Average Error: 31.5 → 0.1
Time: 2.2min
Precision: binary64
Cost: 13376
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\frac{\sin x \cdot \frac{\tan \left(\frac{x}{2}\right)}{x}}{x}\]
\frac{1 - \cos x}{x \cdot x}
↓
\frac{\sin x \cdot \frac{\tan \left(\frac{x}{2}\right)}{x}}{x}
(FPCore (x) :precision binary64 (/ (- 1.0 (cos x)) (* x x)))
↓
(FPCore (x) :precision binary64 (/ (* (sin x) (/ (tan (/ x 2.0)) x)) x))
double code(double x) {
	return (1.0 - cos(x)) / (x * x);
}
↓
double code(double x) {
	return (sin(x) * (tan(x / 2.0) / x)) / x;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Alternatives

Alternative 1
Error0.4
Cost13376
\[\frac{\sin x}{\frac{x}{\frac{\tan \left(\frac{x}{2}\right)}{x}}}\]
Alternative 2
Error0.1
Cost13376
\[\frac{\tan \left(\frac{x}{2}\right)}{x} \cdot \frac{\sin x}{x}\]
Alternative 3
Error0.2
Cost13505
\[\begin{array}{l} \mathbf{if}\;x \leq -0.03383914795959769:\\ \;\;\;\;\left(1 - \cos x\right) \cdot {x}^{-2}\\ \mathbf{elif}\;x \leq 0.028866246474509107:\\ \;\;\;\;{x}^{4} \cdot 0.001388888888888889 + \left(0.5 - \left(x \cdot x\right) \cdot 0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array}\]
Alternative 4
Error0.4
Cost7810
\[\begin{array}{l} \mathbf{if}\;x \leq -0.03383914795959769:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \mathbf{elif}\;x \leq 0.028866246474509107:\\ \;\;\;\;{x}^{4} \cdot 0.001388888888888889 + \left(0.5 - \left(x \cdot x\right) \cdot 0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array}\]
Alternative 5
Error0.4
Cost7490
\[\begin{array}{l} \mathbf{if}\;x \leq -0.005492313491037343:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \mathbf{elif}\;x \leq 0.005279724014658356:\\ \;\;\;\;0.5 - x \cdot \left(x \cdot 0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array}\]
Alternative 6
Error0.6
Cost7176
\[\begin{array}{l} \mathbf{if}\;x \leq -0.005492313491037343 \lor \neg \left(x \leq 0.005279724014658356\right):\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.5 - x \cdot \left(x \cdot 0.041666666666666664\right)\\ \end{array}\]
Alternative 7
Error15.6
Cost706
\[\begin{array}{l} \mathbf{if}\;x \leq -1.2851071242050192 \cdot 10^{+77}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.1175171150878477 \cdot 10^{+77}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]
Alternative 8
Error31.0
Cost64
\[0.5\]

Error

Derivation

  1. Initial program 31.5

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

  3. Applied flip--_binary64_5331.6

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

  4. Simplified16.1

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

  6. Applied *-un-lft-identity_binary64_7816.1

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

  7. Applied times-frac_binary64_8416.1

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

  8. Applied associate-/l*_binary64_2316.3

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

  9. Simplified0.4

    \[\leadsto \frac{\frac{\sin x}{1}}{\color{blue}{\frac{x}{\frac{\tan \left(\frac{x}{2}\right)}{x}}}}\]
  10. Using strategy rm

  11. Applied *-un-lft-identity_binary64_780.4

    \[\leadsto \frac{\frac{\sin x}{1}}{\frac{x}{\frac{\tan \left(\frac{x}{2}\right)}{\color{blue}{1 \cdot x}}}}\]

  12. Applied *-un-lft-identity_binary64_780.4

    \[\leadsto \frac{\frac{\sin x}{1}}{\frac{x}{\frac{\color{blue}{1 \cdot \tan \left(\frac{x}{2}\right)}}{1 \cdot x}}}\]

  13. Applied times-frac_binary64_840.4

    \[\leadsto \frac{\frac{\sin x}{1}}{\frac{x}{\color{blue}{\frac{1}{1} \cdot \frac{\tan \left(\frac{x}{2}\right)}{x}}}}\]

  14. Applied *-un-lft-identity_binary64_780.4

    \[\leadsto \frac{\frac{\sin x}{1}}{\frac{\color{blue}{1 \cdot x}}{\frac{1}{1} \cdot \frac{\tan \left(\frac{x}{2}\right)}{x}}}\]

  15. Applied times-frac_binary64_840.4

    \[\leadsto \frac{\frac{\sin x}{1}}{\color{blue}{\frac{1}{\frac{1}{1}} \cdot \frac{x}{\frac{\tan \left(\frac{x}{2}\right)}{x}}}}\]

  16. Applied associate-/r*_binary64_220.4

    \[\leadsto \color{blue}{\frac{\frac{\frac{\sin x}{1}}{\frac{1}{\frac{1}{1}}}}{\frac{x}{\frac{\tan \left(\frac{x}{2}\right)}{x}}}}\]

  17. Simplified0.4

    \[\leadsto \frac{\color{blue}{\sin x}}{\frac{x}{\frac{\tan \left(\frac{x}{2}\right)}{x}}}\]
  18. Using strategy rm

  19. Applied associate-/r/_binary64_240.4

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

  20. Applied associate-/r*_binary64_220.1

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

  21. Simplified0.1

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

  22. Simplified0.1

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

  23. Final simplification0.1

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

Reproduce

herbie shell --seed 2021014 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))