Average Error: 39.9 → 16.7
Time: 7.1s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -7.2588072145492384 \cdot 10^{-30} \lor \neg \left(\varepsilon \le 2.7743615214540619 \cdot 10^{-6}\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \cos \varepsilon, \sin x \cdot \sin \varepsilon\right) \cdot \left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right), \frac{1}{\mathsf{fma}\left(\cos x, \cos \varepsilon, \sin x \cdot \sin \varepsilon\right)}, -\cos x\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)\\ \end{array}\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -7.2588072145492384 \cdot 10^{-30} \lor \neg \left(\varepsilon \le 2.7743615214540619 \cdot 10^{-6}\right):\\
\;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \cos \varepsilon, \sin x \cdot \sin \varepsilon\right) \cdot \left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right), \frac{1}{\mathsf{fma}\left(\cos x, \cos \varepsilon, \sin x \cdot \sin \varepsilon\right)}, -\cos x\right)\\

\mathbf{else}:\\
\;\;\;\;\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)\\

\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 <= -7.258807214549238e-30) || !(eps <= 2.774361521454062e-06))) {
		VAR = ((double) fma(((double) (((double) fma(((double) cos(x)), ((double) cos(eps)), ((double) (((double) sin(x)) * ((double) sin(eps)))))) * ((double) (((double) (((double) cos(x)) * ((double) cos(eps)))) - ((double) (((double) sin(x)) * ((double) sin(eps)))))))), ((double) (1.0 / ((double) fma(((double) cos(x)), ((double) cos(eps)), ((double) (((double) sin(x)) * ((double) sin(eps)))))))), ((double) -(((double) cos(x))))));
	} else {
		VAR = ((double) (eps * ((double) (((double) (((double) (0.16666666666666666 * ((double) pow(x, 3.0)))) - x)) - ((double) (eps * 0.5))))));
	}
	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 < -7.258807214549238e-30 or 2.774361521454062e-06 < eps

    1. Initial program 32.0

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

    3. Applied cos-sum3.0

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

    5. Applied flip--3.1

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

    6. Simplified3.1

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

    7. Simplified3.0

      \[\leadsto \frac{\mathsf{fma}\left(\cos x, \cos \varepsilon, \sin x \cdot \sin \varepsilon\right) \cdot \left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}{\color{blue}{\mathsf{fma}\left(\cos x, \cos \varepsilon, \sin x \cdot \sin \varepsilon\right)}} - \cos x\]
    8. Using strategy rm

    9. Applied div-inv3.2

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

    10. Applied fma-neg3.1

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

    if -7.258807214549238e-30 < eps < 2.774361521454062e-06

    1. Initial program 48.6

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

    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

      \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification16.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -7.2588072145492384 \cdot 10^{-30} \lor \neg \left(\varepsilon \le 2.7743615214540619 \cdot 10^{-6}\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \cos \varepsilon, \sin x \cdot \sin \varepsilon\right) \cdot \left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right), \frac{1}{\mathsf{fma}\left(\cos x, \cos \varepsilon, \sin x \cdot \sin \varepsilon\right)}, -\cos x\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020121 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))