Toniolo and Linder, Equation (10+)

Average Error: 31.4 → 8.1

Time: 40.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -1.8851239965119804 \cdot 10^{+28}:\\ \;\;\;\;\frac{\frac{\sqrt{2}}{\sin k}}{\frac{t}{\ell}} \cdot \frac{\sqrt{2}}{\frac{t}{\ell} \cdot \left(\frac{\frac{\sin k}{\cos k} \cdot \left(k \cdot k\right)}{t} + \frac{2 \cdot t}{\frac{\cos k}{\sin k}}\right)}\\ \mathbf{elif}\;t \le 8.333143050057771 \cdot 10^{+130}:\\ \;\;\;\;\frac{\frac{2}{\sin k}}{\frac{t}{\ell} \cdot \left(\frac{t \cdot t}{\ell} \cdot \left(\frac{\sin k}{\cos k} \cdot 2\right) + \frac{\sin k}{\frac{\cos k}{k} \cdot \frac{\ell}{k}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{2}{\sin k}}{\left(8 + \left(\left(\frac{k}{t} \cdot \frac{k}{t}\right) \cdot \frac{k}{t}\right) \cdot \left(\left(\frac{k}{t} \cdot \frac{k}{t}\right) \cdot \frac{k}{t}\right)\right) \cdot \left(\frac{t}{\ell} \cdot \left(t \cdot \tan k\right)\right)}}{\frac{t}{\ell}} \cdot \left(\left(4 - 2 \cdot \left(\frac{k}{t} \cdot \frac{k}{t}\right)\right) + \left(\frac{k}{t} \cdot \frac{k}{t}\right) \cdot \left(\frac{k}{t} \cdot \frac{k}{t}\right)\right)\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r3410189 = 2.0;
        double r3410190 = t;
        double r3410191 = 3.0;
        double r3410192 = pow(r3410190, r3410191);
        double r3410193 = l;
        double r3410194 = r3410193 * r3410193;
        double r3410195 = r3410192 / r3410194;
        double r3410196 = k;
        double r3410197 = sin(r3410196);
        double r3410198 = r3410195 * r3410197;
        double r3410199 = tan(r3410196);
        double r3410200 = r3410198 * r3410199;
        double r3410201 = 1.0;
        double r3410202 = r3410196 / r3410190;
        double r3410203 = pow(r3410202, r3410189);
        double r3410204 = r3410201 + r3410203;
        double r3410205 = r3410204 + r3410201;
        double r3410206 = r3410200 * r3410205;
        double r3410207 = r3410189 / r3410206;
        return r3410207;
}

↓

double f(double t, double l, double k) {
        double r3410208 = t;
        double r3410209 = -1.8851239965119804e+28;
        bool r3410210 = r3410208 <= r3410209;
        double r3410211 = 2.0;
        double r3410212 = sqrt(r3410211);
        double r3410213 = k;
        double r3410214 = sin(r3410213);
        double r3410215 = r3410212 / r3410214;
        double r3410216 = l;
        double r3410217 = r3410208 / r3410216;
        double r3410218 = r3410215 / r3410217;
        double r3410219 = cos(r3410213);
        double r3410220 = r3410214 / r3410219;
        double r3410221 = r3410213 * r3410213;
        double r3410222 = r3410220 * r3410221;
        double r3410223 = r3410222 / r3410208;
        double r3410224 = r3410211 * r3410208;
        double r3410225 = r3410219 / r3410214;
        double r3410226 = r3410224 / r3410225;
        double r3410227 = r3410223 + r3410226;
        double r3410228 = r3410217 * r3410227;
        double r3410229 = r3410212 / r3410228;
        double r3410230 = r3410218 * r3410229;
        double r3410231 = 8.333143050057771e+130;
        bool r3410232 = r3410208 <= r3410231;
        double r3410233 = r3410211 / r3410214;
        double r3410234 = r3410208 * r3410208;
        double r3410235 = r3410234 / r3410216;
        double r3410236 = r3410220 * r3410211;
        double r3410237 = r3410235 * r3410236;
        double r3410238 = r3410219 / r3410213;
        double r3410239 = r3410216 / r3410213;
        double r3410240 = r3410238 * r3410239;
        double r3410241 = r3410214 / r3410240;
        double r3410242 = r3410237 + r3410241;
        double r3410243 = r3410217 * r3410242;
        double r3410244 = r3410233 / r3410243;
        double r3410245 = 8.0;
        double r3410246 = r3410213 / r3410208;
        double r3410247 = r3410246 * r3410246;
        double r3410248 = r3410247 * r3410246;
        double r3410249 = r3410248 * r3410248;
        double r3410250 = r3410245 + r3410249;
        double r3410251 = tan(r3410213);
        double r3410252 = r3410208 * r3410251;
        double r3410253 = r3410217 * r3410252;
        double r3410254 = r3410250 * r3410253;
        double r3410255 = r3410233 / r3410254;
        double r3410256 = r3410255 / r3410217;
        double r3410257 = 4.0;
        double r3410258 = r3410211 * r3410247;
        double r3410259 = r3410257 - r3410258;
        double r3410260 = r3410247 * r3410247;
        double r3410261 = r3410259 + r3410260;
        double r3410262 = r3410256 * r3410261;
        double r3410263 = r3410232 ? r3410244 : r3410262;
        double r3410264 = r3410210 ? r3410230 : r3410263;
        return r3410264;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 3 regimes

