Average Error: 54.3 → 0.4
Time: 22.1s
Precision: 64
$\frac{\left(1 - \frac{{x}^{2}}{2}\right) - \cos x}{{x}^{4}}$
$\begin{array}{l} \mathbf{if}\;x \le -0.12153203655906179:\\ \;\;\;\;\frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)} - \left(\frac{\frac{1}{2}}{x \cdot x} + \frac{\frac{\cos x}{x \cdot x}}{x \cdot x}\right)\\ \mathbf{elif}\;x \le 0.09809591180606583:\\ \;\;\;\;\left(\frac{-1}{40320} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) + \frac{x \cdot x}{720}\right) - \frac{1}{24}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)} - \left(\frac{\frac{1}{2}}{x \cdot x} + \frac{\frac{\cos x}{x \cdot x}}{x \cdot x}\right)\\ \end{array}$
\frac{\left(1 - \frac{{x}^{2}}{2}\right) - \cos x}{{x}^{4}}
\begin{array}{l}
\mathbf{if}\;x \le -0.12153203655906179:\\
\;\;\;\;\frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)} - \left(\frac{\frac{1}{2}}{x \cdot x} + \frac{\frac{\cos x}{x \cdot x}}{x \cdot x}\right)\\

\mathbf{elif}\;x \le 0.09809591180606583:\\
\;\;\;\;\left(\frac{-1}{40320} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) + \frac{x \cdot x}{720}\right) - \frac{1}{24}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)} - \left(\frac{\frac{1}{2}}{x \cdot x} + \frac{\frac{\cos x}{x \cdot x}}{x \cdot x}\right)\\

\end{array}
double f(double x) {
double r12610683 = 1.0;
double r12610684 = x;
double r12610685 = 2.0;
double r12610686 = pow(r12610684, r12610685);
double r12610687 = r12610686 / r12610685;
double r12610688 = r12610683 - r12610687;
double r12610689 = cos(r12610684);
double r12610690 = r12610688 - r12610689;
double r12610691 = 4.0;
double r12610692 = pow(r12610684, r12610691);
double r12610693 = r12610690 / r12610692;
return r12610693;
}


double f(double x) {
double r12610694 = x;
double r12610695 = -0.12153203655906179;
bool r12610696 = r12610694 <= r12610695;
double r12610697 = 1.0;
double r12610698 = r12610694 * r12610694;
double r12610699 = r12610698 * r12610698;
double r12610700 = r12610697 / r12610699;
double r12610701 = 0.5;
double r12610702 = r12610701 / r12610698;
double r12610703 = cos(r12610694);
double r12610704 = r12610703 / r12610698;
double r12610705 = r12610704 / r12610698;
double r12610706 = r12610702 + r12610705;
double r12610707 = r12610700 - r12610706;
double r12610708 = 0.09809591180606583;
bool r12610709 = r12610694 <= r12610708;
double r12610710 = -2.48015873015873e-05;
double r12610711 = r12610710 * r12610699;
double r12610712 = 720.0;
double r12610713 = r12610698 / r12610712;
double r12610714 = r12610711 + r12610713;
double r12610715 = 0.041666666666666664;
double r12610716 = r12610714 - r12610715;
double r12610717 = r12610709 ? r12610716 : r12610707;
double r12610718 = r12610696 ? r12610707 : r12610717;
return r12610718;
}



# Try it out

Results

 In Out
Enter valid numbers for all inputs

# Derivation

1. Split input into 2 regimes
2. ## if x < -0.12153203655906179 or 0.09809591180606583 < x

1. Initial program 45.9

$\frac{\left(1 - \frac{{x}^{2}}{2}\right) - \cos x}{{x}^{4}}$
2. Simplified45.9

$\leadsto \color{blue}{\frac{\left(1 - \frac{x \cdot x}{2}\right) - \cos x}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}}$
3. Taylor expanded around -inf 0.8

$\leadsto \color{blue}{\frac{1}{{x}^{4}} - \left(\frac{1}{2} \cdot \frac{1}{{x}^{2}} + \frac{\cos x}{{x}^{4}}\right)}$
4. Simplified0.8

$\leadsto \color{blue}{\frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)} - \left(\frac{\frac{1}{2}}{x \cdot x} + \frac{\frac{\cos x}{x \cdot x}}{x \cdot x}\right)}$

## if -0.12153203655906179 < x < 0.09809591180606583

1. Initial program 62.8

$\frac{\left(1 - \frac{{x}^{2}}{2}\right) - \cos x}{{x}^{4}}$
2. Simplified62.8

$\leadsto \color{blue}{\frac{\left(1 - \frac{x \cdot x}{2}\right) - \cos x}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}}$
3. Taylor expanded around 0 0.0

$\leadsto \color{blue}{\frac{1}{720} \cdot {x}^{2} - \left(\frac{1}{40320} \cdot {x}^{4} + \frac{1}{24}\right)}$
4. Simplified0.1

$\leadsto \color{blue}{\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{-1}{40320} + \frac{x \cdot x}{720}\right) - \frac{1}{24}}$
3. Recombined 2 regimes into one program.
4. Final simplification0.4

$\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.12153203655906179:\\ \;\;\;\;\frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)} - \left(\frac{\frac{1}{2}}{x \cdot x} + \frac{\frac{\cos x}{x \cdot x}}{x \cdot x}\right)\\ \mathbf{elif}\;x \le 0.09809591180606583:\\ \;\;\;\;\left(\frac{-1}{40320} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) + \frac{x \cdot x}{720}\right) - \frac{1}{24}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)} - \left(\frac{\frac{1}{2}}{x \cdot x} + \frac{\frac{\cos x}{x \cdot x}}{x \cdot x}\right)\\ \end{array}$

# Reproduce

herbie shell --seed 1
(FPCore (x)
:name "(1-x^2/2-cos(x))/x^4"
(/ (- (- 1 (/ (pow x 2) 2)) (cos x)) (pow x 4)))