2tan (problem 3.3.2)

Average Error: 37.0 → 15.5

Time: 7.6s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -4.92520080088448257 \cdot 10^{-61}:\\ \;\;\;\;\frac{\left(\tan x + \tan \varepsilon\right) \cdot \cos x + \sin x \cdot \left(\tan x \cdot \tan \varepsilon + -1\right)}{\cos x \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)}\\ \mathbf{elif}\;\varepsilon \le 5.09795473738253029 \cdot 10^{-64}:\\ \;\;\;\;\varepsilon + x \cdot \left(\varepsilon \cdot \left(\varepsilon + x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \left(\tan x \cdot \left(\tan \varepsilon \cdot \tan \varepsilon\right)\right)} \cdot \left(\left(\tan x \cdot \tan \varepsilon + 1\right) \cdot \left(\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \left(\tan x \cdot \left(\tan \varepsilon \cdot \tan \varepsilon\right)\right)} \cdot \left(\tan x \cdot \tan \varepsilon + 1\right)\right)\right) - \tan x \cdot \tan x}{\tan x + \frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \left(\tan x \cdot \left(\tan \varepsilon \cdot \tan \varepsilon\right)\right)} \cdot \left(\tan x \cdot \tan \varepsilon + 1\right)}\\ \end{array}\]

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 <= -4.9252008008844826e-61)) {
		VAR = ((double) (((double) (((double) (((double) (((double) tan(x)) + ((double) tan(eps)))) * ((double) cos(x)))) + ((double) (((double) sin(x)) * ((double) (((double) (((double) tan(x)) * ((double) tan(eps)))) + -1.0)))))) / ((double) (((double) cos(x)) * ((double) (1.0 - ((double) (((double) tan(x)) * ((double) tan(eps))))))))));
	} else {
		double VAR_1;
		if ((eps <= 5.09795473738253e-64)) {
			VAR_1 = ((double) (eps + ((double) (x * ((double) (eps * ((double) (eps + x))))))));
		} else {
			VAR_1 = ((double) (((double) (((double) (((double) (((double) (((double) tan(x)) + ((double) tan(eps)))) / ((double) (1.0 - ((double) (((double) tan(x)) * ((double) (((double) tan(x)) * ((double) (((double) tan(eps)) * ((double) tan(eps)))))))))))) * ((double) (((double) (((double) (((double) tan(x)) * ((double) tan(eps)))) + 1.0)) * ((double) (((double) (((double) (((double) tan(x)) + ((double) tan(eps)))) / ((double) (1.0 - ((double) (((double) tan(x)) * ((double) (((double) tan(x)) * ((double) (((double) tan(eps)) * ((double) tan(eps)))))))))))) * ((double) (((double) (((double) tan(x)) * ((double) tan(eps)))) + 1.0)))))))) - ((double) (((double) tan(x)) * ((double) tan(x)))))) / ((double) (((double) tan(x)) + ((double) (((double) (((double) (((double) tan(x)) + ((double) tan(eps)))) / ((double) (1.0 - ((double) (((double) tan(x)) * ((double) (((double) tan(x)) * ((double) (((double) tan(eps)) * ((double) tan(eps)))))))))))) * ((double) (((double) (((double) tan(x)) * ((double) tan(eps)))) + 1.0))))))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	37.0
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 < -4.92520080088448257e-61

   1. Initial program 30.6

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

   3. Applied tan-quot 30.5

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

   4. Applied tan-sum 4.4

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

   5. Applied frac-sub 4.5

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

   6. Simplified 4.5

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

   if -4.92520080088448257e-61 < eps < 5.09795473738253029e-64

   1. Initial program 47.4

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

   2. Taylor expanded around 0 31.5

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

   3. Simplified 31.3

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

   if 5.09795473738253029e-64 < eps

   1. Initial program 29.5

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

   3. Applied tan-sum 5.5

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

   5. Applied flip-- 5.6

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

   6. Applied associate-/r/ 5.6

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

   7. Simplified 5.6

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

   9. Applied flip-- 5.7

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

   10. Simplified 5.7

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

   11. Simplified 5.7

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

4. Final simplification 15.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -4.92520080088448257 \cdot 10^{-61}:\\ \;\;\;\;\frac{\left(\tan x + \tan \varepsilon\right) \cdot \cos x + \sin x \cdot \left(\tan x \cdot \tan \varepsilon + -1\right)}{\cos x \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)}\\ \mathbf{elif}\;\varepsilon \le 5.09795473738253029 \cdot 10^{-64}:\\ \;\;\;\;\varepsilon + x \cdot \left(\varepsilon \cdot \left(\varepsilon + x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \left(\tan x \cdot \left(\tan \varepsilon \cdot \tan \varepsilon\right)\right)} \cdot \left(\left(\tan x \cdot \tan \varepsilon + 1\right) \cdot \left(\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \left(\tan x \cdot \left(\tan \varepsilon \cdot \tan \varepsilon\right)\right)} \cdot \left(\tan x \cdot \tan \varepsilon + 1\right)\right)\right) - \tan x \cdot \tan x}{\tan x + \frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \left(\tan x \cdot \left(\tan \varepsilon \cdot \tan \varepsilon\right)\right)} \cdot \left(\tan x \cdot \tan \varepsilon + 1\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020185 
(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)))