VandenBroeck and Keller, Equation (6)

Average Error: 16.9 → 8.5

Time: 26.0s

Precision: 64

• could not determine a ground truth for program body

  1. F = -5.6444360326808685e-251
  2. l = -4110355191848148.5

Math

\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]

↓

\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \le -1.141064572853054344318688712228171997078 \cdot 10^{162}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sqrt{1}}{F} \cdot \left(\frac{\sqrt{1}}{F} \cdot \tan \left(\left(\sqrt[3]{\pi \cdot \ell} \cdot \sqrt[3]{\pi \cdot \ell}\right) \cdot \sqrt[3]{\pi \cdot \ell}\right)\right)\\ \mathbf{elif}\;\pi \cdot \ell \le 1.588550440417116988311629408469501691404 \cdot 10^{150}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sqrt{1}}{F} \cdot \frac{\sin \left(\pi \cdot \ell\right) \cdot \sqrt{1}}{\left(\left(\frac{1}{24} \cdot \left({\pi}^{4} \cdot {\ell}^{4}\right) + 1\right) - \frac{1}{2} \cdot \left({\pi}^{2} \cdot {\ell}^{2}\right)\right) \cdot F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sqrt{1}}{F} \cdot \left(\frac{\sqrt{1}}{F} \cdot \tan \left(e^{\log \left(\pi \cdot \ell\right)}\right)\right)\\ \end{array}\]

C

double f(double F, double l) {
        double r24766 = atan2(1.0, 0.0);
        double r24767 = l;
        double r24768 = r24766 * r24767;
        double r24769 = 1.0;
        double r24770 = F;
        double r24771 = r24770 * r24770;
        double r24772 = r24769 / r24771;
        double r24773 = tan(r24768);
        double r24774 = r24772 * r24773;
        double r24775 = r24768 - r24774;
        return r24775;
}

↓

double f(double F, double l) {
        double r24776 = atan2(1.0, 0.0);
        double r24777 = l;
        double r24778 = r24776 * r24777;
        double r24779 = -1.1410645728530543e+162;
        bool r24780 = r24778 <= r24779;
        double r24781 = 1.0;
        double r24782 = sqrt(r24781);
        double r24783 = F;
        double r24784 = r24782 / r24783;
        double r24785 = cbrt(r24778);
        double r24786 = r24785 * r24785;
        double r24787 = r24786 * r24785;
        double r24788 = tan(r24787);
        double r24789 = r24784 * r24788;
        double r24790 = r24784 * r24789;
        double r24791 = r24778 - r24790;
        double r24792 = 1.588550440417117e+150;
        bool r24793 = r24778 <= r24792;
        double r24794 = sin(r24778);
        double r24795 = r24794 * r24782;
        double r24796 = 0.041666666666666664;
        double r24797 = 4.0;
        double r24798 = pow(r24776, r24797);
        double r24799 = pow(r24777, r24797);
        double r24800 = r24798 * r24799;
        double r24801 = r24796 * r24800;
        double r24802 = 1.0;
        double r24803 = r24801 + r24802;
        double r24804 = 0.5;
        double r24805 = 2.0;
        double r24806 = pow(r24776, r24805);
        double r24807 = pow(r24777, r24805);
        double r24808 = r24806 * r24807;
        double r24809 = r24804 * r24808;
        double r24810 = r24803 - r24809;
        double r24811 = r24810 * r24783;
        double r24812 = r24795 / r24811;
        double r24813 = r24784 * r24812;
        double r24814 = r24778 - r24813;
        double r24815 = log(r24778);
        double r24816 = exp(r24815);
        double r24817 = tan(r24816);
        double r24818 = r24784 * r24817;
        double r24819 = r24784 * r24818;
        double r24820 = r24778 - r24819;
        double r24821 = r24793 ? r24814 : r24820;
        double r24822 = r24780 ? r24791 : r24821;
        return r24822;
}

Error

Bits error versus F

Bits error versus l

Derivation

1. Split input into 3 regimes

2. if (* PI l) < -1.1410645728530543e+162

   1. Initial program 20.3

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt 20.3

      \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\sqrt{1} \cdot \sqrt{1}}}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]

   4. Applied times-frac 20.3

      \[\leadsto \pi \cdot \ell - \color{blue}{\left(\frac{\sqrt{1}}{F} \cdot \frac{\sqrt{1}}{F}\right)} \cdot \tan \left(\pi \cdot \ell\right)\]

   5. Applied associate-*l* 20.3

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\sqrt{1}}{F} \cdot \left(\frac{\sqrt{1}}{F} \cdot \tan \left(\pi \cdot \ell\right)\right)}\]
   6. Using strategy rm

   7. Applied add-cube-cbrt 20.3

      \[\leadsto \pi \cdot \ell - \frac{\sqrt{1}}{F} \cdot \left(\frac{\sqrt{1}}{F} \cdot \tan \color{blue}{\left(\left(\sqrt[3]{\pi \cdot \ell} \cdot \sqrt[3]{\pi \cdot \ell}\right) \cdot \sqrt[3]{\pi \cdot \ell}\right)}\right)\]

   if -1.1410645728530543e+162 < (* PI l) < 1.588550440417117e+150

   1. Initial program 15.7

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt 15.7

      \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\sqrt{1} \cdot \sqrt{1}}}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]

   4. Applied times-frac 15.8

      \[\leadsto \pi \cdot \ell - \color{blue}{\left(\frac{\sqrt{1}}{F} \cdot \frac{\sqrt{1}}{F}\right)} \cdot \tan \left(\pi \cdot \ell\right)\]

   5. Applied associate-*l* 9.7

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\sqrt{1}}{F} \cdot \left(\frac{\sqrt{1}}{F} \cdot \tan \left(\pi \cdot \ell\right)\right)}\]
   6. Using strategy rm

