Average Error: 31.9 → 0.5
Time: 4.6s
Precision: binary64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.034533322447672377:\\ \;\;\;\;\left(\sqrt[3]{\log \left(e^{1 - \cos x}\right)} \cdot \frac{\sqrt[3]{1 - \cos x}}{x}\right) \cdot \frac{\sqrt[3]{e^{\log \left(1 - \cos x\right)}}}{x}\\ \mathbf{elif}\;x \le 0.0342323272118313024:\\ \;\;\;\;\frac{1}{2} + \left({x}^{4} \cdot \frac{1}{720} + x \cdot \left(x \cdot \frac{-1}{24}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{1 \cdot 1 + \cos x \cdot \left(1 + \cos x\right)}}{x \cdot x}\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.034533322447672377:\\
\;\;\;\;\left(\sqrt[3]{\log \left(e^{1 - \cos x}\right)} \cdot \frac{\sqrt[3]{1 - \cos x}}{x}\right) \cdot \frac{\sqrt[3]{e^{\log \left(1 - \cos x\right)}}}{x}\\

\mathbf{elif}\;x \le 0.0342323272118313024:\\
\;\;\;\;\frac{1}{2} + \left({x}^{4} \cdot \frac{1}{720} + x \cdot \left(x \cdot \frac{-1}{24}\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{1 \cdot 1 + \cos x \cdot \left(1 + \cos x\right)}}{x \cdot x}\\

\end{array}
double code(double x) {
	return (((double) (1.0 - ((double) cos(x)))) / ((double) (x * x)));
}
↓
double code(double x) {
	double VAR;
	if ((x <= -0.03453332244767238)) {
		VAR = ((double) (((double) (((double) cbrt(((double) log(((double) exp(((double) (1.0 - ((double) cos(x)))))))))) * (((double) cbrt(((double) (1.0 - ((double) cos(x)))))) / x))) * (((double) cbrt(((double) exp(((double) log(((double) (1.0 - ((double) cos(x)))))))))) / x)));
	} else {
		double VAR_1;
		if ((x <= 0.0342323272118313)) {
			VAR_1 = ((double) (0.5 + ((double) (((double) (((double) pow(x, 4.0)) * 0.001388888888888889)) + ((double) (x * ((double) (x * -0.041666666666666664))))))));
		} else {
			VAR_1 = ((((double) (((double) pow(1.0, 3.0)) - ((double) pow(((double) cos(x)), 3.0)))) / ((double) (((double) (1.0 * 1.0)) + ((double) (((double) cos(x)) * ((double) (1.0 + ((double) cos(x))))))))) / ((double) (x * x)));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if x < -0.034533322447672377

    1. Initial program 1.0

      \[\frac{1 - \cos x}{x \cdot x}\]
    2. Using strategy rm

    3. Applied add-cube-cbrt1.4

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

    4. Applied times-frac0.8

      \[\leadsto \color{blue}{\frac{\sqrt[3]{1 - \cos x} \cdot \sqrt[3]{1 - \cos x}}{x} \cdot \frac{\sqrt[3]{1 - \cos x}}{x}}\]

    5. Simplified0.8

      \[\leadsto \color{blue}{\left(\sqrt[3]{1 - \cos x} \cdot \frac{\sqrt[3]{1 - \cos x}}{x}\right)} \cdot \frac{\sqrt[3]{1 - \cos x}}{x}\]
    6. Using strategy rm

    7. Applied add-log-exp0.8

      \[\leadsto \left(\sqrt[3]{1 - \color{blue}{\log \left(e^{\cos x}\right)}} \cdot \frac{\sqrt[3]{1 - \cos x}}{x}\right) \cdot \frac{\sqrt[3]{1 - \cos x}}{x}\]

    8. Applied add-log-exp0.8

      \[\leadsto \left(\sqrt[3]{\color{blue}{\log \left(e^{1}\right)} - \log \left(e^{\cos x}\right)} \cdot \frac{\sqrt[3]{1 - \cos x}}{x}\right) \cdot \frac{\sqrt[3]{1 - \cos x}}{x}\]

    9. Applied diff-log0.9

      \[\leadsto \left(\sqrt[3]{\color{blue}{\log \left(\frac{e^{1}}{e^{\cos x}}\right)}} \cdot \frac{\sqrt[3]{1 - \cos x}}{x}\right) \cdot \frac{\sqrt[3]{1 - \cos x}}{x}\]

    10. Simplified0.8

      \[\leadsto \left(\sqrt[3]{\log \color{blue}{\left(e^{1 - \cos x}\right)}} \cdot \frac{\sqrt[3]{1 - \cos x}}{x}\right) \cdot \frac{\sqrt[3]{1 - \cos x}}{x}\]
    11. Using strategy rm

    12. Applied add-exp-log0.8

      \[\leadsto \left(\sqrt[3]{\log \left(e^{1 - \cos x}\right)} \cdot \frac{\sqrt[3]{1 - \cos x}}{x}\right) \cdot \frac{\sqrt[3]{\color{blue}{e^{\log \left(1 - \cos x\right)}}}}{x}\]

    if -0.034533322447672377 < x < 0.0342323272118313024

    1. Initial program 62.1

      \[\frac{1 - \cos x}{x \cdot x}\]

    2. Taylor expanded around 0 0.0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + \frac{1}{720} \cdot {x}^{4}\right) - \frac{1}{24} \cdot {x}^{2}}\]

    3. Simplified0.0

      \[\leadsto \color{blue}{\frac{1}{2} + \left({x}^{4} \cdot \frac{1}{720} + x \cdot \left(x \cdot \frac{-1}{24}\right)\right)}\]

    if 0.0342323272118313024 < x

    1. Initial program 0.9

      \[\frac{1 - \cos x}{x \cdot x}\]
    2. Using strategy rm

    3. Applied flip3--1.0

      \[\leadsto \frac{\color{blue}{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{1 \cdot 1 + \left(\cos x \cdot \cos x + 1 \cdot \cos x\right)}}}{x \cdot x}\]

    4. Simplified1.0

      \[\leadsto \frac{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\color{blue}{1 \cdot 1 + \cos x \cdot \left(1 + \cos x\right)}}}{x \cdot x}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.034533322447672377:\\ \;\;\;\;\left(\sqrt[3]{\log \left(e^{1 - \cos x}\right)} \cdot \frac{\sqrt[3]{1 - \cos x}}{x}\right) \cdot \frac{\sqrt[3]{e^{\log \left(1 - \cos x\right)}}}{x}\\ \mathbf{elif}\;x \le 0.0342323272118313024:\\ \;\;\;\;\frac{1}{2} + \left({x}^{4} \cdot \frac{1}{720} + x \cdot \left(x \cdot \frac{-1}{24}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{1 \cdot 1 + \cos x \cdot \left(1 + \cos x\right)}}{x \cdot x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020182 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))