Average Error: 39.5 → 15.9
Time: 6.9s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -3.0852323357979476 \cdot 10^{-10} \lor \neg \left(\varepsilon \le 6.27540161566563378 \cdot 10^{-24}\right):\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \log \left(e^{\sin x \cdot \sin \varepsilon + \cos x}\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)\\ \end{array}\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -3.0852323357979476 \cdot 10^{-10} \lor \neg \left(\varepsilon \le 6.27540161566563378 \cdot 10^{-24}\right):\\
\;\;\;\;\cos x \cdot \cos \varepsilon - \log \left(e^{\sin x \cdot \sin \varepsilon + \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 (cos((x + eps)) - cos(x));
}
↓
double code(double x, double eps) {
	double VAR;
	if (((eps <= -3.0852323357979476e-10) || !(eps <= 6.275401615665634e-24))) {
		VAR = ((cos(x) * cos(eps)) - log(exp(((sin(x) * sin(eps)) + cos(x)))));
	} else {
		VAR = (eps * (((0.16666666666666666 * pow(x, 3.0)) - x) - (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 < -3.0852323357979476e-10 or 6.275401615665634e-24 < eps

    1. Initial program 31.8

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

    3. Applied cos-sum2.8

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

    4. Applied associate--l-2.8

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

    6. Applied add-log-exp2.9

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

    7. Applied add-log-exp3.0

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

    8. Applied sum-log3.0

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

    9. Simplified2.9

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

    if -3.0852323357979476e-10 < eps < 6.275401615665634e-24

    1. Initial program 48.7

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

    2. Taylor expanded around 0 31.5

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

    3. Simplified31.5

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

  4. Final simplification15.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -3.0852323357979476 \cdot 10^{-10} \lor \neg \left(\varepsilon \le 6.27540161566563378 \cdot 10^{-24}\right):\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \log \left(e^{\sin x \cdot \sin \varepsilon + \cos x}\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020102 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))