2tan (problem 3.3.2)

Average Error: 36.7 → 15.7

Time: 13.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -7.40491158059780431 \cdot 10^{-25}:\\ \;\;\;\;\frac{\frac{1}{\frac{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}}{\left(\left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon + 1\right)\right) \cdot \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon + 1\right)\right) - 1\right) \cdot \left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)}}}{\left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon + 1\right) - 1\right) \cdot \left(\cos x \cdot \cos \varepsilon\right)} - \tan x\\ \mathbf{elif}\;\varepsilon \le 7.89240441352321796 \cdot 10^{-132}:\\ \;\;\;\;\left(\varepsilon \cdot x\right) \cdot \left(x + \varepsilon\right) + \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \sin x}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}\\ \end{array}\]

C

double f(double x, double eps) {
        double r139601 = x;
        double r139602 = eps;
        double r139603 = r139601 + r139602;
        double r139604 = tan(r139603);
        double r139605 = tan(r139601);
        double r139606 = r139604 - r139605;
        return r139606;
}

↓

double f(double x, double eps) {
        double r139607 = eps;
        double r139608 = -7.404911580597804e-25;
        bool r139609 = r139607 <= r139608;
        double r139610 = 1.0;
        double r139611 = x;
        double r139612 = tan(r139611);
        double r139613 = tan(r139607);
        double r139614 = r139612 * r139613;
        double r139615 = 3.0;
        double r139616 = pow(r139614, r139615);
        double r139617 = r139610 - r139616;
        double r139618 = r139614 + r139610;
        double r139619 = r139614 * r139618;
        double r139620 = r139619 * r139619;
        double r139621 = r139620 - r139610;
        double r139622 = sin(r139611);
        double r139623 = cos(r139607);
        double r139624 = r139622 * r139623;
        double r139625 = cos(r139611);
        double r139626 = sin(r139607);
        double r139627 = r139625 * r139626;
        double r139628 = r139624 + r139627;
        double r139629 = r139621 * r139628;
        double r139630 = r139617 / r139629;
        double r139631 = r139610 / r139630;
        double r139632 = r139619 - r139610;
        double r139633 = r139625 * r139623;
        double r139634 = r139632 * r139633;
        double r139635 = r139631 / r139634;
        double r139636 = r139635 - r139612;
        double r139637 = 7.892404413523218e-132;
        bool r139638 = r139607 <= r139637;
        double r139639 = r139607 * r139611;
        double r139640 = r139611 + r139607;
        double r139641 = r139639 * r139640;
        double r139642 = r139641 + r139607;
        double r139643 = r139612 + r139613;
        double r139644 = r139643 * r139625;
        double r139645 = r139610 - r139614;
        double r139646 = r139645 * r139622;
        double r139647 = r139644 - r139646;
        double r139648 = r139645 * r139625;
        double r139649 = r139647 / r139648;
        double r139650 = r139638 ? r139642 : r139649;
        double r139651 = r139609 ? r139636 : r139650;
        return r139651;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	36.7
Target	15.2
Herbie	15.7

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

Derivation

1. Split input into 3 regimes

2. if eps < -7.404911580597804e-25

   1. Initial program 29.9

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

   3. Applied tan-sum 1.9

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

   5. Applied clear-num 2.0

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

   7. Applied flip3-- 2.0

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

   8. Applied associate-/l/ 2.0

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

   9. Simplified 2.0

      \[\leadsto \frac{1}{\frac{{1}^{3} - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}}{\color{blue}{\left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon + 1\right) + 1\right) \cdot \left(\tan x + \tan \varepsilon\right)}}} - \tan x\]
   10. Using strategy rm

   11. Applied tan-quot 2.1

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

   12. Applied tan-quot 2.2

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

   13. Applied frac-add 2.2

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

   14. Applied flip-+ 2.2

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

   15. Applied frac-times 2.2

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

   16. Applied associate-/r/ 2.3

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

   17. Applied associate-/r* 2.2

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

   18. Simplified 2.2

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

   if -7.404911580597804e-25 < eps < 7.892404413523218e-132

   1. Initial program 46.5

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

   2. Taylor expanded around 0 30.9

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

   3. Simplified 30.6

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

   if 7.892404413523218e-132 < eps

   1. Initial program 31.9

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

   3. Applied tan-quot 31.8

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

   4. Applied tan-sum 10.9

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

   5. Applied frac-sub 11.0

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

4. Final simplification 15.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -7.40491158059780431 \cdot 10^{-25}:\\ \;\;\;\;\frac{\frac{1}{\frac{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}}{\left(\left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon + 1\right)\right) \cdot \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon + 1\right)\right) - 1\right) \cdot \left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)}}}{\left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon + 1\right) - 1\right) \cdot \left(\cos x \cdot \cos \varepsilon\right)} - \tan x\\ \mathbf{elif}\;\varepsilon \le 7.89240441352321796 \cdot 10^{-132}:\\ \;\;\;\;\left(\varepsilon \cdot x\right) \cdot \left(x + \varepsilon\right) + \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \sin x}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020024 
(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)))