Toniolo and Linder, Equation (10-)

Average Error: 48.0 → 9.3

Time: 23.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;k \le -2.9620071106894416 \cdot 10^{153} \lor \neg \left(k \le 2.05572463714506226 \cdot 10^{147}\right):\\ \;\;\;\;\left(\frac{2 \cdot \ell}{{\left({k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)\right)}^{1}} \cdot \frac{\ell}{{\left(\sin k\right)}^{2}}\right) \cdot \cos k\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{2 \cdot \ell}{{\left({k}^{2}\right)}^{1}}}{{\left({t}^{1}\right)}^{1}} \cdot \frac{\ell}{{\left(\sin k\right)}^{2}}\right) \cdot \cos k\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r84299 = 2.0;
        double r84300 = t;
        double r84301 = 3.0;
        double r84302 = pow(r84300, r84301);
        double r84303 = l;
        double r84304 = r84303 * r84303;
        double r84305 = r84302 / r84304;
        double r84306 = k;
        double r84307 = sin(r84306);
        double r84308 = r84305 * r84307;
        double r84309 = tan(r84306);
        double r84310 = r84308 * r84309;
        double r84311 = 1.0;
        double r84312 = r84306 / r84300;
        double r84313 = pow(r84312, r84299);
        double r84314 = r84311 + r84313;
        double r84315 = r84314 - r84311;
        double r84316 = r84310 * r84315;
        double r84317 = r84299 / r84316;
        return r84317;
}

↓

double f(double t, double l, double k) {
        double r84318 = k;
        double r84319 = -2.9620071106894416e+153;
        bool r84320 = r84318 <= r84319;
        double r84321 = 2.0557246371450623e+147;
        bool r84322 = r84318 <= r84321;
        double r84323 = !r84322;
        bool r84324 = r84320 || r84323;
        double r84325 = 2.0;
        double r84326 = l;
        double r84327 = r84325 * r84326;
        double r84328 = 2.0;
        double r84329 = r84325 / r84328;
        double r84330 = pow(r84318, r84329);
        double r84331 = t;
        double r84332 = 1.0;
        double r84333 = pow(r84331, r84332);
        double r84334 = r84330 * r84333;
        double r84335 = r84330 * r84334;
        double r84336 = pow(r84335, r84332);
        double r84337 = r84327 / r84336;
        double r84338 = sin(r84318);
        double r84339 = pow(r84338, r84328);
        double r84340 = r84326 / r84339;
        double r84341 = r84337 * r84340;
        double r84342 = cos(r84318);
        double r84343 = r84341 * r84342;
        double r84344 = pow(r84318, r84325);
        double r84345 = pow(r84344, r84332);
        double r84346 = r84327 / r84345;
        double r84347 = pow(r84333, r84332);
        double r84348 = r84346 / r84347;
        double r84349 = r84348 * r84340;
        double r84350 = r84349 * r84342;
        double r84351 = r84324 ? r84343 : r84350;
        return r84351;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 2 regimes

2. if k < -2.9620071106894416e+153 or 2.0557246371450623e+147 < k

   1. Initial program 39.3

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

   2. Simplified 33.9

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

   3. Taylor expanded around inf 24.1

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

   5. Applied sqr-pow 24.1

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

   6. Applied associate-*l* 19.4

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

   8. Applied associate-*r/ 19.4

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

   9. Applied associate-*l/ 19.4

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

   10. Applied associate-/r/ 19.4

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

   11. Simplified 22.1

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

   13. Applied sqr-pow 22.1

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

   14. Applied associate-*l* 15.1

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

   if -2.9620071106894416e+153 < k < 2.0557246371450623e+147

   1. Initial program 54.1

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

   2. Simplified 44.5

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

   3. Taylor expanded around inf 21.5

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

   5. Applied sqr-pow 21.5

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

   6. Applied associate-*l* 21.5

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

   8. Applied associate-*r/ 21.5

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

   9. Applied associate-*l/ 21.5

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

   10. Applied associate-/r/ 21.5

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

   11. Simplified 10.4

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

   13. Applied unpow-prod-down 10.4

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

   14. Applied associate-/r* 5.2

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

4. Final simplification 9.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \le -2.9620071106894416 \cdot 10^{153} \lor \neg \left(k \le 2.05572463714506226 \cdot 10^{147}\right):\\ \;\;\;\;\left(\frac{2 \cdot \ell}{{\left({k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)\right)}^{1}} \cdot \frac{\ell}{{\left(\sin k\right)}^{2}}\right) \cdot \cos k\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{2 \cdot \ell}{{\left({k}^{2}\right)}^{1}}}{{\left({t}^{1}\right)}^{1}} \cdot \frac{\ell}{{\left(\sin k\right)}^{2}}\right) \cdot \cos k\\ \end{array}\]

Reproduce

herbie shell --seed 2020027 
(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))))