Average Error: 39.5 → 16.3
Time: 6.8s
Precision: binary64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -2.6231847589922844 \cdot 10^{-6}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\sqrt[3]{{\left(\sin x \cdot \sin \varepsilon\right)}^{3}} + \cos x\right)\\ \mathbf{elif}\;\varepsilon \le 1.4423632138238029 \cdot 10^{-25}:\\ \;\;\;\;\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \sqrt[3]{{\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^{3}}\\ \end{array}\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -2.6231847589922844 \cdot 10^{-6}:\\
\;\;\;\;\cos x \cdot \cos \varepsilon - \left(\sqrt[3]{{\left(\sin x \cdot \sin \varepsilon\right)}^{3}} + \cos x\right)\\

\mathbf{elif}\;\varepsilon \le 1.4423632138238029 \cdot 10^{-25}:\\
\;\;\;\;\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)\\

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

\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 <= -2.6231847589922844e-06)) {
		VAR = ((double) (((double) (((double) cos(x)) * ((double) cos(eps)))) - ((double) (((double) cbrt(((double) pow(((double) (((double) sin(x)) * ((double) sin(eps)))), 3.0)))) + ((double) cos(x))))));
	} else {
		double VAR_1;
		if ((eps <= 1.4423632138238029e-25)) {
			VAR_1 = ((double) (eps * ((double) (((double) (((double) (0.16666666666666666 * ((double) pow(x, 3.0)))) - x)) - ((double) (eps * 0.5))))));
		} else {
			VAR_1 = ((double) (((double) (((double) cos(x)) * ((double) cos(eps)))) - ((double) cbrt(((double) pow(((double) (((double) (((double) sin(x)) * ((double) sin(eps)))) + ((double) cos(x)))), 3.0))))));
		}
		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 < -2.6231847589922844e-6

    1. Initial program 30.3

      \[\cos \left(x + \varepsilon\right) - \cos x\]
    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. Applied associate--l-0.9

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

    6. Applied add-cbrt-cube1.0

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

    7. Applied add-cbrt-cube1.0

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

    8. Applied cbrt-unprod1.0

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

    9. Simplified1.0

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

    if -2.6231847589922844e-6 < eps < 1.4423632138238029e-25

    1. Initial program 48.8

      \[\cos \left(x + \varepsilon\right) - \cos x\]

    2. Taylor expanded around 0 31.7

      \[\leadsto \color{blue}{\frac{1}{6} \cdot \left({x}^{3} \cdot \varepsilon\right) - \left(x \cdot \varepsilon + \frac{1}{2} \cdot {\varepsilon}^{2}\right)}\]

    3. Simplified31.7

      \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)}\]

    if 1.4423632138238029e-25 < eps

    1. Initial program 31.8

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

    3. Applied cos-sum3.8

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

    4. Applied associate--l-3.8

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

    6. Applied add-cbrt-cube4.0

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

    7. Simplified3.9

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

  4. Final simplification16.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -2.6231847589922844 \cdot 10^{-6}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\sqrt[3]{{\left(\sin x \cdot \sin \varepsilon\right)}^{3}} + \cos x\right)\\ \mathbf{elif}\;\varepsilon \le 1.4423632138238029 \cdot 10^{-25}:\\ \;\;\;\;\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \sqrt[3]{{\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^{3}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020161 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))