2tan (problem 3.3.2)

Average Error: 36.9 → 13.7

Time: 16.5s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -6.9054160181468085 \cdot 10^{-18}:\\ \;\;\;\;\frac{\left(\left({\left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon + 1\right)\right)}^{3} + 1\right) \cdot \frac{\tan \varepsilon + \tan x}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}}\right) \cdot \cos x - \sin x \cdot \left(\left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon + 1\right)\right) \cdot \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon + 1\right) - 1\right) + 1\right)}{\cos x \cdot \left(\left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon + 1\right)\right) \cdot \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon + 1\right) - 1\right) + 1\right)}\\ \mathbf{elif}\;\varepsilon \le 5.29247912990205075 \cdot 10^{-18}:\\ \;\;\;\;x \cdot {\varepsilon}^{2} + \left(\frac{1}{3} \cdot {\varepsilon}^{3} + \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \sqrt[3]{{\left(\tan x \cdot \tan \varepsilon\right)}^{3}}} - \tan x\\ \end{array}\]

C

double f(double x, double eps) {
        double r119637 = x;
        double r119638 = eps;
        double r119639 = r119637 + r119638;
        double r119640 = tan(r119639);
        double r119641 = tan(r119637);
        double r119642 = r119640 - r119641;
        return r119642;
}

↓

double f(double x, double eps) {
        double r119643 = eps;
        double r119644 = -6.9054160181468085e-18;
        bool r119645 = r119643 <= r119644;
        double r119646 = x;
        double r119647 = tan(r119646);
        double r119648 = tan(r119643);
        double r119649 = r119647 * r119648;
        double r119650 = 1.0;
        double r119651 = r119649 + r119650;
        double r119652 = r119649 * r119651;
        double r119653 = 3.0;
        double r119654 = pow(r119652, r119653);
        double r119655 = r119654 + r119650;
        double r119656 = r119648 + r119647;
        double r119657 = pow(r119649, r119653);
        double r119658 = r119650 - r119657;
        double r119659 = r119656 / r119658;
        double r119660 = r119655 * r119659;
        double r119661 = cos(r119646);
        double r119662 = r119660 * r119661;
        double r119663 = sin(r119646);
        double r119664 = r119652 - r119650;
        double r119665 = r119652 * r119664;
        double r119666 = r119665 + r119650;
        double r119667 = r119663 * r119666;
        double r119668 = r119662 - r119667;
        double r119669 = r119661 * r119666;
        double r119670 = r119668 / r119669;
        double r119671 = 5.292479129902051e-18;
        bool r119672 = r119643 <= r119671;
        double r119673 = 2.0;
        double r119674 = pow(r119643, r119673);
        double r119675 = r119646 * r119674;
        double r119676 = 0.3333333333333333;
        double r119677 = pow(r119643, r119653);
        double r119678 = r119676 * r119677;
        double r119679 = r119678 + r119643;
        double r119680 = r119675 + r119679;
        double r119681 = r119647 + r119648;
        double r119682 = cbrt(r119657);
        double r119683 = r119650 - r119682;
        double r119684 = r119681 / r119683;
        double r119685 = r119684 - r119647;
        double r119686 = r119672 ? r119680 : r119685;
        double r119687 = r119645 ? r119670 : r119686;
        return r119687;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	36.9
Target	15.2
Herbie	13.7

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

Derivation

1. Split input into 3 regimes

2. if eps < -6.9054160181468085e-18

   1. Initial program 29.8

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

   3. Applied tan-sum 1.0

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

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

   9. Applied tan-quot 1.1

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

   10. Applied flip3-+ 1.1

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

   11. Applied associate-*r/ 1.1

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

   12. Applied frac-sub 1.1

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

   13. Simplified 1.2

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

   14. Simplified 1.1

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

   if -6.9054160181468085e-18 < eps < 5.292479129902051e-18

   1. Initial program 45.4

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

   3. Applied tan-sum 45.4

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

   5. Applied flip3-- 45.4

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

   6. Applied associate-/r/ 45.4

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

   7. Simplified 45.4

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

   8. Taylor expanded around 0 28.0

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

   if 5.292479129902051e-18 < eps

   1. Initial program 29.2

      \[\tan \left(x + \varepsilon\right) - \tan x\]
   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 add-cbrt-cube 0.9

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

   6. Applied add-cbrt-cube 0.9

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

   7. Applied cbrt-unprod 0.9

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

   8. Simplified 0.9

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

4. Final simplification 13.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -6.9054160181468085 \cdot 10^{-18}:\\ \;\;\;\;\frac{\left(\left({\left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon + 1\right)\right)}^{3} + 1\right) \cdot \frac{\tan \varepsilon + \tan x}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}}\right) \cdot \cos x - \sin x \cdot \left(\left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon + 1\right)\right) \cdot \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon + 1\right) - 1\right) + 1\right)}{\cos x \cdot \left(\left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon + 1\right)\right) \cdot \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon + 1\right) - 1\right) + 1\right)}\\ \mathbf{elif}\;\varepsilon \le 5.29247912990205075 \cdot 10^{-18}:\\ \;\;\;\;x \cdot {\varepsilon}^{2} + \left(\frac{1}{3} \cdot {\varepsilon}^{3} + \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \sqrt[3]{{\left(\tan x \cdot \tan \varepsilon\right)}^{3}}} - \tan x\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 
(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)))