Toniolo and Linder, Equation (10+)

Average Error: 32.2 → 9.0

Time: 1.8m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -1.0373876915132293 \cdot 10^{+65}:\\ \;\;\;\;\frac{\frac{\sqrt{2}}{\sin k \cdot \frac{t}{\ell}}}{\frac{\frac{\frac{k}{t} \cdot \frac{k}{t} + 2}{\frac{\sqrt{\sqrt{2}}}{\frac{t}{\ell}}}}{\frac{\frac{\sqrt{\sqrt{2}}}{\tan k}}{t}}}\\ \mathbf{elif}\;t \le 368905.46702557366:\\ \;\;\;\;\frac{\frac{\sqrt{2}}{\sin k \cdot \frac{t}{\ell}}}{\frac{\sin k \cdot {k}^{2}}{\left(\sqrt{2} \cdot \ell\right) \cdot \cos k} + 2 \cdot \frac{\sin k \cdot {t}^{2}}{\left(\cos k \cdot \ell\right) \cdot \sqrt{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\sin k \cdot \frac{t}{\ell}}}{\frac{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}{\frac{\frac{1}{\tan k}}{t}}} \cdot \frac{\sqrt{2}}{\frac{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}{\frac{\sqrt{2}}{\frac{t}{\ell}}}}\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r2639225 = 2.0;
        double r2639226 = t;
        double r2639227 = 3.0;
        double r2639228 = pow(r2639226, r2639227);
        double r2639229 = l;
        double r2639230 = r2639229 * r2639229;
        double r2639231 = r2639228 / r2639230;
        double r2639232 = k;
        double r2639233 = sin(r2639232);
        double r2639234 = r2639231 * r2639233;
        double r2639235 = tan(r2639232);
        double r2639236 = r2639234 * r2639235;
        double r2639237 = 1.0;
        double r2639238 = r2639232 / r2639226;
        double r2639239 = pow(r2639238, r2639225);
        double r2639240 = r2639237 + r2639239;
        double r2639241 = r2639240 + r2639237;
        double r2639242 = r2639236 * r2639241;
        double r2639243 = r2639225 / r2639242;
        return r2639243;
}

↓

double f(double t, double l, double k) {
        double r2639244 = t;
        double r2639245 = -1.0373876915132293e+65;
        bool r2639246 = r2639244 <= r2639245;
        double r2639247 = 2.0;
        double r2639248 = sqrt(r2639247);
        double r2639249 = k;
        double r2639250 = sin(r2639249);
        double r2639251 = l;
        double r2639252 = r2639244 / r2639251;
        double r2639253 = r2639250 * r2639252;
        double r2639254 = r2639248 / r2639253;
        double r2639255 = r2639249 / r2639244;
        double r2639256 = r2639255 * r2639255;
        double r2639257 = r2639256 + r2639247;
        double r2639258 = sqrt(r2639248);
        double r2639259 = r2639258 / r2639252;
        double r2639260 = r2639257 / r2639259;
        double r2639261 = tan(r2639249);
        double r2639262 = r2639258 / r2639261;
        double r2639263 = r2639262 / r2639244;
        double r2639264 = r2639260 / r2639263;
        double r2639265 = r2639254 / r2639264;
        double r2639266 = 368905.46702557366;
        bool r2639267 = r2639244 <= r2639266;
        double r2639268 = pow(r2639249, r2639247);
        double r2639269 = r2639250 * r2639268;
        double r2639270 = r2639248 * r2639251;
        double r2639271 = cos(r2639249);
        double r2639272 = r2639270 * r2639271;
        double r2639273 = r2639269 / r2639272;
        double r2639274 = pow(r2639244, r2639247);
        double r2639275 = r2639250 * r2639274;
        double r2639276 = r2639271 * r2639251;
        double r2639277 = r2639276 * r2639248;
        double r2639278 = r2639275 / r2639277;
        double r2639279 = r2639247 * r2639278;
        double r2639280 = r2639273 + r2639279;
        double r2639281 = r2639254 / r2639280;
        double r2639282 = 1.0;
        double r2639283 = r2639282 / r2639253;
        double r2639284 = sqrt(r2639257);
        double r2639285 = r2639282 / r2639261;
        double r2639286 = r2639285 / r2639244;
        double r2639287 = r2639284 / r2639286;
        double r2639288 = r2639283 / r2639287;
        double r2639289 = r2639248 / r2639252;
        double r2639290 = r2639284 / r2639289;
        double r2639291 = r2639248 / r2639290;
        double r2639292 = r2639288 * r2639291;
        double r2639293 = r2639267 ? r2639281 : r2639292;
        double r2639294 = r2639246 ? r2639265 : r2639293;
        return r2639294;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 3 regimes

