Average Error: 39.3 → 16.3
Time: 6.5s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1.184078211908822 \cdot 10^{-13}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos \varepsilon, \cos x, -\mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\right) \cdot \left(\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) + \cos x\right)}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) + \cos x}\\ \mathbf{elif}\;\varepsilon \le 1.69858524052719363 \cdot 10^{-7}:\\ \;\;\;\;\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\mathsf{fma}\left(\cos x, \cos \varepsilon, -\mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\right)}\right)\\ \end{array}\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -1.184078211908822 \cdot 10^{-13}:\\
\;\;\;\;\frac{\mathsf{fma}\left(\cos \varepsilon, \cos x, -\mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\right) \cdot \left(\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) + \cos x\right)}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) + \cos x}\\

\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}:\\
\;\;\;\;\log \left(e^{\mathsf{fma}\left(\cos x, \cos \varepsilon, -\mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\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 <= -1.184078211908822e-13)) {
		VAR = ((fma(cos(eps), cos(x), -fma(sin(x), sin(eps), cos(x))) * (((cos(x) * cos(eps)) - (sin(x) * sin(eps))) + cos(x))) / (((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 = log(exp(fma(cos(x), cos(eps), -fma(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 flip--2.3

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

    6. Simplified1.9

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

    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. Simplified1.1

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

    7. Applied fma-neg1.1

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

    9. Applied add-log-exp1.2

      \[\leadsto \color{blue}{\log \left(e^{\mathsf{fma}\left(\cos x, \cos \varepsilon, -\mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\right)}\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification16.3

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

Reproduce

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