Toniolo and Linder, Equation (10+)

Average Error: 32.2 → 15.0

Time: 16.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \le 1.1118751811811023 \cdot 10^{-267}:\\ \;\;\;\;\frac{2}{\left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3}}} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\\ \mathbf{elif}\;\ell \cdot \ell \le 1.6076584110961083 \cdot 10^{148}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(2, {\left(\frac{1}{{-1}^{3}}\right)}^{1} \cdot \frac{{\left(\sqrt[3]{-1}\right)}^{9} \cdot \left({t}^{3} \cdot {\left(\sin k\right)}^{2}\right)}{\cos k \cdot {\ell}^{2}}, {\left(\frac{1}{{-1}^{3}}\right)}^{1} \cdot \frac{{\left(\sqrt[3]{-1}\right)}^{9} \cdot \left({\left(\sin k\right)}^{2} \cdot \left({k}^{2} \cdot t\right)\right)}{\cos k \cdot {\ell}^{2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\left({\left(\sqrt[3]{t}\right)}^{3} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)\right) \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3}} \cdot \cos k}}\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r107125 = 2.0;
        double r107126 = t;
        double r107127 = 3.0;
        double r107128 = pow(r107126, r107127);
        double r107129 = l;
        double r107130 = r107129 * r107129;
        double r107131 = r107128 / r107130;
        double r107132 = k;
        double r107133 = sin(r107132);
        double r107134 = r107131 * r107133;
        double r107135 = tan(r107132);
        double r107136 = r107134 * r107135;
        double r107137 = 1.0;
        double r107138 = r107132 / r107126;
        double r107139 = pow(r107138, r107125);
        double r107140 = r107137 + r107139;
        double r107141 = r107140 + r107137;
        double r107142 = r107136 * r107141;
        double r107143 = r107125 / r107142;
        return r107143;
}

↓

double f(double t, double l, double k) {
        double r107144 = l;
        double r107145 = r107144 * r107144;
        double r107146 = 1.1118751811811023e-267;
        bool r107147 = r107145 <= r107146;
        double r107148 = 2.0;
        double r107149 = t;
        double r107150 = cbrt(r107149);
        double r107151 = 3.0;
        double r107152 = pow(r107150, r107151);
        double r107153 = r107144 / r107152;
        double r107154 = r107152 / r107153;
        double r107155 = r107152 / r107144;
        double r107156 = k;
        double r107157 = sin(r107156);
        double r107158 = r107155 * r107157;
        double r107159 = r107154 * r107158;
        double r107160 = tan(r107156);
        double r107161 = 1.0;
        double r107162 = r107156 / r107149;
        double r107163 = pow(r107162, r107148);
        double r107164 = r107161 + r107163;
        double r107165 = r107164 + r107161;
        double r107166 = r107160 * r107165;
        double r107167 = r107159 * r107166;
        double r107168 = r107148 / r107167;
        double r107169 = 1.6076584110961083e+148;
        bool r107170 = r107145 <= r107169;
        double r107171 = 1.0;
        double r107172 = -1.0;
        double r107173 = pow(r107172, r107151);
        double r107174 = r107171 / r107173;
        double r107175 = pow(r107174, r107161);
        double r107176 = cbrt(r107172);
        double r107177 = 9.0;
        double r107178 = pow(r107176, r107177);
        double r107179 = 3.0;
        double r107180 = pow(r107149, r107179);
        double r107181 = 2.0;
        double r107182 = pow(r107157, r107181);
        double r107183 = r107180 * r107182;
        double r107184 = r107178 * r107183;
        double r107185 = cos(r107156);
        double r107186 = pow(r107144, r107181);
        double r107187 = r107185 * r107186;
        double r107188 = r107184 / r107187;
        double r107189 = r107175 * r107188;
        double r107190 = pow(r107156, r107181);
        double r107191 = r107190 * r107149;
        double r107192 = r107182 * r107191;
        double r107193 = r107178 * r107192;
        double r107194 = r107193 / r107187;
        double r107195 = r107175 * r107194;
        double r107196 = fma(r107148, r107189, r107195);
        double r107197 = r107148 / r107196;
        double r107198 = r107152 * r107158;
        double r107199 = r107198 * r107157;
        double r107200 = r107199 * r107165;
        double r107201 = r107153 * r107185;
        double r107202 = r107200 / r107201;
        double r107203 = r107148 / r107202;
        double r107204 = r107170 ? r107197 : r107203;
        double r107205 = r107147 ? r107168 : r107204;
        return r107205;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 3 regimes

