Average Error: 37.3 → 0.4
Time: 6.3s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\left(\cos \varepsilon - 1\right) \cdot \sin x + \cos x \cdot \sin \varepsilon\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\left(\cos \varepsilon - 1\right) \cdot \sin x + \cos x \cdot \sin \varepsilon
double f(double x, double eps) {
        double r90592 = x;
        double r90593 = eps;
        double r90594 = r90592 + r90593;
        double r90595 = sin(r90594);
        double r90596 = sin(r90592);
        double r90597 = r90595 - r90596;
        return r90597;
}
↓
double f(double x, double eps) {
        double r90598 = eps;
        double r90599 = cos(r90598);
        double r90600 = 1.0;
        double r90601 = r90599 - r90600;
        double r90602 = x;
        double r90603 = sin(r90602);
        double r90604 = r90601 * r90603;
        double r90605 = cos(r90602);
        double r90606 = sin(r90598);
        double r90607 = r90605 * r90606;
        double r90608 = r90604 + r90607;
        return r90608;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.3
Target14.8
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-sum22.3

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

  4. Taylor expanded around inf 22.3

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

  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 add-log-exp0.4

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

  8. Applied add-log-exp0.4

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

  9. Applied diff-log0.5

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

  10. Simplified0.4

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

  12. Applied fma-udef0.4

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

  13. Simplified0.4

    \[\leadsto \color{blue}{\left(\cos \varepsilon - 1\right) \cdot \sin x} + \cos x \cdot \sin \varepsilon\]

  14. Final simplification0.4

    \[\leadsto \left(\cos \varepsilon - 1\right) \cdot \sin x + \cos x \cdot \sin \varepsilon\]

Reproduce

herbie shell --seed 2020039 +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)))