2cos (problem 3.3.5)

Average Error: 39.5 → 15.6

Time: 7.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -0.00381593506258463963:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x, \cos \varepsilon, \sin x \cdot \sin \varepsilon\right) \cdot \left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}{\mathsf{fma}\left(\cos x, \cos \varepsilon, \sin x \cdot \sin \varepsilon\right)} - \cos x\\ \mathbf{elif}\;\varepsilon \le 3.237071340470319 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{24}, {\varepsilon}^{4}, -\mathsf{fma}\left(x, \varepsilon, \frac{1}{2} \cdot {\varepsilon}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sqrt[3]{{\left(\sin x \cdot \sin \varepsilon\right)}^{3}}\right) - \cos x\\ \end{array}\]

C

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 <= -0.0038159350625846396)) {
		VAR = ((double) (((double) (((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) fma(((double) cos(x)), ((double) cos(eps)), ((double) (((double) sin(x)) * ((double) sin(eps)))))))) - ((double) cos(x))));
	} else {
		double VAR_1;
		if ((eps <= 3.237071340470319e-06)) {
			VAR_1 = ((double) fma(0.041666666666666664, ((double) pow(eps, 4.0)), ((double) -(((double) fma(x, eps, ((double) (0.5 * ((double) pow(eps, 2.0))))))))));
		} else {
			VAR_1 = ((double) (((double) (((double) (((double) cos(x)) * ((double) cos(eps)))) - ((double) cbrt(((double) pow(((double) (((double) sin(x)) * ((double) sin(eps)))), 3.0)))))) - ((double) cos(x))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus eps

Derivation

1. Split input into 3 regimes

2. if eps < -0.0038159350625846396

   1. Initial program 30.1

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

   3. Applied cos-sum 0.9

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

   5. Applied flip-- 1.0

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

   6. Simplified 1.0

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

   7. Simplified 0.9

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

   if -0.0038159350625846396 < eps < 3.237071340470319e-06

   1. Initial program 48.7

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

   3. Applied +-commutative 48.7

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

   4. Applied cos-sum 48.1

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

   5. Applied associate--l- 48.1

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

   6. Simplified 48.1

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

   7. Taylor expanded around 0 30.3

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

   8. Simplified 30.3

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

   if 3.237071340470319e-06 < eps

   1. Initial program 30.7

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

   3. Applied cos-sum 1.0

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

   5. Applied add-cbrt-cube 1.0

      \[\leadsto \left(\cos x \cdot \cos \varepsilon - \sin x \cdot \color{blue}{\sqrt[3]{\left(\sin \varepsilon \cdot \sin \varepsilon\right) \cdot \sin \varepsilon}}\right) - \cos x\]

   6. Applied add-cbrt-cube 1.1

      \[\leadsto \left(\cos x \cdot \cos \varepsilon - \color{blue}{\sqrt[3]{\left(\sin x \cdot \sin x\right) \cdot \sin x}} \cdot \sqrt[3]{\left(\sin \varepsilon \cdot \sin \varepsilon\right) \cdot \sin \varepsilon}\right) - \cos x\]

   7. Applied cbrt-unprod 1.0

      \[\leadsto \left(\cos x \cdot \cos \varepsilon - \color{blue}{\sqrt[3]{\left(\left(\sin x \cdot \sin x\right) \cdot \sin x\right) \cdot \left(\left(\sin \varepsilon \cdot \sin \varepsilon\right) \cdot \sin \varepsilon\right)}}\right) - \cos x\]

   8. Simplified 1.0

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

4. Final simplification 15.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -0.00381593506258463963:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x, \cos \varepsilon, \sin x \cdot \sin \varepsilon\right) \cdot \left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}{\mathsf{fma}\left(\cos x, \cos \varepsilon, \sin x \cdot \sin \varepsilon\right)} - \cos x\\ \mathbf{elif}\;\varepsilon \le 3.237071340470319 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{24}, {\varepsilon}^{4}, -\mathsf{fma}\left(x, \varepsilon, \frac{1}{2} \cdot {\varepsilon}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sqrt[3]{{\left(\sin x \cdot \sin \varepsilon\right)}^{3}}\right) - \cos x\\ \end{array}\]

Reproduce

herbie shell --seed 2020113 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))