Average Error: 31.8 → 0.4
Time: 4.2s
Precision: binary64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -0.03510726007850325:\\ \;\;\;\;\frac{1}{x} \cdot \frac{\frac{{1}^{3} - \sqrt[3]{{\left({\left(\cos x\right)}^{3}\right)}^{3}}}{1 \cdot 1 + \cos x \cdot \left(1 + \cos x\right)}}{x}\\ \mathbf{elif}\;x \leq 0.03609453174808888:\\ \;\;\;\;{x}^{4} \cdot 0.001388888888888889 + \left(0.5 - \left(x \cdot x\right) \cdot 0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\left(1 \cdot 1 + \cos x \cdot \left(1 + \cos x\right)\right) \cdot \left(x \cdot x\right)}\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -0.03510726007850325:\\
\;\;\;\;\frac{1}{x} \cdot \frac{\frac{{1}^{3} - \sqrt[3]{{\left({\left(\cos x\right)}^{3}\right)}^{3}}}{1 \cdot 1 + \cos x \cdot \left(1 + \cos x\right)}}{x}\\

\mathbf{elif}\;x \leq 0.03609453174808888:\\
\;\;\;\;{x}^{4} \cdot 0.001388888888888889 + \left(0.5 - \left(x \cdot x\right) \cdot 0.041666666666666664\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\left(1 \cdot 1 + \cos x \cdot \left(1 + \cos x\right)\right) \cdot \left(x \cdot x\right)}\\

\end{array}
(FPCore (x) :precision binary64 (/ (- 1.0 (cos x)) (* x x)))
↓
(FPCore (x)
 :precision binary64
 (if (<= x -0.03510726007850325)
   (*
    (/ 1.0 x)
    (/
     (/
      (- (pow 1.0 3.0) (cbrt (pow (pow (cos x) 3.0) 3.0)))
      (+ (* 1.0 1.0) (* (cos x) (+ 1.0 (cos x)))))
     x))
   (if (<= x 0.03609453174808888)
     (+
      (* (pow x 4.0) 0.001388888888888889)
      (- 0.5 (* (* x x) 0.041666666666666664)))
     (/
      (- (pow 1.0 3.0) (pow (cos x) 3.0))
      (* (+ (* 1.0 1.0) (* (cos x) (+ 1.0 (cos x)))) (* x x))))))
double code(double x) {
	return (((double) (1.0 - ((double) cos(x)))) / ((double) (x * x)));
}
↓
double code(double x) {
	double tmp;
	if ((x <= -0.03510726007850325)) {
		tmp = ((double) ((1.0 / x) * ((((double) (((double) pow(1.0, 3.0)) - ((double) cbrt(((double) pow(((double) pow(((double) cos(x)), 3.0)), 3.0)))))) / ((double) (((double) (1.0 * 1.0)) + ((double) (((double) cos(x)) * ((double) (1.0 + ((double) cos(x))))))))) / x)));
	} else {
		double tmp_1;
		if ((x <= 0.03609453174808888)) {
			tmp_1 = ((double) (((double) (((double) pow(x, 4.0)) * 0.001388888888888889)) + ((double) (0.5 - ((double) (((double) (x * x)) * 0.041666666666666664))))));
		} else {
			tmp_1 = (((double) (((double) pow(1.0, 3.0)) - ((double) pow(((double) cos(x)), 3.0)))) / ((double) (((double) (((double) (1.0 * 1.0)) + ((double) (((double) cos(x)) * ((double) (1.0 + ((double) cos(x)))))))) * ((double) (x * x)))));
		}
		tmp = tmp_1;
	}
	return tmp;
}

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 < -0.035107260078503247

    1. Initial program 1.2

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

    3. Applied *-un-lft-identity_binary641.2

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

    4. Applied times-frac_binary640.5

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

    6. Applied flip3--_binary640.5

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

    7. Simplified0.5

      \[\leadsto \frac{1}{x} \cdot \frac{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\color{blue}{1 \cdot 1 + \cos x \cdot \left(1 + \cos x\right)}}}{x}\]
    8. Using strategy rm

    9. Applied add-cbrt-cube_binary640.6

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

    10. Simplified0.6

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

    if -0.035107260078503247 < x < 0.036094531748088882

    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

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

    if 0.036094531748088882 < x

    1. Initial program 1.0

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

    3. Applied flip3--_binary641.1

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

    4. Applied associate-/l/_binary641.2

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

    5. Simplified1.2

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.03510726007850325:\\ \;\;\;\;\frac{1}{x} \cdot \frac{\frac{{1}^{3} - \sqrt[3]{{\left({\left(\cos x\right)}^{3}\right)}^{3}}}{1 \cdot 1 + \cos x \cdot \left(1 + \cos x\right)}}{x}\\ \mathbf{elif}\;x \leq 0.03609453174808888:\\ \;\;\;\;{x}^{4} \cdot 0.001388888888888889 + \left(0.5 - \left(x \cdot x\right) \cdot 0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\left(1 \cdot 1 + \cos x \cdot \left(1 + \cos x\right)\right) \cdot \left(x \cdot x\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020204 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))