2tan (problem 3.3.2)

Average Error: 37.0 → 15.1

Time: 10.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1.8570209891564074 \cdot 10^{-31}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\tan \varepsilon, \tan x, 1\right)}{1 + \sqrt{\left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right)}}, \frac{\tan x + \tan \varepsilon}{1 - \sqrt{\left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right)}}, -\tan x\right) + \mathsf{fma}\left(-\tan x, 1, \tan x\right)\\ \mathbf{elif}\;\varepsilon \le 2.6139865205108016 \cdot 10^{-45}:\\ \;\;\;\;\mathsf{fma}\left({\varepsilon}^{2}, x, \mathsf{fma}\left(\varepsilon, {x}^{2}, \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\left(\tan x + \tan \varepsilon\right) \cdot \mathsf{fma}\left(\tan \varepsilon, \tan x, 1\right)}{1 - \sqrt[3]{{\left(\tan \varepsilon\right)}^{3} \cdot \left({\left(\tan x\right)}^{3} \cdot \left(\left(\left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right)\right) \cdot \left(\tan \varepsilon \cdot \tan x\right)\right)\right)}} - \tan x\right) + \mathsf{fma}\left(-\tan x, 1, \tan x\right)\\ \end{array}\]

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 <= -1.8570209891564074e-31)) {
		VAR = ((double) (((double) fma(((double) (((double) fma(((double) tan(eps)), ((double) tan(x)), 1.0)) / ((double) (1.0 + ((double) sqrt(((double) (((double) (((double) tan(eps)) * ((double) tan(x)))) * ((double) (((double) tan(eps)) * ((double) tan(x)))))))))))), ((double) (((double) (((double) tan(x)) + ((double) tan(eps)))) / ((double) (1.0 - ((double) sqrt(((double) (((double) (((double) tan(eps)) * ((double) tan(x)))) * ((double) (((double) tan(eps)) * ((double) tan(x)))))))))))), ((double) -(((double) tan(x)))))) + ((double) fma(((double) -(((double) tan(x)))), 1.0, ((double) tan(x))))));
	} else {
		double VAR_1;
		if ((eps <= 2.6139865205108016e-45)) {
			VAR_1 = ((double) fma(((double) pow(eps, 2.0)), x, ((double) fma(eps, ((double) pow(x, 2.0)), eps))));
		} else {
			VAR_1 = ((double) (((double) (((double) (((double) (((double) (((double) tan(x)) + ((double) tan(eps)))) * ((double) fma(((double) tan(eps)), ((double) tan(x)), 1.0)))) / ((double) (1.0 - ((double) cbrt(((double) (((double) pow(((double) tan(eps)), 3.0)) * ((double) (((double) pow(((double) tan(x)), 3.0)) * ((double) (((double) (((double) (((double) tan(eps)) * ((double) tan(x)))) * ((double) (((double) tan(eps)) * ((double) tan(x)))))) * ((double) (((double) tan(eps)) * ((double) tan(x)))))))))))))))) - ((double) tan(x)))) + ((double) fma(((double) -(((double) tan(x)))), 1.0, ((double) tan(x))))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	37.0
Target	15.0
Herbie	15.1

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

Derivation

1. Split input into 3 regimes

2. if eps < -1.8570209891564074e-31

   1. Initial program 29.8

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

   3. Applied tan-sum 2.4

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

   4. Simplified 2.4

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

   6. Applied add-cube-cbrt 2.7

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

   7. Applied flip-- 2.8

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

   8. Applied associate-/r/ 2.8

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

   9. Applied prod-diff 2.7

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\tan x + \tan \varepsilon}{1 \cdot 1 - \left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right)}, 1 + \tan \varepsilon \cdot \tan x, -\sqrt[3]{\tan x} \cdot \left(\sqrt[3]{\tan x} \cdot \sqrt[3]{\tan x}\right)\right) + \mathsf{fma}\left(-\sqrt[3]{\tan x}, \sqrt[3]{\tan x} \cdot \sqrt[3]{\tan x}, \sqrt[3]{\tan x} \cdot \left(\sqrt[3]{\tan x} \cdot \sqrt[3]{\tan x}\right)\right)}\]

   10. Simplified 2.4

       \[\leadsto \color{blue}{\left(\frac{\left(\tan x + \tan \varepsilon\right) \cdot \mathsf{fma}\left(\tan \varepsilon, \tan x, 1\right)}{1 - \left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right)} - \tan x\right)} + \mathsf{fma}\left(-\sqrt[3]{\tan x}, \sqrt[3]{\tan x} \cdot \sqrt[3]{\tan x}, \sqrt[3]{\tan x} \cdot \left(\sqrt[3]{\tan x} \cdot \sqrt[3]{\tan x}\right)\right)\]

   11. Simplified 2.5

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

   13. Applied add-sqr-sqrt 2.5

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

   14. Applied *-un-lft-identity 2.5

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

   15. Applied difference-of-squares 2.5

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

   16. Applied *-commutative 2.5

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

   17. Applied times-frac 2.4

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

   18. Applied fma-neg 2.4

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

   if -1.8570209891564074e-31 < eps < 2.6139865205108016e-45

   1. Initial program 45.7

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

   2. Taylor expanded around 0 30.2

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

   3. Simplified 30.2

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

   if 2.6139865205108016e-45 < eps

   1. Initial program 30.6

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

   3. Applied tan-sum 3.9

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

   4. Simplified 3.9

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

   6. Applied add-cube-cbrt 4.2

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

   7. Applied flip-- 4.2

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

   8. Applied associate-/r/ 4.3

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

   9. Applied prod-diff 4.2

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\tan x + \tan \varepsilon}{1 \cdot 1 - \left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right)}, 1 + \tan \varepsilon \cdot \tan x, -\sqrt[3]{\tan x} \cdot \left(\sqrt[3]{\tan x} \cdot \sqrt[3]{\tan x}\right)\right) + \mathsf{fma}\left(-\sqrt[3]{\tan x}, \sqrt[3]{\tan x} \cdot \sqrt[3]{\tan x}, \sqrt[3]{\tan x} \cdot \left(\sqrt[3]{\tan x} \cdot \sqrt[3]{\tan x}\right)\right)}\]

