Average Error: 31.7 → 0.5
Time: 4.3s
Precision: binary64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.033986447814389004:\\ \;\;\;\;\left(\sqrt[3]{1 - \cos x} \cdot \frac{\sqrt[3]{1 - \cos x}}{x}\right) \cdot \frac{\sqrt[3]{\frac{1 \cdot 1 - \cos x \cdot \cos x}{1 + \cos x}}}{x}\\ \mathbf{elif}\;x \le 0.031123487775836019:\\ \;\;\;\;{x}^{4} \cdot \frac{1}{720} + \left(\frac{1}{2} + x \cdot \left(x \cdot \frac{-1}{24}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{1 - \cos x}}{x \cdot \frac{x}{\sqrt{1 - \cos x}}}\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.033986447814389004:\\
\;\;\;\;\left(\sqrt[3]{1 - \cos x} \cdot \frac{\sqrt[3]{1 - \cos x}}{x}\right) \cdot \frac{\sqrt[3]{\frac{1 \cdot 1 - \cos x \cdot \cos x}{1 + \cos x}}}{x}\\

\mathbf{elif}\;x \le 0.031123487775836019:\\
\;\;\;\;{x}^{4} \cdot \frac{1}{720} + \left(\frac{1}{2} + x \cdot \left(x \cdot \frac{-1}{24}\right)\right)\\

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

\end{array}
double code(double x) {
	return ((double) (((double) (1.0 - ((double) cos(x)))) / ((double) (x * x))));
}
↓
double code(double x) {
	double VAR;
	if ((x <= -0.033986447814389004)) {
		VAR = ((double) (((double) (((double) cbrt(((double) (1.0 - ((double) cos(x)))))) * ((double) (((double) cbrt(((double) (1.0 - ((double) cos(x)))))) / x)))) * ((double) (((double) cbrt(((double) (((double) (((double) (1.0 * 1.0)) - ((double) (((double) cos(x)) * ((double) cos(x)))))) / ((double) (1.0 + ((double) cos(x)))))))) / x))));
	} else {
		double VAR_1;
		if ((x <= 0.03112348777583602)) {
			VAR_1 = ((double) (((double) (((double) pow(x, 4.0)) * 0.001388888888888889)) + ((double) (0.5 + ((double) (x * ((double) (x * -0.041666666666666664))))))));
		} else {
			VAR_1 = ((double) (((double) sqrt(((double) (1.0 - ((double) cos(x)))))) / ((double) (x * ((double) (x / ((double) sqrt(((double) (1.0 - ((double) cos(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 < -0.033986447814389004

    1. Initial program 1.1

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

    3. Applied add-cube-cbrt1.5

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

    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 flip--0.9

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

    if -0.033986447814389004 < x < 0.031123487775836019

    1. Initial program 62.3

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

    2. Taylor expanded around 0 0.0

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

    3. Simplified0.0

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

    if 0.031123487775836019 < x

    1. Initial program 1.0

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

    3. Applied add-sqr-sqrt1.1

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

    4. Applied associate-/l*1.2

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

    5. Simplified1.2

      \[\leadsto \frac{\sqrt{1 - \cos x}}{\color{blue}{x \cdot \frac{x}{\sqrt{1 - \cos x}}}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.033986447814389004:\\ \;\;\;\;\left(\sqrt[3]{1 - \cos x} \cdot \frac{\sqrt[3]{1 - \cos x}}{x}\right) \cdot \frac{\sqrt[3]{\frac{1 \cdot 1 - \cos x \cdot \cos x}{1 + \cos x}}}{x}\\ \mathbf{elif}\;x \le 0.031123487775836019:\\ \;\;\;\;{x}^{4} \cdot \frac{1}{720} + \left(\frac{1}{2} + x \cdot \left(x \cdot \frac{-1}{24}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{1 - \cos x}}{x \cdot \frac{x}{\sqrt{1 - \cos x}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020179 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))