Average Error: 39.6 → 16.0
Time: 6.3s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1.4619102741755152 \cdot 10^{-4} \lor \neg \left(\varepsilon \le 2.5335197603511498 \cdot 10^{-20}\right):\\ \;\;\;\;\mathsf{fma}\left(\cos \varepsilon, \cos x, -\left(\sin x \cdot \sin \varepsilon + \cos x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{24}, {\varepsilon}^{4}, -\mathsf{fma}\left(x, \varepsilon, \frac{1}{2} \cdot {\varepsilon}^{2}\right)\right)\\ \end{array}\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -1.4619102741755152 \cdot 10^{-4} \lor \neg \left(\varepsilon \le 2.5335197603511498 \cdot 10^{-20}\right):\\
\;\;\;\;\mathsf{fma}\left(\cos \varepsilon, \cos x, -\left(\sin x \cdot \sin \varepsilon + \cos x\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{1}{24}, {\varepsilon}^{4}, -\mathsf{fma}\left(x, \varepsilon, \frac{1}{2} \cdot {\varepsilon}^{2}\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 <= -0.00014619102741755152) || !(eps <= 2.5335197603511498e-20))) {
		VAR = fma(cos(eps), cos(x), -((sin(x) * sin(eps)) + cos(x)));
	} else {
		VAR = fma(0.041666666666666664, pow(eps, 4.0), -fma(x, eps, (0.5 * pow(eps, 2.0))));
	}
	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 < -0.00014619102741755152 or 2.5335197603511498e-20 < eps

    1. Initial program 31.1

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

    3. Applied cos-sum2.0

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

    4. Taylor expanded around inf 2.0

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

    5. Simplified2.0

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

    7. Applied fma-udef2.0

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

    if -0.00014619102741755152 < eps < 2.5335197603511498e-20

    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. Taylor expanded around inf 48.2

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

    5. Simplified48.2

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

    7. Applied fma-udef48.2

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

    8. Taylor expanded around 0 30.9

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

    9. Simplified30.9

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

  4. Final simplification16.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1.4619102741755152 \cdot 10^{-4} \lor \neg \left(\varepsilon \le 2.5335197603511498 \cdot 10^{-20}\right):\\ \;\;\;\;\mathsf{fma}\left(\cos \varepsilon, \cos x, -\left(\sin x \cdot \sin \varepsilon + \cos x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{24}, {\varepsilon}^{4}, -\mathsf{fma}\left(x, \varepsilon, \frac{1}{2} \cdot {\varepsilon}^{2}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020100 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))