Average Error: 37.6 → 0.6
Time: 6.4s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\left(\sqrt[3]{\sin x} \cdot \sqrt[3]{\sin x}\right) \cdot \left(\sqrt[3]{\sin x} \cdot \left(\cos \varepsilon - 1\right)\right) + \cos x \cdot \sin \varepsilon\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\left(\sqrt[3]{\sin x} \cdot \sqrt[3]{\sin x}\right) \cdot \left(\sqrt[3]{\sin x} \cdot \left(\cos \varepsilon - 1\right)\right) + \cos x \cdot \sin \varepsilon
double f(double x, double eps) {
        double r147045 = x;
        double r147046 = eps;
        double r147047 = r147045 + r147046;
        double r147048 = sin(r147047);
        double r147049 = sin(r147045);
        double r147050 = r147048 - r147049;
        return r147050;
}
↓
double f(double x, double eps) {
        double r147051 = x;
        double r147052 = sin(r147051);
        double r147053 = cbrt(r147052);
        double r147054 = r147053 * r147053;
        double r147055 = eps;
        double r147056 = cos(r147055);
        double r147057 = 1.0;
        double r147058 = r147056 - r147057;
        double r147059 = r147053 * r147058;
        double r147060 = r147054 * r147059;
        double r147061 = cos(r147051);
        double r147062 = sin(r147055);
        double r147063 = r147061 * r147062;
        double r147064 = r147060 + r147063;
        return r147064;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.6
Target15.4
Herbie0.6
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Initial program 37.6

    \[\sin \left(x + \varepsilon\right) - \sin x\]
  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. Applied associate--l+22.1

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

  5. Taylor expanded around inf 22.1

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

  6. Simplified0.4

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

  8. Applied add-cube-cbrt0.6

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

  9. Applied associate-*l*0.6

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

  10. Final simplification0.6

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

Reproduce

herbie shell --seed 2020036 
(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)))