Toniolo and Linder, Equation (10-)

Average Error: 48.1 → 12.4

Time: 27.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;k \le -8.35668469717318001 \cdot 10^{123}:\\ \;\;\;\;2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}\right)\right)\\ \mathbf{elif}\;k \le -3.707273909255465 \cdot 10^{-110}:\\ \;\;\;\;2 \cdot \frac{\left({\left(\frac{1}{{k}^{2} \cdot {t}^{1}}\right)}^{1} \cdot \ell\right) \cdot \cos k}{\left|\sin k\right| \cdot \frac{\left|\sin k\right|}{\ell}}\\ \mathbf{elif}\;k \le 1.13294708702049773 \cdot 10^{-76}:\\ \;\;\;\;2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \left(\left(\frac{\cos k}{\left|\sin k\right|} \cdot \frac{\ell}{\left|\sin k\right|}\right) \cdot \ell\right)\right)\\ \mathbf{elif}\;k \le 1.0399004082354796 \cdot 10^{154}:\\ \;\;\;\;2 \cdot \frac{\left({\left(\frac{1}{{k}^{2} \cdot {t}^{1}}\right)}^{1} \cdot \ell\right) \cdot \cos k}{\left|\sin k\right| \cdot \frac{\left|\sin k\right|}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}\right)\right)\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r83004 = 2.0;
        double r83005 = t;
        double r83006 = 3.0;
        double r83007 = pow(r83005, r83006);
        double r83008 = l;
        double r83009 = r83008 * r83008;
        double r83010 = r83007 / r83009;
        double r83011 = k;
        double r83012 = sin(r83011);
        double r83013 = r83010 * r83012;
        double r83014 = tan(r83011);
        double r83015 = r83013 * r83014;
        double r83016 = 1.0;
        double r83017 = r83011 / r83005;
        double r83018 = pow(r83017, r83004);
        double r83019 = r83016 + r83018;
        double r83020 = r83019 - r83016;
        double r83021 = r83015 * r83020;
        double r83022 = r83004 / r83021;
        return r83022;
}

↓

double f(double t, double l, double k) {
        double r83023 = k;
        double r83024 = -8.35668469717318e+123;
        bool r83025 = r83023 <= r83024;
        double r83026 = 2.0;
        double r83027 = 1.0;
        double r83028 = 2.0;
        double r83029 = r83026 / r83028;
        double r83030 = pow(r83023, r83029);
        double r83031 = r83027 / r83030;
        double r83032 = 1.0;
        double r83033 = pow(r83031, r83032);
        double r83034 = t;
        double r83035 = pow(r83034, r83032);
        double r83036 = r83030 * r83035;
        double r83037 = r83027 / r83036;
        double r83038 = pow(r83037, r83032);
        double r83039 = cos(r83023);
        double r83040 = l;
        double r83041 = pow(r83040, r83028);
        double r83042 = r83039 * r83041;
        double r83043 = sin(r83023);
        double r83044 = pow(r83043, r83028);
        double r83045 = r83042 / r83044;
        double r83046 = r83038 * r83045;
        double r83047 = r83033 * r83046;
        double r83048 = r83026 * r83047;
        double r83049 = -3.707273909255465e-110;
        bool r83050 = r83023 <= r83049;
        double r83051 = pow(r83023, r83026);
        double r83052 = r83051 * r83035;
        double r83053 = r83027 / r83052;
        double r83054 = pow(r83053, r83032);
        double r83055 = r83054 * r83040;
        double r83056 = r83055 * r83039;
        double r83057 = fabs(r83043);
        double r83058 = r83057 / r83040;
        double r83059 = r83057 * r83058;
        double r83060 = r83056 / r83059;
        double r83061 = r83026 * r83060;
        double r83062 = 1.1329470870204977e-76;
        bool r83063 = r83023 <= r83062;
        double r83064 = r83030 * r83036;
        double r83065 = r83027 / r83064;
        double r83066 = pow(r83065, r83032);
        double r83067 = r83039 / r83057;
        double r83068 = r83040 / r83057;
        double r83069 = r83067 * r83068;
        double r83070 = r83069 * r83040;
        double r83071 = r83066 * r83070;
        double r83072 = r83026 * r83071;
        double r83073 = 1.0399004082354796e+154;
        bool r83074 = r83023 <= r83073;
        double r83075 = r83074 ? r83061 : r83048;
        double r83076 = r83063 ? r83072 : r83075;
        double r83077 = r83050 ? r83061 : r83076;
        double r83078 = r83025 ? r83048 : r83077;
        return r83078;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 3 regimes

2. if k < -8.35668469717318e+123 or 1.0399004082354796e+154 < k

   1. Initial program 39.8

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

   2. Simplified 34.0

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

   3. Taylor expanded around inf 23.8

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

   5. Applied sqr-pow 23.8

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

   6. Applied associate-*l* 19.3

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

   8. Applied *-un-lft-identity 19.3

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

   9. Applied times-frac 19.1

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

   10. Applied unpow-prod-down 19.1

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

   11. Applied associate-*l* 16.5

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

   if -8.35668469717318e+123 < k < -3.707273909255465e-110 or 1.1329470870204977e-76 < k < 1.0399004082354796e+154

   1. Initial program 52.9

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

   2. Simplified 41.8

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

   3. Taylor expanded around inf 15.7

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

   5. Applied sqr-pow 15.7

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

   6. Applied associate-*l* 15.7

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

   8. Applied add-sqr-sqrt 15.7

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

   9. Applied times-frac 15.7

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

   10. Simplified 15.7

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

   11. Simplified 15.0

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

   13. Applied frac-times 14.4

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

   14. Applied associate-*r/ 7.5

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

   15. Simplified 7.5

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

   if -3.707273909255465e-110 < k < 1.1329470870204977e-76

   1. Initial program 63.9

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

   2. Simplified 63.5

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

   3. Taylor expanded around inf 45.1

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

   5. Applied sqr-pow 45.1

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

   6. Applied associate-*l* 45.0

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

   8. Applied add-sqr-sqrt 45.0

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

   9. Applied times-frac 45.0

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

   10. Simplified 45.0

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

   11. Simplified 26.4

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

   13. Applied associate-/r/ 26.4

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

   14. Applied associate-*r* 16.8

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

4. Final simplification 12.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \le -8.35668469717318001 \cdot 10^{123}:\\ \;\;\;\;2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}\right)\right)\\ \mathbf{elif}\;k \le -3.707273909255465 \cdot 10^{-110}:\\ \;\;\;\;2 \cdot \frac{\left({\left(\frac{1}{{k}^{2} \cdot {t}^{1}}\right)}^{1} \cdot \ell\right) \cdot \cos k}{\left|\sin k\right| \cdot \frac{\left|\sin k\right|}{\ell}}\\ \mathbf{elif}\;k \le 1.13294708702049773 \cdot 10^{-76}:\\ \;\;\;\;2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \left(\left(\frac{\cos k}{\left|\sin k\right|} \cdot \frac{\ell}{\left|\sin k\right|}\right) \cdot \ell\right)\right)\\ \mathbf{elif}\;k \le 1.0399004082354796 \cdot 10^{154}:\\ \;\;\;\;2 \cdot \frac{\left({\left(\frac{1}{{k}^{2} \cdot {t}^{1}}\right)}^{1} \cdot \ell\right) \cdot \cos k}{\left|\sin k\right| \cdot \frac{\left|\sin k\right|}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020034 +o rules:numerics
(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))))