Average Error: 37.3 → 0.4
Time: 6.5s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\mathsf{fma}\left(\sin x, \log \left(e^{\cos \varepsilon - 1}\right), \cos x \cdot \sin \varepsilon\right)\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\mathsf{fma}\left(\sin x, \log \left(e^{\cos \varepsilon - 1}\right), \cos x \cdot \sin \varepsilon\right)
double f(double x, double eps) {
        double r122933 = x;
        double r122934 = eps;
        double r122935 = r122933 + r122934;
        double r122936 = sin(r122935);
        double r122937 = sin(r122933);
        double r122938 = r122936 - r122937;
        return r122938;
}
↓
double f(double x, double eps) {
        double r122939 = x;
        double r122940 = sin(r122939);
        double r122941 = eps;
        double r122942 = cos(r122941);
        double r122943 = 1.0;
        double r122944 = r122942 - r122943;
        double r122945 = exp(r122944);
        double r122946 = log(r122945);
        double r122947 = cos(r122939);
        double r122948 = sin(r122941);
        double r122949 = r122947 * r122948;
        double r122950 = fma(r122940, r122946, r122949);
        return r122950;
}

Error

Bits error versus x

Bits error versus eps

Target

Original37.3
Target15.7
Herbie0.4
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Initial program 37.3

    \[\sin \left(x + \varepsilon\right) - \sin x\]
  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. Applied associate--l+21.6

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

  5. Taylor expanded around inf 21.5

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

  6. Simplified0.4

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

  8. Applied add-log-exp0.4

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

  9. Applied add-log-exp0.4

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

  10. Applied diff-log0.5

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

  11. Simplified0.4

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

  12. Final simplification0.4

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

Reproduce

herbie shell --seed 2020002 +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)))