Average Error: 31.6 → 0.4
Time: 4.7s
Precision: binary64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -0.03375314859115401:\\ \;\;\;\;\frac{1 \cdot 1 - \cos x \cdot \cos x}{x \cdot \left(x \cdot \left(1 + \cos x\right)\right)}\\ \mathbf{elif}\;x \leq 0.03592992967463037:\\ \;\;\;\;{x}^{4} \cdot 0.001388888888888889 + \left(0.5 + x \cdot \left(x \cdot -0.041666666666666664\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{\cos x}{x}}{x}\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -0.03375314859115401:\\
\;\;\;\;\frac{1 \cdot 1 - \cos x \cdot \cos x}{x \cdot \left(x \cdot \left(1 + \cos x\right)\right)}\\

\mathbf{elif}\;x \leq 0.03592992967463037:\\
\;\;\;\;{x}^{4} \cdot 0.001388888888888889 + \left(0.5 + x \cdot \left(x \cdot -0.041666666666666664\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{1}{x} - \frac{\cos x}{x}}{x}\\

\end{array}
(FPCore (x) :precision binary64 (/ (- 1.0 (cos x)) (* x x)))
↓
(FPCore (x)
 :precision binary64
 (if (<= x -0.03375314859115401)
   (/ (- (* 1.0 1.0) (* (cos x) (cos x))) (* x (* x (+ 1.0 (cos x)))))
   (if (<= x 0.03592992967463037)
     (+
      (* (pow x 4.0) 0.001388888888888889)
      (+ 0.5 (* x (* x -0.041666666666666664))))
     (/ (- (/ 1.0 x) (/ (cos x) x)) x))))
double code(double x) {
	return (((double) (1.0 - ((double) cos(x)))) / ((double) (x * x)));
}
↓
double code(double x) {
	double VAR;
	if ((x <= -0.03375314859115401)) {
		VAR = (((double) (((double) (1.0 * 1.0)) - ((double) (((double) cos(x)) * ((double) cos(x)))))) / ((double) (x * ((double) (x * ((double) (1.0 + ((double) cos(x)))))))));
	} else {
		double VAR_1;
		if ((x <= 0.03592992967463037)) {
			VAR_1 = ((double) (((double) (((double) pow(x, 4.0)) * 0.001388888888888889)) + ((double) (0.5 + ((double) (x * ((double) (x * -0.041666666666666664))))))));
		} else {
			VAR_1 = (((double) ((1.0 / x) - (((double) cos(x)) / 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 < -0.033753148591154011

    1. Initial program 0.9

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

    3. Applied flip--1.1

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

    4. Applied associate-/l/1.1

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

    5. Simplified1.1

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

    if -0.033753148591154011 < x < 0.0359299296746303717

    1. Initial program 62.1

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

    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

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

    if 0.0359299296746303717 < x

    1. Initial program 0.9

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

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.03375314859115401:\\ \;\;\;\;\frac{1 \cdot 1 - \cos x \cdot \cos x}{x \cdot \left(x \cdot \left(1 + \cos x\right)\right)}\\ \mathbf{elif}\;x \leq 0.03592992967463037:\\ \;\;\;\;{x}^{4} \cdot 0.001388888888888889 + \left(0.5 + x \cdot \left(x \cdot -0.041666666666666664\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{\cos x}{x}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020198 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))