2tan (problem 3.3.2)

Average Error: 36.7 → 15.3

Time: 8.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -8.1479570759041135 \cdot 10^{-55}:\\ \;\;\;\;\frac{\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \sin x}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}\\ \mathbf{elif}\;\varepsilon \le 1.56534825561633359 \cdot 10^{-66}:\\ \;\;\;\;\left(\varepsilon \cdot x\right) \cdot \left(x + \varepsilon\right) + \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\tan x + \tan \varepsilon\right) \cdot \left(1 + \tan x \cdot \tan \varepsilon\right)\right) \cdot \cos x - \left(1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)\right) \cdot \sin x}{\left(1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)\right) \cdot \cos x}\\ \end{array}\]

C

double code(double x, double eps) {
	return (tan((x + eps)) - tan(x));
}

↓

double code(double x, double eps) {
	double VAR;
	if ((eps <= -8.147957075904114e-55)) {
		VAR = ((((tan(x) + tan(eps)) * cos(x)) - ((1.0 - (tan(x) * tan(eps))) * sin(x))) / ((1.0 - (tan(x) * tan(eps))) * cos(x)));
	} else {
		double VAR_1;
		if ((eps <= 1.5653482556163336e-66)) {
			VAR_1 = (((eps * x) * (x + eps)) + eps);
		} else {
			VAR_1 = (((((tan(x) + tan(eps)) * (1.0 + (tan(x) * tan(eps)))) * cos(x)) - ((1.0 - ((tan(x) * tan(eps)) * (tan(x) * tan(eps)))) * sin(x))) / ((1.0 - ((tan(x) * tan(eps)) * (tan(x) * tan(eps)))) * cos(x)));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	36.7
Target	14.8
Herbie	15.3

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

Derivation

1. Split input into 3 regimes

2. if eps < -8.147957075904114e-55

   1. Initial program 29.6

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

   3. Applied tan-quot 29.5

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

   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

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

   if -8.147957075904114e-55 < eps < 1.5653482556163336e-66

   1. Initial program 47.2

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

   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 1.5653482556163336e-66 < eps

   1. Initial program 29.9

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

   3. Applied tan-sum 5.4

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

   5. Applied flip-- 5.4

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

   6. Applied associate-/r/ 5.4

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

   7. Simplified 5.4

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

   9. Applied tan-quot 5.5

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

   10. Applied associate-*l/ 5.5

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

   11. Applied frac-sub 5.5

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

4. Final simplification 15.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -8.1479570759041135 \cdot 10^{-55}:\\ \;\;\;\;\frac{\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \sin x}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}\\ \mathbf{elif}\;\varepsilon \le 1.56534825561633359 \cdot 10^{-66}:\\ \;\;\;\;\left(\varepsilon \cdot x\right) \cdot \left(x + \varepsilon\right) + \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\tan x + \tan \varepsilon\right) \cdot \left(1 + \tan x \cdot \tan \varepsilon\right)\right) \cdot \cos x - \left(1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)\right) \cdot \sin x}{\left(1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)\right) \cdot \cos x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020105 
(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)))