Average Error: 39.3 → 16.3
Time: 6.8s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1.184078211908822 \cdot 10^{-13}:\\ \;\;\;\;\log \left(e^{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x}\right)\\ \mathbf{elif}\;\varepsilon \le 1.69858524052719363 \cdot 10^{-7}:\\ \;\;\;\;\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \frac{\left(\sin x \cdot \sin \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right) - \cos x \cdot \cos x}{\sin x \cdot \sin \varepsilon - \cos x}\\ \end{array}\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -1.184078211908822 \cdot 10^{-13}:\\
\;\;\;\;\log \left(e^{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x}\right)\\

\mathbf{elif}\;\varepsilon \le 1.69858524052719363 \cdot 10^{-7}:\\
\;\;\;\;\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 - \frac{\left(\sin x \cdot \sin \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right) - \cos x \cdot \cos x}{\sin x \cdot \sin \varepsilon - \cos x}\\

\end{array}
double code(double x, double eps) {
	return (cos((x + eps)) - cos(x));
}
↓
double code(double x, double eps) {
	double VAR;
	if ((eps <= -1.184078211908822e-13)) {
		VAR = log(exp((((cos(x) * cos(eps)) - (sin(x) * sin(eps))) - cos(x))));
	} else {
		double VAR_1;
		if ((eps <= 1.6985852405271936e-07)) {
			VAR_1 = (eps * (((0.16666666666666666 * pow(x, 3.0)) - x) - (eps * 0.5)));
		} else {
			VAR_1 = ((cos(x) * cos(eps)) - ((((sin(x) * sin(eps)) * (sin(x) * sin(eps))) - (cos(x) * cos(x))) / ((sin(x) * sin(eps)) - 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 < -1.184078211908822e-13

    1. Initial program 30.9

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

    3. Applied cos-sum1.9

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

    5. Applied add-log-exp2.0

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

    6. Applied add-log-exp2.0

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

    7. Applied add-log-exp2.2

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

    8. Applied diff-log2.2

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

    9. Applied diff-log2.4

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

    10. Simplified2.0

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

    if -1.184078211908822e-13 < eps < 1.6985852405271936e-07

    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 1.6985852405271936e-07 < eps

    1. Initial program 29.8

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

    3. Applied cos-sum1.1

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

    4. Applied associate--l-1.1

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

    6. Applied flip-+1.2

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

  4. Final simplification16.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1.184078211908822 \cdot 10^{-13}:\\ \;\;\;\;\log \left(e^{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x}\right)\\ \mathbf{elif}\;\varepsilon \le 1.69858524052719363 \cdot 10^{-7}:\\ \;\;\;\;\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \frac{\left(\sin x \cdot \sin \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right) - \cos x \cdot \cos x}{\sin x \cdot \sin \varepsilon - \cos x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020079 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))