Toniolo and Linder, Equation (10+)

Average Error: 32.4 → 12.1

Time: 1.9m

Precision: 64

Math

\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

↓

\[\begin{array}{l} \mathbf{if}\;t \le 2362365607704.0986328125:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(2, \left({\left(\frac{1}{{-1}^{2}}\right)}^{1} \cdot \frac{1}{\cos k}\right) \cdot \frac{\left(\sin k \cdot t\right) \cdot \left(\sin k \cdot t\right)}{\ell}, \frac{{\left(\frac{1}{{-1}^{2}}\right)}^{1}}{\cos k} \cdot \frac{\left(\sin k \cdot k\right) \cdot \left(\sin k \cdot k\right)}{\ell}\right)}{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{\left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right) \cdot {\left(\sqrt[3]{t}\right)}^{3}}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{{\left(\sqrt[3]{t}\right)}^{3}}}} \cdot \frac{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{{\left(\sqrt[3]{t}\right)}^{3}}}}}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{{\left(\sqrt[3]{t}\right)}^{3}}}}}\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r3856550 = 2.0;
        double r3856551 = t;
        double r3856552 = 3.0;
        double r3856553 = pow(r3856551, r3856552);
        double r3856554 = l;
        double r3856555 = r3856554 * r3856554;
        double r3856556 = r3856553 / r3856555;
        double r3856557 = k;
        double r3856558 = sin(r3856557);
        double r3856559 = r3856556 * r3856558;
        double r3856560 = tan(r3856557);
        double r3856561 = r3856559 * r3856560;
        double r3856562 = 1.0;
        double r3856563 = r3856557 / r3856551;
        double r3856564 = pow(r3856563, r3856550);
        double r3856565 = r3856562 + r3856564;
        double r3856566 = r3856565 + r3856562;
        double r3856567 = r3856561 * r3856566;
        double r3856568 = r3856550 / r3856567;
        return r3856568;
}

↓

double f(double t, double l, double k) {
        double r3856569 = t;
        double r3856570 = 2362365607704.0986;
        bool r3856571 = r3856569 <= r3856570;
        double r3856572 = 2.0;
        double r3856573 = 1.0;
        double r3856574 = -1.0;
        double r3856575 = pow(r3856574, r3856572);
        double r3856576 = r3856573 / r3856575;
        double r3856577 = 1.0;
        double r3856578 = pow(r3856576, r3856577);
        double r3856579 = k;
        double r3856580 = cos(r3856579);
        double r3856581 = r3856573 / r3856580;
        double r3856582 = r3856578 * r3856581;
        double r3856583 = sin(r3856579);
        double r3856584 = r3856583 * r3856569;
        double r3856585 = r3856584 * r3856584;
        double r3856586 = l;
        double r3856587 = r3856585 / r3856586;
        double r3856588 = r3856582 * r3856587;
        double r3856589 = r3856578 / r3856580;
        double r3856590 = r3856583 * r3856579;
        double r3856591 = r3856590 * r3856590;
        double r3856592 = r3856591 / r3856586;
        double r3856593 = r3856589 * r3856592;
        double r3856594 = fma(r3856572, r3856588, r3856593);
        double r3856595 = cbrt(r3856569);
        double r3856596 = 3.0;
        double r3856597 = pow(r3856595, r3856596);
        double r3856598 = r3856586 / r3856597;
        double r3856599 = r3856594 / r3856598;
        double r3856600 = r3856572 / r3856599;
        double r3856601 = r3856597 / r3856586;
        double r3856602 = r3856601 * r3856583;
        double r3856603 = r3856602 * r3856597;
        double r3856604 = cbrt(r3856586);
        double r3856605 = cbrt(r3856597);
        double r3856606 = r3856604 / r3856605;
        double r3856607 = r3856603 / r3856606;
        double r3856608 = r3856579 / r3856569;
        double r3856609 = pow(r3856608, r3856572);
        double r3856610 = r3856577 + r3856609;
        double r3856611 = r3856610 + r3856577;
        double r3856612 = tan(r3856579);
        double r3856613 = r3856611 * r3856612;
        double r3856614 = r3856613 / r3856606;
        double r3856615 = r3856607 * r3856614;
        double r3856616 = r3856615 / r3856606;
        double r3856617 = r3856572 / r3856616;
        double r3856618 = r3856571 ? r3856600 : r3856617;
        return r3856618;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 2 regimes

