VandenBroeck and Keller, Equation (6)

Average Error: 16.2 → 8.7

Time: 24.5s

Precision: 64

• could not determine a ground truth for program body

  1. F = -4.6929347775450423e+185
  2. l = 694825683354768.5

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \le -6.329011492119437021952109634828818118288 \cdot 10^{159}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sqrt{1}}{F} \cdot \frac{\sin \left(\pi \cdot \ell\right) \cdot \sqrt{1}}{\cos \left(\left(\sqrt[3]{\pi \cdot \ell} \cdot \sqrt[3]{\pi \cdot \ell}\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{\pi \cdot \ell}} \cdot \sqrt[3]{\sqrt[3]{\pi \cdot \ell}}\right) \cdot \sqrt[3]{\sqrt[3]{\pi \cdot \ell}}\right)\right) \cdot F}\\ \mathbf{elif}\;\pi \cdot \ell \le 7.515140750009139392686817039051963899626 \cdot 10^{135}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sqrt{1}}{F} \cdot \frac{\sin \left(\pi \cdot \ell\right) \cdot \sqrt{1}}{\mathsf{fma}\left(\left(\pi \cdot \ell\right) \cdot \left(\pi \cdot \ell\right), \frac{-1}{2}, \mathsf{fma}\left(\frac{1}{24}, {\pi}^{4} \cdot {\ell}^{4}, 1\right)\right) \cdot F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sqrt{1}}{F} \cdot \left(\frac{\sqrt{1}}{F} \cdot \tan \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \ell\right)\right)\right)\right)\\ \end{array}\]

C

double f(double F, double l) {
        double r28617 = atan2(1.0, 0.0);
        double r28618 = l;
        double r28619 = r28617 * r28618;
        double r28620 = 1.0;
        double r28621 = F;
        double r28622 = r28621 * r28621;
        double r28623 = r28620 / r28622;
        double r28624 = tan(r28619);
        double r28625 = r28623 * r28624;
        double r28626 = r28619 - r28625;
        return r28626;
}

↓

double f(double F, double l) {
        double r28627 = atan2(1.0, 0.0);
        double r28628 = l;
        double r28629 = r28627 * r28628;
        double r28630 = -6.329011492119437e+159;
        bool r28631 = r28629 <= r28630;
        double r28632 = 1.0;
        double r28633 = sqrt(r28632);
        double r28634 = F;
        double r28635 = r28633 / r28634;
        double r28636 = sin(r28629);
        double r28637 = r28636 * r28633;
        double r28638 = cbrt(r28629);
        double r28639 = r28638 * r28638;
        double r28640 = cbrt(r28638);
        double r28641 = r28640 * r28640;
        double r28642 = r28641 * r28640;
        double r28643 = r28639 * r28642;
        double r28644 = cos(r28643);
        double r28645 = r28644 * r28634;
        double r28646 = r28637 / r28645;
        double r28647 = r28635 * r28646;
        double r28648 = r28629 - r28647;
        double r28649 = 7.51514075000914e+135;
        bool r28650 = r28629 <= r28649;
        double r28651 = r28629 * r28629;
        double r28652 = -0.5;
        double r28653 = 0.041666666666666664;
        double r28654 = 4.0;
        double r28655 = pow(r28627, r28654);
        double r28656 = pow(r28628, r28654);
        double r28657 = r28655 * r28656;
        double r28658 = 1.0;
        double r28659 = fma(r28653, r28657, r28658);
        double r28660 = fma(r28651, r28652, r28659);
        double r28661 = r28660 * r28634;
        double r28662 = r28637 / r28661;
        double r28663 = r28635 * r28662;
        double r28664 = r28629 - r28663;
        double r28665 = log1p(r28629);
        double r28666 = expm1(r28665);
        double r28667 = tan(r28666);
        double r28668 = r28635 * r28667;
        double r28669 = r28635 * r28668;
        double r28670 = r28629 - r28669;
        double r28671 = r28650 ? r28664 : r28670;
        double r28672 = r28631 ? r28648 : r28671;
        return r28672;
}

Error

Bits error versus F

Bits error versus l

Derivation

1. Split input into 3 regimes

2. if (* PI l) < -6.329011492119437e+159

   1. Initial program 18.4

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

   3. Applied add-sqr-sqrt 18.4

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

   4. Applied times-frac 18.4

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

   5. Applied associate-*l* 18.4

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

   7. Applied tan-quot 18.4

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

   8. Applied frac-times 18.4

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

   9. Simplified 18.4

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

   10. Simplified 18.4

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

   12. Applied add-cube-cbrt 18.5

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

   14. Applied add-cube-cbrt 18.4

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

   if -6.329011492119437e+159 < (* PI l) < 7.51514075000914e+135

   1. Initial program 14.8

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

   3. Applied add-sqr-sqrt 14.8

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

   4. Applied times-frac 14.8

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

   5. Applied associate-*l* 9.4

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

   7. Applied tan-quot 9.4

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

   8. Applied frac-times 9.3

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

   9. Simplified 9.3

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

   10. Simplified 9.3

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

   11. Taylor expanded around 0 4.4

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

   12. Simplified 4.4

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

   if 7.51514075000914e+135 < (* PI l)

   1. Initial program 20.9

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

   3. Applied add-sqr-sqrt 20.9

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

   4. Applied times-frac 20.9

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

   5. Applied associate-*l* 20.9

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

   7. Applied expm1-log1p-u 20.9

      \[\leadsto \pi \cdot \ell - \frac{\sqrt{1}}{F} \cdot \left(\frac{\sqrt{1}}{F} \cdot \tan \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \ell\right)\right)\right)}\right)\]
3. Recombined 3 regimes into one program.

4. Final simplification 8.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \le -6.329011492119437021952109634828818118288 \cdot 10^{159}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sqrt{1}}{F} \cdot \frac{\sin \left(\pi \cdot \ell\right) \cdot \sqrt{1}}{\cos \left(\left(\sqrt[3]{\pi \cdot \ell} \cdot \sqrt[3]{\pi \cdot \ell}\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{\pi \cdot \ell}} \cdot \sqrt[3]{\sqrt[3]{\pi \cdot \ell}}\right) \cdot \sqrt[3]{\sqrt[3]{\pi \cdot \ell}}\right)\right) \cdot F}\\ \mathbf{elif}\;\pi \cdot \ell \le 7.515140750009139392686817039051963899626 \cdot 10^{135}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sqrt{1}}{F} \cdot \frac{\sin \left(\pi \cdot \ell\right) \cdot \sqrt{1}}{\mathsf{fma}\left(\left(\pi \cdot \ell\right) \cdot \left(\pi \cdot \ell\right), \frac{-1}{2}, \mathsf{fma}\left(\frac{1}{24}, {\pi}^{4} \cdot {\ell}^{4}, 1\right)\right) \cdot F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sqrt{1}}{F} \cdot \left(\frac{\sqrt{1}}{F} \cdot \tan \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \ell\right)\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019209 +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))