Average Error: 31.5 → 0.3
Time: 4.0s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.0340458149574629151 \lor \neg \left(x \le 0.032087851221962699\right):\\ \;\;\;\;\frac{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\mathsf{fma}\left(1, 1, \mathsf{fma}\left(\cos x, \cos x, 1 \cdot \cos x\right)\right) \cdot x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left({x}^{4}, \frac{1}{720}, \frac{1}{2} - \frac{1}{24} \cdot {x}^{2}\right)\right)\right)\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.0340458149574629151 \lor \neg \left(x \le 0.032087851221962699\right):\\
\;\;\;\;\frac{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\mathsf{fma}\left(1, 1, \mathsf{fma}\left(\cos x, \cos x, 1 \cdot \cos x\right)\right) \cdot x}}{x}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left({x}^{4}, \frac{1}{720}, \frac{1}{2} - \frac{1}{24} \cdot {x}^{2}\right)\right)\right)\\

\end{array}
double code(double x) {
	return ((double) (((double) (1.0 - ((double) cos(x)))) / ((double) (x * x))));
}
↓
double code(double x) {
	double VAR;
	if (((x <= -0.034045814957462915) || !(x <= 0.0320878512219627))) {
		VAR = ((double) (((double) (((double) (((double) pow(1.0, 3.0)) - ((double) pow(((double) cos(x)), 3.0)))) / ((double) (((double) fma(1.0, 1.0, ((double) fma(((double) cos(x)), ((double) cos(x)), ((double) (1.0 * ((double) cos(x)))))))) * x)))) / x));
	} else {
		VAR = ((double) expm1(((double) log1p(((double) fma(((double) pow(x, 4.0)), 0.001388888888888889, ((double) (0.5 - ((double) (0.041666666666666664 * ((double) pow(x, 2.0))))))))))));
	}
	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 2 regimes

  2. if x < -0.034045814957462915 or 0.0320878512219627 < x

    1. Initial program 1.1

      \[\frac{1 - \cos x}{x \cdot x}\]
    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 flip3--0.5

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

    6. Applied associate-/l/0.5

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

    7. Simplified0.5

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

    if -0.034045814957462915 < x < 0.0320878512219627

    1. Initial program 62.3

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

    2. Taylor expanded around 0 0.0

      \[\leadsto \color{blue}{\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}}\]

    3. Simplified0.0

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{4}, \frac{1}{720}, \frac{1}{2} - \frac{1}{24} \cdot {x}^{2}\right)}\]
    4. Using strategy rm

    5. Applied expm1-log1p-u0.0

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left({x}^{4}, \frac{1}{720}, \frac{1}{2} - \frac{1}{24} \cdot {x}^{2}\right)\right)\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.0340458149574629151 \lor \neg \left(x \le 0.032087851221962699\right):\\ \;\;\;\;\frac{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\mathsf{fma}\left(1, 1, \mathsf{fma}\left(\cos x, \cos x, 1 \cdot \cos x\right)\right) \cdot x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left({x}^{4}, \frac{1}{720}, \frac{1}{2} - \frac{1}{24} \cdot {x}^{2}\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020120 +o rules:numerics
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1 (cos x)) (* x x)))