Toniolo and Linder, Equation (10-)

Average Error: 46.7 → 8.4

Time: 1.1m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;k \le -1.3736813118155453 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{-2}{t \cdot \left(\left(\frac{k}{\ell} \cdot \frac{t}{\ell}\right) \cdot \frac{k}{t}\right)}}{\left(-\tan k\right) \cdot \sin k}\\ \mathbf{elif}\;k \le 6.197353443504037 \cdot 10^{+128}:\\ \;\;\;\;\frac{\frac{-2}{\frac{t}{\frac{\ell}{k \cdot k}}}}{-\frac{\tan k \cdot \sin k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-2}{\frac{\frac{\frac{t \cdot k}{t}}{\frac{\ell}{t}} \cdot \left(t \cdot k\right)}{t}}}{-\frac{\tan k \cdot \sin k}{\ell}}\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r2076767 = 2.0;
        double r2076768 = t;
        double r2076769 = 3.0;
        double r2076770 = pow(r2076768, r2076769);
        double r2076771 = l;
        double r2076772 = r2076771 * r2076771;
        double r2076773 = r2076770 / r2076772;
        double r2076774 = k;
        double r2076775 = sin(r2076774);
        double r2076776 = r2076773 * r2076775;
        double r2076777 = tan(r2076774);
        double r2076778 = r2076776 * r2076777;
        double r2076779 = 1.0;
        double r2076780 = r2076774 / r2076768;
        double r2076781 = pow(r2076780, r2076767);
        double r2076782 = r2076779 + r2076781;
        double r2076783 = r2076782 - r2076779;
        double r2076784 = r2076778 * r2076783;
        double r2076785 = r2076767 / r2076784;
        return r2076785;
}

↓

double f(double t, double l, double k) {
        double r2076786 = k;
        double r2076787 = -1.3736813118155453e+154;
        bool r2076788 = r2076786 <= r2076787;
        double r2076789 = -2.0;
        double r2076790 = t;
        double r2076791 = l;
        double r2076792 = r2076786 / r2076791;
        double r2076793 = r2076790 / r2076791;
        double r2076794 = r2076792 * r2076793;
        double r2076795 = r2076786 / r2076790;
        double r2076796 = r2076794 * r2076795;
        double r2076797 = r2076790 * r2076796;
        double r2076798 = r2076789 / r2076797;
        double r2076799 = tan(r2076786);
        double r2076800 = -r2076799;
        double r2076801 = sin(r2076786);
        double r2076802 = r2076800 * r2076801;
        double r2076803 = r2076798 / r2076802;
        double r2076804 = 6.197353443504037e+128;
        bool r2076805 = r2076786 <= r2076804;
        double r2076806 = r2076786 * r2076786;
        double r2076807 = r2076791 / r2076806;
        double r2076808 = r2076790 / r2076807;
        double r2076809 = r2076789 / r2076808;
        double r2076810 = r2076799 * r2076801;
        double r2076811 = r2076810 / r2076791;
        double r2076812 = -r2076811;
        double r2076813 = r2076809 / r2076812;
        double r2076814 = r2076790 * r2076786;
        double r2076815 = r2076814 / r2076790;
        double r2076816 = r2076791 / r2076790;
        double r2076817 = r2076815 / r2076816;
        double r2076818 = r2076817 * r2076814;
        double r2076819 = r2076818 / r2076790;
        double r2076820 = r2076789 / r2076819;
        double r2076821 = r2076820 / r2076812;
        double r2076822 = r2076805 ? r2076813 : r2076821;
        double r2076823 = r2076788 ? r2076803 : r2076822;
        return r2076823;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 3 regimes

2. if k < -1.3736813118155453e+154

   1. Initial program 36.6

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

   2. Simplified 16.9

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

   4. Applied frac-2neg 16.9

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

   5. Simplified 12.0

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

   7. Applied associate-*r* 11.5

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

   8. Taylor expanded around 0 11.5

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

   if -1.3736813118155453e+154 < k < 6.197353443504037e+128

   1. Initial program 53.5

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

   2. Simplified 25.3

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

   4. Applied frac-2neg 25.3

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

   5. Simplified 20.0

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

   7. Applied associate-*r* 19.5

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

   9. Applied associate-*l/ 19.5

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

   10. Applied associate-*l/ 19.5

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

   11. Applied associate-*r/ 17.9

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

   12. Applied associate-*l/ 12.8

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

   13. Applied associate-/r/ 12.5

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

   14. Applied associate-/l* 11.7

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

   15. Taylor expanded around 0 10.2

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

   16. Simplified 5.0

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

   if 6.197353443504037e+128 < k

   1. Initial program 38.3

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

   2. Simplified 18.1

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

   4. Applied frac-2neg 18.1

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

   5. Simplified 13.4

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

   7. Applied associate-*r* 12.7

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

   9. Applied associate-*l/ 12.7

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

   10. Applied associate-*l/ 12.7

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

   11. Applied associate-*r/ 13.7

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

   12. Applied associate-*l/ 17.2

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

   13. Applied associate-/r/ 17.2

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

   14. Applied associate-/l* 17.2

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

   16. Applied associate-*l/ 17.2

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

   17. Applied associate-*l/ 18.3

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

   18. Simplified 14.1

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

4. Final simplification 8.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \le -1.3736813118155453 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{-2}{t \cdot \left(\left(\frac{k}{\ell} \cdot \frac{t}{\ell}\right) \cdot \frac{k}{t}\right)}}{\left(-\tan k\right) \cdot \sin k}\\ \mathbf{elif}\;k \le 6.197353443504037 \cdot 10^{+128}:\\ \;\;\;\;\frac{\frac{-2}{\frac{t}{\frac{\ell}{k \cdot k}}}}{-\frac{\tan k \cdot \sin k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-2}{\frac{\frac{\frac{t \cdot k}{t}}{\frac{\ell}{t}} \cdot \left(t \cdot k\right)}{t}}}{-\frac{\tan k \cdot \sin k}{\ell}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019152 +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))))