103 lines
3.2 KiB
Go
103 lines
3.2 KiB
Go
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// Copyright (c) 2026 Petr Balvín <opensource@petrbalvin.org> (https://petrbalvin.org)
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// SPDX-License-Identifier: MIT
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package core
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import (
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"math"
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"math/big"
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"testing"
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)
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// cosm1Big evaluates cos(x) − 1 by the Taylor series in 256-bit
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// arithmetic, the exact referent the float64 implementation is held
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// against across the crossover.
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func cosm1Big(x float64) *big.Float {
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const prec = 256
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xf := new(big.Float).SetPrec(prec).SetFloat64(x)
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sq := new(big.Float).SetPrec(prec).Mul(xf, xf)
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sum := new(big.Float).SetPrec(prec).SetInt64(1)
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term := new(big.Float).SetPrec(prec).SetInt64(1)
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for k := 1; k <= 60; k++ {
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term.Mul(term, sq)
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term.Quo(term, new(big.Float).SetPrec(prec).SetInt64(int64(2*k-1)))
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term.Quo(term, new(big.Float).SetPrec(prec).SetInt64(int64(2*k)))
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term.Neg(term)
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sum.Add(sum, term)
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}
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return sum.Sub(sum, new(big.Float).SetPrec(prec).SetInt64(1))
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}
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// TestCosm1AgainstSeries holds the implementation against the exact
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// referent on both sides of the crossover, from arguments whose answer
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// is −x²/2 as far as the format can see up to ones where the direct
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// subtraction carries it alone.
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func TestCosm1AgainstSeries(t *testing.T) {
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points := []float64{
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1e-300, 1e-200, 5e-12, 1e-10, 1e-9, 1e-8, 1e-6, 1e-4, 0.01, 0.1,
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0.3, 0.5, 0.78, math.Pi / 4, 0.79, 1, 2, -0.3, -0.78, -2,
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}
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for _, x := range points {
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got, err := Cosm1(mustFloats(t, []float64{x}))
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if err != nil {
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t.Fatalf("Cosm1(%v): %v", x, err)
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}
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want, _ := cosm1Big(x).Float64()
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v := got.FloatAt(0)
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if want == 0 {
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if v != 0 {
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t.Fatalf("Cosm1(%v) = %v, want 0", x, v)
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}
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continue
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}
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if d := math.Abs(v-want) / math.Abs(want); d > 6e-16 {
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t.Fatalf("Cosm1(%v) = %.17g, want %.17g (relative %.3g)", x, v, want, d)
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}
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}
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}
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// TestCosm1Pins pins the behaviour the referent cannot speak for: the
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// exact zero at the origin, the underflowed answer keeping the minus
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// sign the function's range promises, the bit-equality with the direct
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// subtraction above the crossover, and the integer promotion.
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func TestCosm1Pins(t *testing.T) {
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zero, err := Cosm1(mustFloats(t, []float64{0}))
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if err != nil {
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t.Fatal(err)
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}
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if zero.FloatAt(0) != 0 {
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t.Fatalf("Cosm1(0) = %v, want 0", zero.FloatAt(0))
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}
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tiny, err := Cosm1(mustFloats(t, []float64{1e-300}))
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if err != nil {
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t.Fatal(err)
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}
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if tiny.FloatAt(0) != 0 || !math.Signbit(tiny.FloatAt(0)) {
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t.Fatalf("Cosm1(1e-300) = %v, want the negative zero the true answer underflows to", tiny.FloatAt(0))
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}
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for _, x := range []float64{0.79, 1, 2, 10, 100, 1e6, -3.5} {
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got, err := Cosm1(mustFloats(t, []float64{x}))
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if err != nil {
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t.Fatalf("Cosm1(%v): %v", x, err)
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}
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if want := math.Cos(x) - 1; got.FloatAt(0) != want {
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t.Fatalf("Cosm1(%v) = %.17g, want the direct %.17g", x, got.FloatAt(0), want)
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}
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}
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ints, err := FromInts([]int64{0, 1}, 2)
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if err != nil {
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t.Fatal(err)
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}
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promoted, err := Cosm1(ints)
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if err != nil {
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t.Fatalf("Cosm1 over ints: %v", err)
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}
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if promoted.FloatAt(1) != math.Cos(1)-1 {
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t.Fatalf("Cosm1(int 1) = %.17g, want %.17g", promoted.FloatAt(1), math.Cos(1)-1)
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}
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cx, _ := FromComplexes([]complex128{1}, 1)
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if _, err := Cosm1(cx); err == nil {
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t.Fatal("Cosm1 over complex: want an error")
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}
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}
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