Average Error: 37.6 → 0.4
Time: 16.7s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\mathsf{fma}\left(\left(\sin \varepsilon\right), \left(\cos x\right), \left(\mathsf{fma}\left(\left(\sin x\right), \left(\cos \varepsilon\right), \left(-\sin x\right)\right)\right)\right)\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\mathsf{fma}\left(\left(\sin \varepsilon\right), \left(\cos x\right), \left(\mathsf{fma}\left(\left(\sin x\right), \left(\cos \varepsilon\right), \left(-\sin x\right)\right)\right)\right)
double f(double x, double eps) {
        double r2911353 = x;
        double r2911354 = eps;
        double r2911355 = r2911353 + r2911354;
        double r2911356 = sin(r2911355);
        double r2911357 = sin(r2911353);
        double r2911358 = r2911356 - r2911357;
        return r2911358;
}
↓
double f(double x, double eps) {
        double r2911359 = eps;
        double r2911360 = sin(r2911359);
        double r2911361 = x;
        double r2911362 = cos(r2911361);
        double r2911363 = sin(r2911361);
        double r2911364 = cos(r2911359);
        double r2911365 = -r2911363;
        double r2911366 = fma(r2911363, r2911364, r2911365);
        double r2911367 = fma(r2911360, r2911362, r2911366);
        return r2911367;
}

Error

Bits error versus x

Bits error versus eps

Target

Original37.6
Target15.2
Herbie0.4
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Initial program 37.6

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

  3. Applied sin-sum22.2

    \[\leadsto \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \sin x\]
  4. Using strategy rm

  5. Applied log1p-expm1-u22.3

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

  6. Simplified22.3

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

  7. Taylor expanded around -inf 22.2

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

  8. Simplified0.4

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

  10. Applied fma-neg0.4

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

  11. Final simplification0.4

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

Reproduce

herbie shell --seed 2019128 +o rules:numerics
(FPCore (x eps)
  :name "2sin (example 3.3)"

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

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