2tan (problem 3.3.2)

Average Error: 37.4 → 13.8

Time: 29.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -5.135442363002646 \cdot 10^{-18}:\\ \;\;\;\;\frac{\tan \varepsilon + \tan x}{1 - \left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \left(\tan x \cdot \left(\tan \varepsilon \cdot \tan x\right)\right)\right)} \cdot \left(\left(\tan \varepsilon \cdot \tan x + \left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right)\right) + 1\right) - \tan x\\ \mathbf{elif}\;\varepsilon \le 2.106612565673929 \cdot 10^{-51}:\\ \;\;\;\;\left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \frac{1}{3} + x \cdot \left(\varepsilon \cdot \varepsilon\right)\right) + \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{\tan \varepsilon + \tan x}{1 - \left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \left(\tan x \cdot \left(\tan \varepsilon \cdot \tan x\right)\right)\right)} \cdot \left(\left(\tan \varepsilon \cdot \tan x + \left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right)\right) + 1\right)\right) \cdot \left(\frac{\tan \varepsilon + \tan x}{1 - \left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \left(\tan x \cdot \left(\tan \varepsilon \cdot \tan x\right)\right)\right)} \cdot \left(\left(\tan \varepsilon \cdot \tan x + \left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right)\right) + 1\right)\right) - \tan x \cdot \tan x}{\frac{\tan \varepsilon + \tan x}{1 - \left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \left(\tan x \cdot \left(\tan \varepsilon \cdot \tan x\right)\right)\right)} \cdot \left(\left(\tan \varepsilon \cdot \tan x + \left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right)\right) + 1\right) + \tan x}\\ \end{array}\]

C

double f(double x, double eps) {
        double r5964587 = x;
        double r5964588 = eps;
        double r5964589 = r5964587 + r5964588;
        double r5964590 = tan(r5964589);
        double r5964591 = tan(r5964587);
        double r5964592 = r5964590 - r5964591;
        return r5964592;
}

↓

double f(double x, double eps) {
        double r5964593 = eps;
        double r5964594 = -5.135442363002646e-18;
        bool r5964595 = r5964593 <= r5964594;
        double r5964596 = tan(r5964593);
        double r5964597 = x;
        double r5964598 = tan(r5964597);
        double r5964599 = r5964596 + r5964598;
        double r5964600 = 1.0;
        double r5964601 = r5964596 * r5964598;
        double r5964602 = r5964598 * r5964601;
        double r5964603 = r5964596 * r5964602;
        double r5964604 = r5964601 * r5964603;
        double r5964605 = r5964600 - r5964604;
        double r5964606 = r5964599 / r5964605;
        double r5964607 = r5964601 * r5964601;
        double r5964608 = r5964601 + r5964607;
        double r5964609 = r5964608 + r5964600;
        double r5964610 = r5964606 * r5964609;
        double r5964611 = r5964610 - r5964598;
        double r5964612 = 2.106612565673929e-51;
        bool r5964613 = r5964593 <= r5964612;
        double r5964614 = r5964593 * r5964593;
        double r5964615 = r5964593 * r5964614;
        double r5964616 = 0.3333333333333333;
        double r5964617 = r5964615 * r5964616;
        double r5964618 = r5964597 * r5964614;
        double r5964619 = r5964617 + r5964618;
        double r5964620 = r5964619 + r5964593;
        double r5964621 = r5964610 * r5964610;
        double r5964622 = r5964598 * r5964598;
        double r5964623 = r5964621 - r5964622;
        double r5964624 = r5964610 + r5964598;
        double r5964625 = r5964623 / r5964624;
        double r5964626 = r5964613 ? r5964620 : r5964625;
        double r5964627 = r5964595 ? r5964611 : r5964626;
        return r5964627;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	37.4
Target	15.6
Herbie	13.8

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

Derivation

1. Split input into 3 regimes

2. if eps < -5.135442363002646e-18

   1. Initial program 30.7

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

   3. Applied tan-sum 0.9

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

   5. Applied flip3-- 1.0

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

   6. Applied associate-/r/ 1.0

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

   7. Simplified 1.0

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

   9. Applied associate-*l* 1.0

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

   if -5.135442363002646e-18 < eps < 2.106612565673929e-51

   1. Initial program 45.7

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

   3. Applied tan-sum 45.7

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

   5. Applied flip3-- 45.7

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

   6. Applied associate-/r/ 45.7

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

   7. Simplified 45.7

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

   9. Applied associate-*l* 45.7

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

   10. Taylor expanded around 0 27.7

       \[\leadsto \color{blue}{x \cdot {\varepsilon}^{2} + \left(\frac{1}{3} \cdot {\varepsilon}^{3} + \varepsilon\right)}\]

   11. Simplified 27.7

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

   if 2.106612565673929e-51 < eps

   1. Initial program 30.8

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

   3. Applied tan-sum 4.0

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

   5. Applied flip3-- 4.1

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

   6. Applied associate-/r/ 4.1

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

   7. Simplified 4.1

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

   9. Applied associate-*l* 4.1

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

   11. Applied flip-- 4.2

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

4. Final simplification 13.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -5.135442363002646 \cdot 10^{-18}:\\ \;\;\;\;\frac{\tan \varepsilon + \tan x}{1 - \left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \left(\tan x \cdot \left(\tan \varepsilon \cdot \tan x\right)\right)\right)} \cdot \left(\left(\tan \varepsilon \cdot \tan x + \left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right)\right) + 1\right) - \tan x\\ \mathbf{elif}\;\varepsilon \le 2.106612565673929 \cdot 10^{-51}:\\ \;\;\;\;\left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \frac{1}{3} + x \cdot \left(\varepsilon \cdot \varepsilon\right)\right) + \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{\tan \varepsilon + \tan x}{1 - \left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \left(\tan x \cdot \left(\tan \varepsilon \cdot \tan x\right)\right)\right)} \cdot \left(\left(\tan \varepsilon \cdot \tan x + \left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right)\right) + 1\right)\right) \cdot \left(\frac{\tan \varepsilon + \tan x}{1 - \left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \left(\tan x \cdot \left(\tan \varepsilon \cdot \tan x\right)\right)\right)} \cdot \left(\left(\tan \varepsilon \cdot \tan x + \left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right)\right) + 1\right)\right) - \tan x \cdot \tan x}{\frac{\tan \varepsilon + \tan x}{1 - \left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \left(\tan x \cdot \left(\tan \varepsilon \cdot \tan x\right)\right)\right)} \cdot \left(\left(\tan \varepsilon \cdot \tan x + \left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right)\right) + 1\right) + \tan x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019162 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"

  :herbie-target
  (/ (sin eps) (* (cos x) (cos (+ x eps))))

  (- (tan (+ x eps)) (tan x)))