2tan (problem 3.3.2)

Average Error: 36.8 → 16.0

Time: 10.8s

Precision: binary64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -3.87985933356410738 \cdot 10^{-34} \lor \neg \left(\varepsilon \le 8.96237258412460441 \cdot 10^{-99}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{\frac{1 - \sqrt[3]{{\left({\left(\tan x \cdot \tan \varepsilon\right)}^{3}\right)}^{3}}}{\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(1 + \tan x \cdot \tan \varepsilon\right) + 1}} - \tan x\\ \mathbf{else}:\\ \;\;\;\;x \cdot {\varepsilon}^{2} + \left(\varepsilon + {x}^{2} \cdot \varepsilon\right)\\ \end{array}\]

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 <= -3.879859333564107e-34) || !(eps <= 8.962372584124604e-99))) {
		VAR = ((double) ((((double) (((double) tan(x)) + ((double) tan(eps)))) / (((double) (1.0 - ((double) cbrt(((double) pow(((double) pow(((double) (((double) tan(x)) * ((double) tan(eps)))), 3.0)), 3.0)))))) / ((double) (((double) (((double) (((double) tan(x)) * ((double) tan(eps)))) * ((double) (1.0 + ((double) (((double) tan(x)) * ((double) tan(eps)))))))) + 1.0)))) - ((double) tan(x))));
	} else {
		VAR = ((double) (((double) (x * ((double) pow(eps, 2.0)))) + ((double) (eps + ((double) (((double) pow(x, 2.0)) * eps))))));
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	36.8
Target	14.9
Herbie	16.0

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

Derivation

1. Split input into 2 regimes

2. if eps < -3.87985933356410738e-34 or 8.96237258412460441e-99 < eps

   1. Initial program 29.9

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

   3. Applied tan-sum 5.4

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

   5. Applied flip3-- 5.5

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

   6. Simplified 5.5

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

   7. Simplified 5.5

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

   9. Applied add-cbrt-cube 5.5

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

   10. Simplified 5.5

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

   if -3.87985933356410738e-34 < eps < 8.96237258412460441e-99

   1. Initial program 47.6

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

   3. Applied tan-sum 47.6

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

   5. Applied flip3-- 47.6

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

   6. Simplified 47.6

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

   7. Simplified 47.6

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

   8. Taylor expanded around 0 32.5

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

4. Final simplification 16.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -3.87985933356410738 \cdot 10^{-34} \lor \neg \left(\varepsilon \le 8.96237258412460441 \cdot 10^{-99}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{\frac{1 - \sqrt[3]{{\left({\left(\tan x \cdot \tan \varepsilon\right)}^{3}\right)}^{3}}}{\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(1 + \tan x \cdot \tan \varepsilon\right) + 1}} - \tan x\\ \mathbf{else}:\\ \;\;\;\;x \cdot {\varepsilon}^{2} + \left(\varepsilon + {x}^{2} \cdot \varepsilon\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020182 
(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)))