2. if t < -1.0373876915132293e+65

   1. Initial program 23.0

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

   2. Simplified 10.4

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

   4. Applied associate-*l* 9.8

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

   6. Applied associate-*r* 7.1

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

   8. Applied *-un-lft-identity 7.1

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

   9. Applied add-sqr-sqrt 7.2

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

   10. Applied times-frac 7.2

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

   11. Applied times-frac 6.6

       \[\leadsto \frac{\color{blue}{\frac{\frac{\sqrt{2}}{1}}{\sin k \cdot \frac{t}{\ell}} \cdot \frac{\frac{\sqrt{2}}{\tan k}}{\frac{t}{\ell} \cdot t}}}{\frac{k}{t} \cdot \frac{k}{t} + 2}\]

   12. Applied associate-/l* 6.5

       \[\leadsto \color{blue}{\frac{\frac{\frac{\sqrt{2}}{1}}{\sin k \cdot \frac{t}{\ell}}}{\frac{\frac{k}{t} \cdot \frac{k}{t} + 2}{\frac{\frac{\sqrt{2}}{\tan k}}{\frac{t}{\ell} \cdot t}}}}\]
   13. Using strategy rm

   14. Applied *-un-lft-identity 6.5

       \[\leadsto \frac{\frac{\frac{\sqrt{2}}{1}}{\sin k \cdot \frac{t}{\ell}}}{\frac{\frac{k}{t} \cdot \frac{k}{t} + 2}{\frac{\frac{\sqrt{2}}{\color{blue}{1 \cdot \tan k}}}{\frac{t}{\ell} \cdot t}}}\]

   15. Applied add-sqr-sqrt 6.4

       \[\leadsto \frac{\frac{\frac{\sqrt{2}}{1}}{\sin k \cdot \frac{t}{\ell}}}{\frac{\frac{k}{t} \cdot \frac{k}{t} + 2}{\frac{\frac{\color{blue}{\sqrt{\sqrt{2}} \cdot \sqrt{\sqrt{2}}}}{1 \cdot \tan k}}{\frac{t}{\ell} \cdot t}}}\]

   16. Applied times-frac 6.4

       \[\leadsto \frac{\frac{\frac{\sqrt{2}}{1}}{\sin k \cdot \frac{t}{\ell}}}{\frac{\frac{k}{t} \cdot \frac{k}{t} + 2}{\frac{\color{blue}{\frac{\sqrt{\sqrt{2}}}{1} \cdot \frac{\sqrt{\sqrt{2}}}{\tan k}}}{\frac{t}{\ell} \cdot t}}}\]

   17. Applied times-frac 1.2

       \[\leadsto \frac{\frac{\frac{\sqrt{2}}{1}}{\sin k \cdot \frac{t}{\ell}}}{\frac{\frac{k}{t} \cdot \frac{k}{t} + 2}{\color{blue}{\frac{\frac{\sqrt{\sqrt{2}}}{1}}{\frac{t}{\ell}} \cdot \frac{\frac{\sqrt{\sqrt{2}}}{\tan k}}{t}}}}\]

   18. Applied associate-/r* 1.2

       \[\leadsto \frac{\frac{\frac{\sqrt{2}}{1}}{\sin k \cdot \frac{t}{\ell}}}{\color{blue}{\frac{\frac{\frac{k}{t} \cdot \frac{k}{t} + 2}{\frac{\frac{\sqrt{\sqrt{2}}}{1}}{\frac{t}{\ell}}}}{\frac{\frac{\sqrt{\sqrt{2}}}{\tan k}}{t}}}}\]

   if -1.0373876915132293e+65 < t < 368905.46702557366

   1. Initial program 44.3

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

   2. Simplified 34.5

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

   4. Applied associate-*l* 33.8

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

   6. Applied associate-*r* 31.8

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

   8. Applied *-un-lft-identity 31.8

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

   9. Applied add-sqr-sqrt 31.9

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

   10. Applied times-frac 31.9

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

   11. Applied times-frac 31.2

       \[\leadsto \frac{\color{blue}{\frac{\frac{\sqrt{2}}{1}}{\sin k \cdot \frac{t}{\ell}} \cdot \frac{\frac{\sqrt{2}}{\tan k}}{\frac{t}{\ell} \cdot t}}}{\frac{k}{t} \cdot \frac{k}{t} + 2}\]

   12. Applied associate-/l* 27.8

       \[\leadsto \color{blue}{\frac{\frac{\frac{\sqrt{2}}{1}}{\sin k \cdot \frac{t}{\ell}}}{\frac{\frac{k}{t} \cdot \frac{k}{t} + 2}{\frac{\frac{\sqrt{2}}{\tan k}}{\frac{t}{\ell} \cdot t}}}}\]

