VandenBroeck and Keller, Equation (6)

Average Error: 17.3 → 8.5

Time: 9.3s

Precision: 64

• could not determine a ground truth for program body

  1. F = 1.0925393300280745e+126
  2. l = -716357237560921.5

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \le -9.718573105215283439369659162476488255481 \cdot 10^{155}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sqrt{1}}{F} \cdot \left(\frac{\sqrt{1}}{F} \cdot \tan \left(\sqrt{\pi} \cdot \left(\sqrt{\pi} \cdot \ell\right)\right)\right)\\ \mathbf{elif}\;\pi \cdot \ell \le 8.939014277543762652462397904252204810382 \cdot 10^{150}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sqrt{1}}{F} \cdot \frac{\sqrt{1} \cdot \sin \left(\pi \cdot \ell\right)}{F \cdot \left(\left(\frac{1}{24} \cdot \left({\pi}^{4} \cdot {\ell}^{4}\right) + 1\right) - \frac{1}{2} \cdot \left({\pi}^{2} \cdot {\ell}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sqrt{1}}{F} \cdot \frac{\sqrt{1} \cdot \sin \left(\pi \cdot \ell\right)}{F \cdot \cos \left(\left(\left(\sqrt[3]{\sqrt{\pi}} \cdot \sqrt[3]{\sqrt{\pi}}\right) \cdot \left(\sqrt[3]{\sqrt{\pi}} \cdot \sqrt[3]{\sqrt{\pi}}\right)\right) \cdot \left(\sqrt[3]{\pi} \cdot \ell\right)\right)}\\ \end{array}\]

C

double f(double F, double l) {
        double r14745 = atan2(1.0, 0.0);
        double r14746 = l;
        double r14747 = r14745 * r14746;
        double r14748 = 1.0;
        double r14749 = F;
        double r14750 = r14749 * r14749;
        double r14751 = r14748 / r14750;
        double r14752 = tan(r14747);
        double r14753 = r14751 * r14752;
        double r14754 = r14747 - r14753;
        return r14754;
}

↓

double f(double F, double l) {
        double r14755 = atan2(1.0, 0.0);
        double r14756 = l;
        double r14757 = r14755 * r14756;
        double r14758 = -9.718573105215283e+155;
        bool r14759 = r14757 <= r14758;
        double r14760 = 1.0;
        double r14761 = sqrt(r14760);
        double r14762 = F;
        double r14763 = r14761 / r14762;
        double r14764 = sqrt(r14755);
        double r14765 = r14764 * r14756;
        double r14766 = r14764 * r14765;
        double r14767 = tan(r14766);
        double r14768 = r14763 * r14767;
        double r14769 = r14763 * r14768;
        double r14770 = r14757 - r14769;
        double r14771 = 8.939014277543763e+150;
        bool r14772 = r14757 <= r14771;
        double r14773 = sin(r14757);
        double r14774 = r14761 * r14773;
        double r14775 = 0.041666666666666664;
        double r14776 = 4.0;
        double r14777 = pow(r14755, r14776);
        double r14778 = pow(r14756, r14776);
        double r14779 = r14777 * r14778;
        double r14780 = r14775 * r14779;
        double r14781 = 1.0;
        double r14782 = r14780 + r14781;
        double r14783 = 0.5;
        double r14784 = 2.0;
        double r14785 = pow(r14755, r14784);
        double r14786 = pow(r14756, r14784);
        double r14787 = r14785 * r14786;
        double r14788 = r14783 * r14787;
        double r14789 = r14782 - r14788;
        double r14790 = r14762 * r14789;
        double r14791 = r14774 / r14790;
        double r14792 = r14763 * r14791;
        double r14793 = r14757 - r14792;
        double r14794 = cbrt(r14764);
        double r14795 = r14794 * r14794;
        double r14796 = r14795 * r14795;
        double r14797 = cbrt(r14755);
        double r14798 = r14797 * r14756;
        double r14799 = r14796 * r14798;
        double r14800 = cos(r14799);
        double r14801 = r14762 * r14800;
        double r14802 = r14774 / r14801;
        double r14803 = r14763 * r14802;
        double r14804 = r14757 - r14803;
        double r14805 = r14772 ? r14793 : r14804;
        double r14806 = r14759 ? r14770 : r14805;
        return r14806;
}

Error

Bits error versus F

Bits error versus l

Derivation

1. Split input into 3 regimes

2. if (* PI l) < -9.718573105215283e+155

   1. Initial program 21.5

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

   3. Applied add-sqr-sqrt 21.5

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

   4. Applied times-frac 21.5

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

   5. Applied associate-*l* 21.5

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

   7. Applied add-sqr-sqrt 21.5

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

   8. Applied associate-*l* 21.6

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

   if -9.718573105215283e+155 < (* PI l) < 8.939014277543763e+150

   1. Initial program 15.8

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

   3. Applied add-sqr-sqrt 15.8

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

   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* 9.8

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

   7. Applied tan-quot 9.8

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

   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. Taylor expanded around 0 3.9

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

   if 8.939014277543763e+150 < (* PI l)

   1. Initial program 21.5

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

   3. Applied add-sqr-sqrt 21.5

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

   4. Applied times-frac 21.5

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

   5. Applied associate-*l* 21.5

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

   7. Applied tan-quot 21.5

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

   8. Applied frac-times 21.5

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

   10. Applied add-cube-cbrt 21.4

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

   11. Applied associate-*l* 21.4

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

   13. Applied add-sqr-sqrt 21.5

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

   14. Applied cbrt-prod 21.4

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

   15. Applied add-sqr-sqrt 21.4

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

   16. Applied cbrt-prod 21.4

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

   17. Applied swap-sqr 21.4

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

4. Final simplification 8.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \le -9.718573105215283439369659162476488255481 \cdot 10^{155}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sqrt{1}}{F} \cdot \left(\frac{\sqrt{1}}{F} \cdot \tan \left(\sqrt{\pi} \cdot \left(\sqrt{\pi} \cdot \ell\right)\right)\right)\\ \mathbf{elif}\;\pi \cdot \ell \le 8.939014277543762652462397904252204810382 \cdot 10^{150}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sqrt{1}}{F} \cdot \frac{\sqrt{1} \cdot \sin \left(\pi \cdot \ell\right)}{F \cdot \left(\left(\frac{1}{24} \cdot \left({\pi}^{4} \cdot {\ell}^{4}\right) + 1\right) - \frac{1}{2} \cdot \left({\pi}^{2} \cdot {\ell}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sqrt{1}}{F} \cdot \frac{\sqrt{1} \cdot \sin \left(\pi \cdot \ell\right)}{F \cdot \cos \left(\left(\left(\sqrt[3]{\sqrt{\pi}} \cdot \sqrt[3]{\sqrt{\pi}}\right) \cdot \left(\sqrt[3]{\sqrt{\pi}} \cdot \sqrt[3]{\sqrt{\pi}}\right)\right) \cdot \left(\sqrt[3]{\pi} \cdot \ell\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019362 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))