2tan (problem 3.3.2)

Average Error: 37.0 → 0.5

Time: 10.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -3.6106432805517226 \cdot 10^{-6} \lor \neg \left(\varepsilon \le 4.30447233934794542 \cdot 10^{-19}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - {\left(\tan \varepsilon \cdot \tan x\right)}^{3}} \cdot \left(\left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right) + 1 \cdot \left(\tan \varepsilon \cdot \tan x\right)\right) + \left(1 \cdot \frac{\tan x + \tan \varepsilon}{1 - {\left(\sqrt[3]{\tan \varepsilon} \cdot \sqrt[3]{\tan \varepsilon}\right)}^{3} \cdot {\left(\sqrt[3]{\tan \varepsilon} \cdot \tan x\right)}^{3}} - \tan x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - {\left(\tan \varepsilon \cdot \tan x\right)}^{3}} \cdot \left(\left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right) + 1 \cdot \left(\tan \varepsilon \cdot \tan x\right)\right) + \left(\varepsilon + \left(\frac{1}{3} \cdot {\varepsilon}^{3} + \frac{2}{15} \cdot {\varepsilon}^{5}\right)\right)\\ \end{array}\]

C

double code(double x, double eps) {
	return ((double) (((double) tan(((double) (x + eps)))) - ((double) tan(x))));
}

↓

double code(double x, double eps) {
	double VAR;
	if (((eps <= -3.6106432805517226e-06) || !(eps <= 4.3044723393479454e-19))) {
		VAR = ((double) (((double) (((double) (((double) (((double) tan(x)) + ((double) tan(eps)))) / ((double) (1.0 - ((double) pow(((double) (((double) tan(eps)) * ((double) tan(x)))), 3.0)))))) * ((double) (((double) (((double) (((double) tan(eps)) * ((double) tan(x)))) * ((double) (((double) tan(eps)) * ((double) tan(x)))))) + ((double) (1.0 * ((double) (((double) tan(eps)) * ((double) tan(x)))))))))) + ((double) (((double) (1.0 * ((double) (((double) (((double) tan(x)) + ((double) tan(eps)))) / ((double) (1.0 - ((double) (((double) pow(((double) (((double) cbrt(((double) tan(eps)))) * ((double) cbrt(((double) tan(eps)))))), 3.0)) * ((double) pow(((double) (((double) cbrt(((double) tan(eps)))) * ((double) tan(x)))), 3.0)))))))))) - ((double) tan(x))))));
	} else {
		VAR = ((double) (((double) (((double) (((double) (((double) tan(x)) + ((double) tan(eps)))) / ((double) (1.0 - ((double) pow(((double) (((double) tan(eps)) * ((double) tan(x)))), 3.0)))))) * ((double) (((double) (((double) (((double) tan(eps)) * ((double) tan(x)))) * ((double) (((double) tan(eps)) * ((double) tan(x)))))) + ((double) (1.0 * ((double) (((double) tan(eps)) * ((double) tan(x)))))))))) + ((double) (eps + ((double) (((double) (0.3333333333333333 * ((double) pow(eps, 3.0)))) + ((double) (0.13333333333333333 * ((double) pow(eps, 5.0))))))))));
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	37.0
Target	15.0
Herbie	0.5

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

Derivation

1. Split input into 2 regimes

2. if eps < -3.6106432805517226e-06 or 4.3044723393479454e-19 < eps

   1. Initial program 29.7

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

   3. Applied tan-sum 0.8

      \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x\]

   4. Simplified 0.8

      \[\leadsto \frac{\tan x + \tan \varepsilon}{\color{blue}{1 - \tan \varepsilon \cdot \tan x}} - \tan x\]
   5. Using strategy rm

   6. Applied flip3-- 0.8

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

   7. Applied associate-/r/ 0.8

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

   8. Simplified 0.8

      \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - {\left(\tan \varepsilon \cdot \tan x\right)}^{3}}} \cdot \left(1 \cdot 1 + \left(\left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right) + 1 \cdot \left(\tan \varepsilon \cdot \tan x\right)\right)\right) - \tan x\]
   9. Using strategy rm

   10. Applied +-commutative 0.8

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

   11. Applied distribute-lft-in 0.8

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

   12. Applied associate--l+ 0.8

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

   13. Simplified 0.8

       \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - {\left(\tan \varepsilon \cdot \tan x\right)}^{3}} \cdot \left(\left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right) + 1 \cdot \left(\tan \varepsilon \cdot \tan x\right)\right) + \color{blue}{\left(1 \cdot \frac{\tan x + \tan \varepsilon}{1 - {\left(\tan \varepsilon \cdot \tan x\right)}^{3}} - \tan x\right)}\]
   14. Using strategy rm

   15. Applied add-cube-cbrt 0.8

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

   16. Applied associate-*l* 0.8

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

   17. Applied unpow-prod-down 0.8

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

   if -3.6106432805517226e-06 < eps < 4.3044723393479454e-19

   1. Initial program 44.8

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

   3. Applied tan-sum 44.4

      \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x\]

   4. Simplified 44.4

      \[\leadsto \frac{\tan x + \tan \varepsilon}{\color{blue}{1 - \tan \varepsilon \cdot \tan x}} - \tan x\]
   5. Using strategy rm

   6. Applied flip3-- 44.4

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

   7. Applied associate-/r/ 44.4

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

   8. Simplified 44.4

      \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - {\left(\tan \varepsilon \cdot \tan x\right)}^{3}}} \cdot \left(1 \cdot 1 + \left(\left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right) + 1 \cdot \left(\tan \varepsilon \cdot \tan x\right)\right)\right) - \tan x\]
   9. Using strategy rm

   10. Applied +-commutative 44.4

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

   11. Applied distribute-lft-in 44.4

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

   12. Applied associate--l+ 39.9

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

   13. Simplified 39.9

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

   14. Taylor expanded around 0 0.2

       \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - {\left(\tan \varepsilon \cdot \tan x\right)}^{3}} \cdot \left(\left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right) + 1 \cdot \left(\tan \varepsilon \cdot \tan x\right)\right) + \color{blue}{\left(\varepsilon + \left(\frac{1}{3} \cdot {\varepsilon}^{3} + \frac{2}{15} \cdot {\varepsilon}^{5}\right)\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -3.6106432805517226 \cdot 10^{-6} \lor \neg \left(\varepsilon \le 4.30447233934794542 \cdot 10^{-19}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - {\left(\tan \varepsilon \cdot \tan x\right)}^{3}} \cdot \left(\left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right) + 1 \cdot \left(\tan \varepsilon \cdot \tan x\right)\right) + \left(1 \cdot \frac{\tan x + \tan \varepsilon}{1 - {\left(\sqrt[3]{\tan \varepsilon} \cdot \sqrt[3]{\tan \varepsilon}\right)}^{3} \cdot {\left(\sqrt[3]{\tan \varepsilon} \cdot \tan x\right)}^{3}} - \tan x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - {\left(\tan \varepsilon \cdot \tan x\right)}^{3}} \cdot \left(\left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right) + 1 \cdot \left(\tan \varepsilon \cdot \tan x\right)\right) + \left(\varepsilon + \left(\frac{1}{3} \cdot {\varepsilon}^{3} + \frac{2}{15} \cdot {\varepsilon}^{5}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020113 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"
  :precision binary64

  :herbie-target
  (/ (sin eps) (* (cos x) (cos (+ x eps))))

  (- (tan (+ x eps)) (tan x)))