr = r1 - t2;                                      // r1 - t2 = rr + e2 (2Sum)
    const float bv0 = rr - r1;
    const float e2 = (r1 - (rr - bv0)) + (-t2 - bv0);
    const float re = e2 - kf * -1.2905320041778356e-11f;
    const float sp = rr * 4097.0f;                                 // rr^2 = zh + zl
    const float rh = sp - (sp - rr);
    const float rl = rr - rh;
    const float zh = rr * rr;
    const float zl = ((rh * rh - zh) + (rh * rl + rh * rl)) + rl * rl;
    const float h = 0.1666666567325592f + rr * (0.04166644066572189f + rr * (0.008333569392561913f
                    + rr * (0.0013937620678916574f + rr * 0.0001974606711883098f)));
    const float c = 0.5f * zl + ((re + re * rr) + (zh * rr) * h);
    const float hz = 0.5f * zh;                                    // rr + zh/2 = s1 + e1 (2Sum)
    const float s1 = rr + hz;
    const float bv1 = s1 - rr;
    const float e1 = (rr - (s1 - bv1)) + (hz - bv1);
    const float t1 = 1.0f + s1;                                    // 1 + s1 = t1 + e3 (Fast2Sum, |s1| < 1)
    const float e3 = (1.0f - t1) + s1;
    const float e = BraggScale2(t1 + (e3 + (e1 + c)), k);
    return BraggBitsToFloat((BraggFloatToBits(e) & in_range) | (static_cast<int>(over) * 0x7f800000));
}