Average Error: 36.7 → 0.4
Time: 6.2s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\mathsf{fma}\left(\sin x, \frac{{\left(\cos \varepsilon\right)}^{3} - {1}^{3}}{\mathsf{fma}\left(\cos \varepsilon, \cos \varepsilon + 1, 1\right)}, \cos x \cdot \sin \varepsilon\right)\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\mathsf{fma}\left(\sin x, \frac{{\left(\cos \varepsilon\right)}^{3} - {1}^{3}}{\mathsf{fma}\left(\cos \varepsilon, \cos \varepsilon + 1, 1\right)}, \cos x \cdot \sin \varepsilon\right)
double f(double x, double eps) {
        double r135941 = x;
        double r135942 = eps;
        double r135943 = r135941 + r135942;
        double r135944 = sin(r135943);
        double r135945 = sin(r135941);
        double r135946 = r135944 - r135945;
        return r135946;
}
↓
double f(double x, double eps) {
        double r135947 = x;
        double r135948 = sin(r135947);
        double r135949 = eps;
        double r135950 = cos(r135949);
        double r135951 = 3.0;
        double r135952 = pow(r135950, r135951);
        double r135953 = 1.0;
        double r135954 = pow(r135953, r135951);
        double r135955 = r135952 - r135954;
        double r135956 = r135950 + r135953;
        double r135957 = fma(r135950, r135956, r135953);
        double r135958 = r135955 / r135957;
        double r135959 = cos(r135947);
        double r135960 = sin(r135949);
        double r135961 = r135959 * r135960;
        double r135962 = fma(r135948, r135958, r135961);
        return r135962;
}

Error

Bits error versus x

Bits error versus eps

Target

Original36.7
Target15.0
Herbie0.4
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Initial program 36.7

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

  3. Applied sin-sum21.5

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

  4. Taylor expanded around inf 21.5

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

  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 flip3--0.4

    \[\leadsto \mathsf{fma}\left(\sin x, \color{blue}{\frac{{\left(\cos \varepsilon\right)}^{3} - {1}^{3}}{\cos \varepsilon \cdot \cos \varepsilon + \left(1 \cdot 1 + \cos \varepsilon \cdot 1\right)}}, \cos x \cdot \sin \varepsilon\right)\]

  8. Simplified0.4

    \[\leadsto \mathsf{fma}\left(\sin x, \frac{{\left(\cos \varepsilon\right)}^{3} - {1}^{3}}{\color{blue}{\mathsf{fma}\left(\cos \varepsilon, \cos \varepsilon + 1, 1\right)}}, \cos x \cdot \sin \varepsilon\right)\]

  9. Final simplification0.4

    \[\leadsto \mathsf{fma}\left(\sin x, \frac{{\left(\cos \varepsilon\right)}^{3} - {1}^{3}}{\mathsf{fma}\left(\cos \varepsilon, \cos \varepsilon + 1, 1\right)}, \cos x \cdot \sin \varepsilon\right)\]

Reproduce

herbie shell --seed 2020024 +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)))