Toniolo and Linder, Equation (10-)

Average Error: 48.5 → 14.9

Time: 1.1m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\ell \le -1.350083645795687900514049661869243401303 \cdot 10^{154}:\\ \;\;\;\;\frac{\frac{2}{\left(\left(\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}}{\ell} \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell}\right) \cdot \sin k\right) \cdot \tan k}}{{\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{elif}\;\ell \le -1.926172283746208905126600187598424399764 \cdot 10^{-88}:\\ \;\;\;\;2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left(\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1} \cdot \left(\cos k \cdot {\ell}^{2}\right)\right) \cdot \frac{1}{{\left(\sin k\right)}^{2}}\right)\right)\\ \mathbf{elif}\;\ell \le 8.125799631108940674547963257039853431087 \cdot 10^{-130}:\\ \;\;\;\;2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \mathsf{fma}\left(\frac{\ell}{k}, \frac{\ell}{k}, {\ell}^{2} \cdot \frac{-1}{6}\right)\right)\\ \mathbf{elif}\;\ell \le 1.982834535788033321623200081388133861373 \cdot 10^{134}:\\ \;\;\;\;2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \left(\left(\sqrt[3]{\frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}} \cdot \sqrt[3]{\frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}}\right) \cdot \sqrt[3]{\frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\left(\left(\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}}{\ell} \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell}\right) \cdot \sin k\right) \cdot \tan k}}{{\left(\frac{k}{t}\right)}^{2}}\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r93966 = 2.0;
        double r93967 = t;
        double r93968 = 3.0;
        double r93969 = pow(r93967, r93968);
        double r93970 = l;
        double r93971 = r93970 * r93970;
        double r93972 = r93969 / r93971;
        double r93973 = k;
        double r93974 = sin(r93973);
        double r93975 = r93972 * r93974;
        double r93976 = tan(r93973);
        double r93977 = r93975 * r93976;
        double r93978 = 1.0;
        double r93979 = r93973 / r93967;
        double r93980 = pow(r93979, r93966);
        double r93981 = r93978 + r93980;
        double r93982 = r93981 - r93978;
        double r93983 = r93977 * r93982;
        double r93984 = r93966 / r93983;
        return r93984;
}

↓

double f(double t, double l, double k) {
        double r93985 = l;
        double r93986 = -1.350083645795688e+154;
        bool r93987 = r93985 <= r93986;
        double r93988 = 2.0;
        double r93989 = t;
        double r93990 = cbrt(r93989);
        double r93991 = r93990 * r93990;
        double r93992 = 3.0;
        double r93993 = pow(r93991, r93992);
        double r93994 = r93993 / r93985;
        double r93995 = pow(r93990, r93992);
        double r93996 = r93995 / r93985;
        double r93997 = r93994 * r93996;
        double r93998 = k;
        double r93999 = sin(r93998);
        double r94000 = r93997 * r93999;
        double r94001 = tan(r93998);
        double r94002 = r94000 * r94001;
        double r94003 = r93988 / r94002;
        double r94004 = r93998 / r93989;
        double r94005 = pow(r94004, r93988);
        double r94006 = r94003 / r94005;
        double r94007 = -1.926172283746209e-88;
        bool r94008 = r93985 <= r94007;
        double r94009 = 1.0;
        double r94010 = 2.0;
        double r94011 = r93988 / r94010;
        double r94012 = pow(r93998, r94011);
        double r94013 = r94009 / r94012;
        double r94014 = 1.0;
        double r94015 = pow(r94013, r94014);
        double r94016 = pow(r93989, r94014);
        double r94017 = r94012 * r94016;
        double r94018 = r94009 / r94017;
        double r94019 = pow(r94018, r94014);
        double r94020 = cos(r93998);
        double r94021 = pow(r93985, r94010);
        double r94022 = r94020 * r94021;
        double r94023 = r94019 * r94022;
        double r94024 = pow(r93999, r94010);
        double r94025 = r94009 / r94024;
        double r94026 = r94023 * r94025;
        double r94027 = r94015 * r94026;
        double r94028 = r93988 * r94027;
        double r94029 = 8.12579963110894e-130;
        bool r94030 = r93985 <= r94029;
        double r94031 = r94012 * r94017;
        double r94032 = r94009 / r94031;
        double r94033 = pow(r94032, r94014);
        double r94034 = r93985 / r93998;
        double r94035 = -0.16666666666666666;
        double r94036 = r94021 * r94035;
        double r94037 = fma(r94034, r94034, r94036);
        double r94038 = r94033 * r94037;
        double r94039 = r93988 * r94038;
        double r94040 = 1.9828345357880333e+134;
        bool r94041 = r93985 <= r94040;
        double r94042 = r94022 / r94024;
        double r94043 = cbrt(r94042);
        double r94044 = r94043 * r94043;
        double r94045 = r94044 * r94043;
        double r94046 = r94033 * r94045;
        double r94047 = r93988 * r94046;
        double r94048 = r94041 ? r94047 : r94006;
        double r94049 = r94030 ? r94039 : r94048;
        double r94050 = r94008 ? r94028 : r94049;
        double r94051 = r93987 ? r94006 : r94050;
        return r94051;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 4 regimes

