VandenBroeck and Keller, Equation (6)

Average Error: 16.1 → 8.2

Time: 8.7s

Precision: 64

• could not determine a ground truth for program body

  1. F = 4.3015410020898555e-127
  2. l = -1030240412319637.5

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \le -2.7182725096522959 \cdot 10^{155}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \frac{\sqrt[3]{1} \cdot \sin \left(\pi \cdot \ell\right)}{F \cdot \cos \left(\left(\left(\sqrt[3]{\sqrt{\pi}} \cdot \sqrt[3]{\sqrt{\pi}}\right) \cdot \left(\sqrt[3]{\sqrt{\pi}} \cdot \sqrt[3]{\sqrt{\pi}}\right)\right) \cdot \left(\sqrt[3]{\pi} \cdot \ell\right)\right)}\\ \mathbf{elif}\;\pi \cdot \ell \le 4.30574485349055633 \cdot 10^{139}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \frac{\sqrt[3]{1} \cdot \sin \left(\pi \cdot \ell\right)}{F \cdot \mathsf{fma}\left(\frac{1}{24} \cdot {\pi}^{4}, {\ell}^{4}, 1 - \frac{1}{2} \cdot \left({\pi}^{2} \cdot {\ell}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \ell\right)\right)\right)\\ \end{array}\]

C

double f(double F, double l) {
        double r13577 = atan2(1.0, 0.0);
        double r13578 = l;
        double r13579 = r13577 * r13578;
        double r13580 = 1.0;
        double r13581 = F;
        double r13582 = r13581 * r13581;
        double r13583 = r13580 / r13582;
        double r13584 = tan(r13579);
        double r13585 = r13583 * r13584;
        double r13586 = r13579 - r13585;
        return r13586;
}

↓

double f(double F, double l) {
        double r13587 = atan2(1.0, 0.0);
        double r13588 = l;
        double r13589 = r13587 * r13588;
        double r13590 = -2.718272509652296e+155;
        bool r13591 = r13589 <= r13590;
        double r13592 = 1.0;
        double r13593 = cbrt(r13592);
        double r13594 = r13593 * r13593;
        double r13595 = F;
        double r13596 = r13594 / r13595;
        double r13597 = sin(r13589);
        double r13598 = r13593 * r13597;
        double r13599 = sqrt(r13587);
        double r13600 = cbrt(r13599);
        double r13601 = r13600 * r13600;
        double r13602 = r13601 * r13601;
        double r13603 = cbrt(r13587);
        double r13604 = r13603 * r13588;
        double r13605 = r13602 * r13604;
        double r13606 = cos(r13605);
        double r13607 = r13595 * r13606;
        double r13608 = r13598 / r13607;
        double r13609 = r13596 * r13608;
        double r13610 = r13589 - r13609;
        double r13611 = 4.3057448534905563e+139;
        bool r13612 = r13589 <= r13611;
        double r13613 = 0.041666666666666664;
        double r13614 = 4.0;
        double r13615 = pow(r13587, r13614);
        double r13616 = r13613 * r13615;
        double r13617 = pow(r13588, r13614);
        double r13618 = 1.0;
        double r13619 = 0.5;
        double r13620 = 2.0;
        double r13621 = pow(r13587, r13620);
        double r13622 = pow(r13588, r13620);
        double r13623 = r13621 * r13622;
        double r13624 = r13619 * r13623;
        double r13625 = r13618 - r13624;
        double r13626 = fma(r13616, r13617, r13625);
        double r13627 = r13595 * r13626;
        double r13628 = r13598 / r13627;
        double r13629 = r13596 * r13628;
        double r13630 = r13589 - r13629;
        double r13631 = r13595 * r13595;
        double r13632 = r13592 / r13631;
        double r13633 = log1p(r13589);
        double r13634 = expm1(r13633);
        double r13635 = tan(r13634);
        double r13636 = r13632 * r13635;
        double r13637 = r13589 - r13636;
        double r13638 = r13612 ? r13630 : r13637;
        double r13639 = r13591 ? r13610 : r13638;
        return r13639;
}

Error

Bits error versus F

Bits error versus l

Derivation

1. Split input into 3 regimes

2. if (* PI l) < -2.718272509652296e+155

   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-cube-cbrt 20.0

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

   4. Applied times-frac 20.0

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

   5. Applied associate-*l* 20.0

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

   7. Applied tan-quot 20.0

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

   8. Applied frac-times 20.0

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

   10. Applied add-cube-cbrt 20.0

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

   11. Applied associate-*l* 20.0

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

   13. Applied add-sqr-sqrt 20.0

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

   14. Applied cbrt-prod 20.0

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

   15. Applied add-sqr-sqrt 20.0

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

   16. Applied cbrt-prod 20.0

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

   17. Applied swap-sqr 20.0

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

   if -2.718272509652296e+155 < (* PI l) < 4.3057448534905563e+139

   1. Initial program 14.6

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

   3. Applied add-cube-cbrt 14.6

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

   4. Applied times-frac 14.7

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

   5. Applied associate-*l* 9.2

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

   7. Applied tan-quot 9.2

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

   8. Applied frac-times 9.2

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

   9. Taylor expanded around 0 3.7

      \[\leadsto \pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \frac{\sqrt[3]{1} \cdot \sin \left(\pi \cdot \ell\right)}{F \cdot \color{blue}{\left(\left(\frac{1}{24} \cdot \left({\pi}^{4} \cdot {\ell}^{4}\right) + 1\right) - \frac{1}{2} \cdot \left({\pi}^{2} \cdot {\ell}^{2}\right)\right)}}\]

   10. Simplified 3.7

       \[\leadsto \pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \frac{\sqrt[3]{1} \cdot \sin \left(\pi \cdot \ell\right)}{F \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{24} \cdot {\pi}^{4}, {\ell}^{4}, 1 - \frac{1}{2} \cdot \left({\pi}^{2} \cdot {\ell}^{2}\right)\right)}}\]

   if 4.3057448534905563e+139 < (* PI l)

   1. Initial program 19.6

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

   3. Applied expm1-log1p-u 19.6

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

4. Final simplification 8.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \le -2.7182725096522959 \cdot 10^{155}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \frac{\sqrt[3]{1} \cdot \sin \left(\pi \cdot \ell\right)}{F \cdot \cos \left(\left(\left(\sqrt[3]{\sqrt{\pi}} \cdot \sqrt[3]{\sqrt{\pi}}\right) \cdot \left(\sqrt[3]{\sqrt{\pi}} \cdot \sqrt[3]{\sqrt{\pi}}\right)\right) \cdot \left(\sqrt[3]{\pi} \cdot \ell\right)\right)}\\ \mathbf{elif}\;\pi \cdot \ell \le 4.30574485349055633 \cdot 10^{139}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \frac{\sqrt[3]{1} \cdot \sin \left(\pi \cdot \ell\right)}{F \cdot \mathsf{fma}\left(\frac{1}{24} \cdot {\pi}^{4}, {\ell}^{4}, 1 - \frac{1}{2} \cdot \left({\pi}^{2} \cdot {\ell}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \ell\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020018 +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))