Toniolo and Linder, Equation (10-)

Average Error: 48.1 → 9.3

Time: 1.2m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;k \le -1.33481975672023252010467643903032149745 \cdot 10^{154} \lor \neg \left(k \le -1.35218524776890705644444846209239839645 \cdot 10^{-123} \lor \neg \left(k \le 1.094978693998942963552062400840003390623 \cdot 10^{-100} \lor \neg \left(k \le 1.570388734911729430952442810681471278398 \cdot 10^{163}\right)\right)\right):\\ \;\;\;\;\frac{\cos k}{\frac{\sin k}{\ell}} \cdot \frac{2 \cdot {\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1}}{\frac{\sin k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\left(\left({\left(\frac{\sqrt{1}}{{k}^{2}}\right)}^{1} \cdot \left({\left(\frac{\sqrt{1}}{{t}^{1}}\right)}^{1} \cdot \frac{1}{\frac{\sin k}{\ell}}\right)\right) \cdot \cos k\right) \cdot \ell\right)}{\sin k}\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r109710 = 2.0;
        double r109711 = t;
        double r109712 = 3.0;
        double r109713 = pow(r109711, r109712);
        double r109714 = l;
        double r109715 = r109714 * r109714;
        double r109716 = r109713 / r109715;
        double r109717 = k;
        double r109718 = sin(r109717);
        double r109719 = r109716 * r109718;
        double r109720 = tan(r109717);
        double r109721 = r109719 * r109720;
        double r109722 = 1.0;
        double r109723 = r109717 / r109711;
        double r109724 = pow(r109723, r109710);
        double r109725 = r109722 + r109724;
        double r109726 = r109725 - r109722;
        double r109727 = r109721 * r109726;
        double r109728 = r109710 / r109727;
        return r109728;
}

↓

double f(double t, double l, double k) {
        double r109729 = k;
        double r109730 = -1.3348197567202325e+154;
        bool r109731 = r109729 <= r109730;
        double r109732 = -1.352185247768907e-123;
        bool r109733 = r109729 <= r109732;
        double r109734 = 1.094978693998943e-100;
        bool r109735 = r109729 <= r109734;
        double r109736 = 1.5703887349117294e+163;
        bool r109737 = r109729 <= r109736;
        double r109738 = !r109737;
        bool r109739 = r109735 || r109738;
        double r109740 = !r109739;
        bool r109741 = r109733 || r109740;
        double r109742 = !r109741;
        bool r109743 = r109731 || r109742;
        double r109744 = cos(r109729);
        double r109745 = sin(r109729);
        double r109746 = l;
        double r109747 = r109745 / r109746;
        double r109748 = r109744 / r109747;
        double r109749 = 2.0;
        double r109750 = 1.0;
        double r109751 = 2.0;
        double r109752 = r109749 / r109751;
        double r109753 = pow(r109729, r109752);
        double r109754 = t;
        double r109755 = 1.0;
        double r109756 = pow(r109754, r109755);
        double r109757 = r109753 * r109756;
        double r109758 = r109753 * r109757;
        double r109759 = r109750 / r109758;
        double r109760 = pow(r109759, r109755);
        double r109761 = r109749 * r109760;
        double r109762 = r109761 / r109747;
        double r109763 = r109748 * r109762;
        double r109764 = sqrt(r109750);
        double r109765 = pow(r109729, r109749);
        double r109766 = r109764 / r109765;
        double r109767 = pow(r109766, r109755);
        double r109768 = r109764 / r109756;
        double r109769 = pow(r109768, r109755);
        double r109770 = r109750 / r109747;
        double r109771 = r109769 * r109770;
        double r109772 = r109767 * r109771;
        double r109773 = r109772 * r109744;
        double r109774 = r109773 * r109746;
        double r109775 = r109749 * r109774;
        double r109776 = r109775 / r109745;
        double r109777 = r109743 ? r109763 : r109776;
        return r109777;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 2 regimes

2. if k < -1.3348197567202325e+154 or -1.352185247768907e-123 < k < 1.094978693998943e-100 or 1.5703887349117294e+163 < k

   1. Initial program 43.1

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

   2. Simplified 38.9

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

   3. Taylor expanded around inf 29.1

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

   5. Applied unpow2 29.1

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

   6. Applied associate-/r* 29.1

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

   7. Simplified 27.9

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

   9. Applied *-un-lft-identity 27.9

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

   10. Applied div-inv 27.9

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

   11. Applied *-un-lft-identity 27.9

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

   12. Applied times-frac 27.9

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

   13. Applied times-frac 26.8

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

   14. Applied associate-*r* 25.0

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

   15. Simplified 25.0

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

   17. Applied sqr-pow 25.0

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

   18. Applied associate-*l* 15.1

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

   if -1.3348197567202325e+154 < k < -1.352185247768907e-123 or 1.094978693998943e-100 < k < 1.5703887349117294e+163

   1. Initial program 52.5

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

   2. Simplified 41.2

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

   3. Taylor expanded around inf 16.5

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

   5. Applied unpow2 16.5

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

   6. Applied associate-/r* 16.5

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

   7. Simplified 15.6

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

   9. Applied *-un-lft-identity 15.6

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

   10. Applied div-inv 15.6

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

   11. Applied *-un-lft-identity 15.6

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

   12. Applied times-frac 15.5

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

   13. Applied times-frac 14.4

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

   14. Applied associate-*r* 7.9

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

   15. Simplified 7.9

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

   17. Applied add-sqr-sqrt 7.9

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

   18. Applied times-frac 7.7

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

   19. Applied unpow-prod-down 7.7

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

   20. Applied associate-*l* 3.7

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

4. Final simplification 9.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \le -1.33481975672023252010467643903032149745 \cdot 10^{154} \lor \neg \left(k \le -1.35218524776890705644444846209239839645 \cdot 10^{-123} \lor \neg \left(k \le 1.094978693998942963552062400840003390623 \cdot 10^{-100} \lor \neg \left(k \le 1.570388734911729430952442810681471278398 \cdot 10^{163}\right)\right)\right):\\ \;\;\;\;\frac{\cos k}{\frac{\sin k}{\ell}} \cdot \frac{2 \cdot {\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1}}{\frac{\sin k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\left(\left({\left(\frac{\sqrt{1}}{{k}^{2}}\right)}^{1} \cdot \left({\left(\frac{\sqrt{1}}{{t}^{1}}\right)}^{1} \cdot \frac{1}{\frac{\sin k}{\ell}}\right)\right) \cdot \cos k\right) \cdot \ell\right)}{\sin k}\\ \end{array}\]

Reproduce

herbie shell --seed 2019294 
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10-)"
  :precision binary64
  (/ 2 (* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (- (+ 1 (pow (/ k t) 2)) 1))))