2tan (problem 3.3.2)

Average Error: 41.1 → 17.1

Time: 6.8s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1.1327425688405658 \cdot 10^{-82}:\\ \;\;\;\;\left(\sqrt[3]{{\left(\frac{\sin x}{\left(1 - \frac{\sin x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon}\right) \cdot \cos x}\right)}^{3}} + \frac{\sin \varepsilon}{\left(1 - \frac{\sin x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon}\right) \cdot \cos \varepsilon}\right) - \frac{\sin x}{\cos x}\\ \mathbf{elif}\;\varepsilon \le 1.06382706172173794 \cdot 10^{-30}:\\ \;\;\;\;\left(\varepsilon \cdot x\right) \cdot \left(x + \varepsilon\right) + \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \left(1 - \frac{\sin x \cdot \tan \varepsilon}{\cos x}\right) \cdot \sin x}{\left(1 - \frac{\sin x \cdot \tan \varepsilon}{\cos x}\right) \cdot \cos x}\\ \end{array}\]

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 <= -1.1327425688405658e-82)) {
		VAR = ((double) (((double) (((double) cbrt(((double) pow(((double) (((double) sin(x)) / ((double) (((double) (1.0 - ((double) (((double) (((double) sin(x)) * ((double) sin(eps)))) / ((double) (((double) cos(x)) * ((double) cos(eps)))))))) * ((double) cos(x)))))), 3.0)))) + ((double) (((double) sin(eps)) / ((double) (((double) (1.0 - ((double) (((double) (((double) sin(x)) * ((double) sin(eps)))) / ((double) (((double) cos(x)) * ((double) cos(eps)))))))) * ((double) cos(eps)))))))) - ((double) (((double) sin(x)) / ((double) cos(x))))));
	} else {
		double VAR_1;
		if ((eps <= 1.063827061721738e-30)) {
			VAR_1 = ((double) (((double) (((double) (eps * x)) * ((double) (x + eps)))) + eps));
		} else {
			VAR_1 = ((double) (((double) (((double) (((double) (((double) tan(x)) + ((double) tan(eps)))) * ((double) cos(x)))) - ((double) (((double) (1.0 - ((double) (((double) (((double) sin(x)) * ((double) tan(eps)))) / ((double) cos(x)))))) * ((double) sin(x)))))) / ((double) (((double) (1.0 - ((double) (((double) (((double) sin(x)) * ((double) tan(eps)))) / ((double) cos(x)))))) * ((double) cos(x))))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	41.1
Target	16.6
Herbie	17.1

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

Derivation

1. Split input into 3 regimes

2. if eps < -1.1327425688405658e-82

   1. Initial program 41.0

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

   3. Applied tan-sum 8.5

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

   4. Taylor expanded around inf 8.7

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

   6. Applied add-cbrt-cube 8.7

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

   7. Applied add-cbrt-cube 8.7

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

   8. Applied cbrt-unprod 8.7

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

   9. Applied add-cbrt-cube 8.7

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

   10. Applied cbrt-undiv 8.6

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

   11. Simplified 8.6

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

   if -1.1327425688405658e-82 < eps < 1.06382706172173794e-30

   1. Initial program 46.6

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

   2. Taylor expanded around 0 30.5

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

   3. Simplified 30.3

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

   if 1.06382706172173794e-30 < eps

   1. Initial program 28.9

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

   3. Applied tan-sum 2.1

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

   5. Applied tan-quot 2.1

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

   6. Applied associate-*l/ 2.1

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

   8. Applied tan-quot 2.2

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

   9. Applied frac-sub 2.2

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

4. Final simplification 17.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1.1327425688405658 \cdot 10^{-82}:\\ \;\;\;\;\left(\sqrt[3]{{\left(\frac{\sin x}{\left(1 - \frac{\sin x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon}\right) \cdot \cos x}\right)}^{3}} + \frac{\sin \varepsilon}{\left(1 - \frac{\sin x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon}\right) \cdot \cos \varepsilon}\right) - \frac{\sin x}{\cos x}\\ \mathbf{elif}\;\varepsilon \le 1.06382706172173794 \cdot 10^{-30}:\\ \;\;\;\;\left(\varepsilon \cdot x\right) \cdot \left(x + \varepsilon\right) + \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \left(1 - \frac{\sin x \cdot \tan \varepsilon}{\cos x}\right) \cdot \sin x}{\left(1 - \frac{\sin x \cdot \tan \varepsilon}{\cos x}\right) \cdot \cos x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020156 
(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)))