tensor/internal/core/fresnel.go

// Copyright (c) 2026 Petr Balvín <opensource@petrbalvin.org> (https://petrbalvin.org)
// SPDX-License-Identifier: MIT

package core

import (
	"math"
	"math/big"
)

// Fresnel integrals, the diffraction pair C(x) = ∫₀^x cos(πt²/2) dt
// and S(x) = ∫₀^x sin(πt²/2) dt.
//
// The power series converge for every real x, but their intermediate
// terms grow like e^(πx²/2) before cancelling down to the ½ limit.
// Within |x| ≤ 4 the cancellation is mild and plain float64 holds full
// accuracy. Above that the same series is summed in extended
// precision (math/big), with the working bit size scaled to the
// largest intermediate term, so the float64 result stays correct
// instead of drowning in cancellation noise. Both integrals are odd
// functions.

// FresnelC returns the Fresnel cosine integral C(x) of each element.
func FresnelC(a *Array) (*Array, error) { return a.realFunc("FresnelC", fresnelC) }

// FresnelS returns the Fresnel sine integral S(x) of each element.
func FresnelS(a *Array) (*Array, error) { return a.realFunc("FresnelS", fresnelS) }

const fresnelSeriesCutoff = 3

// fresnelAsymptoticFrom is the argument above which the erfc
// asymptotic series answers the Fresnel pair. There the terms of that
// series fall by a factor 2πx² per order, so a handful of them reach
// double precision, where the power series would need a working
// precision growing with x² (and its term count with x²).
const fresnelAsymptoticFrom = 10

func fresnelC(x float64) float64 {
	if x < 0 {
		return -fresnelC(-x)
	}
	if x > fresnelAsymptoticFrom {
		return fresnelAsym(x, false)
	}
	if x > fresnelSeriesCutoff {
		return fresnelBig(x, false)
	}
	return fresnelCSeries(x)
}

func fresnelS(x float64) float64 {
	if x < 0 {
		return -fresnelS(-x)
	}
	if x > fresnelAsymptoticFrom {
		return fresnelAsym(x, true)
	}
	if x > fresnelSeriesCutoff {
		return fresnelBig(x, true)
	}
	return fresnelSSeries(x)
}

// fresnelCSeries sums C(x) = Σ (−1)^k (π/2)^{2k} x^{4k+1}/((2k)!(4k+1))
// by the term-ratio recurrence. Accurate while the largest
// intermediate term stays comfortably inside float64, which holds
// through |x| ≈ 4.
func fresnelCSeries(x float64) float64 {
	sum := 0.0
	term := x // the k = 0 term
	for k := range 200 {
		sum += term
		next := -term * (math.Pi / 2) * (math.Pi / 2) * x * x * x * x /
			float64((2*k+1)*(2*k+2)) * float64(4*k+1) / float64(4*k+5)
		if math.Abs(next) < 1e-18*math.Abs(sum) {
			break
		}
		term = next
	}
	return sum
}

// fresnelSSeries sums S(x) = Σ (−1)^k (π/2)^{2k+1} x^{4k+3}/((2k+1)!(4k+3))
// by the term-ratio recurrence, with the same validity range as the
// cosine branch.
func fresnelSSeries(x float64) float64 {
	sum := 0.0
	term := (math.Pi / 2) * x * x * x / 3 // the k = 0 term
	for k := range 200 {
		sum += term
		next := -term * (math.Pi / 2) * (math.Pi / 2) * x * x * x * x /
			float64((2*k+2)*(2*k+3)) * float64(4*k+3) / float64(4*k+7)
		if math.Abs(next) < 1e-18*math.Abs(sum) {
			break
		}
		term = next
	}
	return sum
}

// fresnelAsym evaluates C or S from the auxiliary functions above.
// x must be positive and at least fresnelAsymptoticFrom.
func fresnelAsym(x float64, sine bool) float64 {
	f, g := fresnelAux(x)
	u := math.Pi * x * x / 2
	cosu, sinu := math.Cos(u), math.Sin(u)
	if sine {
		return 0.5 - f*cosu - g*sinu
	}
	return 0.5 + f*sinu - g*cosu
}

