Average Error: 37.0 → 0.5
Time: 6.1s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\mathsf{fma}\left(\sin x, \cos \varepsilon - 1, \mathsf{log1p}\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{expm1}\left(\cos x \cdot \sin \varepsilon\right)\right)\right)\right)\right)\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\mathsf{fma}\left(\sin x, \cos \varepsilon - 1, \mathsf{log1p}\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{expm1}\left(\cos x \cdot \sin \varepsilon\right)\right)\right)\right)\right)
double code(double x, double eps) {
	return (sin((x + eps)) - sin(x));
}
↓
double code(double x, double eps) {
	return fma(sin(x), (cos(eps) - 1.0), log1p(log1p(expm1(expm1((cos(x) * sin(eps)))))));
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.0
Target15.5
Herbie0.5
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Initial program 37.0

    \[\sin \left(x + \varepsilon\right) - \sin x\]
  2. Using strategy rm

  3. Applied sin-sum21.4

    \[\leadsto \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \sin x\]

  4. Taylor expanded around inf 21.4

    \[\leadsto \color{blue}{\left(\sin \varepsilon \cdot \cos x + \sin x \cdot \cos \varepsilon\right) - \sin x}\]

  5. Simplified0.4

    \[\leadsto \color{blue}{\mathsf{fma}\left(\sin x, \cos \varepsilon - 1, \cos x \cdot \sin \varepsilon\right)}\]
  6. Using strategy rm

  7. Applied log1p-expm1-u0.4

    \[\leadsto \mathsf{fma}\left(\sin x, \cos \varepsilon - 1, \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos x \cdot \sin \varepsilon\right)\right)}\right)\]
  8. Using strategy rm

  9. Applied log1p-expm1-u0.5

    \[\leadsto \mathsf{fma}\left(\sin x, \cos \varepsilon - 1, \mathsf{log1p}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{expm1}\left(\cos x \cdot \sin \varepsilon\right)\right)\right)}\right)\right)\]

  10. Final simplification0.5

    \[\leadsto \mathsf{fma}\left(\sin x, \cos \varepsilon - 1, \mathsf{log1p}\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{expm1}\left(\cos x \cdot \sin \varepsilon\right)\right)\right)\right)\right)\]

Reproduce

herbie shell --seed 2020060 +o rules:numerics
(FPCore (x eps)
  :name "2sin (example 3.3)"
  :precision binary64

  :herbie-target
  (* 2 (* (cos (+ x (/ eps 2))) (sin (/ eps 2))))

  (- (sin (+ x eps)) (sin x)))