   13. Taylor expanded around inf 17.7

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

   if 368905.46702557366 < t

   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 10.6

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

   4. Applied associate-*l* 10.2

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

   6. Applied associate-*r* 7.3

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

   8. Applied *-un-lft-identity 7.3

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

   9. Applied add-sqr-sqrt 7.4

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

   10. Applied times-frac 7.4

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

   11. Applied times-frac 6.6

       \[\leadsto \frac{\color{blue}{\frac{\frac{\sqrt{2}}{1}}{\sin k \cdot \frac{t}{\ell}} \cdot \frac{\frac{\sqrt{2}}{\tan k}}{\frac{t}{\ell} \cdot t}}}{\frac{k}{t} \cdot \frac{k}{t} + 2}\]

   12. Applied associate-/l* 6.2

       \[\leadsto \color{blue}{\frac{\frac{\frac{\sqrt{2}}{1}}{\sin k \cdot \frac{t}{\ell}}}{\frac{\frac{k}{t} \cdot \frac{k}{t} + 2}{\frac{\frac{\sqrt{2}}{\tan k}}{\frac{t}{\ell} \cdot t}}}}\]
   13. Using strategy rm

   14. Applied div-inv 6.3

       \[\leadsto \frac{\frac{\frac{\sqrt{2}}{1}}{\sin k \cdot \frac{t}{\ell}}}{\frac{\frac{k}{t} \cdot \frac{k}{t} + 2}{\frac{\color{blue}{\sqrt{2} \cdot \frac{1}{\tan k}}}{\frac{t}{\ell} \cdot t}}}\]

   15. Applied times-frac 2.1

       \[\leadsto \frac{\frac{\frac{\sqrt{2}}{1}}{\sin k \cdot \frac{t}{\ell}}}{\frac{\frac{k}{t} \cdot \frac{k}{t} + 2}{\color{blue}{\frac{\sqrt{2}}{\frac{t}{\ell}} \cdot \frac{\frac{1}{\tan k}}{t}}}}\]

   16. Applied add-sqr-sqrt 2.1

       \[\leadsto \frac{\frac{\frac{\sqrt{2}}{1}}{\sin k \cdot \frac{t}{\ell}}}{\frac{\color{blue}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2} \cdot \sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}}{\frac{\sqrt{2}}{\frac{t}{\ell}} \cdot \frac{\frac{1}{\tan k}}{t}}}\]

   17. Applied times-frac 2.0

       \[\leadsto \frac{\frac{\frac{\sqrt{2}}{1}}{\sin k \cdot \frac{t}{\ell}}}{\color{blue}{\frac{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}{\frac{\sqrt{2}}{\frac{t}{\ell}}} \cdot \frac{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}{\frac{\frac{1}{\tan k}}{t}}}}\]

   18. Applied div-inv 2.0

       \[\leadsto \frac{\color{blue}{\frac{\sqrt{2}}{1} \cdot \frac{1}{\sin k \cdot \frac{t}{\ell}}}}{\frac{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}{\frac{\sqrt{2}}{\frac{t}{\ell}}} \cdot \frac{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}{\frac{\frac{1}{\tan k}}{t}}}\]

   19. Applied times-frac 2.4

       \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{1}}{\frac{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}{\frac{\sqrt{2}}{\frac{t}{\ell}}}} \cdot \frac{\frac{1}{\sin k \cdot \frac{t}{\ell}}}{\frac{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}{\frac{\frac{1}{\tan k}}{t}}}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 9.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.0373876915132293 \cdot 10^{+65}:\\ \;\;\;\;\frac{\frac{\sqrt{2}}{\sin k \cdot \frac{t}{\ell}}}{\frac{\frac{\frac{k}{t} \cdot \frac{k}{t} + 2}{\frac{\sqrt{\sqrt{2}}}{\frac{t}{\ell}}}}{\frac{\frac{\sqrt{\sqrt{2}}}{\tan k}}{t}}}\\ \mathbf{elif}\;t \le 368905.46702557366:\\ \;\;\;\;\frac{\frac{\sqrt{2}}{\sin k \cdot \frac{t}{\ell}}}{\frac{\sin k \cdot {k}^{2}}{\left(\sqrt{2} \cdot \ell\right) \cdot \cos k} + 2 \cdot \frac{\sin k \cdot {t}^{2}}{\left(\cos k \cdot \ell\right) \cdot \sqrt{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\sin k \cdot \frac{t}{\ell}}}{\frac{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}{\frac{\frac{1}{\tan k}}{t}}} \cdot \frac{\sqrt{2}}{\frac{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}{\frac{\sqrt{2}}{\frac{t}{\ell}}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019153 
(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))))