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;
}



# Try it out

Results

 In Out
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

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