2tan (problem 3.3.2)

Average Error: 36.8 → 15.2

Time: 10.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -5.97198873028037458 \cdot 10^{-27} \lor \neg \left(\varepsilon \le 4.74228187640458693 \cdot 10^{-23}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{\log \left({e}^{\left(1 - \tan x \cdot \tan \varepsilon\right)} \cdot {e}^{\left(\mathsf{fma}\left(-\tan \varepsilon, \tan x, \tan \varepsilon \cdot \tan x\right)\right)}\right)} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({\varepsilon}^{2}, x, \mathsf{fma}\left(\varepsilon, {x}^{2}, \varepsilon\right)\right)\\ \end{array}\]

C

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

↓

double code(double x, double eps) {
	double temp;
	if (((eps <= -5.9719887302803746e-27) || !(eps <= 4.742281876404587e-23))) {
		temp = (((tan(x) + tan(eps)) / log((pow(((double) M_E), (1.0 - (tan(x) * tan(eps)))) * pow(((double) M_E), fma(-tan(eps), tan(x), (tan(eps) * tan(x))))))) - tan(x));
	} else {
		temp = fma(pow(eps, 2.0), x, fma(eps, pow(x, 2.0), eps));
	}
	return temp;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	36.8
Target	15.2
Herbie	15.2

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

Derivation

1. Split input into 2 regimes

2. if eps < -5.9719887302803746e-27 or 4.742281876404587e-23 < eps

   1. Initial program 29.8

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

   3. Applied tan-sum 1.7

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

   5. Applied add-log-exp 1.8

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

   6. Applied add-log-exp 1.8

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

   7. Applied diff-log 1.9

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

   8. Simplified 1.8

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

   10. Applied *-un-lft-identity 1.8

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

   11. Applied exp-prod 1.9

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

   12. Simplified 1.9

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

   14. Applied add-cube-cbrt 1.9

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

   15. Applied prod-diff 1.9

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

   16. Applied unpow-prod-up 1.9

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

   17. Simplified 1.9

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

   if -5.9719887302803746e-27 < eps < 4.742281876404587e-23

   1. Initial program 45.1

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

   2. Taylor expanded around 0 30.9

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

   3. Simplified 30.9

      \[\leadsto \color{blue}{\mathsf{fma}\left({\varepsilon}^{2}, x, \mathsf{fma}\left(\varepsilon, {x}^{2}, \varepsilon\right)\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 15.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -5.97198873028037458 \cdot 10^{-27} \lor \neg \left(\varepsilon \le 4.74228187640458693 \cdot 10^{-23}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{\log \left({e}^{\left(1 - \tan x \cdot \tan \varepsilon\right)} \cdot {e}^{\left(\mathsf{fma}\left(-\tan \varepsilon, \tan x, \tan \varepsilon \cdot \tan x\right)\right)}\right)} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({\varepsilon}^{2}, x, \mathsf{fma}\left(\varepsilon, {x}^{2}, \varepsilon\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020057 +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)))