Average Error: 31.3 → 0.1
Time: 5.1s
Precision: binary64
Cost: 13376
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\frac{\tan \left(\frac{x}{2}\right) \cdot \frac{\sin x}{x}}{x}\]
\frac{1 - \cos x}{x \cdot x}
↓
\frac{\tan \left(\frac{x}{2}\right) \cdot \frac{\sin x}{x}}{x}
(FPCore (x) :precision binary64 (/ (- 1.0 (cos x)) (* x x)))
↓
(FPCore (x) :precision binary64 (/ (* (tan (/ x 2.0)) (/ (sin x) x)) x))
double code(double x) {
	return (1.0 - cos(x)) / (x * x);
}
↓
double code(double x) {
	return (tan(x / 2.0) * (sin(x) / x)) / x;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Alternatives

Alternative 1
Error0.3
Cost13697
\[\begin{array}{l} \mathbf{if}\;x \leq -0.0004457821252976667:\\ \;\;\;\;\sin x \cdot \frac{\tan \left(\frac{x}{2}\right)}{x \cdot x}\\ \mathbf{elif}\;x \leq 0.0054783708862614616:\\ \;\;\;\;0.5 - \left(x \cdot x\right) \cdot 0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x} \cdot \frac{1}{x}\\ \end{array}\]
Alternative 2
Error0.3
Cost7618
\[\begin{array}{l} \mathbf{if}\;x \leq -0.005012177499423301:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \mathbf{elif}\;x \leq 0.0054783708862614616:\\ \;\;\;\;0.5 - \left(x \cdot x\right) \cdot 0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x} \cdot \frac{1}{x}\\ \end{array}\]
Alternative 3
Error0.3
Cost7176
\[\begin{array}{l} \mathbf{if}\;x \leq -0.005012177499423301 \lor \neg \left(x \leq 0.0054783708862614616\right):\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5 - \left(x \cdot x\right) \cdot 0.041666666666666664\\ \end{array}\]
Alternative 4
Error0.6
Cost7176
\[\begin{array}{l} \mathbf{if}\;x \leq -0.005012177499423301 \lor \neg \left(x \leq 0.0054783708862614616\right):\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.5 - \left(x \cdot x\right) \cdot 0.041666666666666664\\ \end{array}\]
Alternative 5
Error15.6
Cost1090
\[\begin{array}{l} \mathbf{if}\;x \leq -3.4968886079870694:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 3.5071823034446923:\\ \;\;\;\;0.5 - \left(x \cdot x\right) \cdot 0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]
Alternative 6
Error15.5
Cost706
\[\begin{array}{l} \mathbf{if}\;x \leq -1.2257923376035007 \cdot 10^{+77}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.1081643599956554 \cdot 10^{+77}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]
Alternative 7
Error41.0
Cost706
\[\begin{array}{l} \mathbf{if}\;x \leq -1.2257923376035007 \cdot 10^{+77}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.1081643599956554 \cdot 10^{+77}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]
Alternative 8
Error56.7
Cost64
\[1\]

Error

Derivation

  1. Initial program 31.3

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

  3. Applied flip--_binary64_5331.4

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

  4. Simplified15.9

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

  6. Applied associate-/r*_binary64_2215.2

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

  7. Simplified0.1

    \[\leadsto \frac{\color{blue}{\frac{\sin x}{x} \cdot \tan \left(\frac{x}{2}\right)}}{x}\]

  8. Simplified0.1

    \[\leadsto \color{blue}{\frac{\tan \left(\frac{x}{2}\right) \cdot \frac{\sin x}{x}}{x}}\]

  9. Final simplification0.1

    \[\leadsto \frac{\tan \left(\frac{x}{2}\right) \cdot \frac{\sin x}{x}}{x}\]

Reproduce

herbie shell --seed 2021044 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))