Average Error: 31.3 → 0.3
Time: 3.9s
Precision: binary64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -0.035212767529547204 \lor \neg \left(x \leq 0.03047569278769371\right):\\ \;\;\;\;\left(\frac{1}{x} - \frac{\cos x}{x}\right) \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;{x}^{4} \cdot 0.001388888888888889 + \left(0.5 + x \cdot \left(x \cdot -0.041666666666666664\right)\right)\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -0.035212767529547204 \lor \neg \left(x \leq 0.03047569278769371\right):\\
\;\;\;\;\left(\frac{1}{x} - \frac{\cos x}{x}\right) \cdot \frac{1}{x}\\

\mathbf{else}:\\
\;\;\;\;{x}^{4} \cdot 0.001388888888888889 + \left(0.5 + x \cdot \left(x \cdot -0.041666666666666664\right)\right)\\

\end{array}
double code(double x) {
	return (((double) (1.0 - ((double) cos(x)))) / ((double) (x * x)));
}
↓
double code(double x) {
	double VAR;
	if (((x <= -0.035212767529547204) || !(x <= 0.03047569278769371))) {
		VAR = ((double) (((double) ((1.0 / x) - (((double) cos(x)) / x))) * (1.0 / 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 < -0.035212767529547204 or 0.030475692787693711 < x

    1. Initial program 1.1

      \[\frac{1 - \cos x}{x \cdot x}\]
    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 div-sub0.6

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

    7. Applied div-inv0.7

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

    if -0.035212767529547204 < x < 0.030475692787693711

    1. Initial program 62.3

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

    2. Taylor expanded around 0 0.0

      \[\leadsto \color{blue}{\left(0.001388888888888889 \cdot {x}^{4} + 0.5\right) - 0.041666666666666664 \cdot {x}^{2}}\]

    3. Simplified0.0

      \[\leadsto \color{blue}{{x}^{4} \cdot 0.001388888888888889 + \left(0.5 + x \cdot \left(x \cdot -0.041666666666666664\right)\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.035212767529547204 \lor \neg \left(x \leq 0.03047569278769371\right):\\ \;\;\;\;\left(\frac{1}{x} - \frac{\cos x}{x}\right) \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;{x}^{4} \cdot 0.001388888888888889 + \left(0.5 + x \cdot \left(x \cdot -0.041666666666666664\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020196 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))