2. if l < -1.350083645795688e+154 or 1.9828345357880333e+134 < l

   1. Initial program 63.0

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

   2. Simplified 62.5

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

   4. Applied add-cube-cbrt 62.6

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

   5. Applied unpow-prod-down 62.6

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

   6. Applied times-frac 47.6

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

   if -1.350083645795688e+154 < l < -1.926172283746209e-88

   1. Initial program 45.6

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

   2. Simplified 36.7

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

   3. Taylor expanded around inf 13.2

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

   5. Applied sqr-pow 13.2

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

   6. Applied associate-*l* 8.0

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

   8. Applied *-un-lft-identity 8.0

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

   9. Applied times-frac 7.6

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

   10. Applied unpow-prod-down 7.6

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

   11. Applied associate-*l* 4.5

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

   13. Applied div-inv 4.5

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

   14. Applied associate-*r* 3.9

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

   if -1.926172283746209e-88 < l < 8.12579963110894e-130

   1. Initial program 45.6

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

   2. Simplified 37.1

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

   3. Taylor expanded around inf 17.1

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

   5. Applied sqr-pow 17.1

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

   6. Applied associate-*l* 17.0

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

   7. Taylor expanded around 0 18.7

      \[\leadsto 2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} - \frac{1}{6} \cdot {\ell}^{2}\right)}\right)\]

   8. Simplified 10.7

      \[\leadsto 2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \color{blue}{\mathsf{fma}\left(\frac{\ell}{k}, \frac{\ell}{k}, {\ell}^{2} \cdot \frac{-1}{6}\right)}\right)\]

   if 8.12579963110894e-130 < l < 1.9828345357880333e+134

   1. Initial program 45.4

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

   2. Simplified 36.2

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

   3. Taylor expanded around inf 12.1

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

   5. Applied sqr-pow 12.1

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

   6. Applied associate-*l* 7.4

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

   8. Applied add-cube-cbrt 7.7

      \[\leadsto 2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}} \cdot \sqrt[3]{\frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}}\right) \cdot \sqrt[3]{\frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}}\right)}\right)\]
3. Recombined 4 regimes into one program.

4. Final simplification 14.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \le -1.350083645795687900514049661869243401303 \cdot 10^{154}:\\ \;\;\;\;\frac{\frac{2}{\left(\left(\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}}{\ell} \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell}\right) \cdot \sin k\right) \cdot \tan k}}{{\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{elif}\;\ell \le -1.926172283746208905126600187598424399764 \cdot 10^{-88}:\\ \;\;\;\;2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left(\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1} \cdot \left(\cos k \cdot {\ell}^{2}\right)\right) \cdot \frac{1}{{\left(\sin k\right)}^{2}}\right)\right)\\ \mathbf{elif}\;\ell \le 8.125799631108940674547963257039853431087 \cdot 10^{-130}:\\ \;\;\;\;2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \mathsf{fma}\left(\frac{\ell}{k}, \frac{\ell}{k}, {\ell}^{2} \cdot \frac{-1}{6}\right)\right)\\ \mathbf{elif}\;\ell \le 1.982834535788033321623200081388133861373 \cdot 10^{134}:\\ \;\;\;\;2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \left(\left(\sqrt[3]{\frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}} \cdot \sqrt[3]{\frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}}\right) \cdot \sqrt[3]{\frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\left(\left(\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}}{\ell} \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell}\right) \cdot \sin k\right) \cdot \tan k}}{{\left(\frac{k}{t}\right)}^{2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 +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))))