2. if (* l l) < 1.1118751811811023e-267

   1. Initial program 24.5

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

   3. Applied add-cube-cbrt 24.5

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

   4. Applied unpow-prod-down 24.5

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

   5. Applied times-frac 18.0

      \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}}{\ell} \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
   6. Using strategy rm

   7. Applied unpow-prod-down 18.0

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

   8. Applied associate-/l* 12.6

      \[\leadsto \frac{2}{\left(\left(\left(\color{blue}{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3}}}} \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
   9. Using strategy rm

   10. Applied associate-*l* 9.9

       \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3}}} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
   11. Using strategy rm

   12. Applied associate-*l* 9.5

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

   if 1.1118751811811023e-267 < (* l l) < 1.6076584110961083e+148

   1. Initial program 24.2

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

   3. Applied add-cube-cbrt 24.5

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

   4. Applied unpow-prod-down 24.5

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

   5. Applied times-frac 22.6

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

   6. Taylor expanded around -inf 17.1

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \left({\left(\frac{1}{{-1}^{3}}\right)}^{1} \cdot \frac{{\left(\sqrt[3]{-1}\right)}^{9} \cdot \left({t}^{3} \cdot {\left(\sin k\right)}^{2}\right)}{\cos k \cdot {\ell}^{2}}\right) + {\left(\frac{1}{{-1}^{3}}\right)}^{1} \cdot \frac{{\left(\sqrt[3]{-1}\right)}^{9} \cdot \left({\left(\sin k\right)}^{2} \cdot \left({k}^{2} \cdot t\right)\right)}{\cos k \cdot {\ell}^{2}}}}\]

   7. Simplified 17.1

      \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(2, {\left(\frac{1}{{-1}^{3}}\right)}^{1} \cdot \frac{{\left(\sqrt[3]{-1}\right)}^{9} \cdot \left({t}^{3} \cdot {\left(\sin k\right)}^{2}\right)}{\cos k \cdot {\ell}^{2}}, {\left(\frac{1}{{-1}^{3}}\right)}^{1} \cdot \frac{{\left(\sqrt[3]{-1}\right)}^{9} \cdot \left({\left(\sin k\right)}^{2} \cdot \left({k}^{2} \cdot t\right)\right)}{\cos k \cdot {\ell}^{2}}\right)}}\]

   if 1.6076584110961083e+148 < (* l l)

   1. Initial program 51.7

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

   3. Applied add-cube-cbrt 51.8

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

   4. Applied unpow-prod-down 51.9

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

   5. Applied times-frac 37.3

      \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}}{\ell} \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
   6. Using strategy rm

   7. Applied unpow-prod-down 37.3

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

   8. Applied associate-/l* 26.3

      \[\leadsto \frac{2}{\left(\left(\left(\color{blue}{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3}}}} \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
   9. Using strategy rm

   10. Applied associate-*l* 25.0

       \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3}}} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
   11. Using strategy rm

   12. Applied tan-quot 25.0

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

   13. Applied associate-*l/ 23.0

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

   14. Applied frac-times 21.5

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

   15. Applied associate-*l/ 19.6

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

4. Final simplification 15.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \le 1.1118751811811023 \cdot 10^{-267}:\\ \;\;\;\;\frac{2}{\left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3}}} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\\ \mathbf{elif}\;\ell \cdot \ell \le 1.6076584110961083 \cdot 10^{148}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(2, {\left(\frac{1}{{-1}^{3}}\right)}^{1} \cdot \frac{{\left(\sqrt[3]{-1}\right)}^{9} \cdot \left({t}^{3} \cdot {\left(\sin k\right)}^{2}\right)}{\cos k \cdot {\ell}^{2}}, {\left(\frac{1}{{-1}^{3}}\right)}^{1} \cdot \frac{{\left(\sqrt[3]{-1}\right)}^{9} \cdot \left({\left(\sin k\right)}^{2} \cdot \left({k}^{2} \cdot t\right)\right)}{\cos k \cdot {\ell}^{2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\left({\left(\sqrt[3]{t}\right)}^{3} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)\right) \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3}} \cdot \cos k}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020062 +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))))