   10. Simplified 3.9

       \[\leadsto \color{blue}{\left(\frac{\left(\tan x + \tan \varepsilon\right) \cdot \mathsf{fma}\left(\tan \varepsilon, \tan x, 1\right)}{1 - \left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right)} - \tan x\right)} + \mathsf{fma}\left(-\sqrt[3]{\tan x}, \sqrt[3]{\tan x} \cdot \sqrt[3]{\tan x}, \sqrt[3]{\tan x} \cdot \left(\sqrt[3]{\tan x} \cdot \sqrt[3]{\tan x}\right)\right)\]

   11. Simplified 4.0

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

   13. Applied add-cbrt-cube 4.0

       \[\leadsto \left(\frac{\left(\tan x + \tan \varepsilon\right) \cdot \mathsf{fma}\left(\tan \varepsilon, \tan x, 1\right)}{1 - \left(\tan \varepsilon \cdot \tan x\right) \cdot \color{blue}{\sqrt[3]{\left(\left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right)\right) \cdot \left(\tan \varepsilon \cdot \tan x\right)}}} - \tan x\right) + \mathsf{fma}\left(-\tan x, 1, \tan x\right)\]

   14. Applied add-cbrt-cube 4.0

       \[\leadsto \left(\frac{\left(\tan x + \tan \varepsilon\right) \cdot \mathsf{fma}\left(\tan \varepsilon, \tan x, 1\right)}{1 - \left(\tan \varepsilon \cdot \color{blue}{\sqrt[3]{\left(\tan x \cdot \tan x\right) \cdot \tan x}}\right) \cdot \sqrt[3]{\left(\left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right)\right) \cdot \left(\tan \varepsilon \cdot \tan x\right)}} - \tan x\right) + \mathsf{fma}\left(-\tan x, 1, \tan x\right)\]

   15. Applied add-cbrt-cube 4.0

       \[\leadsto \left(\frac{\left(\tan x + \tan \varepsilon\right) \cdot \mathsf{fma}\left(\tan \varepsilon, \tan x, 1\right)}{1 - \left(\color{blue}{\sqrt[3]{\left(\tan \varepsilon \cdot \tan \varepsilon\right) \cdot \tan \varepsilon}} \cdot \sqrt[3]{\left(\tan x \cdot \tan x\right) \cdot \tan x}\right) \cdot \sqrt[3]{\left(\left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right)\right) \cdot \left(\tan \varepsilon \cdot \tan x\right)}} - \tan x\right) + \mathsf{fma}\left(-\tan x, 1, \tan x\right)\]

   16. Applied cbrt-unprod 4.0

       \[\leadsto \left(\frac{\left(\tan x + \tan \varepsilon\right) \cdot \mathsf{fma}\left(\tan \varepsilon, \tan x, 1\right)}{1 - \color{blue}{\sqrt[3]{\left(\left(\tan \varepsilon \cdot \tan \varepsilon\right) \cdot \tan \varepsilon\right) \cdot \left(\left(\tan x \cdot \tan x\right) \cdot \tan x\right)}} \cdot \sqrt[3]{\left(\left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right)\right) \cdot \left(\tan \varepsilon \cdot \tan x\right)}} - \tan x\right) + \mathsf{fma}\left(-\tan x, 1, \tan x\right)\]

   17. Applied cbrt-unprod 4.0

       \[\leadsto \left(\frac{\left(\tan x + \tan \varepsilon\right) \cdot \mathsf{fma}\left(\tan \varepsilon, \tan x, 1\right)}{1 - \color{blue}{\sqrt[3]{\left(\left(\left(\tan \varepsilon \cdot \tan \varepsilon\right) \cdot \tan \varepsilon\right) \cdot \left(\left(\tan x \cdot \tan x\right) \cdot \tan x\right)\right) \cdot \left(\left(\left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right)\right) \cdot \left(\tan \varepsilon \cdot \tan x\right)\right)}}} - \tan x\right) + \mathsf{fma}\left(-\tan x, 1, \tan x\right)\]

   18. Simplified 4.0

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

4. Final simplification 15.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1.8570209891564074 \cdot 10^{-31}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\tan \varepsilon, \tan x, 1\right)}{1 + \sqrt{\left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right)}}, \frac{\tan x + \tan \varepsilon}{1 - \sqrt{\left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right)}}, -\tan x\right) + \mathsf{fma}\left(-\tan x, 1, \tan x\right)\\ \mathbf{elif}\;\varepsilon \le 2.6139865205108016 \cdot 10^{-45}:\\ \;\;\;\;\mathsf{fma}\left({\varepsilon}^{2}, x, \mathsf{fma}\left(\varepsilon, {x}^{2}, \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\left(\tan x + \tan \varepsilon\right) \cdot \mathsf{fma}\left(\tan \varepsilon, \tan x, 1\right)}{1 - \sqrt[3]{{\left(\tan \varepsilon\right)}^{3} \cdot \left({\left(\tan x\right)}^{3} \cdot \left(\left(\left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right)\right) \cdot \left(\tan \varepsilon \cdot \tan x\right)\right)\right)}} - \tan x\right) + \mathsf{fma}\left(-\tan x, 1, \tan x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020113 +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)))