Average Error: 31.3 → 7.4
Time: 19.3s
Precision: 64
\[\frac{\sin \left(x + 10^{-05}\right) - \sin x}{10^{-05}}\]
\[\frac{\sqrt[3]{\left(\cos x \cdot \sin \left( 10^{-05} \right) + \cos \left( 10^{-05} \right) \cdot \sin x\right) - \sin x} \cdot \sqrt[3]{\left(\cos x \cdot \sin \left( 10^{-05} \right) + \cos \left( 10^{-05} \right) \cdot \sin x\right) - \sin x}}{\frac{10^{-05}}{\sqrt[3]{\left(\cos x \cdot \sin \left( 10^{-05} \right) + \cos \left( 10^{-05} \right) \cdot \sin x\right) - \sin x}}}\]
\frac{\sin \left(x + 10^{-05}\right) - \sin x}{10^{-05}}
\frac{\sqrt[3]{\left(\cos x \cdot \sin \left( 10^{-05} \right) + \cos \left( 10^{-05} \right) \cdot \sin x\right) - \sin x} \cdot \sqrt[3]{\left(\cos x \cdot \sin \left( 10^{-05} \right) + \cos \left( 10^{-05} \right) \cdot \sin x\right) - \sin x}}{\frac{10^{-05}}{\sqrt[3]{\left(\cos x \cdot \sin \left( 10^{-05} \right) + \cos \left( 10^{-05} \right) \cdot \sin x\right) - \sin x}}}
double f(double x) {
        double r5232515 = x;
        double r5232516 = 1e-05;
        double r5232517 = r5232515 + r5232516;
        double r5232518 = sin(r5232517);
        double r5232519 = sin(r5232515);
        double r5232520 = r5232518 - r5232519;
        double r5232521 = r5232520 / r5232516;
        return r5232521;
}

double f(double x) {
        double r5232522 = x;
        double r5232523 = cos(r5232522);
        double r5232524 = 1e-05;
        double r5232525 = sin(r5232524);
        double r5232526 = r5232523 * r5232525;
        double r5232527 = cos(r5232524);
        double r5232528 = sin(r5232522);
        double r5232529 = r5232527 * r5232528;
        double r5232530 = r5232526 + r5232529;
        double r5232531 = r5232530 - r5232528;
        double r5232532 = cbrt(r5232531);
        double r5232533 = r5232532 * r5232532;
        double r5232534 = r5232524 / r5232532;
        double r5232535 = r5232533 / r5232534;
        return r5232535;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.3

    \[\frac{\sin \left(x + 10^{-05}\right) - \sin x}{10^{-05}}\]
  2. Using strategy rm
  3. Applied sin-sum7.8

    \[\leadsto \frac{\color{blue}{\left(\sin x \cdot \cos \left( 10^{-05} \right) + \cos x \cdot \sin \left( 10^{-05} \right)\right)} - \sin x}{10^{-05}}\]
  4. Using strategy rm
  5. Applied add-cube-cbrt7.9

    \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\left(\sin x \cdot \cos \left( 10^{-05} \right) + \cos x \cdot \sin \left( 10^{-05} \right)\right) - \sin x} \cdot \sqrt[3]{\left(\sin x \cdot \cos \left( 10^{-05} \right) + \cos x \cdot \sin \left( 10^{-05} \right)\right) - \sin x}\right) \cdot \sqrt[3]{\left(\sin x \cdot \cos \left( 10^{-05} \right) + \cos x \cdot \sin \left( 10^{-05} \right)\right) - \sin x}}}{10^{-05}}\]
  6. Applied associate-/l*7.4

    \[\leadsto \color{blue}{\frac{\sqrt[3]{\left(\sin x \cdot \cos \left( 10^{-05} \right) + \cos x \cdot \sin \left( 10^{-05} \right)\right) - \sin x} \cdot \sqrt[3]{\left(\sin x \cdot \cos \left( 10^{-05} \right) + \cos x \cdot \sin \left( 10^{-05} \right)\right) - \sin x}}{\frac{10^{-05}}{\sqrt[3]{\left(\sin x \cdot \cos \left( 10^{-05} \right) + \cos x \cdot \sin \left( 10^{-05} \right)\right) - \sin x}}}}\]
  7. Final simplification7.4

    \[\leadsto \frac{\sqrt[3]{\left(\cos x \cdot \sin \left( 10^{-05} \right) + \cos \left( 10^{-05} \right) \cdot \sin x\right) - \sin x} \cdot \sqrt[3]{\left(\cos x \cdot \sin \left( 10^{-05} \right) + \cos \left( 10^{-05} \right) \cdot \sin x\right) - \sin x}}{\frac{10^{-05}}{\sqrt[3]{\left(\cos x \cdot \sin \left( 10^{-05} \right) + \cos \left( 10^{-05} \right) \cdot \sin x\right) - \sin x}}}\]

Reproduce

herbie shell --seed 1 
(FPCore (x)
  :name "(sin(x+0.00001)-sin(x))/0.00001"
  (/ (- (sin (+ x 1e-05)) (sin x)) 1e-05))