2tan (problem 3.3.2)

Average Error: 37.5 → 15.3

Time: 7.6s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1.3369941042866534 \cdot 10^{-17}:\\ \;\;\;\;\frac{\left(\tan x + \tan \varepsilon\right) \cdot \left(\left(1 - \tan x \cdot \left(\tan x \cdot \left(\tan \varepsilon \cdot \tan \varepsilon\right)\right)\right) \cdot \cos x\right) + \left(1 - \tan x \cdot \left(\tan x \cdot \left(\tan \varepsilon \cdot \tan \varepsilon\right)\right)\right) \cdot \left(\sin x \cdot \left(\tan x \cdot \tan \varepsilon + -1\right)\right)}{\left(1 - \tan x \cdot \left(\tan x \cdot \left(\tan \varepsilon \cdot \tan \varepsilon\right)\right)\right) \cdot \left(\cos x \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)\right)}\\ \mathbf{elif}\;\varepsilon \le 1.2407832468725051 \cdot 10^{-72}:\\ \;\;\;\;\varepsilon + x \cdot \left(\varepsilon \cdot \left(\varepsilon + x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}\right)}^{3}} - \tan x\\ \end{array}\]

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 <= -1.3369941042866534e-17)) {
		VAR = ((double) (((double) (((double) (((double) (((double) tan(x)) + ((double) tan(eps)))) * ((double) (((double) (1.0 - ((double) (((double) tan(x)) * ((double) (((double) tan(x)) * ((double) (((double) tan(eps)) * ((double) tan(eps)))))))))) * ((double) cos(x)))))) + ((double) (((double) (1.0 - ((double) (((double) tan(x)) * ((double) (((double) tan(x)) * ((double) (((double) tan(eps)) * ((double) tan(eps)))))))))) * ((double) (((double) sin(x)) * ((double) (((double) (((double) tan(x)) * ((double) tan(eps)))) + -1.0)))))))) / ((double) (((double) (1.0 - ((double) (((double) tan(x)) * ((double) (((double) tan(x)) * ((double) (((double) tan(eps)) * ((double) tan(eps)))))))))) * ((double) (((double) cos(x)) * ((double) (1.0 - ((double) (((double) tan(x)) * ((double) tan(eps))))))))))));
	} else {
		double VAR_1;
		if ((eps <= 1.240783246872505e-72)) {
			VAR_1 = ((double) (eps + ((double) (x * ((double) (eps * ((double) (eps + x))))))));
		} else {
			VAR_1 = ((double) (((double) cbrt(((double) pow(((double) (((double) (((double) tan(x)) + ((double) tan(eps)))) / ((double) (1.0 - ((double) (((double) tan(x)) * ((double) tan(eps)))))))), 3.0)))) - ((double) tan(x))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	37.5
Target	15.9
Herbie	15.3

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

Derivation

1. Split input into 3 regimes

2. if eps < -1.3369941042866534e-17

   1. Initial program 30.0

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

   3. Applied tan-sum 0.9

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

   5. Applied flip-- 0.9

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

   6. Applied associate-/r/ 0.9

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

   7. Simplified 0.9

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

   9. Applied tan-quot 1.0

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

   10. Applied flip-+ 1.0

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

   11. Applied frac-times 1.0

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

   12. Applied frac-sub 1.0

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

   13. Simplified 1.0

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

   14. Simplified 1.0

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

   if -1.3369941042866534e-17 < eps < 1.2407832468725051e-72

   1. Initial program 46.0

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

   2. Taylor expanded around 0 31.0

      \[\leadsto \color{blue}{x \cdot {\varepsilon}^{2} + \left(\varepsilon + {x}^{2} \cdot \varepsilon\right)}\]

   3. Simplified 30.8

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

   if 1.2407832468725051e-72 < eps

   1. Initial program 32.6

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

   3. Applied tan-sum 6.1

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

   5. Applied add-cbrt-cube 6.2

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

   6. Applied add-cbrt-cube 6.4

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

   7. Applied cbrt-undiv 6.4

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

   8. Simplified 6.4

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

4. Final simplification 15.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1.3369941042866534 \cdot 10^{-17}:\\ \;\;\;\;\frac{\left(\tan x + \tan \varepsilon\right) \cdot \left(\left(1 - \tan x \cdot \left(\tan x \cdot \left(\tan \varepsilon \cdot \tan \varepsilon\right)\right)\right) \cdot \cos x\right) + \left(1 - \tan x \cdot \left(\tan x \cdot \left(\tan \varepsilon \cdot \tan \varepsilon\right)\right)\right) \cdot \left(\sin x \cdot \left(\tan x \cdot \tan \varepsilon + -1\right)\right)}{\left(1 - \tan x \cdot \left(\tan x \cdot \left(\tan \varepsilon \cdot \tan \varepsilon\right)\right)\right) \cdot \left(\cos x \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)\right)}\\ \mathbf{elif}\;\varepsilon \le 1.2407832468725051 \cdot 10^{-72}:\\ \;\;\;\;\varepsilon + x \cdot \left(\varepsilon \cdot \left(\varepsilon + x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}\right)}^{3}} - \tan x\\ \end{array}\]

Reproduce

herbie shell --seed 2020184 
(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)))