2tan (problem 3.3.2)

Average Error: 36.6 → 14.5

Time: 7.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -3.6640770997684913 \cdot 10^{-52} \lor \neg \left(\varepsilon \le 9.0088976236056553 \cdot 10^{-105}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right)} \cdot \left(\tan \varepsilon \cdot \tan x\right) + \left(1 \cdot \frac{\tan x + \tan \varepsilon}{1 - \left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right)} - \tan x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \varepsilon\right) \cdot \left(\varepsilon + x\right) + \varepsilon\\ \end{array}\]

C

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

↓

double code(double x, double eps) {
	double VAR;
	if (((eps <= -3.664077099768491e-52) || !(eps <= 9.008897623605655e-105))) {
		VAR = ((((tan(x) + tan(eps)) / (1.0 - ((tan(eps) * tan(x)) * (tan(eps) * tan(x))))) * (tan(eps) * tan(x))) + ((1.0 * ((tan(x) + tan(eps)) / (1.0 - ((tan(eps) * tan(x)) * (tan(eps) * tan(x)))))) - tan(x)));
	} else {
		VAR = (((x * eps) * (eps + x)) + eps);
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	36.6
Target	15.0
Herbie	14.5

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

Derivation

1. Split input into 2 regimes

2. if eps < -3.664077099768491e-52 or 9.008897623605655e-105 < eps

   1. Initial program 29.8

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

   3. Applied tan-sum 5.9

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

   4. Simplified 5.9

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

   6. Applied flip-- 5.9

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

   7. Applied associate-/r/ 5.9

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

   8. Simplified 5.9

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

   10. Applied +-commutative 5.9

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

   11. Applied distribute-lft-in 5.9

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

   12. Applied associate--l+ 5.1

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

   13. Simplified 5.1

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

   if -3.664077099768491e-52 < eps < 9.008897623605655e-105

   1. Initial program 48.1

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

   3. Applied tan-sum 48.1

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

   4. Simplified 48.1

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

   5. Taylor expanded around 0 30.7

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

   6. Simplified 30.4

      \[\leadsto \color{blue}{\left(x \cdot \varepsilon\right) \cdot \left(\varepsilon + x\right) + \varepsilon}\]
3. Recombined 2 regimes into one program.

4. Final simplification 14.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -3.6640770997684913 \cdot 10^{-52} \lor \neg \left(\varepsilon \le 9.0088976236056553 \cdot 10^{-105}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right)} \cdot \left(\tan \varepsilon \cdot \tan x\right) + \left(1 \cdot \frac{\tan x + \tan \varepsilon}{1 - \left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right)} - \tan x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \varepsilon\right) \cdot \left(\varepsilon + x\right) + \varepsilon\\ \end{array}\]

Reproduce

herbie shell --seed 2020071 
(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)))