   7. Applied tan-quot 9.7

      \[\leadsto \pi \cdot \ell - \frac{\sqrt{1}}{F} \cdot \left(\frac{\sqrt{1}}{F} \cdot \color{blue}{\frac{\sin \left(\pi \cdot \ell\right)}{\cos \left(\pi \cdot \ell\right)}}\right)\]

   8. Applied frac-times 9.7

      \[\leadsto \pi \cdot \ell - \frac{\sqrt{1}}{F} \cdot \color{blue}{\frac{\sqrt{1} \cdot \sin \left(\pi \cdot \ell\right)}{F \cdot \cos \left(\pi \cdot \ell\right)}}\]

   9. Simplified 9.7

      \[\leadsto \pi \cdot \ell - \frac{\sqrt{1}}{F} \cdot \frac{\color{blue}{\sin \left(\pi \cdot \ell\right) \cdot \sqrt{1}}}{F \cdot \cos \left(\pi \cdot \ell\right)}\]

   10. Simplified 9.7

       \[\leadsto \pi \cdot \ell - \frac{\sqrt{1}}{F} \cdot \frac{\sin \left(\pi \cdot \ell\right) \cdot \sqrt{1}}{\color{blue}{\cos \left(\pi \cdot \ell\right) \cdot F}}\]
   11. Using strategy rm

   12. Applied add-cube-cbrt 9.7

       \[\leadsto \pi \cdot \ell - \frac{\sqrt{1}}{F} \cdot \frac{\sin \left(\pi \cdot \ell\right) \cdot \sqrt{1}}{\cos \left(\pi \cdot \color{blue}{\left(\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}\right)}\right) \cdot F}\]

   13. Applied associate-*r* 9.7

       \[\leadsto \pi \cdot \ell - \frac{\sqrt{1}}{F} \cdot \frac{\sin \left(\pi \cdot \ell\right) \cdot \sqrt{1}}{\cos \color{blue}{\left(\left(\pi \cdot \left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right)\right) \cdot \sqrt[3]{\ell}\right)} \cdot F}\]

   14. Taylor expanded around 0 4.3

       \[\leadsto \pi \cdot \ell - \frac{\sqrt{1}}{F} \cdot \frac{\sin \left(\pi \cdot \ell\right) \cdot \sqrt{1}}{\color{blue}{\left(\left(\frac{1}{24} \cdot \left({\pi}^{4} \cdot {\ell}^{4}\right) + 1\right) - \frac{1}{2} \cdot \left({\pi}^{2} \cdot {\ell}^{2}\right)\right)} \cdot F}\]

   if 1.588550440417117e+150 < (* PI l)

   1. Initial program 20.0

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt 20.0

      \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\sqrt{1} \cdot \sqrt{1}}}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]

   4. Applied times-frac 20.0

      \[\leadsto \pi \cdot \ell - \color{blue}{\left(\frac{\sqrt{1}}{F} \cdot \frac{\sqrt{1}}{F}\right)} \cdot \tan \left(\pi \cdot \ell\right)\]

   5. Applied associate-*l* 20.0

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\sqrt{1}}{F} \cdot \left(\frac{\sqrt{1}}{F} \cdot \tan \left(\pi \cdot \ell\right)\right)}\]
   6. Using strategy rm

   7. Applied add-exp-log 20.0

      \[\leadsto \pi \cdot \ell - \frac{\sqrt{1}}{F} \cdot \left(\frac{\sqrt{1}}{F} \cdot \tan \left(\pi \cdot \color{blue}{e^{\log \ell}}\right)\right)\]

   8. Applied add-exp-log 20.0

      \[\leadsto \pi \cdot \ell - \frac{\sqrt{1}}{F} \cdot \left(\frac{\sqrt{1}}{F} \cdot \tan \left(\color{blue}{e^{\log \pi}} \cdot e^{\log \ell}\right)\right)\]

   9. Applied prod-exp 20.0

      \[\leadsto \pi \cdot \ell - \frac{\sqrt{1}}{F} \cdot \left(\frac{\sqrt{1}}{F} \cdot \tan \color{blue}{\left(e^{\log \pi + \log \ell}\right)}\right)\]

   10. Simplified 20.0

       \[\leadsto \pi \cdot \ell - \frac{\sqrt{1}}{F} \cdot \left(\frac{\sqrt{1}}{F} \cdot \tan \left(e^{\color{blue}{\log \left(\pi \cdot \ell\right)}}\right)\right)\]
3. Recombined 3 regimes into one program.

4. Final simplification 8.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \le -1.141064572853054344318688712228171997078 \cdot 10^{162}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sqrt{1}}{F} \cdot \left(\frac{\sqrt{1}}{F} \cdot \tan \left(\left(\sqrt[3]{\pi \cdot \ell} \cdot \sqrt[3]{\pi \cdot \ell}\right) \cdot \sqrt[3]{\pi \cdot \ell}\right)\right)\\ \mathbf{elif}\;\pi \cdot \ell \le 1.588550440417116988311629408469501691404 \cdot 10^{150}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sqrt{1}}{F} \cdot \frac{\sin \left(\pi \cdot \ell\right) \cdot \sqrt{1}}{\left(\left(\frac{1}{24} \cdot \left({\pi}^{4} \cdot {\ell}^{4}\right) + 1\right) - \frac{1}{2} \cdot \left({\pi}^{2} \cdot {\ell}^{2}\right)\right) \cdot F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sqrt{1}}{F} \cdot \left(\frac{\sqrt{1}}{F} \cdot \tan \left(e^{\log \left(\pi \cdot \ell\right)}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))