Average Error: 31.3 → 0.3
Time: 4.4s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.031288658239448007 \lor \neg \left(x \le 0.0246736691289138228\right):\\ \;\;\;\;\frac{\frac{1}{x}}{x} \cdot \left(1 - \cos x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({x}^{4}, \frac{1}{720}, \frac{1}{2} - \frac{1}{24} \cdot {x}^{2}\right)\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.031288658239448007 \lor \neg \left(x \le 0.0246736691289138228\right):\\
\;\;\;\;\frac{\frac{1}{x}}{x} \cdot \left(1 - \cos x\right)\\

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

\end{array}
double code(double x) {
	return ((1.0 - cos(x)) / (x * x));
}
↓
double code(double x) {
	double temp;
	if (((x <= -0.03128865823944801) || !(x <= 0.024673669128913823))) {
		temp = (((1.0 / x) / x) * (1.0 - cos(x)));
	} else {
		temp = fma(pow(x, 4.0), 0.001388888888888889, (0.5 - (0.041666666666666664 * pow(x, 2.0))));
	}
	return temp;
}

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 < -0.03128865823944801 or 0.024673669128913823 < x

    1. Initial program 0.9

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

    3. Applied *-un-lft-identity0.9

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

    4. Applied times-frac0.5

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

    6. Applied clear-num0.5

      \[\leadsto \frac{1}{x} \cdot \color{blue}{\frac{1}{\frac{x}{1 - \cos x}}}\]
    7. Using strategy rm

    8. Applied associate-/r/0.6

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

    9. Applied associate-*r*0.6

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

    10. Simplified0.5

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

    if -0.03128865823944801 < x < 0.024673669128913823

    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

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.031288658239448007 \lor \neg \left(x \le 0.0246736691289138228\right):\\ \;\;\;\;\frac{\frac{1}{x}}{x} \cdot \left(1 - \cos x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({x}^{4}, \frac{1}{720}, \frac{1}{2} - \frac{1}{24} \cdot {x}^{2}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 +o rules:numerics
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1 (cos x)) (* x x)))