2tan (problem 3.3.2)

Average Error: 36.6 → 15.8

Time: 11.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1.4968426811240258 \cdot 10^{-90}:\\ \;\;\;\;\frac{1}{\frac{1 - \tan x \cdot \tan \varepsilon}{\tan x + \tan \varepsilon}} - \tan x\\ \mathbf{elif}\;\varepsilon \le 5.7485271720269307 \cdot 10^{-167}:\\ \;\;\;\;\mathsf{fma}\left({\varepsilon}^{2}, x, \mathsf{fma}\left(\varepsilon, {x}^{2}, \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\tan x + \tan \varepsilon}{{1}^{3} - {\left(\frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}\right)}^{3}}, 1 \cdot 1 + \left(\frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon} \cdot \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon} + 1 \cdot \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}\right), -\tan x\right)\\ \end{array}\]

C

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

↓

double code(double x, double eps) {
	double temp;
	if ((eps <= -1.4968426811240258e-90)) {
		temp = ((1.0 / ((1.0 - (tan(x) * tan(eps))) / (tan(x) + tan(eps)))) - tan(x));
	} else {
		double temp_1;
		if ((eps <= 5.748527172026931e-167)) {
			temp_1 = fma(pow(eps, 2.0), x, fma(eps, pow(x, 2.0), eps));
		} else {
			temp_1 = fma(((tan(x) + tan(eps)) / (pow(1.0, 3.0) - pow(((tan(x) * sin(eps)) / cos(eps)), 3.0))), ((1.0 * 1.0) + ((((tan(x) * sin(eps)) / cos(eps)) * ((tan(x) * sin(eps)) / cos(eps))) + (1.0 * ((tan(x) * sin(eps)) / cos(eps))))), -tan(x));
		}
		temp = temp_1;
	}
	return temp;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	36.6
Target	15.4
Herbie	15.8

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

Derivation

1. Split input into 3 regimes

2. if eps < -1.4968426811240258e-90

   1. Initial program 30.9

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

   3. Applied tan-sum 6.8

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

   5. Applied clear-num 6.9

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

   if -1.4968426811240258e-90 < eps < 5.748527172026931e-167

   1. Initial program 48.9

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

   2. Taylor expanded around 0 30.0

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

   3. Simplified 30.0

      \[\leadsto \color{blue}{\mathsf{fma}\left({\varepsilon}^{2}, x, \mathsf{fma}\left(\varepsilon, {x}^{2}, \varepsilon\right)\right)}\]

   if 5.748527172026931e-167 < eps

   1. Initial program 32.4

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

   3. Applied tan-sum 13.0

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

   5. Applied tan-quot 13.1

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

   6. Applied associate-*r/ 13.1

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

   8. Applied flip3-- 13.1

      \[\leadsto \frac{\tan x + \tan \varepsilon}{\color{blue}{\frac{{1}^{3} - {\left(\frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}\right)}^{3}}{1 \cdot 1 + \left(\frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon} \cdot \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon} + 1 \cdot \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}\right)}}} - \tan x\]

   9. Applied associate-/r/ 13.1

      \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{{1}^{3} - {\left(\frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}\right)}^{3}} \cdot \left(1 \cdot 1 + \left(\frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon} \cdot \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon} + 1 \cdot \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}\right)\right)} - \tan x\]

   10. Applied fma-neg 13.1

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

4. Final simplification 15.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1.4968426811240258 \cdot 10^{-90}:\\ \;\;\;\;\frac{1}{\frac{1 - \tan x \cdot \tan \varepsilon}{\tan x + \tan \varepsilon}} - \tan x\\ \mathbf{elif}\;\varepsilon \le 5.7485271720269307 \cdot 10^{-167}:\\ \;\;\;\;\mathsf{fma}\left({\varepsilon}^{2}, x, \mathsf{fma}\left(\varepsilon, {x}^{2}, \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\tan x + \tan \varepsilon}{{1}^{3} - {\left(\frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}\right)}^{3}}, 1 \cdot 1 + \left(\frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon} \cdot \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon} + 1 \cdot \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}\right), -\tan x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 +o rules:numerics
(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)))