Average Error: 31.0 → 0.5
Time: 4.1s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.0304258770789509851:\\ \;\;\;\;\frac{1}{\frac{x \cdot x}{1 - \cos x}}\\ \mathbf{elif}\;x \le 0.029985657879302199:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{1 - \cos x}}{x} \cdot \left(\left|\sqrt[3]{1 - \cos x}\right| \cdot \frac{\sqrt{\sqrt[3]{\frac{1 \cdot 1 - \cos x \cdot \cos x}{1 + \cos x}}}}{x}\right)\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.0304258770789509851:\\
\;\;\;\;\frac{1}{\frac{x \cdot x}{1 - \cos x}}\\

\mathbf{elif}\;x \le 0.029985657879302199:\\
\;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\sqrt{1 - \cos x}}{x} \cdot \left(\left|\sqrt[3]{1 - \cos x}\right| \cdot \frac{\sqrt{\sqrt[3]{\frac{1 \cdot 1 - \cos x \cdot \cos x}{1 + \cos x}}}}{x}\right)\\

\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 <= -0.030425877078950985)) {
		VAR = ((double) (1.0 / ((double) (((double) (x * x)) / ((double) (1.0 - ((double) cos(x))))))));
	} else {
		double VAR_1;
		if ((x <= 0.0299856578793022)) {
			VAR_1 = ((double) (((double) (((double) (0.001388888888888889 * ((double) pow(x, 4.0)))) + 0.5)) - ((double) (0.041666666666666664 * ((double) pow(x, 2.0))))));
		} else {
			VAR_1 = ((double) (((double) (((double) sqrt(((double) (1.0 - ((double) cos(x)))))) / x)) * ((double) (((double) fabs(((double) cbrt(((double) (1.0 - ((double) cos(x)))))))) * ((double) (((double) sqrt(((double) cbrt(((double) (((double) (((double) (1.0 * 1.0)) - ((double) (((double) cos(x)) * ((double) cos(x)))))) / ((double) (1.0 + ((double) cos(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 < -0.030425877078950985

    1. Initial program 1.0

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

    3. Applied clear-num1.1

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

    if -0.030425877078950985 < x < 0.0299856578793022

    1. Initial program 62.3

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

    2. Taylor expanded around 0 0.0

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

    if 0.0299856578793022 < x

    1. Initial program 0.9

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

    3. Applied add-sqr-sqrt1.0

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

    4. Applied times-frac0.6

      \[\leadsto \color{blue}{\frac{\sqrt{1 - \cos x}}{x} \cdot \frac{\sqrt{1 - \cos x}}{x}}\]
    5. Using strategy rm

    6. Applied *-un-lft-identity0.6

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

    7. Applied add-cube-cbrt0.7

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

    8. Applied sqrt-prod0.7

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

    9. Applied times-frac0.7

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

    10. Simplified0.7

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

    12. Applied flip--0.7

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.0304258770789509851:\\ \;\;\;\;\frac{1}{\frac{x \cdot x}{1 - \cos x}}\\ \mathbf{elif}\;x \le 0.029985657879302199:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{1 - \cos x}}{x} \cdot \left(\left|\sqrt[3]{1 - \cos x}\right| \cdot \frac{\sqrt{\sqrt[3]{\frac{1 \cdot 1 - \cos x \cdot \cos x}{1 + \cos x}}}}{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020131 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))