Toniolo and Linder, Equation (10+)

Average Error: 32.2 → 10.0

Time: 22.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -5.55809296410159264 \cdot 10^{-38} \lor \neg \left(t \le 7.03820609424746626 \cdot 10^{-94}\right):\\ \;\;\;\;\frac{2}{\left(\left(\left({\left(\sqrt[3]{t}\right)}^{3} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)\right) \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \frac{1}{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3}} \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{{k}^{2} \cdot {\left(\sin k\right)}^{2}}{\ell} + 2 \cdot \frac{{t}^{2} \cdot {\left(\sin k\right)}^{2}}{\ell}}{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3}} \cdot \cos k}}\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r134559 = 2.0;
        double r134560 = t;
        double r134561 = 3.0;
        double r134562 = pow(r134560, r134561);
        double r134563 = l;
        double r134564 = r134563 * r134563;
        double r134565 = r134562 / r134564;
        double r134566 = k;
        double r134567 = sin(r134566);
        double r134568 = r134565 * r134567;
        double r134569 = tan(r134566);
        double r134570 = r134568 * r134569;
        double r134571 = 1.0;
        double r134572 = r134566 / r134560;
        double r134573 = pow(r134572, r134559);
        double r134574 = r134571 + r134573;
        double r134575 = r134574 + r134571;
        double r134576 = r134570 * r134575;
        double r134577 = r134559 / r134576;
        return r134577;
}

↓

double f(double t, double l, double k) {
        double r134578 = t;
        double r134579 = -5.558092964101593e-38;
        bool r134580 = r134578 <= r134579;
        double r134581 = 7.038206094247466e-94;
        bool r134582 = r134578 <= r134581;
        double r134583 = !r134582;
        bool r134584 = r134580 || r134583;
        double r134585 = 2.0;
        double r134586 = cbrt(r134578);
        double r134587 = 3.0;
        double r134588 = pow(r134586, r134587);
        double r134589 = l;
        double r134590 = r134588 / r134589;
        double r134591 = k;
        double r134592 = sin(r134591);
        double r134593 = r134590 * r134592;
        double r134594 = r134588 * r134593;
        double r134595 = r134594 * r134592;
        double r134596 = 1.0;
        double r134597 = r134591 / r134578;
        double r134598 = pow(r134597, r134585);
        double r134599 = r134596 + r134598;
        double r134600 = r134599 + r134596;
        double r134601 = r134595 * r134600;
        double r134602 = 1.0;
        double r134603 = r134589 / r134588;
        double r134604 = cos(r134591);
        double r134605 = r134603 * r134604;
        double r134606 = r134602 / r134605;
        double r134607 = r134601 * r134606;
        double r134608 = r134585 / r134607;
        double r134609 = 2.0;
        double r134610 = pow(r134591, r134609);
        double r134611 = pow(r134592, r134609);
        double r134612 = r134610 * r134611;
        double r134613 = r134612 / r134589;
        double r134614 = pow(r134578, r134609);
        double r134615 = r134614 * r134611;
        double r134616 = r134615 / r134589;
        double r134617 = r134585 * r134616;
        double r134618 = r134613 + r134617;
        double r134619 = r134618 / r134605;
        double r134620 = r134585 / r134619;
        double r134621 = r134584 ? r134608 : r134620;
        return r134621;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 2 regimes

2. if t < -5.558092964101593e-38 or 7.038206094247466e-94 < 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 22.7

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

   4. Applied unpow-prod-down 22.7

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

   5. Applied times-frac 16.4

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

   6. Applied associate-*l* 14.2

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

   8. Applied unpow-prod-down 14.2

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

   9. Applied associate-/l* 9.2

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

   11. Applied tan-quot 9.2

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

   12. Applied associate-*l/ 8.0

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

   13. Applied frac-times 5.8

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

   14. Applied associate-*l/ 5.2

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

   16. Applied div-inv 5.2

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

   if -5.558092964101593e-38 < t < 7.038206094247466e-94

   1. Initial program 57.0

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

   3. Applied add-cube-cbrt 57.0

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

   4. Applied unpow-prod-down 57.0

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

   5. Applied times-frac 48.8

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

   6. Applied associate-*l* 48.4

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

   8. Applied unpow-prod-down 48.4

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

   9. Applied associate-/l* 41.1

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

   11. Applied tan-quot 41.1

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

   12. Applied associate-*l/ 41.1

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

   13. Applied frac-times 42.4

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

   14. Applied associate-*l/ 39.1

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

   15. Taylor expanded around inf 22.5

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

4. Final simplification 10.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -5.55809296410159264 \cdot 10^{-38} \lor \neg \left(t \le 7.03820609424746626 \cdot 10^{-94}\right):\\ \;\;\;\;\frac{2}{\left(\left(\left({\left(\sqrt[3]{t}\right)}^{3} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)\right) \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \frac{1}{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3}} \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{{k}^{2} \cdot {\left(\sin k\right)}^{2}}{\ell} + 2 \cdot \frac{{t}^{2} \cdot {\left(\sin k\right)}^{2}}{\ell}}{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3}} \cdot \cos k}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020018 
(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))))