Average Error: 36.8 → 0.3
Time: 19.0s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\mathsf{fma}\left(\sin \varepsilon, \cos x, \mathsf{fma}\left(\sin x, \cos \varepsilon, -\sin x\right)\right)\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\mathsf{fma}\left(\sin \varepsilon, \cos x, \mathsf{fma}\left(\sin x, \cos \varepsilon, -\sin x\right)\right)
double f(double x, double eps) {
        double r4548898 = x;
        double r4548899 = eps;
        double r4548900 = r4548898 + r4548899;
        double r4548901 = sin(r4548900);
        double r4548902 = sin(r4548898);
        double r4548903 = r4548901 - r4548902;
        return r4548903;
}
↓
double f(double x, double eps) {
        double r4548904 = eps;
        double r4548905 = sin(r4548904);
        double r4548906 = x;
        double r4548907 = cos(r4548906);
        double r4548908 = sin(r4548906);
        double r4548909 = cos(r4548904);
        double r4548910 = -r4548908;
        double r4548911 = fma(r4548908, r4548909, r4548910);
        double r4548912 = fma(r4548905, r4548907, r4548911);
        return r4548912;
}

Error

Bits error versus x

Bits error versus eps

Target

Original36.8
Target15.0
Herbie0.3
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Initial program 36.8

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

  3. Applied sin-sum21.8

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

  5. Applied add-cube-cbrt21.9

    \[\leadsto \left(\sin x \cdot \cos \varepsilon + \color{blue}{\left(\left(\sqrt[3]{\cos x} \cdot \sqrt[3]{\cos x}\right) \cdot \sqrt[3]{\cos x}\right)} \cdot \sin \varepsilon\right) - \sin x\]

  6. Applied associate-*l*21.9

    \[\leadsto \left(\sin x \cdot \cos \varepsilon + \color{blue}{\left(\sqrt[3]{\cos x} \cdot \sqrt[3]{\cos x}\right) \cdot \left(\sqrt[3]{\cos x} \cdot \sin \varepsilon\right)}\right) - \sin x\]

  7. Taylor expanded around inf 21.8

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

  8. Simplified0.4

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

  10. Applied fma-neg0.3

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

  11. Final simplification0.3

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

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (x eps)
  :name "2sin (example 3.3)"

  :herbie-target
  (* 2.0 (* (cos (+ x (/ eps 2.0))) (sin (/ eps 2.0))))

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