2cos (problem 3.3.5)

Average Error: 39.6 → 16.4

Time: 6.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1.7962576357457494 \cdot 10^{-34}:\\ \;\;\;\;\cos \varepsilon \cdot \cos x - \sqrt[3]{{\left(\mathsf{fma}\left(\sin \varepsilon, \sin x, \cos x\right)\right)}^{3}}\\ \mathbf{elif}\;\varepsilon \le 3.54463964995298486 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{24}, {\varepsilon}^{4}, -\mathsf{fma}\left(x, \varepsilon, \frac{1}{2} \cdot {\varepsilon}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \varepsilon \cdot \cos x - \mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(\sin \varepsilon, \sin x, \cos x\right)\right)\right)\\ \end{array}\]

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 <= -1.7962576357457494e-34)) {
		VAR = ((double) (((double) (((double) cos(eps)) * ((double) cos(x)))) - ((double) cbrt(((double) pow(((double) fma(((double) sin(eps)), ((double) sin(x)), ((double) cos(x)))), 3.0))))));
	} else {
		double VAR_1;
		if ((eps <= 3.544639649952985e-07)) {
			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) cos(eps)) * ((double) cos(x)))) - ((double) log1p(((double) expm1(((double) fma(((double) sin(eps)), ((double) sin(x)), ((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 < -1.7962576357457494e-34

   1. Initial program 32.7

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

   3. Applied +-commutative 32.7

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

   4. Applied cos-sum 5.7

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

   5. Applied associate--l- 5.7

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

   6. Simplified 5.7

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

   8. Applied add-cbrt-cube 5.9

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

   9. Simplified 5.9

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

   if -1.7962576357457494e-34 < eps < 3.544639649952985e-07

   1. Initial program 49.2

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

   3. Applied +-commutative 49.2

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

   4. Applied cos-sum 48.9

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

   5. Applied associate--l- 48.9

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

   6. Simplified 48.9

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

   7. Taylor expanded around 0 31.0

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

   8. Simplified 31.0

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

   if 3.544639649952985e-07 < eps

   1. Initial program 29.8

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

   3. Applied +-commutative 29.8

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

   4. Applied cos-sum 1.1

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

   5. Applied associate--l- 1.2

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

   6. Simplified 1.2

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

   8. Applied log1p-expm1-u 1.2

      \[\leadsto \cos \varepsilon \cdot \cos x - \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(\sin \varepsilon, \sin x, \cos x\right)\right)\right)}\]
3. Recombined 3 regimes into one program.

4. Final simplification 16.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1.7962576357457494 \cdot 10^{-34}:\\ \;\;\;\;\cos \varepsilon \cdot \cos x - \sqrt[3]{{\left(\mathsf{fma}\left(\sin \varepsilon, \sin x, \cos x\right)\right)}^{3}}\\ \mathbf{elif}\;\varepsilon \le 3.54463964995298486 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{24}, {\varepsilon}^{4}, -\mathsf{fma}\left(x, \varepsilon, \frac{1}{2} \cdot {\varepsilon}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \varepsilon \cdot \cos x - \mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(\sin \varepsilon, \sin x, \cos x\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020114 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))