Average Error: 30.5 → 0.3
Time: 26.8s
Precision: 64
$\frac{1 - \frac{\sin x}{x}}{{x}^{2}}$
$\begin{array}{l} \mathbf{if}\;x \le -0.009272911760413088:\\ \;\;\;\;\left(\sqrt[3]{1 - \frac{\sin x}{x}} \cdot \frac{\sqrt[3]{1 - \frac{\sin x}{x}}}{x}\right) \cdot \frac{\sqrt[3]{1 - \frac{\sin x}{x}}}{x}\\ \mathbf{elif}\;x \le 0.009462403866380574:\\ \;\;\;\;\frac{1}{6} + \left(x \cdot x\right) \cdot \left(\left(\left(x \cdot \left(\frac{1}{5040} \cdot x\right)\right)\right) - \frac{1}{120}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{\sin x}{x}}{x \cdot x}\\ \end{array}$
\frac{1 - \frac{\sin x}{x}}{{x}^{2}}
\begin{array}{l}
\mathbf{if}\;x \le -0.009272911760413088:\\
\;\;\;\;\left(\sqrt[3]{1 - \frac{\sin x}{x}} \cdot \frac{\sqrt[3]{1 - \frac{\sin x}{x}}}{x}\right) \cdot \frac{\sqrt[3]{1 - \frac{\sin x}{x}}}{x}\\

\mathbf{elif}\;x \le 0.009462403866380574:\\
\;\;\;\;\frac{1}{6} + \left(x \cdot x\right) \cdot \left(\left(\left(x \cdot \left(\frac{1}{5040} \cdot x\right)\right)\right) - \frac{1}{120}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{1 - \frac{\sin x}{x}}{x \cdot x}\\

\end{array}
double f(double x) {
double r14531001 = 1.0;
double r14531002 = x;
double r14531003 = sin(r14531002);
double r14531004 = r14531003 / r14531002;
double r14531005 = r14531001 - r14531004;
double r14531006 = 2.0;
double r14531007 = pow(r14531002, r14531006);
double r14531008 = r14531005 / r14531007;
return r14531008;
}


double f(double x) {
double r14531009 = x;
double r14531010 = -0.009272911760413088;
bool r14531011 = r14531009 <= r14531010;
double r14531012 = 1.0;
double r14531013 = sin(r14531009);
double r14531014 = r14531013 / r14531009;
double r14531015 = r14531012 - r14531014;
double r14531016 = cbrt(r14531015);
double r14531017 = r14531016 / r14531009;
double r14531018 = r14531016 * r14531017;
double r14531019 = r14531018 * r14531017;
double r14531020 = 0.009462403866380574;
bool r14531021 = r14531009 <= r14531020;
double r14531022 = 0.16666666666666666;
double r14531023 = r14531009 * r14531009;
double r14531024 = 0.0001984126984126984;
double r14531025 = r14531024 * r14531009;
double r14531026 = r14531009 * r14531025;
double r14531027 = /* ERROR: no posit support in C */;
double r14531028 = /* ERROR: no posit support in C */;
double r14531029 = 0.008333333333333333;
double r14531030 = r14531028 - r14531029;
double r14531031 = r14531023 * r14531030;
double r14531032 = r14531022 + r14531031;
double r14531033 = r14531015 / r14531023;
double r14531034 = r14531021 ? r14531032 : r14531033;
double r14531035 = r14531011 ? r14531019 : r14531034;
return r14531035;
}



# Derivation

1. Split input into 3 regimes
2. ## if x < -0.009272911760413088

1. Initial program 0.8

$\frac{1 - \frac{\sin x}{x}}{{x}^{2}}$
2. Simplified0.8

$\leadsto \color{blue}{\frac{1 - \frac{\sin x}{x}}{x \cdot x}}$
3. Using strategy rm

$\leadsto \frac{\color{blue}{\left(\sqrt[3]{1 - \frac{\sin x}{x}} \cdot \sqrt[3]{1 - \frac{\sin x}{x}}\right) \cdot \sqrt[3]{1 - \frac{\sin x}{x}}}}{x \cdot x}$
5. Applied times-frac0.4

$\leadsto \color{blue}{\frac{\sqrt[3]{1 - \frac{\sin x}{x}} \cdot \sqrt[3]{1 - \frac{\sin x}{x}}}{x} \cdot \frac{\sqrt[3]{1 - \frac{\sin x}{x}}}{x}}$
6. Simplified0.4

$\leadsto \color{blue}{\left(\frac{\sqrt[3]{1 - \frac{\sin x}{x}}}{x} \cdot \sqrt[3]{1 - \frac{\sin x}{x}}\right)} \cdot \frac{\sqrt[3]{1 - \frac{\sin x}{x}}}{x}$

## if -0.009272911760413088 < x < 0.009462403866380574

1. Initial program 61.3

$\frac{1 - \frac{\sin x}{x}}{{x}^{2}}$
2. Simplified61.3

$\leadsto \color{blue}{\frac{1 - \frac{\sin x}{x}}{x \cdot x}}$
3. Taylor expanded around 0 0.0

$\leadsto \color{blue}{\left(\frac{1}{5040} \cdot {x}^{4} + \frac{1}{6}\right) - \frac{1}{120} \cdot {x}^{2}}$
4. Simplified0.0

$\leadsto \color{blue}{\frac{1}{6} + \left(x \cdot x\right) \cdot \left(\left(x \cdot \frac{1}{5040}\right) \cdot x - \frac{1}{120}\right)}$
5. Using strategy rm
6. Applied insert-posit160.1

$\leadsto \frac{1}{6} + \left(x \cdot x\right) \cdot \left(\color{blue}{\left(\left(\left(x \cdot \frac{1}{5040}\right) \cdot x\right)\right)} - \frac{1}{120}\right)$

## if 0.009462403866380574 < x

1. Initial program 0.9

$\frac{1 - \frac{\sin x}{x}}{{x}^{2}}$
2. Simplified0.9

$\leadsto \color{blue}{\frac{1 - \frac{\sin x}{x}}{x \cdot x}}$
3. Taylor expanded around -inf 0.9

$\leadsto \frac{\color{blue}{1 - \frac{\sin x}{x}}}{x \cdot x}$
3. Recombined 3 regimes into one program.
4. Final simplification0.3

$\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.009272911760413088:\\ \;\;\;\;\left(\sqrt[3]{1 - \frac{\sin x}{x}} \cdot \frac{\sqrt[3]{1 - \frac{\sin x}{x}}}{x}\right) \cdot \frac{\sqrt[3]{1 - \frac{\sin x}{x}}}{x}\\ \mathbf{elif}\;x \le 0.009462403866380574:\\ \;\;\;\;\frac{1}{6} + \left(x \cdot x\right) \cdot \left(\left(\left(x \cdot \left(\frac{1}{5040} \cdot x\right)\right)\right) - \frac{1}{120}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{\sin x}{x}}{x \cdot x}\\ \end{array}$

# Reproduce

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