VandenBroeck and Keller, Equation (6)

Average Error: 16.8 → 8.9

Time: 9.4s

Precision: 64

• could not determine a ground truth for program body

  1. F = 1.196399875273985e-114
  2. l = -293211462110594.5

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \le -3.131028332018703448139277285469040119312 \cdot 10^{156}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \left(\left(\sqrt[3]{\frac{\sqrt[3]{1}}{F}} \cdot \sqrt[3]{\frac{\sqrt[3]{1}}{F}}\right) \cdot \left(\sqrt[3]{\frac{\sqrt[3]{1}}{F}} \cdot \tan \left(\pi \cdot \ell\right)\right)\right)\\ \mathbf{elif}\;\pi \cdot \ell \le 1.786958944027669917833146492636069850813 \cdot 10^{135}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \frac{\sqrt[3]{1} \cdot \sin \left(\pi \cdot \ell\right)}{F \cdot \mathsf{fma}\left(\frac{1}{24} \cdot {\pi}^{4}, {\ell}^{4}, 1 - \frac{1}{2} \cdot \left({\pi}^{2} \cdot {\ell}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \frac{\sqrt[3]{1} \cdot \sin \left(\pi \cdot \ell\right)}{F \cdot \cos \left(\left(\pi \cdot \left(\sqrt[3]{\ell} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\ell}\right)\right)\right)\right) \cdot \sqrt[3]{\ell}\right)}\\ \end{array}\]

C

double f(double F, double l) {
        double r16514 = atan2(1.0, 0.0);
        double r16515 = l;
        double r16516 = r16514 * r16515;
        double r16517 = 1.0;
        double r16518 = F;
        double r16519 = r16518 * r16518;
        double r16520 = r16517 / r16519;
        double r16521 = tan(r16516);
        double r16522 = r16520 * r16521;
        double r16523 = r16516 - r16522;
        return r16523;
}

↓

double f(double F, double l) {
        double r16524 = atan2(1.0, 0.0);
        double r16525 = l;
        double r16526 = r16524 * r16525;
        double r16527 = -3.1310283320187034e+156;
        bool r16528 = r16526 <= r16527;
        double r16529 = 1.0;
        double r16530 = cbrt(r16529);
        double r16531 = r16530 * r16530;
        double r16532 = F;
        double r16533 = r16531 / r16532;
        double r16534 = r16530 / r16532;
        double r16535 = cbrt(r16534);
        double r16536 = r16535 * r16535;
        double r16537 = tan(r16526);
        double r16538 = r16535 * r16537;
        double r16539 = r16536 * r16538;
        double r16540 = r16533 * r16539;
        double r16541 = r16526 - r16540;
        double r16542 = 1.78695894402767e+135;
        bool r16543 = r16526 <= r16542;
        double r16544 = sin(r16526);
        double r16545 = r16530 * r16544;
        double r16546 = 0.041666666666666664;
        double r16547 = 4.0;
        double r16548 = pow(r16524, r16547);
        double r16549 = r16546 * r16548;
        double r16550 = pow(r16525, r16547);
        double r16551 = 1.0;
        double r16552 = 0.5;
        double r16553 = 2.0;
        double r16554 = pow(r16524, r16553);
        double r16555 = pow(r16525, r16553);
        double r16556 = r16554 * r16555;
        double r16557 = r16552 * r16556;
        double r16558 = r16551 - r16557;
        double r16559 = fma(r16549, r16550, r16558);
        double r16560 = r16532 * r16559;
        double r16561 = r16545 / r16560;
        double r16562 = r16533 * r16561;
        double r16563 = r16526 - r16562;
        double r16564 = cbrt(r16525);
        double r16565 = log1p(r16564);
        double r16566 = expm1(r16565);
        double r16567 = r16564 * r16566;
        double r16568 = r16524 * r16567;
        double r16569 = r16568 * r16564;
        double r16570 = cos(r16569);
        double r16571 = r16532 * r16570;
        double r16572 = r16545 / r16571;
        double r16573 = r16533 * r16572;
        double r16574 = r16526 - r16573;
        double r16575 = r16543 ? r16563 : r16574;
        double r16576 = r16528 ? r16541 : r16575;
        return r16576;
}

Error

Bits error versus F

Bits error versus l

Derivation

1. Split input into 3 regimes

2. if (* PI l) < -3.1310283320187034e+156

   1. Initial program 20.8

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

   3. Applied add-cube-cbrt 20.8

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

   4. Applied times-frac 20.8

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

   5. Applied associate-*l* 20.8

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

   7. Applied add-cube-cbrt 20.8

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

   8. Applied associate-*l* 20.8

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

   if -3.1310283320187034e+156 < (* PI l) < 1.78695894402767e+135

   1. Initial program 15.1

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

   3. Applied add-cube-cbrt 15.1

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

   4. Applied times-frac 15.2

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

   5. Applied associate-*l* 9.1

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

   7. Applied tan-quot 9.1

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

   8. Applied frac-times 9.1

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

   9. Taylor expanded around 0 4.1

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

   10. Simplified 4.1

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

   if 1.78695894402767e+135 < (* PI l)

   1. Initial program 20.8

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

   3. Applied add-cube-cbrt 20.8

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

   4. Applied times-frac 20.8

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

   5. Applied associate-*l* 20.8

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

   7. Applied tan-quot 20.8

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

   8. Applied frac-times 20.8

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

   10. Applied add-cube-cbrt 20.8

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

   11. Applied associate-*r* 20.8

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

   13. Applied expm1-log1p-u 20.8

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

4. Final simplification 8.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \le -3.131028332018703448139277285469040119312 \cdot 10^{156}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \left(\left(\sqrt[3]{\frac{\sqrt[3]{1}}{F}} \cdot \sqrt[3]{\frac{\sqrt[3]{1}}{F}}\right) \cdot \left(\sqrt[3]{\frac{\sqrt[3]{1}}{F}} \cdot \tan \left(\pi \cdot \ell\right)\right)\right)\\ \mathbf{elif}\;\pi \cdot \ell \le 1.786958944027669917833146492636069850813 \cdot 10^{135}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \frac{\sqrt[3]{1} \cdot \sin \left(\pi \cdot \ell\right)}{F \cdot \mathsf{fma}\left(\frac{1}{24} \cdot {\pi}^{4}, {\ell}^{4}, 1 - \frac{1}{2} \cdot \left({\pi}^{2} \cdot {\ell}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \frac{\sqrt[3]{1} \cdot \sin \left(\pi \cdot \ell\right)}{F \cdot \cos \left(\left(\pi \cdot \left(\sqrt[3]{\ell} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\ell}\right)\right)\right)\right) \cdot \sqrt[3]{\ell}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019346 +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))