2tan (problem 3.3.2)

Average Error: 36.9 → 15.2

Time: 7.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1.44770063686003076 \cdot 10^{-29}:\\ \;\;\;\;\frac{\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \frac{1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)}{1 + \tan x \cdot \tan \varepsilon} \cdot \sin x}{\frac{1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)}{1 + \tan x \cdot \tan \varepsilon} \cdot \cos x}\\ \mathbf{elif}\;\varepsilon \le 1.26608976468615072 \cdot 10^{-42}:\\ \;\;\;\;\left(\varepsilon \cdot x\right) \cdot \left(x + \varepsilon\right) + \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{\frac{1 - \frac{\left(\tan x \cdot \tan x\right) \cdot {\left(\sin \varepsilon\right)}^{2}}{{\left(\cos \varepsilon\right)}^{2}}}{1 + \tan x \cdot \tan \varepsilon}} - \tan x\\ \end{array}\]

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 <= -1.4477006368600308e-29)) {
		VAR = ((double) (((double) (((double) (((double) (((double) tan(x)) + ((double) tan(eps)))) * ((double) cos(x)))) - ((double) (((double) (((double) (1.0 - ((double) (((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) sin(x)))))) / ((double) (((double) (((double) (1.0 - ((double) (((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) cos(x))))));
	} else {
		double VAR_1;
		if ((eps <= 1.2660897646861507e-42)) {
			VAR_1 = ((double) (((double) (((double) (eps * x)) * ((double) (x + eps)))) + eps));
		} else {
			VAR_1 = ((double) (((double) (((double) (((double) tan(x)) + ((double) tan(eps)))) / ((double) (((double) (1.0 - ((double) (((double) (((double) (((double) tan(x)) * ((double) tan(x)))) * ((double) pow(((double) sin(eps)), 2.0)))) / ((double) pow(((double) cos(eps)), 2.0)))))) / ((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	36.9
Target	15.1
Herbie	15.2

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

Derivation

1. Split input into 3 regimes

2. if eps < -1.4477006368600308e-29

   1. Initial program 28.8

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

   3. Applied tan-sum 2.0

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

   5. Applied flip-- 2.0

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

   6. Simplified 2.0

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

   8. Applied tan-quot 2.1

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

   9. Applied frac-sub 2.1

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

   if -1.4477006368600308e-29 < eps < 1.2660897646861507e-42

   1. Initial program 45.8

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

   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

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

   if 1.2660897646861507e-42 < eps

   1. Initial program 30.8

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

   3. Applied tan-sum 3.4

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

   5. Applied flip-- 3.4

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

   6. Simplified 3.4

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

   8. Applied tan-quot 3.4

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

   9. Applied associate-*r/ 3.5

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

   10. Applied tan-quot 3.5

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

   11. Applied associate-*r/ 3.5

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

   12. Applied frac-times 3.5

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

   13. Simplified 3.5

       \[\leadsto \frac{\tan x + \tan \varepsilon}{\frac{1 - \frac{\color{blue}{\left(\tan x \cdot \tan x\right) \cdot {\left(\sin \varepsilon\right)}^{2}}}{\cos \varepsilon \cdot \cos \varepsilon}}{1 + \tan x \cdot \tan \varepsilon}} - \tan x\]

   14. Simplified 3.5

       \[\leadsto \frac{\tan x + \tan \varepsilon}{\frac{1 - \frac{\left(\tan x \cdot \tan x\right) \cdot {\left(\sin \varepsilon\right)}^{2}}{\color{blue}{{\left(\cos \varepsilon\right)}^{2}}}}{1 + \tan x \cdot \tan \varepsilon}} - \tan x\]
3. Recombined 3 regimes into one program.

4. Final simplification 15.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1.44770063686003076 \cdot 10^{-29}:\\ \;\;\;\;\frac{\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \frac{1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)}{1 + \tan x \cdot \tan \varepsilon} \cdot \sin x}{\frac{1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)}{1 + \tan x \cdot \tan \varepsilon} \cdot \cos x}\\ \mathbf{elif}\;\varepsilon \le 1.26608976468615072 \cdot 10^{-42}:\\ \;\;\;\;\left(\varepsilon \cdot x\right) \cdot \left(x + \varepsilon\right) + \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{\frac{1 - \frac{\left(\tan x \cdot \tan x\right) \cdot {\left(\sin \varepsilon\right)}^{2}}{{\left(\cos \varepsilon\right)}^{2}}}{1 + \tan x \cdot \tan \varepsilon}} - \tan x\\ \end{array}\]

Reproduce

herbie shell --seed 2020130 
(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)))