Average Error: 31.1 → 0.3
Time: 8.5s
Precision: binary64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.0282202505047884043:\\ \;\;\;\;\frac{1}{x} \cdot \frac{\left(1 \cdot 1 - \frac{1}{2}\right) - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{x \cdot \frac{{1}^{3} + {\left(\cos x\right)}^{3}}{\cos x \cdot \left(\cos x - 1\right) + 1 \cdot 1}}\\ \mathbf{elif}\;x \le 0.0340362813457630239:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{1 - \cos x}{x}\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.0282202505047884043:\\
\;\;\;\;\frac{1}{x} \cdot \frac{\left(1 \cdot 1 - \frac{1}{2}\right) - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{x \cdot \frac{{1}^{3} + {\left(\cos x\right)}^{3}}{\cos x \cdot \left(\cos x - 1\right) + 1 \cdot 1}}\\

\mathbf{elif}\;x \le 0.0340362813457630239:\\
\;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{x} \cdot \frac{1 - \cos x}{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 <= -0.028220250504788404)) {
		VAR = ((double) (((double) (1.0 / x)) * ((double) (((double) (((double) (((double) (1.0 * 1.0)) - 0.5)) - ((double) (0.5 * ((double) cos(((double) (2.0 * x)))))))) / ((double) (x * ((double) (((double) (((double) pow(1.0, 3.0)) + ((double) pow(((double) cos(x)), 3.0)))) / ((double) (((double) (((double) cos(x)) * ((double) (((double) cos(x)) - 1.0)))) + ((double) (1.0 * 1.0))))))))))));
	} else {
		double VAR_1;
		if ((x <= 0.034036281345763024)) {
			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) (1.0 / x)) * ((double) (((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 < -0.0282202505047884043

    1. Initial program 1.0

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

    3. Applied *-un-lft-identity1.0

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

    4. Applied times-frac0.5

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

    6. Applied flip--0.7

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

    7. Applied associate-/l/0.7

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

    9. Applied sqr-cos0.7

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

    10. Applied associate--r+0.7

      \[\leadsto \frac{1}{x} \cdot \frac{\color{blue}{\left(1 \cdot 1 - \frac{1}{2}\right) - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}}{x \cdot \left(1 + \cos x\right)}\]
    11. Using strategy rm

    12. Applied flip3-+0.7

      \[\leadsto \frac{1}{x} \cdot \frac{\left(1 \cdot 1 - \frac{1}{2}\right) - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{x \cdot \color{blue}{\frac{{1}^{3} + {\left(\cos x\right)}^{3}}{1 \cdot 1 + \left(\cos x \cdot \cos x - 1 \cdot \cos x\right)}}}\]

    13. Simplified0.7

      \[\leadsto \frac{1}{x} \cdot \frac{\left(1 \cdot 1 - \frac{1}{2}\right) - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{x \cdot \frac{{1}^{3} + {\left(\cos x\right)}^{3}}{\color{blue}{\cos x \cdot \left(\cos x - 1\right) + 1 \cdot 1}}}\]

    if -0.0282202505047884043 < x < 0.0340362813457630239

    1. Initial program 62.2

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

    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 0.0340362813457630239 < x

    1. Initial program 1.1

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

    3. Applied *-un-lft-identity1.1

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

    4. Applied times-frac0.5

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.0282202505047884043:\\ \;\;\;\;\frac{1}{x} \cdot \frac{\left(1 \cdot 1 - \frac{1}{2}\right) - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{x \cdot \frac{{1}^{3} + {\left(\cos x\right)}^{3}}{\cos x \cdot \left(\cos x - 1\right) + 1 \cdot 1}}\\ \mathbf{elif}\;x \le 0.0340362813457630239:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{1 - \cos x}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020163 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))