Toniolo and Linder, Equation (10-)

Average Error: 48.8 → 9.0

Time: 2.0m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \le 1.399163382008016659275762795122350368347 \cdot 10^{-311}:\\ \;\;\;\;\left(\left(\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \frac{\sqrt[3]{\cos k} \cdot \sqrt[3]{\cos k}}{\frac{\sin k}{\ell}}\right) \cdot \frac{\sqrt[3]{\cos k}}{\frac{\sin k}{\ell}}\right) \cdot {\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1}\right) \cdot 2\\ \mathbf{elif}\;\ell \cdot \ell \le 9.045380543102303888898108269892584715769 \cdot 10^{114}:\\ \;\;\;\;2 \cdot \frac{{\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1} \cdot \left(\cos k \cdot {\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}\right)}{\frac{\sin k}{\ell} \cdot \frac{\sin k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \frac{\cos k}{\sin k \cdot \frac{\sin k}{\ell}}\right) \cdot \ell\right) \cdot {\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1}\right)\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r7843686 = 2.0;
        double r7843687 = t;
        double r7843688 = 3.0;
        double r7843689 = pow(r7843687, r7843688);
        double r7843690 = l;
        double r7843691 = r7843690 * r7843690;
        double r7843692 = r7843689 / r7843691;
        double r7843693 = k;
        double r7843694 = sin(r7843693);
        double r7843695 = r7843692 * r7843694;
        double r7843696 = tan(r7843693);
        double r7843697 = r7843695 * r7843696;
        double r7843698 = 1.0;
        double r7843699 = r7843693 / r7843687;
        double r7843700 = pow(r7843699, r7843686);
        double r7843701 = r7843698 + r7843700;
        double r7843702 = r7843701 - r7843698;
        double r7843703 = r7843697 * r7843702;
        double r7843704 = r7843686 / r7843703;
        return r7843704;
}

↓

double f(double t, double l, double k) {
        double r7843705 = l;
        double r7843706 = r7843705 * r7843705;
        double r7843707 = 1.399163382008e-311;
        bool r7843708 = r7843706 <= r7843707;
        double r7843709 = 1.0;
        double r7843710 = k;
        double r7843711 = 2.0;
        double r7843712 = 2.0;
        double r7843713 = r7843711 / r7843712;
        double r7843714 = pow(r7843710, r7843713);
        double r7843715 = r7843709 / r7843714;
        double r7843716 = 1.0;
        double r7843717 = pow(r7843715, r7843716);
        double r7843718 = cos(r7843710);
        double r7843719 = cbrt(r7843718);
        double r7843720 = r7843719 * r7843719;
        double r7843721 = sin(r7843710);
        double r7843722 = r7843721 / r7843705;
        double r7843723 = r7843720 / r7843722;
        double r7843724 = r7843717 * r7843723;
        double r7843725 = r7843719 / r7843722;
        double r7843726 = r7843724 * r7843725;
        double r7843727 = t;
        double r7843728 = pow(r7843727, r7843716);
        double r7843729 = r7843714 * r7843728;
        double r7843730 = r7843709 / r7843729;
        double r7843731 = pow(r7843730, r7843716);
        double r7843732 = r7843726 * r7843731;
        double r7843733 = r7843732 * r7843711;
        double r7843734 = 9.045380543102304e+114;
        bool r7843735 = r7843706 <= r7843734;
        double r7843736 = r7843718 * r7843717;
        double r7843737 = r7843731 * r7843736;
        double r7843738 = r7843722 * r7843722;
        double r7843739 = r7843737 / r7843738;
        double r7843740 = r7843711 * r7843739;
        double r7843741 = r7843721 * r7843722;
        double r7843742 = r7843718 / r7843741;
        double r7843743 = r7843717 * r7843742;
        double r7843744 = r7843743 * r7843705;
        double r7843745 = r7843744 * r7843731;
        double r7843746 = r7843711 * r7843745;
        double r7843747 = r7843735 ? r7843740 : r7843746;
        double r7843748 = r7843708 ? r7843733 : r7843747;
        return r7843748;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 3 regimes

2. if (* l l) < 1.399163382008e-311

   1. Initial program 46.8

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

   2. Simplified 37.6

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

   3. Taylor expanded around inf 20.3

      \[\leadsto \color{blue}{2 \cdot \left({\left(\frac{1}{{t}^{1} \cdot {k}^{2}}\right)}^{1} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}\right)}\]
   4. Using strategy rm

   5. Applied sqr-pow 20.3

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

   6. Applied associate-*r* 20.3

      \[\leadsto 2 \cdot \left({\left(\frac{1}{\color{blue}{\left({t}^{1} \cdot {k}^{\left(\frac{2}{2}\right)}\right) \cdot {k}^{\left(\frac{2}{2}\right)}}}\right)}^{1} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}\right)\]
   7. Using strategy rm

