Average Error: 37.4 → 0.4
Time: 6.4s
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 r138806 = x;
        double r138807 = eps;
        double r138808 = r138806 + r138807;
        double r138809 = sin(r138808);
        double r138810 = sin(r138806);
        double r138811 = r138809 - r138810;
        return r138811;
}
↓
double f(double x, double eps) {
        double r138812 = eps;
        double r138813 = cos(r138812);
        double r138814 = 1.0;
        double r138815 = r138813 - r138814;
        double r138816 = x;
        double r138817 = sin(r138816);
        double r138818 = r138815 * r138817;
        double r138819 = cos(r138816);
        double r138820 = sin(r138812);
        double r138821 = r138819 * r138820;
        double r138822 = r138818 + r138821;
        return r138822;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.4
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 37.4

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

  3. Applied sin-sum22.1

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

  5. Applied *-un-lft-identity22.1

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

  6. Applied *-un-lft-identity22.1

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

  7. Applied distribute-lft-out--22.1

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

  8. Simplified0.4

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

  10. Applied add-log-exp0.4

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

  11. Applied add-log-exp0.4

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

  12. Applied diff-log0.5

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

  13. Simplified0.4

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

  15. Applied fma-udef0.4

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

  16. Simplified0.4

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

  17. Final simplification0.4

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

Reproduce

herbie shell --seed 2020100 +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)))