Average Error: 32.1 → 0.6
Time: 3.7s
Precision: binary64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.029842379482625983:\\ \;\;\;\;\frac{1}{x \cdot \frac{x}{1 - \cos x}}\\ \mathbf{elif}\;x \le 0.0366604219404516665:\\ \;\;\;\;\left({x}^{4} \cdot \frac{1}{720} + \frac{1}{2}\right) + x \cdot \left(x \cdot \frac{-1}{24}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot x} - \frac{\cos x}{x \cdot x}\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.029842379482625983:\\
\;\;\;\;\frac{1}{x \cdot \frac{x}{1 - \cos x}}\\

\mathbf{elif}\;x \le 0.0366604219404516665:\\
\;\;\;\;\left({x}^{4} \cdot \frac{1}{720} + \frac{1}{2}\right) + x \cdot \left(x \cdot \frac{-1}{24}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{x \cdot x} - \frac{\cos x}{x \cdot 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 <= -0.029842379482625983)) {
		VAR = ((double) (1.0 / ((double) (x * ((double) (x / ((double) (1.0 - ((double) cos(x))))))))));
	} else {
		double VAR_1;
		if ((x <= 0.036660421940451667)) {
			VAR_1 = ((double) (((double) (((double) (((double) pow(x, 4.0)) * 0.001388888888888889)) + 0.5)) + ((double) (x * ((double) (x * -0.041666666666666664))))));
		} else {
			VAR_1 = ((double) (((double) (1.0 / ((double) (x * x)))) - ((double) (((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 < -0.029842379482625983

    1. Initial program 1.0

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

    3. Applied clear-num1.0

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

    4. Simplified1.0

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

    if -0.029842379482625983 < x < 0.0366604219404516665

    1. Initial program 62.4

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

    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

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

    5. Applied associate-+r+0.0

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

    if 0.0366604219404516665 < x

    1. Initial program 1.2

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

    3. Applied div-sub1.2

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.029842379482625983:\\ \;\;\;\;\frac{1}{x \cdot \frac{x}{1 - \cos x}}\\ \mathbf{elif}\;x \le 0.0366604219404516665:\\ \;\;\;\;\left({x}^{4} \cdot \frac{1}{720} + \frac{1}{2}\right) + x \cdot \left(x \cdot \frac{-1}{24}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot x} - \frac{\cos x}{x \cdot x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020184 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))