Toniolo and Linder, Equation (10-)

Average Error: 47.5 → 11.6

Time: 1.4m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;k \le -3.2333473137924527 \cdot 10^{+127}:\\ \;\;\;\;\frac{\frac{-2}{t \cdot \left(\left(\frac{k}{\ell} \cdot \frac{t}{\ell}\right) \cdot \frac{k}{t}\right)}}{\tan k \cdot \left(-\sin k\right)}\\ \mathbf{elif}\;k \le 1.3164499310012228 \cdot 10^{+141}:\\ \;\;\;\;\frac{\frac{-2}{\frac{t}{\ell} \cdot \left(\frac{t \cdot k}{t} \cdot \frac{t \cdot k}{t}\right)}}{-\frac{\tan k \cdot \sin k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin k} \cdot \frac{\frac{-2}{\left(\left(\left(\frac{t}{\ell} \cdot \frac{k}{t}\right) \cdot \frac{t}{\ell}\right) \cdot \frac{k}{t}\right) \cdot t}}{-\tan k}\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r1943781 = 2.0;
        double r1943782 = t;
        double r1943783 = 3.0;
        double r1943784 = pow(r1943782, r1943783);
        double r1943785 = l;
        double r1943786 = r1943785 * r1943785;
        double r1943787 = r1943784 / r1943786;
        double r1943788 = k;
        double r1943789 = sin(r1943788);
        double r1943790 = r1943787 * r1943789;
        double r1943791 = tan(r1943788);
        double r1943792 = r1943790 * r1943791;
        double r1943793 = 1.0;
        double r1943794 = r1943788 / r1943782;
        double r1943795 = pow(r1943794, r1943781);
        double r1943796 = r1943793 + r1943795;
        double r1943797 = r1943796 - r1943793;
        double r1943798 = r1943792 * r1943797;
        double r1943799 = r1943781 / r1943798;
        return r1943799;
}

↓

double f(double t, double l, double k) {
        double r1943800 = k;
        double r1943801 = -3.2333473137924527e+127;
        bool r1943802 = r1943800 <= r1943801;
        double r1943803 = -2.0;
        double r1943804 = t;
        double r1943805 = l;
        double r1943806 = r1943800 / r1943805;
        double r1943807 = r1943804 / r1943805;
        double r1943808 = r1943806 * r1943807;
        double r1943809 = r1943800 / r1943804;
        double r1943810 = r1943808 * r1943809;
        double r1943811 = r1943804 * r1943810;
        double r1943812 = r1943803 / r1943811;
        double r1943813 = tan(r1943800);
        double r1943814 = sin(r1943800);
        double r1943815 = -r1943814;
        double r1943816 = r1943813 * r1943815;
        double r1943817 = r1943812 / r1943816;
        double r1943818 = 1.3164499310012228e+141;
        bool r1943819 = r1943800 <= r1943818;
        double r1943820 = r1943804 * r1943800;
        double r1943821 = r1943820 / r1943804;
        double r1943822 = r1943821 * r1943821;
        double r1943823 = r1943807 * r1943822;
        double r1943824 = r1943803 / r1943823;
        double r1943825 = r1943813 * r1943814;
        double r1943826 = r1943825 / r1943805;
        double r1943827 = -r1943826;
        double r1943828 = r1943824 / r1943827;
        double r1943829 = 1.0;
        double r1943830 = r1943829 / r1943814;
        double r1943831 = r1943807 * r1943809;
        double r1943832 = r1943831 * r1943807;
        double r1943833 = r1943832 * r1943809;
        double r1943834 = r1943833 * r1943804;
        double r1943835 = r1943803 / r1943834;
        double r1943836 = -r1943813;
        double r1943837 = r1943835 / r1943836;
        double r1943838 = r1943830 * r1943837;
        double r1943839 = r1943819 ? r1943828 : r1943838;
        double r1943840 = r1943802 ? r1943817 : r1943839;
        return r1943840;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 3 regimes

2. if k < -3.2333473137924527e+127

   1. Initial program 39.7

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

   2. Simplified 18.8

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

   4. Applied frac-2neg 18.8

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

   5. Simplified 14.7

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

   7. Applied associate-*r* 14.2

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

   8. Taylor expanded around inf 14.1

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

   if -3.2333473137924527e+127 < k < 1.3164499310012228e+141

   1. Initial program 54.5

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

   2. Simplified 24.9

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

   4. Applied frac-2neg 24.9

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

   5. Simplified 20.4

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

   7. Applied associate-*r* 19.8

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

   9. Applied associate-*l/ 19.7

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

   10. Applied associate-*l/ 19.7

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

   11. Applied associate-*r/ 17.8

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

   12. Applied associate-*l/ 13.2

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

   13. Applied associate-/r/ 12.9

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

   14. Applied associate-/l* 12.2

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

   16. Applied pow1 12.2

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

   17. Applied pow1 12.2

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

   18. Applied pow-prod-down 12.2

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

   19. Simplified 10.6

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

   if 1.3164499310012228e+141 < k

   1. Initial program 38.5

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

   2. Simplified 16.3

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

   4. Applied frac-2neg 16.3

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

   5. Simplified 11.9

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

   7. Applied associate-*r* 11.4

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

   9. Applied distribute-rgt-neg-in 11.4

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

   10. Applied *-un-lft-identity 11.4

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

   11. Applied times-frac 11.4

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

4. Final simplification 11.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \le -3.2333473137924527 \cdot 10^{+127}:\\ \;\;\;\;\frac{\frac{-2}{t \cdot \left(\left(\frac{k}{\ell} \cdot \frac{t}{\ell}\right) \cdot \frac{k}{t}\right)}}{\tan k \cdot \left(-\sin k\right)}\\ \mathbf{elif}\;k \le 1.3164499310012228 \cdot 10^{+141}:\\ \;\;\;\;\frac{\frac{-2}{\frac{t}{\ell} \cdot \left(\frac{t \cdot k}{t} \cdot \frac{t \cdot k}{t}\right)}}{-\frac{\tan k \cdot \sin k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin k} \cdot \frac{\frac{-2}{\left(\left(\left(\frac{t}{\ell} \cdot \frac{k}{t}\right) \cdot \frac{t}{\ell}\right) \cdot \frac{k}{t}\right) \cdot t}}{-\tan k}\\ \end{array}\]

Reproduce

herbie shell --seed 2019142 +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))))