2tan (problem 3.3.2)

Average Error: 37.5 → 15.6

Time: 8.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -6.4124348219606893 \cdot 10^{-18} \lor \neg \left(\varepsilon \le 1.23249914217843071 \cdot 10^{-89}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\sin x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon}} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot x\right) \cdot \left(x + \varepsilon\right) + \varepsilon\\ \end{array}\]

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 <= -6.412434821960689e-18) || !(eps <= 1.2324991421784307e-89))) {
		VAR = ((double) (((double) (((double) (((double) tan(x)) + ((double) tan(eps)))) / ((double) (1.0 - ((double) (((double) (((double) sin(x)) * ((double) sin(eps)))) / ((double) (((double) cos(x)) * ((double) cos(eps)))))))))) - ((double) tan(x))));
	} else {
		VAR = ((double) (((double) (((double) (eps * x)) * ((double) (x + eps)))) + eps));
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	37.5
Target	15.3
Herbie	15.6

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

Derivation

1. Split input into 2 regimes

2. if eps < -6.4124348219606893e-18 or 1.23249914217843071e-89 < eps

   1. Initial program 31.2

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

   3. Applied tan-sum 4.5

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

   5. Applied tan-quot 4.5

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

   6. Applied tan-quot 4.5

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

   7. Applied frac-times 4.5

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

   if -6.4124348219606893e-18 < eps < 1.23249914217843071e-89

   1. Initial program 46.0

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

   2. Taylor expanded around 0 31.1

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

   3. Simplified 30.9

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

4. Final simplification 15.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -6.4124348219606893 \cdot 10^{-18} \lor \neg \left(\varepsilon \le 1.23249914217843071 \cdot 10^{-89}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\sin x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon}} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot x\right) \cdot \left(x + \varepsilon\right) + \varepsilon\\ \end{array}\]

Reproduce

herbie shell --seed 2020177 
(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)))