2. if t < 2362365607704.0986

   1. Initial program 36.6

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 36.7

      \[\leadsto \frac{2}{\left(\left(\frac{{\color{blue}{\left(\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}\right)}}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

   4. Applied unpow-prod-down 36.7

      \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3} \cdot {\left(\sqrt[3]{t}\right)}^{3}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

   5. Applied times-frac 29.6

      \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}}{\ell} \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

   6. Applied associate-*l* 28.0

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}}{\ell} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
   7. Using strategy rm

   8. Applied unpow-prod-down 28.0

      \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{\left(\sqrt[3]{t}\right)}^{3} \cdot {\left(\sqrt[3]{t}\right)}^{3}}}{\ell} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

   9. Applied associate-/l* 22.5

      \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3}}}} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
   10. Using strategy rm

   11. Applied associate-*l/ 21.9

       \[\leadsto \frac{2}{\left(\color{blue}{\frac{{\left(\sqrt[3]{t}\right)}^{3} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)}{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3}}}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

   12. Applied associate-*l/ 21.5

       \[\leadsto \frac{2}{\color{blue}{\frac{\left({\left(\sqrt[3]{t}\right)}^{3} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)\right) \cdot \tan k}{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3}}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

   13. Applied associate-*l/ 19.5

       \[\leadsto \frac{2}{\color{blue}{\frac{\left(\left({\left(\sqrt[3]{t}\right)}^{3} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3}}}}}\]

   14. Taylor expanded around -inf 21.3

       \[\leadsto \frac{2}{\frac{\color{blue}{{\left(\frac{1}{{-1}^{2}}\right)}^{1} \cdot \frac{{k}^{2} \cdot {\left(\sin k\right)}^{2}}{\cos k \cdot \ell} + 2 \cdot \left({\left(\frac{1}{{-1}^{2}}\right)}^{1} \cdot \frac{{\left(\sqrt[3]{-1}\right)}^{6} \cdot \left({t}^{2} \cdot {\left(\sin k\right)}^{2}\right)}{\cos k \cdot \ell}\right)}}{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3}}}}\]

   15. Simplified 15.7

       \[\leadsto \frac{2}{\frac{\color{blue}{\mathsf{fma}\left(2, \left({\left(\frac{1}{{-1}^{2}}\right)}^{1} \cdot \frac{-1 \cdot -1}{\cos k}\right) \cdot \frac{\left(t \cdot \sin k\right) \cdot \left(t \cdot \sin k\right)}{\ell}, \frac{{\left(\frac{1}{{-1}^{2}}\right)}^{1}}{\cos k} \cdot \frac{\left(k \cdot \sin k\right) \cdot \left(k \cdot \sin k\right)}{\ell}\right)}}{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3}}}}\]

   if 2362365607704.0986 < t

   1. Initial program 22.5

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 22.6

      \[\leadsto \frac{2}{\left(\left(\frac{{\color{blue}{\left(\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}\right)}}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

   4. Applied unpow-prod-down 22.6

      \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3} \cdot {\left(\sqrt[3]{t}\right)}^{3}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

   5. Applied times-frac 16.3

      \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}}{\ell} \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

   6. Applied associate-*l* 14.4

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}}{\ell} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
   7. Using strategy rm

   8. Applied unpow-prod-down 14.4

      \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{\left(\sqrt[3]{t}\right)}^{3} \cdot {\left(\sqrt[3]{t}\right)}^{3}}}{\ell} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

   9. Applied associate-/l* 8.2

      \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3}}}} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
   10. Using strategy rm

   11. Applied associate-*l/ 6.8

       \[\leadsto \frac{2}{\left(\color{blue}{\frac{{\left(\sqrt[3]{t}\right)}^{3} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)}{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3}}}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

