2tan (problem 3.3.2)

Average Error: 37.2 → 15.9

Time: 7.0s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -7.29880302270468717 \cdot 10^{-70}:\\ \;\;\;\;\frac{\cos x + \sin x \cdot \frac{\tan x \cdot \tan \varepsilon + -1}{\tan x + \tan \varepsilon}}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \frac{\cos x}{\tan x + \tan \varepsilon}}\\ \mathbf{elif}\;\varepsilon \le 6.4523991848902762 \cdot 10^{-56}:\\ \;\;\;\;\varepsilon + x \cdot \left(\varepsilon \cdot \left(\varepsilon + x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} \cdot \left(\tan x - \tan \varepsilon\right)\right) \cdot \frac{1}{\tan x - \tan \varepsilon} - \tan x\\ \end{array}\]

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 <= -7.298803022704687e-70)) {
		VAR = ((double) (((double) (((double) cos(x)) + ((double) (((double) sin(x)) * ((double) (((double) (((double) (((double) tan(x)) * ((double) tan(eps)))) + -1.0)) / ((double) (((double) tan(x)) + ((double) tan(eps)))))))))) / ((double) (((double) (1.0 - ((double) (((double) tan(x)) * ((double) tan(eps)))))) * ((double) (((double) cos(x)) / ((double) (((double) tan(x)) + ((double) tan(eps))))))))));
	} else {
		double VAR_1;
		if ((eps <= 6.452399184890276e-56)) {
			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) tan(eps)))))))) * ((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	37.2
Target	14.9
Herbie	15.9

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

Derivation

1. Split input into 3 regimes

2. if eps < -7.29880302270468717e-70

   1. Initial program 30.1

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

   3. Applied tan-sum 6.0

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

   5. Applied clear-num 6.1

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

   7. Applied tan-quot 6.1

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

   8. Applied frac-sub 6.1

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

   9. Simplified 6.1

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

   10. Simplified 6.2

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

   if -7.29880302270468717e-70 < eps < 6.4523991848902762e-56

   1. Initial program 47.1

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

   2. Taylor expanded around 0 31.5

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

   3. Simplified 31.2

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

   if 6.4523991848902762e-56 < eps

   1. Initial program 30.8

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

   3. Applied tan-sum 4.3

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

   5. Applied clear-num 4.4

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

   7. Applied flip-+ 4.5

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

   8. Applied associate-/r/ 4.5

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

   9. Applied add-sqr-sqrt 4.5

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

   10. Applied times-frac 4.5

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

   11. Simplified 4.4

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

   12. Simplified 4.4

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

4. Final simplification 15.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -7.29880302270468717 \cdot 10^{-70}:\\ \;\;\;\;\frac{\cos x + \sin x \cdot \frac{\tan x \cdot \tan \varepsilon + -1}{\tan x + \tan \varepsilon}}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \frac{\cos x}{\tan x + \tan \varepsilon}}\\ \mathbf{elif}\;\varepsilon \le 6.4523991848902762 \cdot 10^{-56}:\\ \;\;\;\;\varepsilon + x \cdot \left(\varepsilon \cdot \left(\varepsilon + x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} \cdot \left(\tan x - \tan \varepsilon\right)\right) \cdot \frac{1}{\tan x - \tan \varepsilon} - \tan x\\ \end{array}\]

Reproduce

herbie shell --seed 2020179 
(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)))