2. if t < -1.8851239965119804e+28

   1. Initial program 21.7

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

   2. Simplified 10.3

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

   4. Applied div-inv 10.3

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

   5. Applied times-frac 10.3

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

   6. Applied associate-/l* 10.4

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

   7. Simplified 7.9

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

   8. Taylor expanded around inf 9.9

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

   9. Simplified 9.9

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

   11. Applied associate-*r* 8.0

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

   13. Applied *-un-lft-identity 8.0

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

   14. Applied add-sqr-sqrt 8.1

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

   15. Applied times-frac 8.1

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

   16. Applied times-frac 4.5

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

   17. Simplified 4.6

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

   if -1.8851239965119804e+28 < t < 8.333143050057771e+130

   1. Initial program 40.3

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

   2. Simplified 30.1

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

   4. Applied div-inv 30.1

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

   5. Applied times-frac 30.1

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

   6. Applied associate-/l* 29.8

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

   7. Simplified 26.4

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

   8. Taylor expanded around inf 24.2

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

   9. Simplified 23.7

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

   11. Applied associate-*r* 17.7

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

   12. Taylor expanded around inf 15.8

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

   13. Simplified 11.4

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

   if 8.333143050057771e+130 < t

   1. Initial program 21.4

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

   2. Simplified 9.6

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

   4. Applied div-inv 9.6

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

   5. Applied times-frac 9.6

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

   6. Applied associate-/l* 9.7

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

   7. Simplified 6.6

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

   9. Applied flip3-+ 9.2

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

   10. Applied associate-*l/ 9.2

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

   11. Applied associate-*l/ 9.7

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

   12. Applied associate-*l/ 9.7

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

   13. Applied associate-/r/ 9.7

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

   14. Simplified 4.4

       \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\sin k}}{\left(8 + \left(\frac{k}{t} \cdot \left(\frac{k}{t} \cdot \frac{k}{t}\right)\right) \cdot \left(\frac{k}{t} \cdot \left(\frac{k}{t} \cdot \frac{k}{t}\right)\right)\right) \cdot \left(\left(\tan k \cdot t\right) \cdot \frac{t}{\ell}\right)}}{\frac{t}{\ell}}} \cdot \left(\left(\frac{k}{t} \cdot \frac{k}{t}\right) \cdot \left(\frac{k}{t} \cdot \frac{k}{t}\right) + \left(2 \cdot 2 - \left(\frac{k}{t} \cdot \frac{k}{t}\right) \cdot 2\right)\right)\]
3. Recombined 3 regimes into one program.

4. Final simplification 8.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.8851239965119804 \cdot 10^{+28}:\\ \;\;\;\;\frac{\frac{\sqrt{2}}{\sin k}}{\frac{t}{\ell}} \cdot \frac{\sqrt{2}}{\frac{t}{\ell} \cdot \left(\frac{\frac{\sin k}{\cos k} \cdot \left(k \cdot k\right)}{t} + \frac{2 \cdot t}{\frac{\cos k}{\sin k}}\right)}\\ \mathbf{elif}\;t \le 8.333143050057771 \cdot 10^{+130}:\\ \;\;\;\;\frac{\frac{2}{\sin k}}{\frac{t}{\ell} \cdot \left(\frac{t \cdot t}{\ell} \cdot \left(\frac{\sin k}{\cos k} \cdot 2\right) + \frac{\sin k}{\frac{\cos k}{k} \cdot \frac{\ell}{k}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{2}{\sin k}}{\left(8 + \left(\left(\frac{k}{t} \cdot \frac{k}{t}\right) \cdot \frac{k}{t}\right) \cdot \left(\left(\frac{k}{t} \cdot \frac{k}{t}\right) \cdot \frac{k}{t}\right)\right) \cdot \left(\frac{t}{\ell} \cdot \left(t \cdot \tan k\right)\right)}}{\frac{t}{\ell}} \cdot \left(\left(4 - 2 \cdot \left(\frac{k}{t} \cdot \frac{k}{t}\right)\right) + \left(\frac{k}{t} \cdot \frac{k}{t}\right) \cdot \left(\frac{k}{t} \cdot \frac{k}{t}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019158 
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10+)"
  (/ 2 (* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1))))