Average Error: 37.0 → 0.8
Time: 6.1s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\sin x \cdot \left(\cos \varepsilon - 1\right) + \left(\sqrt[3]{\cos x} \cdot \sqrt[3]{\cos x}\right) \cdot \left(\log \left(e^{\sqrt[3]{\cos x}}\right) \cdot \sin \varepsilon\right)\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\sin x \cdot \left(\cos \varepsilon - 1\right) + \left(\sqrt[3]{\cos x} \cdot \sqrt[3]{\cos x}\right) \cdot \left(\log \left(e^{\sqrt[3]{\cos x}}\right) \cdot \sin \varepsilon\right)
double f(double x, double eps) {
        double r105825 = x;
        double r105826 = eps;
        double r105827 = r105825 + r105826;
        double r105828 = sin(r105827);
        double r105829 = sin(r105825);
        double r105830 = r105828 - r105829;
        return r105830;
}
↓
double f(double x, double eps) {
        double r105831 = x;
        double r105832 = sin(r105831);
        double r105833 = eps;
        double r105834 = cos(r105833);
        double r105835 = 1.0;
        double r105836 = r105834 - r105835;
        double r105837 = r105832 * r105836;
        double r105838 = cos(r105831);
        double r105839 = cbrt(r105838);
        double r105840 = r105839 * r105839;
        double r105841 = exp(r105839);
        double r105842 = log(r105841);
        double r105843 = sin(r105833);
        double r105844 = r105842 * r105843;
        double r105845 = r105840 * r105844;
        double r105846 = r105837 + r105845;
        return r105846;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.0
Target15.2
Herbie0.8
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Initial program 37.0

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

  3. Applied sin-sum21.7

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

  4. Taylor expanded around inf 21.7

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

  5. Simplified0.4

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

  7. Applied add-cube-cbrt0.8

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

  8. Applied associate-*l*0.8

    \[\leadsto \sin x \cdot \left(\cos \varepsilon - 1\right) + \color{blue}{\left(\sqrt[3]{\cos x} \cdot \sqrt[3]{\cos x}\right) \cdot \left(\sqrt[3]{\cos x} \cdot \sin \varepsilon\right)}\]
  9. Using strategy rm

  10. Applied add-log-exp0.8

    \[\leadsto \sin x \cdot \left(\cos \varepsilon - 1\right) + \left(\sqrt[3]{\cos x} \cdot \sqrt[3]{\cos x}\right) \cdot \left(\color{blue}{\log \left(e^{\sqrt[3]{\cos x}}\right)} \cdot \sin \varepsilon\right)\]

  11. Final simplification0.8

    \[\leadsto \sin x \cdot \left(\cos \varepsilon - 1\right) + \left(\sqrt[3]{\cos x} \cdot \sqrt[3]{\cos x}\right) \cdot \left(\log \left(e^{\sqrt[3]{\cos x}}\right) \cdot \sin \varepsilon\right)\]

Reproduce

herbie shell --seed 2020064 
(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)))