Average Error: 31.4 → 0.2
Time: 4.8s
Precision: binary64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.02996085913414137 \lor \neg \left(x \le 0.0329094092307169114\right):\\ \;\;\;\;\frac{\frac{1}{\frac{x}{1 - \cos x}}}{x}\\ \mathbf{else}:\\ \;\;\;\;{x}^{4} \cdot \frac{1}{720} + \left(\frac{1}{2} - x \cdot \left(x \cdot \frac{1}{24}\right)\right)\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.02996085913414137 \lor \neg \left(x \le 0.0329094092307169114\right):\\
\;\;\;\;\frac{\frac{1}{\frac{x}{1 - \cos x}}}{x}\\

\mathbf{else}:\\
\;\;\;\;{x}^{4} \cdot \frac{1}{720} + \left(\frac{1}{2} - x \cdot \left(x \cdot \frac{1}{24}\right)\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 <= -0.02996085913414137) || !(x <= 0.03290940923071691))) {
		VAR = ((double) (((double) (1.0 / ((double) (x / ((double) (1.0 - ((double) cos(x)))))))) / x));
	} else {
		VAR = ((double) (((double) (((double) pow(x, 4.0)) * 0.001388888888888889)) + ((double) (0.5 - ((double) (x * ((double) (x * 0.041666666666666664))))))));
	}
	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 2 regimes

  2. if x < -0.02996085913414137 or 0.0329094092307169114 < x

    1. Initial program 1.0

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

    3. Applied associate-/r*0.5

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

    5. Applied clear-num0.5

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

    if -0.02996085913414137 < x < 0.0329094092307169114

    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

      \[\leadsto \color{blue}{{x}^{4} \cdot \frac{1}{720} + \left(\frac{1}{2} - x \cdot \left(x \cdot \frac{1}{24}\right)\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.02996085913414137 \lor \neg \left(x \le 0.0329094092307169114\right):\\ \;\;\;\;\frac{\frac{1}{\frac{x}{1 - \cos x}}}{x}\\ \mathbf{else}:\\ \;\;\;\;{x}^{4} \cdot \frac{1}{720} + \left(\frac{1}{2} - x \cdot \left(x \cdot \frac{1}{24}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020181 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))