2cos (problem 3.3.5)

Average Error: 39.5 → 16.1

Time: 6.6s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -7.0063297743260128 \cdot 10^{-6}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\sqrt[3]{{\left(\sin x \cdot \sin \varepsilon\right)}^{3}} + \cos x\right)\\ \mathbf{elif}\;\varepsilon \le 3.1841549442056059 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)}\right)\\ \end{array}\]

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 <= -7.006329774326013e-06)) {
		VAR = ((double) (((double) (((double) cos(x)) * ((double) cos(eps)))) - ((double) (((double) cbrt(((double) pow(((double) (((double) sin(x)) * ((double) sin(eps)))), 3.0)))) + ((double) cos(x))))));
	} else {
		double VAR_1;
		if ((eps <= 3.184154944205606e-09)) {
			VAR_1 = ((double) (eps * ((double) (((double) (((double) (0.16666666666666666 * ((double) pow(x, 3.0)))) - x)) - ((double) (eps * 0.5))))));
		} else {
			VAR_1 = ((double) log(((double) exp(((double) (((double) (((double) cos(x)) * ((double) cos(eps)))) - ((double) (((double) (((double) sin(x)) * ((double) sin(eps)))) + ((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 < -7.0063297743260128e-6

   1. Initial program 29.8

      \[\cos \left(x + \varepsilon\right) - \cos x\]
   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 add-cbrt-cube 1.1

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

   7. Applied add-cbrt-cube 1.2

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

   8. Applied cbrt-unprod 1.1

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

   9. Simplified 1.1

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

   if -7.0063297743260128e-6 < eps < 3.1841549442056059e-9

   1. Initial program 49.0

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

   2. Taylor expanded around 0 31.9

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

   3. Simplified 31.9

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

   if 3.1841549442056059e-9 < eps

   1. Initial program 31.1

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

   3. Applied cos-sum 1.2

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

   4. Applied associate--l- 1.2

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

   6. Applied add-log-exp 1.3

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

   7. Applied add-log-exp 1.4

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

   8. Applied sum-log 1.4

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

   9. Applied add-log-exp 1.5

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

   10. Applied diff-log 1.6

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

   11. Simplified 1.3

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

4. Final simplification 16.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -7.0063297743260128 \cdot 10^{-6}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\sqrt[3]{{\left(\sin x \cdot \sin \varepsilon\right)}^{3}} + \cos x\right)\\ \mathbf{elif}\;\varepsilon \le 3.1841549442056059 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020175 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))