Toniolo and Linder, Equation (10-)

Average Error: 48.1 → 13.6

Time: 1.1m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\ell \le -1.395768281843777546279887385243220191786 \cdot 10^{142}:\\ \;\;\;\;\frac{\frac{2}{\left(\left(\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}}{\ell} \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell}\right) \cdot \sin k\right) \cdot \tan k}}{{\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{elif}\;\ell \le -2.954141845484123063075571007654470856433 \cdot 10^{-133}:\\ \;\;\;\;2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \frac{{\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1} \cdot \left(\cos k \cdot {\ell}^{2}\right)}{{\left(\sin k\right)}^{2}}\right)\\ \mathbf{elif}\;\ell \le 6.579520119656325980067007236374335260974 \cdot 10^{-209}:\\ \;\;\;\;2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \mathsf{fma}\left(\frac{\ell}{k}, \frac{\ell}{k}, \frac{-1}{6} \cdot {\ell}^{2}\right)\right)\\ \mathbf{elif}\;\ell \le 1.283154238585831613414878640305835438884 \cdot 10^{154}:\\ \;\;\;\;2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1} \cdot \frac{\cos k}{\frac{{\left(\sin k\right)}^{2}}{{\ell}^{2}}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\left(\left(\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}}{\ell} \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell}\right) \cdot \sin k\right) \cdot \tan k}}{{\left(\frac{k}{t}\right)}^{2}}\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r108106 = 2.0;
        double r108107 = t;
        double r108108 = 3.0;
        double r108109 = pow(r108107, r108108);
        double r108110 = l;
        double r108111 = r108110 * r108110;
        double r108112 = r108109 / r108111;
        double r108113 = k;
        double r108114 = sin(r108113);
        double r108115 = r108112 * r108114;
        double r108116 = tan(r108113);
        double r108117 = r108115 * r108116;
        double r108118 = 1.0;
        double r108119 = r108113 / r108107;
        double r108120 = pow(r108119, r108106);
        double r108121 = r108118 + r108120;
        double r108122 = r108121 - r108118;
        double r108123 = r108117 * r108122;
        double r108124 = r108106 / r108123;
        return r108124;
}

↓

double f(double t, double l, double k) {
        double r108125 = l;
        double r108126 = -1.3957682818437775e+142;
        bool r108127 = r108125 <= r108126;
        double r108128 = 2.0;
        double r108129 = t;
        double r108130 = cbrt(r108129);
        double r108131 = r108130 * r108130;
        double r108132 = 3.0;
        double r108133 = pow(r108131, r108132);
        double r108134 = r108133 / r108125;
        double r108135 = pow(r108130, r108132);
        double r108136 = r108135 / r108125;
        double r108137 = r108134 * r108136;
        double r108138 = k;
        double r108139 = sin(r108138);
        double r108140 = r108137 * r108139;
        double r108141 = tan(r108138);
        double r108142 = r108140 * r108141;
        double r108143 = r108128 / r108142;
        double r108144 = r108138 / r108129;
        double r108145 = pow(r108144, r108128);
        double r108146 = r108143 / r108145;
        double r108147 = -2.954141845484123e-133;
        bool r108148 = r108125 <= r108147;
        double r108149 = 1.0;
        double r108150 = 2.0;
        double r108151 = r108128 / r108150;
        double r108152 = pow(r108138, r108151);
        double r108153 = r108149 / r108152;
        double r108154 = 1.0;
        double r108155 = pow(r108153, r108154);
        double r108156 = pow(r108129, r108154);
        double r108157 = r108152 * r108156;
        double r108158 = r108149 / r108157;
        double r108159 = pow(r108158, r108154);
        double r108160 = cos(r108138);
        double r108161 = pow(r108125, r108150);
        double r108162 = r108160 * r108161;
        double r108163 = r108159 * r108162;
        double r108164 = pow(r108139, r108150);
        double r108165 = r108163 / r108164;
        double r108166 = r108155 * r108165;
        double r108167 = r108128 * r108166;
        double r108168 = 6.579520119656326e-209;
        bool r108169 = r108125 <= r108168;
        double r108170 = r108152 * r108157;
        double r108171 = r108149 / r108170;
        double r108172 = pow(r108171, r108154);
        double r108173 = r108125 / r108138;
        double r108174 = -0.16666666666666666;
        double r108175 = r108174 * r108161;
        double r108176 = fma(r108173, r108173, r108175);
        double r108177 = r108172 * r108176;
        double r108178 = r108128 * r108177;
        double r108179 = 1.2831542385858316e+154;
        bool r108180 = r108125 <= r108179;
        double r108181 = r108164 / r108161;
        double r108182 = r108160 / r108181;
        double r108183 = r108159 * r108182;
        double r108184 = r108155 * r108183;
        double r108185 = r108128 * r108184;
        double r108186 = r108180 ? r108185 : r108146;
        double r108187 = r108169 ? r108178 : r108186;
        double r108188 = r108148 ? r108167 : r108187;
        double r108189 = r108127 ? r108146 : r108188;
        return r108189;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 4 regimes

