Average Error: 39.4 → 15.1
Time: 7.0s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -2.7905907270542693 \cdot 10^{-8}:\\ \;\;\;\;\left(\log \left(e^{\cos x \cdot \cos \varepsilon}\right) - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \le 2.17337424111151834 \cdot 10^{-4}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{24}, {\varepsilon}^{4}, -\mathsf{fma}\left(x, \varepsilon, \frac{1}{2} \cdot {\varepsilon}^{2}\right)\right) \cdot \left(\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) + \cos x\right)}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) + \cos x}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{{\left(\cos x \cdot \cos \varepsilon\right)}^{3}} - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array}\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -2.7905907270542693 \cdot 10^{-8}:\\
\;\;\;\;\left(\log \left(e^{\cos x \cdot \cos \varepsilon}\right) - \sin x \cdot \sin \varepsilon\right) - \cos x\\

\mathbf{elif}\;\varepsilon \le 2.17337424111151834 \cdot 10^{-4}:\\
\;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{24}, {\varepsilon}^{4}, -\mathsf{fma}\left(x, \varepsilon, \frac{1}{2} \cdot {\varepsilon}^{2}\right)\right) \cdot \left(\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) + \cos x\right)}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) + \cos x}\\

\mathbf{else}:\\
\;\;\;\;\left(\sqrt[3]{{\left(\cos x \cdot \cos \varepsilon\right)}^{3}} - \sin x \cdot \sin \varepsilon\right) - \cos x\\

\end{array}
double code(double x, double eps) {
	return ((double) (((double) cos(((double) (x + eps)))) - ((double) cos(x))));
}
↓
double code(double x, double eps) {
	double VAR;
	if ((eps <= -2.7905907270542693e-08)) {
		VAR = ((double) (((double) (((double) log(((double) exp(((double) (((double) cos(x)) * ((double) cos(eps)))))))) - ((double) (((double) sin(x)) * ((double) sin(eps)))))) - ((double) cos(x))));
	} else {
		double VAR_1;
		if ((eps <= 0.00021733742411115183)) {
			VAR_1 = ((double) (((double) (((double) fma(0.041666666666666664, ((double) pow(eps, 4.0)), ((double) -(((double) fma(x, eps, ((double) (0.5 * ((double) pow(eps, 2.0)))))))))) * ((double) (((double) (((double) (((double) cos(x)) * ((double) cos(eps)))) - ((double) (((double) sin(x)) * ((double) sin(eps)))))) + ((double) cos(x)))))) / ((double) (((double) (((double) (((double) cos(x)) * ((double) cos(eps)))) - ((double) (((double) sin(x)) * ((double) sin(eps)))))) + ((double) cos(x))))));
		} else {
			VAR_1 = ((double) (((double) (((double) cbrt(((double) pow(((double) (((double) cos(x)) * ((double) cos(eps)))), 3.0)))) - ((double) (((double) sin(x)) * ((double) sin(eps)))))) - ((double) cos(x))));
		}
		VAR = VAR_1;
	}
	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 3 regimes

  2. if eps < -2.7905907270542693e-08

    1. Initial program 30.8

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

    3. Applied cos-sum1.2

      \[\leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
    4. Using strategy rm

    5. Applied add-log-exp1.4

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

    if -2.7905907270542693e-08 < eps < 0.00021733742411115183

    1. Initial program 48.6

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

    3. Applied cos-sum48.2

      \[\leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
    4. Using strategy rm

    5. Applied flip--48.2

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

    6. Simplified48.2

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

    7. Taylor expanded around 0 29.9

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

    8. Simplified29.8

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

    if 0.00021733742411115183 < eps

    1. Initial program 30.6

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

    3. Applied cos-sum0.9

      \[\leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
    4. Using strategy rm

    5. Applied add-cbrt-cube1.2

      \[\leadsto \left(\cos x \cdot \color{blue}{\sqrt[3]{\left(\cos \varepsilon \cdot \cos \varepsilon\right) \cdot \cos \varepsilon}} - \sin x \cdot \sin \varepsilon\right) - \cos x\]

    6. Applied add-cbrt-cube1.2

      \[\leadsto \left(\color{blue}{\sqrt[3]{\left(\cos x \cdot \cos x\right) \cdot \cos x}} \cdot \sqrt[3]{\left(\cos \varepsilon \cdot \cos \varepsilon\right) \cdot \cos \varepsilon} - \sin x \cdot \sin \varepsilon\right) - \cos x\]

    7. Applied cbrt-unprod1.2

      \[\leadsto \left(\color{blue}{\sqrt[3]{\left(\left(\cos x \cdot \cos x\right) \cdot \cos x\right) \cdot \left(\left(\cos \varepsilon \cdot \cos \varepsilon\right) \cdot \cos \varepsilon\right)}} - \sin x \cdot \sin \varepsilon\right) - \cos x\]

    8. Simplified1.2

      \[\leadsto \left(\sqrt[3]{\color{blue}{{\left(\cos x \cdot \cos \varepsilon\right)}^{3}}} - \sin x \cdot \sin \varepsilon\right) - \cos x\]
  3. Recombined 3 regimes into one program.

  4. Final simplification15.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -2.7905907270542693 \cdot 10^{-8}:\\ \;\;\;\;\left(\log \left(e^{\cos x \cdot \cos \varepsilon}\right) - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \le 2.17337424111151834 \cdot 10^{-4}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{24}, {\varepsilon}^{4}, -\mathsf{fma}\left(x, \varepsilon, \frac{1}{2} \cdot {\varepsilon}^{2}\right)\right) \cdot \left(\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) + \cos x\right)}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) + \cos x}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{{\left(\cos x \cdot \cos \varepsilon\right)}^{3}} - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array}\]

Reproduce

herbie shell --seed 2020120 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))