Average Error: 37.1 → 0.4
Time: 6.0s
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 r128030 = x;
        double r128031 = eps;
        double r128032 = r128030 + r128031;
        double r128033 = sin(r128032);
        double r128034 = sin(r128030);
        double r128035 = r128033 - r128034;
        return r128035;
}
↓
double f(double x, double eps) {
        double r128036 = eps;
        double r128037 = cos(r128036);
        double r128038 = 1.0;
        double r128039 = r128037 - r128038;
        double r128040 = x;
        double r128041 = sin(r128040);
        double r128042 = r128039 * r128041;
        double r128043 = cos(r128040);
        double r128044 = sin(r128036);
        double r128045 = r128043 * r128044;
        double r128046 = r128042 + r128045;
        return r128046;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.1
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 37.1

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

  3. Applied sin-sum22.0

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

  5. Applied *-un-lft-identity22.0

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

  6. Applied *-un-lft-identity22.0

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

  7. Applied distribute-lft-out--22.0

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

  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 2020062 +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)))