Average Error: 39.6 → 15.7
Time: 6.2s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1.71614933612317511 \cdot 10^{-6} \lor \neg \left(\varepsilon \le 3.75427990328525584 \cdot 10^{-17}\right):\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left({\varepsilon}^{3} \cdot \frac{1}{24} - \mathsf{fma}\left(\frac{1}{2}, \varepsilon, x\right)\right)\\ \end{array}\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -1.71614933612317511 \cdot 10^{-6} \lor \neg \left(\varepsilon \le 3.75427990328525584 \cdot 10^{-17}\right):\\
\;\;\;\;\cos x \cdot \cos \varepsilon - \mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\\

\mathbf{else}:\\
\;\;\;\;\varepsilon \cdot \left({\varepsilon}^{3} \cdot \frac{1}{24} - \mathsf{fma}\left(\frac{1}{2}, \varepsilon, x\right)\right)\\

\end{array}
double code(double x, double eps) {
	return (cos((x + eps)) - cos(x));
}
↓
double code(double x, double eps) {
	double VAR;
	if (((eps <= -1.7161493361231751e-06) || !(eps <= 3.754279903285256e-17))) {
		VAR = ((cos(x) * cos(eps)) - fma(sin(x), sin(eps), cos(x)));
	} else {
		VAR = (eps * ((pow(eps, 3.0) * 0.041666666666666664) - fma(0.5, eps, x)));
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if eps < -1.7161493361231751e-06 or 3.754279903285256e-17 < eps

    1. Initial program 30.5

      \[\cos \left(x + \varepsilon\right) - \cos x\]
    2. Using strategy rm

    3. Applied cos-sum1.9

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

    4. Applied associate--l-1.9

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

    5. Simplified1.9

      \[\leadsto \cos x \cdot \cos \varepsilon - \color{blue}{\mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)}\]

    if -1.7161493361231751e-06 < eps < 3.754279903285256e-17

    1. Initial program 49.7

      \[\cos \left(x + \varepsilon\right) - \cos x\]
    2. Using strategy rm

    3. Applied cos-sum49.5

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

    4. Taylor expanded around inf 49.5

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

    5. Simplified49.5

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \varepsilon, \cos x, -\mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\right)}\]

    6. Taylor expanded around 0 31.1

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

    7. Simplified31.1

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

  4. Final simplification15.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1.71614933612317511 \cdot 10^{-6} \lor \neg \left(\varepsilon \le 3.75427990328525584 \cdot 10^{-17}\right):\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left({\varepsilon}^{3} \cdot \frac{1}{24} - \mathsf{fma}\left(\frac{1}{2}, \varepsilon, x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020106 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))