2tan (problem 3.3.2)

Average Error: 36.6 → 15.5

Time: 8.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -2.22121376769650082 \cdot 10^{-17}:\\ \;\;\;\;\frac{\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \sin x}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}\\ \mathbf{elif}\;\varepsilon \le 3.48145867597643924 \cdot 10^{-112}:\\ \;\;\;\;\left(\varepsilon \cdot x\right) \cdot \left(x + \varepsilon\right) + \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x \cdot \tan x - \tan \varepsilon \cdot \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} \cdot \frac{1}{\tan x - \tan \varepsilon} - \tan x\\ \end{array}\]

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 <= -2.2212137676965008e-17)) {
		VAR = ((double) (((double) (((double) (((double) (((double) tan(x)) + ((double) tan(eps)))) * ((double) cos(x)))) - ((double) (((double) (1.0 - ((double) (((double) tan(x)) * ((double) tan(eps)))))) * ((double) sin(x)))))) / ((double) (((double) (1.0 - ((double) (((double) tan(x)) * ((double) tan(eps)))))) * ((double) cos(x))))));
	} else {
		double VAR_1;
		if ((eps <= 3.4814586759764392e-112)) {
			VAR_1 = ((double) (((double) (((double) (eps * x)) * ((double) (x + eps)))) + eps));
		} else {
			VAR_1 = ((double) (((double) (((double) (((double) (((double) (((double) tan(x)) * ((double) tan(x)))) - ((double) (((double) tan(eps)) * ((double) tan(eps)))))) / ((double) (1.0 - ((double) (((double) tan(x)) * ((double) tan(eps)))))))) * ((double) (1.0 / ((double) (((double) tan(x)) - ((double) tan(eps)))))))) - ((double) tan(x))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	36.6
Target	15.0
Herbie	15.5

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

Derivation

1. Split input into 3 regimes

2. if eps < -2.2212137676965008e-17

   1. Initial program 28.7

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

   3. Applied tan-quot 28.6

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

   4. Applied tan-sum 1.0

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

   5. Applied frac-sub 1.1

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

   if -2.2212137676965008e-17 < eps < 3.4814586759764392e-112

   1. Initial program 46.3

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

   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

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

   if 3.4814586759764392e-112 < eps

   1. Initial program 31.4

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

   3. Applied tan-sum 8.7

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

   5. Applied clear-num 8.8

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

   7. Applied flip-+ 8.9

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

   8. Applied associate-/r/ 9.0

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

   9. Applied add-cube-cbrt 9.0

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

   10. Applied times-frac 9.0

       \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\frac{1 - \tan x \cdot \tan \varepsilon}{\tan x \cdot \tan x - \tan \varepsilon \cdot \tan \varepsilon}} \cdot \frac{\sqrt[3]{1}}{\tan x - \tan \varepsilon}} - \tan x\]

   11. Simplified 8.9

       \[\leadsto \color{blue}{\frac{\tan x \cdot \tan x - \tan \varepsilon \cdot \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} \cdot \frac{\sqrt[3]{1}}{\tan x - \tan \varepsilon} - \tan x\]

   12. Simplified 8.9

       \[\leadsto \frac{\tan x \cdot \tan x - \tan \varepsilon \cdot \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} \cdot \color{blue}{\frac{1}{\tan x - \tan \varepsilon}} - \tan x\]
3. Recombined 3 regimes into one program.

4. Final simplification 15.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -2.22121376769650082 \cdot 10^{-17}:\\ \;\;\;\;\frac{\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \sin x}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}\\ \mathbf{elif}\;\varepsilon \le 3.48145867597643924 \cdot 10^{-112}:\\ \;\;\;\;\left(\varepsilon \cdot x\right) \cdot \left(x + \varepsilon\right) + \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x \cdot \tan x - \tan \varepsilon \cdot \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} \cdot \frac{1}{\tan x - \tan \varepsilon} - \tan x\\ \end{array}\]

Reproduce

herbie shell --seed 2020140 
(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)))