Average Error: 31.2 → 0.3
Time: 4.1s
Precision: binary64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -0.010529840751750804:\\ \;\;\;\;\frac{\frac{\sin x \cdot \tan \left(\frac{x}{2}\right)}{x}}{x}\\ \mathbf{elif}\;x \leq 5.530578574506921 \cdot 10^{-70}:\\ \;\;\;\;{x}^{4} \cdot 0.001388888888888889 + \left(0.5 - \left(x \cdot x\right) \cdot 0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \frac{\tan \left(\frac{x}{2}\right)}{x \cdot x}\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -0.010529840751750804:\\
\;\;\;\;\frac{\frac{\sin x \cdot \tan \left(\frac{x}{2}\right)}{x}}{x}\\

\mathbf{elif}\;x \leq 5.530578574506921 \cdot 10^{-70}:\\
\;\;\;\;{x}^{4} \cdot 0.001388888888888889 + \left(0.5 - \left(x \cdot x\right) \cdot 0.041666666666666664\right)\\

\mathbf{else}:\\
\;\;\;\;\sin x \cdot \frac{\tan \left(\frac{x}{2}\right)}{x \cdot x}\\

\end{array}
(FPCore (x) :precision binary64 (/ (- 1.0 (cos x)) (* x x)))
↓
(FPCore (x)
 :precision binary64
 (if (<= x -0.010529840751750804)
   (/ (/ (* (sin x) (tan (/ x 2.0))) x) x)
   (if (<= x 5.530578574506921e-70)
     (+
      (* (pow x 4.0) 0.001388888888888889)
      (- 0.5 (* (* x x) 0.041666666666666664)))
     (* (sin x) (/ (tan (/ x 2.0)) (* x x))))))
double code(double x) {
	return (1.0 - cos(x)) / (x * x);
}
↓
double code(double x) {
	double tmp;
	if (x <= -0.010529840751750804) {
		tmp = ((sin(x) * tan(x / 2.0)) / x) / x;
	} else if (x <= 5.530578574506921e-70) {
		tmp = (pow(x, 4.0) * 0.001388888888888889) + (0.5 - ((x * x) * 0.041666666666666664));
	} else {
		tmp = sin(x) * (tan(x / 2.0) / (x * x));
	}
	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 < -0.0105298407517508043

    1. Initial program 0.9

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

    3. Applied flip--_binary64_521.2

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

    4. Simplified1.0

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

    6. Applied associate-/r*_binary64_230.6

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

    7. Simplified0.2

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

    9. Applied associate-*l/_binary64_220.2

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

    if -0.0105298407517508043 < x < 5.5305785745069212e-70

    1. Initial program 62.7

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

    2. Taylor expanded around 0 0.0

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

    3. Simplified0.0

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

    if 5.5305785745069212e-70 < x

    1. Initial program 11.2

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

    3. Applied flip--_binary64_5211.4

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

    4. Simplified1.2

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

    6. Applied associate-/r*_binary64_230.5

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

    7. Simplified0.2

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

    9. Applied associate-*l/_binary64_220.2

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

    11. Applied *-un-lft-identity_binary64_770.2

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

    12. Applied *-un-lft-identity_binary64_770.2

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

    13. Applied times-frac_binary64_830.2

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

    14. Applied times-frac_binary64_830.2

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

    15. Simplified0.2

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

    16. Simplified0.9

      \[\leadsto \sin x \cdot \color{blue}{\frac{\tan \left(\frac{x}{2}\right)}{x \cdot x}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.010529840751750804:\\ \;\;\;\;\frac{\frac{\sin x \cdot \tan \left(\frac{x}{2}\right)}{x}}{x}\\ \mathbf{elif}\;x \leq 5.530578574506921 \cdot 10^{-70}:\\ \;\;\;\;{x}^{4} \cdot 0.001388888888888889 + \left(0.5 - \left(x \cdot x\right) \cdot 0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \frac{\tan \left(\frac{x}{2}\right)}{x \cdot x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020280 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))