Toniolo and Linder, Equation (10+)

Average Error: 31.7 → 5.0

Time: 45.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;k \le -3.313813655101207 \cdot 10^{-57}:\\ \;\;\;\;\frac{1}{t} \cdot \frac{\frac{2}{\sin k}}{\mathsf{fma}\left(\frac{\sin k}{\cos k} \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right), 2, \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{\sin k}{\cos k}\right)}\\ \mathbf{elif}\;k \le 1.6490058462058476 \cdot 10^{-140}:\\ \;\;\;\;\frac{\frac{1}{t}}{\frac{\tan k}{\frac{\ell}{t}}} \cdot \frac{\frac{2}{\sin k}}{\frac{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}{\frac{\ell}{t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{t} \cdot \frac{\frac{\sqrt{2}}{\sin k}}{\mathsf{fma}\left(\frac{\sin k}{\cos k} \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right), 2, \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{\sin k}{\cos k}\right)}\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r2089600 = 2.0;
        double r2089601 = t;
        double r2089602 = 3.0;
        double r2089603 = pow(r2089601, r2089602);
        double r2089604 = l;
        double r2089605 = r2089604 * r2089604;
        double r2089606 = r2089603 / r2089605;
        double r2089607 = k;
        double r2089608 = sin(r2089607);
        double r2089609 = r2089606 * r2089608;
        double r2089610 = tan(r2089607);
        double r2089611 = r2089609 * r2089610;
        double r2089612 = 1.0;
        double r2089613 = r2089607 / r2089601;
        double r2089614 = pow(r2089613, r2089600);
        double r2089615 = r2089612 + r2089614;
        double r2089616 = r2089615 + r2089612;
        double r2089617 = r2089611 * r2089616;
        double r2089618 = r2089600 / r2089617;
        return r2089618;
}

↓

double f(double t, double l, double k) {
        double r2089619 = k;
        double r2089620 = -3.313813655101207e-57;
        bool r2089621 = r2089619 <= r2089620;
        double r2089622 = 1.0;
        double r2089623 = t;
        double r2089624 = r2089622 / r2089623;
        double r2089625 = 2.0;
        double r2089626 = sin(r2089619);
        double r2089627 = r2089625 / r2089626;
        double r2089628 = cos(r2089619);
        double r2089629 = r2089626 / r2089628;
        double r2089630 = l;
        double r2089631 = r2089623 / r2089630;
        double r2089632 = r2089631 * r2089631;
        double r2089633 = r2089629 * r2089632;
        double r2089634 = r2089619 / r2089630;
        double r2089635 = r2089634 * r2089634;
        double r2089636 = r2089635 * r2089629;
        double r2089637 = fma(r2089633, r2089625, r2089636);
        double r2089638 = r2089627 / r2089637;
        double r2089639 = r2089624 * r2089638;
        double r2089640 = 1.6490058462058476e-140;
        bool r2089641 = r2089619 <= r2089640;
        double r2089642 = tan(r2089619);
        double r2089643 = r2089630 / r2089623;
        double r2089644 = r2089642 / r2089643;
        double r2089645 = r2089624 / r2089644;
        double r2089646 = r2089619 / r2089623;
        double r2089647 = fma(r2089646, r2089646, r2089625);
        double r2089648 = r2089647 / r2089643;
        double r2089649 = r2089627 / r2089648;
        double r2089650 = r2089645 * r2089649;
        double r2089651 = sqrt(r2089625);
        double r2089652 = r2089651 / r2089623;
        double r2089653 = r2089651 / r2089626;
        double r2089654 = r2089653 / r2089637;
        double r2089655 = r2089652 * r2089654;
        double r2089656 = r2089641 ? r2089650 : r2089655;
        double r2089657 = r2089621 ? r2089639 : r2089656;
        return r2089657;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 3 regimes

2. if k < -3.313813655101207e-57

   1. Initial program 31.1

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

   2. Simplified 19.6

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

   4. Applied associate-*l/ 19.5

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

   5. Applied associate-/r/ 19.4

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

   6. Applied associate-/l* 17.2

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

   7. Taylor expanded around inf 22.2

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

   8. Simplified 4.9

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

   10. Applied *-un-lft-identity 4.9

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

   11. Applied *-un-lft-identity 4.9

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

   12. Applied times-frac 4.9

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

   13. Applied times-frac 4.9

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

   14. Simplified 4.9

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

   if -3.313813655101207e-57 < k < 1.6490058462058476e-140

   1. Initial program 35.2

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

   2. Simplified 25.6

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

   4. Applied associate-*l/ 19.1

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

   5. Applied associate-/r/ 19.0

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

   6. Applied associate-/l* 18.3

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

   8. Applied times-frac 6.6

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

   9. Applied *-un-lft-identity 6.6

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

   10. Applied times-frac 6.6

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

   11. Applied times-frac 3.0

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

   if 1.6490058462058476e-140 < k

   1. Initial program 30.4

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

   2. Simplified 18.4

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

   4. Applied associate-*l/ 17.8

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

   5. Applied associate-/r/ 17.7

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

   6. Applied associate-/l* 15.8

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

   7. Taylor expanded around inf 22.8

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

   8. Simplified 5.7

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

   10. Applied *-un-lft-identity 5.7

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

   11. Applied add-sqr-sqrt 5.9

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

   12. Applied times-frac 5.8

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

   13. Applied times-frac 6.1

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

   14. Simplified 6.1

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

4. Final simplification 5.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \le -3.313813655101207 \cdot 10^{-57}:\\ \;\;\;\;\frac{1}{t} \cdot \frac{\frac{2}{\sin k}}{\mathsf{fma}\left(\frac{\sin k}{\cos k} \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right), 2, \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{\sin k}{\cos k}\right)}\\ \mathbf{elif}\;k \le 1.6490058462058476 \cdot 10^{-140}:\\ \;\;\;\;\frac{\frac{1}{t}}{\frac{\tan k}{\frac{\ell}{t}}} \cdot \frac{\frac{2}{\sin k}}{\frac{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}{\frac{\ell}{t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{t} \cdot \frac{\frac{\sqrt{2}}{\sin k}}{\mathsf{fma}\left(\frac{\sin k}{\cos k} \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right), 2, \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{\sin k}{\cos k}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019143 +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))))