2cos (problem 3.3.5)

Average Error: 39.9 → 16.7

Time: 6.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -7.2588072145492384 \cdot 10^{-30}:\\ \;\;\;\;\log \left(e^{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x}\right)\\ \mathbf{elif}\;\varepsilon \le 2.7743615214540619 \cdot 10^{-6}:\\ \;\;\;\;\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \frac{{\left(\sin x \cdot \sin \varepsilon\right)}^{3} + {\left(\cos x\right)}^{3}}{\cos x \cdot \left(\cos x - \sin x \cdot \sin \varepsilon\right) + \left(\sin x \cdot \sin \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)}\\ \end{array}\]

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 <= -7.258807214549238e-30)) {
		VAR = ((double) log(((double) exp(((double) (((double) (((double) (((double) cos(x)) * ((double) cos(eps)))) - ((double) (((double) sin(x)) * ((double) sin(eps)))))) - ((double) cos(x))))))));
	} else {
		double VAR_1;
		if ((eps <= 2.774361521454062e-06)) {
			VAR_1 = ((double) (eps * ((double) (((double) (((double) (0.16666666666666666 * ((double) pow(x, 3.0)))) - x)) - ((double) (eps * 0.5))))));
		} else {
			VAR_1 = ((double) (((double) (((double) cos(x)) * ((double) cos(eps)))) - ((double) (((double) (((double) pow(((double) (((double) sin(x)) * ((double) sin(eps)))), 3.0)) + ((double) pow(((double) cos(x)), 3.0)))) / ((double) (((double) (((double) cos(x)) * ((double) (((double) cos(x)) - ((double) (((double) sin(x)) * ((double) sin(eps)))))))) + ((double) (((double) (((double) sin(x)) * ((double) sin(eps)))) * ((double) (((double) sin(x)) * ((double) sin(eps))))))))))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus eps

Derivation

1. Split input into 3 regimes

2. if eps < -7.258807214549238e-30

   1. Initial program 32.6

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

   3. Applied cos-sum 4.8

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

   5. Applied add-log-exp 4.9

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

   6. Applied add-log-exp 5.0

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

   7. Applied add-log-exp 5.1

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

   8. Applied diff-log 5.2

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

   9. Applied diff-log 5.2

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

   10. Simplified 4.9

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

   if -7.258807214549238e-30 < eps < 2.774361521454062e-06

   1. Initial program 48.6

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

   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. Simplified 31.7

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

   if 2.774361521454062e-06 < eps

   1. Initial program 31.3

      \[\cos \left(x + \varepsilon\right) - \cos x\]
   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. Applied associate--l- 1.0

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

   6. Applied flip3-+ 1.1

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

   7. Simplified 1.1

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

4. Final simplification 16.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -7.2588072145492384 \cdot 10^{-30}:\\ \;\;\;\;\log \left(e^{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x}\right)\\ \mathbf{elif}\;\varepsilon \le 2.7743615214540619 \cdot 10^{-6}:\\ \;\;\;\;\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \frac{{\left(\sin x \cdot \sin \varepsilon\right)}^{3} + {\left(\cos x\right)}^{3}}{\cos x \cdot \left(\cos x - \sin x \cdot \sin \varepsilon\right) + \left(\sin x \cdot \sin \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020121 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))