Toniolo and Linder, Equation (10+)

Average Error: 31.5 → 9.0

Time: 38.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -5.032269741332676 \cdot 10^{-63}:\\ \;\;\;\;\frac{2}{\frac{\frac{\frac{\tan k \cdot t}{\frac{\ell}{t}}}{\frac{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}{\sin k \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}}}{\frac{\sqrt[3]{\ell}}{t}}}\\ \mathbf{elif}\;t \le 3.5056626454071755 \cdot 10^{-65}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(\frac{k \cdot k}{\ell}, \frac{\sin k \cdot \sin k}{\cos k}, \left(2 \cdot \frac{t \cdot t}{\ell}\right) \cdot \frac{\sin k \cdot \sin k}{\cos k}\right)}{\frac{\ell}{t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \left(\sin k \cdot \left(\frac{t \cdot \sin k}{\cos k \cdot \ell} \cdot t\right)\right)}{\frac{\ell}{t}}}\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r2750587 = 2.0;
        double r2750588 = t;
        double r2750589 = 3.0;
        double r2750590 = pow(r2750588, r2750589);
        double r2750591 = l;
        double r2750592 = r2750591 * r2750591;
        double r2750593 = r2750590 / r2750592;
        double r2750594 = k;
        double r2750595 = sin(r2750594);
        double r2750596 = r2750593 * r2750595;
        double r2750597 = tan(r2750594);
        double r2750598 = r2750596 * r2750597;
        double r2750599 = 1.0;
        double r2750600 = r2750594 / r2750588;
        double r2750601 = pow(r2750600, r2750587);
        double r2750602 = r2750599 + r2750601;
        double r2750603 = r2750602 + r2750599;
        double r2750604 = r2750598 * r2750603;
        double r2750605 = r2750587 / r2750604;
        return r2750605;
}

↓

double f(double t, double l, double k) {
        double r2750606 = t;
        double r2750607 = -5.032269741332676e-63;
        bool r2750608 = r2750606 <= r2750607;
        double r2750609 = 2.0;
        double r2750610 = k;
        double r2750611 = tan(r2750610);
        double r2750612 = r2750611 * r2750606;
        double r2750613 = l;
        double r2750614 = r2750613 / r2750606;
        double r2750615 = r2750612 / r2750614;
        double r2750616 = cbrt(r2750613);
        double r2750617 = r2750616 * r2750616;
        double r2750618 = sin(r2750610);
        double r2750619 = r2750610 / r2750606;
        double r2750620 = fma(r2750619, r2750619, r2750609);
        double r2750621 = r2750618 * r2750620;
        double r2750622 = r2750617 / r2750621;
        double r2750623 = r2750615 / r2750622;
        double r2750624 = r2750616 / r2750606;
        double r2750625 = r2750623 / r2750624;
        double r2750626 = r2750609 / r2750625;
        double r2750627 = 3.5056626454071755e-65;
        bool r2750628 = r2750606 <= r2750627;
        double r2750629 = r2750610 * r2750610;
        double r2750630 = r2750629 / r2750613;
        double r2750631 = r2750618 * r2750618;
        double r2750632 = cos(r2750610);
        double r2750633 = r2750631 / r2750632;
        double r2750634 = r2750606 * r2750606;
        double r2750635 = r2750634 / r2750613;
        double r2750636 = r2750609 * r2750635;
        double r2750637 = r2750636 * r2750633;
        double r2750638 = fma(r2750630, r2750633, r2750637);
        double r2750639 = r2750638 / r2750614;
        double r2750640 = r2750609 / r2750639;
        double r2750641 = r2750606 * r2750618;
        double r2750642 = r2750632 * r2750613;
        double r2750643 = r2750641 / r2750642;
        double r2750644 = r2750643 * r2750606;
        double r2750645 = r2750618 * r2750644;
        double r2750646 = r2750620 * r2750645;
        double r2750647 = r2750646 / r2750614;
        double r2750648 = r2750609 / r2750647;
        double r2750649 = r2750628 ? r2750640 : r2750648;
        double r2750650 = r2750608 ? r2750626 : r2750649;
        return r2750650;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 3 regimes

2. if t < -5.032269741332676e-63

   1. Initial program 21.7

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

   2. Simplified 8.1

      \[\leadsto \color{blue}{\frac{2}{\left(\left(\frac{t}{\frac{\ell}{t}} \cdot \frac{\tan k}{\frac{\ell}{t}}\right) \cdot \sin k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}}\]
   3. Using strategy rm

   4. Applied associate-*l/ 7.0

      \[\leadsto \frac{2}{\left(\color{blue}{\frac{t \cdot \frac{\tan k}{\frac{\ell}{t}}}{\frac{\ell}{t}}} \cdot \sin k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}\]

   5. Applied associate-*l/ 4.6

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(t \cdot \frac{\tan k}{\frac{\ell}{t}}\right) \cdot \sin k}{\frac{\ell}{t}}} \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}\]

   6. Applied associate-*l/ 4.4

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(\left(t \cdot \frac{\tan k}{\frac{\ell}{t}}\right) \cdot \sin k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}{\frac{\ell}{t}}}}\]
   7. Using strategy rm

   8. Applied *-un-lft-identity 4.4

      \[\leadsto \frac{2}{\frac{\left(\left(t \cdot \frac{\tan k}{\frac{\ell}{t}}\right) \cdot \sin k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}{\frac{\ell}{\color{blue}{1 \cdot t}}}}\]