// fresnelAux returns the auxiliary functions f and g of the Fresnel
// asymptotics,
//
//	S(x) = ½ − f·cos(πx²/2) − g·sin(πx²/2)
//	C(x) = ½ + f·sin(πx²/2) − g·cos(πx²/2)
//
// as the classical expansions
//
//	f(x) = (1/(πx))·Σ (−1)^k (4k−1)!!·t^{2k}
//	g(x) = (1/(πx))·Σ (−1)^k (4k+1)!!·t^{2k+1},   t = 1/(πx²),
//
// whose coefficients follow from (4k+1)(4k+3). x must be positive and
// at least fresnelAsymptoticFrom: there t ≤ 1/314 and the terms fall
// fast enough for four of them to pass double precision. The
// coefficients and the reconstruction were checked against mpmath to
// 1e-20 at x = 8, 12, 20, 100, 1000.
func fresnelAux(x float64) (f, g float64) {
	t := 1 / (math.Pi * x * x)
	cf, cg := 1.0, t
	sumF, sumG := 1.0, t
	for k := range 20 {
		cf = -cf * float64(4*k+1) * float64(4*k+3) * t * t
		cg = -cg * float64(4*k+3) * float64(4*k+5) * t * t
		sumF += cf
		sumG += cg
		if math.Abs(cf)+math.Abs(cg) < 1e-19 {
			break
		}
	}
	inv := 1 / (math.Pi * x)
	return inv * sumF, inv * sumG
}

// beyond the float64 cancellation ceiling. The working bit size grows
// with the largest intermediate term; the tail is cut once the terms
// fall below a few 1e-40 relative to the running sum, which bounds the
// final float64 rounding far below the 1e-15 relative level.
func fresnelBig(x float64, sine bool) float64 {
	// The largest intermediate term of the series grows like e^{πx²/2},
	// so the working precision must grow by π/(2·ln2) ≈ 2.266 bits per
	// unit of x²; a smaller coefficient silently loses the cancellation
	// and the result turns to noise around x ≈ 22.
	bits := uint(64 + int(2.2663*x*x) + 96)
	xf := new(big.Float).SetPrec(bits).SetFloat64(x)
	halfPi := new(big.Float).SetPrec(bits).SetFloat64(math.Pi / 2)
	hh := new(big.Float).SetPrec(bits).Mul(halfPi, halfPi)
	x4 := new(big.Float).SetPrec(bits).Mul(xf, xf)
	x4.Mul(x4, xf)
	x4.Mul(x4, xf)

	term := new(big.Float).SetPrec(bits)
	if sine {
		// S seed: (π/2)·x³/3.
		term.Mul(halfPi, xf)
		term.Mul(term, xf)
		term.Mul(term, xf)
		term.Quo(term, big.NewFloat(3))
	} else {
		term.Set(xf)
	}
	sum := new(big.Float).SetPrec(bits)

	tiny := new(big.Float).SetPrec(bits).SetFloat64(1e-40)
	tinyNeg := new(big.Float).SetPrec(bits).SetFloat64(-1e-40)
	for k := range 200000 {
		sum.Add(sum, term)
		an := 4*k + 1
		d1, d2, d3 := 2*k+1, 2*k+2, 4*k+5
		if sine {
			an = 4*k + 3
			d1, d2, d3 = 2*k+2, 2*k+3, 4*k+7
		}
		next := new(big.Float).SetPrec(bits).Neg(term)
		next.Mul(next, hh)
		next.Mul(next, x4)
		next.Mul(next, big.NewFloat(float64(an)))
		next.Quo(next, big.NewFloat(float64(d1)))
		next.Quo(next, big.NewFloat(float64(d2)))
		next.Quo(next, big.NewFloat(float64(d3)))
		if next.Cmp(tiny) < 0 && next.Cmp(tinyNeg) > 0 {
			break
		}
		term = next
	}
	out, _ := sum.Float64()
	return out
}