Average Error: 37.1 → 0.4
Time: 43.3s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\left(\left(\cos x \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right) - \log \left(e^{\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x}\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot 2\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\left(\left(\cos x \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right) - \log \left(e^{\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x}\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot 2
double f(double x, double eps) {
        double r5060888 = x;
        double r5060889 = eps;
        double r5060890 = r5060888 + r5060889;
        double r5060891 = sin(r5060890);
        double r5060892 = sin(r5060888);
        double r5060893 = r5060891 - r5060892;
        return r5060893;
}
↓
double f(double x, double eps) {
        double r5060894 = x;
        double r5060895 = cos(r5060894);
        double r5060896 = 0.5;
        double r5060897 = eps;
        double r5060898 = r5060896 * r5060897;
        double r5060899 = cos(r5060898);
        double r5060900 = r5060895 * r5060899;
        double r5060901 = sin(r5060898);
        double r5060902 = sin(r5060894);
        double r5060903 = r5060901 * r5060902;
        double r5060904 = exp(r5060903);
        double r5060905 = log(r5060904);
        double r5060906 = r5060900 - r5060905;
        double r5060907 = r5060906 * r5060901;
        double r5060908 = 2.0;
        double r5060909 = r5060907 * r5060908;
        return r5060909;
}

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 diff-sin37.4

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

  4. Simplified15.0

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

  5. Taylor expanded around inf 15.0

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

  6. Simplified15.0

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

  8. Applied fma-udef15.0

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

  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-log-exp0.4

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

  12. Simplified0.4

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

  13. Final simplification0.4

    \[\leadsto \left(\left(\cos x \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right) - \log \left(e^{\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x}\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot 2\]

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(FPCore (x eps)
  :name "2sin (example 3.3)"

  :herbie-target
  (* 2.0 (* (cos (+ x (/ eps 2.0))) (sin (/ eps 2.0))))

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