2. if l < -1.3957682818437775e+142 or 1.2831542385858316e+154 < l

   1. Initial program 63.2

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

   2. Simplified 63.0

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

   4. Applied add-cube-cbrt 63.0

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

   5. Applied unpow-prod-down 63.0

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

   6. Applied times-frac 49.6

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

   if -1.3957682818437775e+142 < l < -2.954141845484123e-133

   1. Initial program 45.2

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

   2. Simplified 35.3

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

   3. Taylor expanded around inf 11.1

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

   5. Applied sqr-pow 11.1

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

   6. Applied associate-*l* 6.5

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

   8. Applied *-un-lft-identity 6.5

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

   9. Applied times-frac 6.4

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

   10. Applied unpow-prod-down 6.4

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

   11. Applied associate-*l* 3.3

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

   13. Applied associate-*r/ 3.1

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

   if -2.954141845484123e-133 < l < 6.579520119656326e-209

   1. Initial program 46.1

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

   2. Simplified 37.4

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

   3. Taylor expanded around inf 18.4

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

   5. Applied sqr-pow 18.4

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

   6. Applied associate-*l* 18.4

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

   7. Taylor expanded around 0 19.0

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

   8. Simplified 10.2

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

   if 6.579520119656326e-209 < l < 1.2831542385858316e+154

   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 35.5

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

   3. Taylor expanded around inf 13.0

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

   5. Applied sqr-pow 13.0

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

   6. Applied associate-*l* 9.6

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

   8. Applied *-un-lft-identity 9.6

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

   9. Applied times-frac 9.3

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

   10. Applied unpow-prod-down 9.3

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

   11. Applied associate-*l* 6.8

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

   13. Applied associate-/l* 6.9

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

4. Final simplification 13.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \le -1.395768281843777546279887385243220191786 \cdot 10^{142}:\\ \;\;\;\;\frac{\frac{2}{\left(\left(\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}}{\ell} \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell}\right) \cdot \sin k\right) \cdot \tan k}}{{\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{elif}\;\ell \le -2.954141845484123063075571007654470856433 \cdot 10^{-133}:\\ \;\;\;\;2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \frac{{\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1} \cdot \left(\cos k \cdot {\ell}^{2}\right)}{{\left(\sin k\right)}^{2}}\right)\\ \mathbf{elif}\;\ell \le 6.579520119656325980067007236374335260974 \cdot 10^{-209}:\\ \;\;\;\;2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \mathsf{fma}\left(\frac{\ell}{k}, \frac{\ell}{k}, \frac{-1}{6} \cdot {\ell}^{2}\right)\right)\\ \mathbf{elif}\;\ell \le 1.283154238585831613414878640305835438884 \cdot 10^{154}:\\ \;\;\;\;2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1} \cdot \frac{\cos k}{\frac{{\left(\sin k\right)}^{2}}{{\ell}^{2}}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\left(\left(\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}}{\ell} \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell}\right) \cdot \sin k\right) \cdot \tan k}}{{\left(\frac{k}{t}\right)}^{2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 +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))))