   12. Applied associate-*l/ 3.8

       \[\leadsto \frac{2}{\color{blue}{\frac{\left({\left(\sqrt[3]{t}\right)}^{3} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)\right) \cdot \tan k}{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3}}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

   13. Applied associate-*l/ 3.5

       \[\leadsto \frac{2}{\color{blue}{\frac{\left(\left({\left(\sqrt[3]{t}\right)}^{3} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3}}}}}\]
   14. Using strategy rm

   15. Applied add-cube-cbrt 3.5

       \[\leadsto \frac{2}{\frac{\left(\left({\left(\sqrt[3]{t}\right)}^{3} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}{\frac{\ell}{\color{blue}{\left(\sqrt[3]{{\left(\sqrt[3]{t}\right)}^{3}} \cdot \sqrt[3]{{\left(\sqrt[3]{t}\right)}^{3}}\right) \cdot \sqrt[3]{{\left(\sqrt[3]{t}\right)}^{3}}}}}}\]

   16. Applied add-cube-cbrt 3.5

       \[\leadsto \frac{2}{\frac{\left(\left({\left(\sqrt[3]{t}\right)}^{3} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}{\frac{\color{blue}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}{\left(\sqrt[3]{{\left(\sqrt[3]{t}\right)}^{3}} \cdot \sqrt[3]{{\left(\sqrt[3]{t}\right)}^{3}}\right) \cdot \sqrt[3]{{\left(\sqrt[3]{t}\right)}^{3}}}}}\]

   17. Applied times-frac 3.5

       \[\leadsto \frac{2}{\frac{\left(\left({\left(\sqrt[3]{t}\right)}^{3} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}{\color{blue}{\frac{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}{\sqrt[3]{{\left(\sqrt[3]{t}\right)}^{3}} \cdot \sqrt[3]{{\left(\sqrt[3]{t}\right)}^{3}}} \cdot \frac{\sqrt[3]{\ell}}{\sqrt[3]{{\left(\sqrt[3]{t}\right)}^{3}}}}}}\]

   18. Applied associate-/r* 3.5

       \[\leadsto \frac{2}{\color{blue}{\frac{\frac{\left(\left({\left(\sqrt[3]{t}\right)}^{3} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}{\frac{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}{\sqrt[3]{{\left(\sqrt[3]{t}\right)}^{3}} \cdot \sqrt[3]{{\left(\sqrt[3]{t}\right)}^{3}}}}}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{{\left(\sqrt[3]{t}\right)}^{3}}}}}}\]

   19. Simplified 3.5

       \[\leadsto \frac{2}{\frac{\color{blue}{\frac{\left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right) \cdot {\left(\sqrt[3]{t}\right)}^{3}}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{{\left(\sqrt[3]{t}\right)}^{3}}}} \cdot \frac{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{{\left(\sqrt[3]{t}\right)}^{3}}}}}}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{{\left(\sqrt[3]{t}\right)}^{3}}}}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 12.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le 2362365607704.0986328125:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(2, \left({\left(\frac{1}{{-1}^{2}}\right)}^{1} \cdot \frac{1}{\cos k}\right) \cdot \frac{\left(\sin k \cdot t\right) \cdot \left(\sin k \cdot t\right)}{\ell}, \frac{{\left(\frac{1}{{-1}^{2}}\right)}^{1}}{\cos k} \cdot \frac{\left(\sin k \cdot k\right) \cdot \left(\sin k \cdot k\right)}{\ell}\right)}{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{\left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right) \cdot {\left(\sqrt[3]{t}\right)}^{3}}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{{\left(\sqrt[3]{t}\right)}^{3}}}} \cdot \frac{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{{\left(\sqrt[3]{t}\right)}^{3}}}}}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{{\left(\sqrt[3]{t}\right)}^{3}}}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10+)"
  (/ 2.0 (* (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k)) (+ (+ 1.0 (pow (/ k t) 2.0)) 1.0))))