Average Error: 37.4 → 0.4
Time: 22.1s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[2 \cdot \left(\left(\cos x \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right) - \sqrt[3]{\left(\sin x \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot \left(\left(\sin x \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot \left(\sin x \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)\right)}\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)\]
\sin \left(x + \varepsilon\right) - \sin x
↓
2 \cdot \left(\left(\cos x \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right) - \sqrt[3]{\left(\sin x \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot \left(\left(\sin x \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot \left(\sin x \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)\right)}\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)
double f(double x, double eps) {
        double r4137827 = x;
        double r4137828 = eps;
        double r4137829 = r4137827 + r4137828;
        double r4137830 = sin(r4137829);
        double r4137831 = sin(r4137827);
        double r4137832 = r4137830 - r4137831;
        return r4137832;
}
↓
double f(double x, double eps) {
        double r4137833 = 2.0;
        double r4137834 = x;
        double r4137835 = cos(r4137834);
        double r4137836 = 0.5;
        double r4137837 = eps;
        double r4137838 = r4137836 * r4137837;
        double r4137839 = cos(r4137838);
        double r4137840 = r4137835 * r4137839;
        double r4137841 = sin(r4137834);
        double r4137842 = sin(r4137838);
        double r4137843 = r4137841 * r4137842;
        double r4137844 = r4137843 * r4137843;
        double r4137845 = r4137843 * r4137844;
        double r4137846 = cbrt(r4137845);
        double r4137847 = r4137840 - r4137846;
        double r4137848 = r4137847 * r4137842;
        double r4137849 = r4137833 * r4137848;
        return r4137849;
}

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
Target15.3
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 diff-sin37.8

    \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)}\]

  4. Simplified15.3

    \[\leadsto 2 \cdot \color{blue}{\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)\right)}\]

  5. Taylor expanded around -inf 15.3

    \[\leadsto 2 \cdot \color{blue}{\left(\cos \left(\frac{1}{2} \cdot \left(2 \cdot x + \varepsilon\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)}\]

  6. Simplified15.3

    \[\leadsto 2 \cdot \color{blue}{\left(\cos \left(\mathsf{fma}\left(\varepsilon, \frac{1}{2}, x\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)}\]
  7. Using strategy rm

  8. Applied fma-udef15.3

    \[\leadsto 2 \cdot \left(\cos \color{blue}{\left(\varepsilon \cdot \frac{1}{2} + x\right)} \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)\]

  9. Applied cos-sum0.3

    \[\leadsto 2 \cdot \left(\color{blue}{\left(\cos \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \cos x - \sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin x\right)} \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)\]
  10. Using strategy rm

  11. Applied add-cbrt-cube0.4

    \[\leadsto 2 \cdot \left(\left(\cos \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \cos x - \color{blue}{\sqrt[3]{\left(\left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin x\right) \cdot \left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin x\right)\right) \cdot \left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin x\right)}}\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)\]

  12. Final simplification0.4

    \[\leadsto 2 \cdot \left(\left(\cos x \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right) - \sqrt[3]{\left(\sin x \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot \left(\left(\sin x \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot \left(\sin x \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)\right)}\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)\]

Reproduce

herbie shell --seed 2019139 +o rules:numerics
(FPCore (x eps)
  :name "2sin (example 3.3)"

  :herbie-target
  (* 2 (* (cos (+ x (/ eps 2))) (sin (/ eps 2))))

  (- (sin (+ x eps)) (sin x)))