Toniolo and Linder, Equation (10+)

Average Error: 33.4 → 13.7

Time: 26.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -4.695605514262814169073413975852826605095 \cdot 10^{-38} \lor \neg \left(t \le 2.172967761371277534577367562818039914418 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{2}{\left(\left({\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \cdot \frac{\ell}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{1}{{-1}^{2}}\right)}^{1} \cdot \frac{{k}^{2} \cdot {\left(\sin k\right)}^{2}}{\cos k \cdot \ell} + 2 \cdot \left({\left(\frac{1}{{-1}^{2}}\right)}^{1} \cdot \frac{{\left(\sqrt[3]{-1}\right)}^{6} \cdot \left({t}^{2} \cdot {\left(\sin k\right)}^{2}\right)}{\cos k \cdot \ell}\right)} \cdot \frac{\ell}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r128561 = 2.0;
        double r128562 = t;
        double r128563 = 3.0;
        double r128564 = pow(r128562, r128563);
        double r128565 = l;
        double r128566 = r128565 * r128565;
        double r128567 = r128564 / r128566;
        double r128568 = k;
        double r128569 = sin(r128568);
        double r128570 = r128567 * r128569;
        double r128571 = tan(r128568);
        double r128572 = r128570 * r128571;
        double r128573 = 1.0;
        double r128574 = r128568 / r128562;
        double r128575 = pow(r128574, r128561);
        double r128576 = r128573 + r128575;
        double r128577 = r128576 + r128573;
        double r128578 = r128572 * r128577;
        double r128579 = r128561 / r128578;
        return r128579;
}

↓

double f(double t, double l, double k) {
        double r128580 = t;
        double r128581 = -4.695605514262814e-38;
        bool r128582 = r128580 <= r128581;
        double r128583 = 2.1729677613712775e-05;
        bool r128584 = r128580 <= r128583;
        double r128585 = !r128584;
        bool r128586 = r128582 || r128585;
        double r128587 = 2.0;
        double r128588 = cbrt(r128580);
        double r128589 = r128588 * r128588;
        double r128590 = 3.0;
        double r128591 = 2.0;
        double r128592 = r128590 / r128591;
        double r128593 = pow(r128589, r128592);
        double r128594 = pow(r128588, r128590);
        double r128595 = l;
        double r128596 = r128594 / r128595;
        double r128597 = k;
        double r128598 = sin(r128597);
        double r128599 = r128596 * r128598;
        double r128600 = r128593 * r128599;
        double r128601 = tan(r128597);
        double r128602 = r128600 * r128601;
        double r128603 = 1.0;
        double r128604 = r128597 / r128580;
        double r128605 = pow(r128604, r128587);
        double r128606 = r128603 + r128605;
        double r128607 = r128606 + r128603;
        double r128608 = r128602 * r128607;
        double r128609 = r128587 / r128608;
        double r128610 = r128595 / r128593;
        double r128611 = r128609 * r128610;
        double r128612 = 1.0;
        double r128613 = -1.0;
        double r128614 = pow(r128613, r128587);
        double r128615 = r128612 / r128614;
        double r128616 = pow(r128615, r128603);
        double r128617 = pow(r128597, r128591);
        double r128618 = pow(r128598, r128591);
        double r128619 = r128617 * r128618;
        double r128620 = cos(r128597);
        double r128621 = r128620 * r128595;
        double r128622 = r128619 / r128621;
        double r128623 = r128616 * r128622;
        double r128624 = cbrt(r128613);
        double r128625 = 6.0;
        double r128626 = pow(r128624, r128625);
        double r128627 = pow(r128580, r128591);
        double r128628 = r128627 * r128618;
        double r128629 = r128626 * r128628;
        double r128630 = r128629 / r128621;
        double r128631 = r128616 * r128630;
        double r128632 = r128587 * r128631;
        double r128633 = r128623 + r128632;
        double r128634 = r128587 / r128633;
        double r128635 = r128634 * r128610;
        double r128636 = r128586 ? r128611 : r128635;
        return r128636;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 2 regimes

2. if t < -4.695605514262814e-38 or 2.1729677613712775e-05 < t

   1. Initial program 23.4

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

   3. Applied add-cube-cbrt 23.6

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

   4. Applied unpow-prod-down 23.6

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

   5. Applied times-frac 17.0

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

   7. Applied sqr-pow 17.0

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

   8. Applied associate-/l* 11.8

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

   10. Applied associate-*l/ 11.8

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

   11. Applied associate-*l/ 9.0

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

   12. Applied associate-*l/ 8.8

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

   13. Applied associate-*l/ 8.5

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

   14. Applied associate-/r/ 8.4

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

   16. Applied associate-*l* 4.1

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

   if -4.695605514262814e-38 < t < 2.1729677613712775e-05

   1. Initial program 51.7

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

   3. Applied add-cube-cbrt 51.8

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

   4. Applied unpow-prod-down 51.8

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

   5. Applied times-frac 44.1

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

   7. Applied sqr-pow 44.1

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

   8. Applied associate-/l* 37.7

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

   10. Applied associate-*l/ 37.7

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

   11. Applied associate-*l/ 36.8

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

   12. Applied associate-*l/ 38.3

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

   13. Applied associate-*l/ 35.3

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

   14. Applied associate-/r/ 35.5

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

   15. Taylor expanded around -inf 31.2

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

4. Final simplification 13.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -4.695605514262814169073413975852826605095 \cdot 10^{-38} \lor \neg \left(t \le 2.172967761371277534577367562818039914418 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{2}{\left(\left({\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \cdot \frac{\ell}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{1}{{-1}^{2}}\right)}^{1} \cdot \frac{{k}^{2} \cdot {\left(\sin k\right)}^{2}}{\cos k \cdot \ell} + 2 \cdot \left({\left(\frac{1}{{-1}^{2}}\right)}^{1} \cdot \frac{{\left(\sqrt[3]{-1}\right)}^{6} \cdot \left({t}^{2} \cdot {\left(\sin k\right)}^{2}\right)}{\cos k \cdot \ell}\right)} \cdot \frac{\ell}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019346 
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10+)"
  :precision binary64
  (/ 2 (* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1))))