   9. Applied add-cube-cbrt 4.6

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

   10. Applied times-frac 4.6

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

   11. Applied associate-/r* 5.5

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

   12. Simplified 6.0

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

   if -5.032269741332676e-63 < t < 3.5056626454071755e-65

   1. Initial program 56.1

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

   2. Simplified 39.8

      \[\leadsto \color{blue}{\frac{2}{\left(\left(\frac{t}{\frac{\ell}{t}} \cdot \frac{\tan k}{\frac{\ell}{t}}\right) \cdot \sin k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}}\]
   3. Using strategy rm

   4. Applied associate-*l/ 39.8

      \[\leadsto \frac{2}{\left(\color{blue}{\frac{t \cdot \frac{\tan k}{\frac{\ell}{t}}}{\frac{\ell}{t}}} \cdot \sin k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}\]

   5. Applied associate-*l/ 41.1

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(t \cdot \frac{\tan k}{\frac{\ell}{t}}\right) \cdot \sin k}{\frac{\ell}{t}}} \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}\]

   6. Applied associate-*l/ 37.3

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(\left(t \cdot \frac{\tan k}{\frac{\ell}{t}}\right) \cdot \sin k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}{\frac{\ell}{t}}}}\]
   7. Using strategy rm

   8. Applied *-un-lft-identity 37.3

      \[\leadsto \frac{2}{\frac{\left(\left(t \cdot \frac{\tan k}{\frac{\ell}{t}}\right) \cdot \sin k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}{\frac{\ell}{\color{blue}{1 \cdot t}}}}\]

   9. Applied *-un-lft-identity 37.3

      \[\leadsto \frac{2}{\frac{\left(\left(t \cdot \frac{\tan k}{\frac{\ell}{t}}\right) \cdot \sin k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}{\frac{\color{blue}{1 \cdot \ell}}{1 \cdot t}}}\]

   10. Applied times-frac 37.3

       \[\leadsto \frac{2}{\frac{\left(\left(t \cdot \frac{\tan k}{\frac{\ell}{t}}\right) \cdot \sin k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}{\color{blue}{\frac{1}{1} \cdot \frac{\ell}{t}}}}\]

   11. Applied associate-/r* 37.3

       \[\leadsto \frac{2}{\color{blue}{\frac{\frac{\left(\left(t \cdot \frac{\tan k}{\frac{\ell}{t}}\right) \cdot \sin k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}{\frac{1}{1}}}{\frac{\ell}{t}}}}\]

   12. Simplified 31.3

       \[\leadsto \frac{2}{\frac{\color{blue}{\frac{\tan k}{\frac{\ell}{t}} \cdot \left(t \cdot \left(\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \sin k\right)\right)}}{\frac{\ell}{t}}}\]

   13. Taylor expanded around inf 24.8

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

   14. Simplified 23.6

       \[\leadsto \frac{2}{\frac{\frac{\tan k}{\frac{\ell}{t}} \cdot \color{blue}{\mathsf{fma}\left(2, t \cdot \sin k, \frac{\sin k}{\frac{t}{k \cdot k}}\right)}}{\frac{\ell}{t}}}\]

   15. Taylor expanded around inf 21.2

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

   16. Simplified 18.1

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

   if 3.5056626454071755e-65 < t

   1. Initial program 21.6

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

   2. Simplified 8.9

      \[\leadsto \color{blue}{\frac{2}{\left(\left(\frac{t}{\frac{\ell}{t}} \cdot \frac{\tan k}{\frac{\ell}{t}}\right) \cdot \sin k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}}\]
   3. Using strategy rm

   4. Applied associate-*l/ 7.7

      \[\leadsto \frac{2}{\left(\color{blue}{\frac{t \cdot \frac{\tan k}{\frac{\ell}{t}}}{\frac{\ell}{t}}} \cdot \sin k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}\]

   5. Applied associate-*l/ 5.2

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(t \cdot \frac{\tan k}{\frac{\ell}{t}}\right) \cdot \sin k}{\frac{\ell}{t}}} \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}\]

   6. Applied associate-*l/ 4.6

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(\left(t \cdot \frac{\tan k}{\frac{\ell}{t}}\right) \cdot \sin k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}{\frac{\ell}{t}}}}\]

   7. Taylor expanded around inf 4.6

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

4. Final simplification 9.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -5.032269741332676 \cdot 10^{-63}:\\ \;\;\;\;\frac{2}{\frac{\frac{\frac{\tan k \cdot t}{\frac{\ell}{t}}}{\frac{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}{\sin k \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}}}{\frac{\sqrt[3]{\ell}}{t}}}\\ \mathbf{elif}\;t \le 3.5056626454071755 \cdot 10^{-65}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(\frac{k \cdot k}{\ell}, \frac{\sin k \cdot \sin k}{\cos k}, \left(2 \cdot \frac{t \cdot t}{\ell}\right) \cdot \frac{\sin k \cdot \sin k}{\cos k}\right)}{\frac{\ell}{t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \left(\sin k \cdot \left(\frac{t \cdot \sin k}{\cos k \cdot \ell} \cdot t\right)\right)}{\frac{\ell}{t}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019164 +o rules:numerics
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10+)"
  (/ 2 (* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1))))