   8. Applied *-un-lft-identity 20.3

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

   9. Applied times-frac 20.3

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

   10. Applied unpow-prod-down 20.3

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

   11. Applied associate-*l* 20.3

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

   12. Simplified 8.1

       \[\leadsto 2 \cdot \left({\left(\frac{1}{{t}^{1} \cdot {k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \color{blue}{\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \frac{\cos k}{\frac{\sin k}{\ell} \cdot \frac{\sin k}{\ell}}\right)}\right)\]
   13. Using strategy rm

   14. Applied add-cube-cbrt 8.1

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

   15. Applied times-frac 7.7

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

   16. Applied associate-*r* 5.8

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

   if 1.399163382008e-311 < (* l l) < 9.045380543102304e+114

   1. Initial program 44.7

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

   2. Simplified 34.4

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

   3. Taylor expanded around inf 6.4

      \[\leadsto \color{blue}{2 \cdot \left({\left(\frac{1}{{t}^{1} \cdot {k}^{2}}\right)}^{1} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}\right)}\]
   4. Using strategy rm

   5. Applied sqr-pow 6.4

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

   6. Applied associate-*r* 3.2

      \[\leadsto 2 \cdot \left({\left(\frac{1}{\color{blue}{\left({t}^{1} \cdot {k}^{\left(\frac{2}{2}\right)}\right) \cdot {k}^{\left(\frac{2}{2}\right)}}}\right)}^{1} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}\right)\]
   7. Using strategy rm

   8. Applied *-un-lft-identity 3.2

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

   9. Applied times-frac 3.1

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

   10. Applied unpow-prod-down 3.1

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

   11. Applied associate-*l* 3.5

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

   12. Simplified 3.0

       \[\leadsto 2 \cdot \left({\left(\frac{1}{{t}^{1} \cdot {k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \color{blue}{\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \frac{\cos k}{\frac{\sin k}{\ell} \cdot \frac{\sin k}{\ell}}\right)}\right)\]
   13. Using strategy rm

   14. Applied associate-*r/ 3.0

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

   15. Applied associate-*r/ 2.3

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

   if 9.045380543102304e+114 < (* l l)

   1. Initial program 56.8

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

   2. Simplified 54.1

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

   3. Taylor expanded around inf 47.1

      \[\leadsto \color{blue}{2 \cdot \left({\left(\frac{1}{{t}^{1} \cdot {k}^{2}}\right)}^{1} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}\right)}\]
   4. Using strategy rm

   5. Applied sqr-pow 47.1

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

   6. Applied associate-*r* 43.7

      \[\leadsto 2 \cdot \left({\left(\frac{1}{\color{blue}{\left({t}^{1} \cdot {k}^{\left(\frac{2}{2}\right)}\right) \cdot {k}^{\left(\frac{2}{2}\right)}}}\right)}^{1} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}\right)\]
   7. Using strategy rm

   8. Applied *-un-lft-identity 43.7

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

   9. Applied times-frac 43.2

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

   10. Applied unpow-prod-down 43.2

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

   11. Applied associate-*l* 38.0

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

   12. Simplified 38.0

       \[\leadsto 2 \cdot \left({\left(\frac{1}{{t}^{1} \cdot {k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \color{blue}{\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \frac{\cos k}{\frac{\sin k}{\ell} \cdot \frac{\sin k}{\ell}}\right)}\right)\]
   13. Using strategy rm

   14. Applied associate-*l/ 38.0

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

   15. Applied associate-/r/ 38.0

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

   16. Applied associate-*r* 21.9

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

4. Final simplification 9.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \le 1.399163382008016659275762795122350368347 \cdot 10^{-311}:\\ \;\;\;\;\left(\left(\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \frac{\sqrt[3]{\cos k} \cdot \sqrt[3]{\cos k}}{\frac{\sin k}{\ell}}\right) \cdot \frac{\sqrt[3]{\cos k}}{\frac{\sin k}{\ell}}\right) \cdot {\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1}\right) \cdot 2\\ \mathbf{elif}\;\ell \cdot \ell \le 9.045380543102303888898108269892584715769 \cdot 10^{114}:\\ \;\;\;\;2 \cdot \frac{{\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1} \cdot \left(\cos k \cdot {\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}\right)}{\frac{\sin k}{\ell} \cdot \frac{\sin k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \frac{\cos k}{\sin k \cdot \frac{\sin k}{\ell}}\right) \cdot \ell\right) \cdot {\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10-)"
  (/ 2.0 (* (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k)) (- (+ 1.0 (pow (/ k t) 2.0)) 1.0))))