Toniolo and Linder, Equation (10-)

Average Error: 47.3 → 8.7

Time: 1.4m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;k \le -1.658703261686034 \cdot 10^{+150}:\\ \;\;\;\;\frac{-2}{\frac{t \cdot \left(k \cdot \left(\left(k \cdot \frac{t}{\ell}\right) \cdot \frac{t}{\ell}\right)\right)}{t}} \cdot \frac{t}{-\tan k \cdot \sin k}\\ \mathbf{elif}\;k \le 1.7572156172981588 \cdot 10^{+142}:\\ \;\;\;\;\frac{\ell}{\tan k \cdot \sin k} \cdot \frac{\frac{-2}{t \cdot \frac{k \cdot k}{\ell}}}{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-2}{t \cdot \left(\frac{k}{t} \cdot \left(\frac{t}{\ell} \cdot \left(\frac{k}{t} \cdot \frac{t}{\ell}\right)\right)\right)}}{\sqrt[3]{-\tan k \cdot \sin k}} \cdot \frac{1}{\sqrt[3]{-\tan k \cdot \sin k} \cdot \sqrt[3]{-\tan k \cdot \sin k}}\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r2196671 = 2.0;
        double r2196672 = t;
        double r2196673 = 3.0;
        double r2196674 = pow(r2196672, r2196673);
        double r2196675 = l;
        double r2196676 = r2196675 * r2196675;
        double r2196677 = r2196674 / r2196676;
        double r2196678 = k;
        double r2196679 = sin(r2196678);
        double r2196680 = r2196677 * r2196679;
        double r2196681 = tan(r2196678);
        double r2196682 = r2196680 * r2196681;
        double r2196683 = 1.0;
        double r2196684 = r2196678 / r2196672;
        double r2196685 = pow(r2196684, r2196671);
        double r2196686 = r2196683 + r2196685;
        double r2196687 = r2196686 - r2196683;
        double r2196688 = r2196682 * r2196687;
        double r2196689 = r2196671 / r2196688;
        return r2196689;
}

↓

double f(double t, double l, double k) {
        double r2196690 = k;
        double r2196691 = -1.658703261686034e+150;
        bool r2196692 = r2196690 <= r2196691;
        double r2196693 = -2.0;
        double r2196694 = t;
        double r2196695 = l;
        double r2196696 = r2196694 / r2196695;
        double r2196697 = r2196690 * r2196696;
        double r2196698 = r2196697 * r2196696;
        double r2196699 = r2196690 * r2196698;
        double r2196700 = r2196694 * r2196699;
        double r2196701 = r2196700 / r2196694;
        double r2196702 = r2196693 / r2196701;
        double r2196703 = tan(r2196690);
        double r2196704 = sin(r2196690);
        double r2196705 = r2196703 * r2196704;
        double r2196706 = -r2196705;
        double r2196707 = r2196694 / r2196706;
        double r2196708 = r2196702 * r2196707;
        double r2196709 = 1.7572156172981588e+142;
        bool r2196710 = r2196690 <= r2196709;
        double r2196711 = r2196695 / r2196705;
        double r2196712 = r2196690 * r2196690;
        double r2196713 = r2196712 / r2196695;
        double r2196714 = r2196694 * r2196713;
        double r2196715 = r2196693 / r2196714;
        double r2196716 = -1.0;
        double r2196717 = r2196715 / r2196716;
        double r2196718 = r2196711 * r2196717;
        double r2196719 = r2196690 / r2196694;
        double r2196720 = r2196719 * r2196696;
        double r2196721 = r2196696 * r2196720;
        double r2196722 = r2196719 * r2196721;
        double r2196723 = r2196694 * r2196722;
        double r2196724 = r2196693 / r2196723;
        double r2196725 = cbrt(r2196706);
        double r2196726 = r2196724 / r2196725;
        double r2196727 = 1.0;
        double r2196728 = r2196725 * r2196725;
        double r2196729 = r2196727 / r2196728;
        double r2196730 = r2196726 * r2196729;
        double r2196731 = r2196710 ? r2196718 : r2196730;
        double r2196732 = r2196692 ? r2196708 : r2196731;
        return r2196732;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 3 regimes

2. if k < -1.658703261686034e+150

   1. Initial program 38.4

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

   2. Simplified 17.6

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

   4. Applied frac-2neg 17.6

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

   5. Simplified 12.9

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

   7. Applied *-un-lft-identity 12.9

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

   8. Applied associate-*r/ 12.9

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

   9. Applied associate-*l/ 12.9

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

   10. Applied associate-*l/ 16.2

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

   11. Applied associate-*r/ 17.0

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

   12. Applied associate-/r/ 17.0

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

   13. Applied times-frac 17.0

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

   14. Simplified 13.5

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

   if -1.658703261686034e+150 < k < 1.7572156172981588e+142

   1. Initial program 53.2

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

   2. Simplified 24.3

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

   4. Applied frac-2neg 24.3

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

   5. Simplified 19.9

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

   7. Applied associate-*r* 19.0

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

   9. Applied neg-mul-1 19.0

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

   10. Applied associate-*l/ 19.0

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

   11. Applied associate-*l/ 19.0

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

   12. Applied associate-*r/ 17.4

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

   13. Applied associate-*l/ 13.3

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

   14. Applied associate-/r/ 13.1

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

   15. Applied times-frac 12.3

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

   16. Taylor expanded around -inf 5.7

       \[\leadsto \frac{\frac{-2}{\color{blue}{\frac{{k}^{2}}{\ell}} \cdot t}}{-1} \cdot \frac{\ell}{\sin k \cdot \tan k}\]

   17. Simplified 5.7

       \[\leadsto \frac{\frac{-2}{\color{blue}{\frac{k \cdot k}{\ell}} \cdot t}}{-1} \cdot \frac{\ell}{\sin k \cdot \tan k}\]

   if 1.7572156172981588e+142 < k

   1. Initial program 40.0

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

   2. Simplified 16.4

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

   4. Applied frac-2neg 16.4

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

   5. Simplified 12.6

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

   7. Applied associate-*r* 12.0

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

   9. Applied add-cube-cbrt 12.1

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

   10. Applied *-un-lft-identity 12.1

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

   11. Applied times-frac 12.1

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

4. Final simplification 8.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \le -1.658703261686034 \cdot 10^{+150}:\\ \;\;\;\;\frac{-2}{\frac{t \cdot \left(k \cdot \left(\left(k \cdot \frac{t}{\ell}\right) \cdot \frac{t}{\ell}\right)\right)}{t}} \cdot \frac{t}{-\tan k \cdot \sin k}\\ \mathbf{elif}\;k \le 1.7572156172981588 \cdot 10^{+142}:\\ \;\;\;\;\frac{\ell}{\tan k \cdot \sin k} \cdot \frac{\frac{-2}{t \cdot \frac{k \cdot k}{\ell}}}{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-2}{t \cdot \left(\frac{k}{t} \cdot \left(\frac{t}{\ell} \cdot \left(\frac{k}{t} \cdot \frac{t}{\ell}\right)\right)\right)}}{\sqrt[3]{-\tan k \cdot \sin k}} \cdot \frac{1}{\sqrt[3]{-\tan k \cdot \sin k} \cdot \sqrt[3]{-\tan k \cdot \sin k}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019149 +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))))