Toniolo and Linder, Equation (10-)

Average Error: 48.2 → 9.2

Time: 24.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;k \le -7.116427264207922396198059745161573311828 \cdot 10^{154} \lor \neg \left(k \le 1.596016421741683330355613183588958719568 \cdot 10^{147}\right):\\ \;\;\;\;\left(\frac{2 \cdot \ell}{{\left({k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)\right)}^{1}} \cdot \frac{\ell}{{\left(\sin k\right)}^{2}}\right) \cdot \cos k\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{2 \cdot \ell}{{\left({k}^{2}\right)}^{1}}}{{\left({t}^{1}\right)}^{1}} \cdot \frac{\ell}{{\left(\sin k\right)}^{2}}\right) \cdot \cos k\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r89514 = 2.0;
        double r89515 = t;
        double r89516 = 3.0;
        double r89517 = pow(r89515, r89516);
        double r89518 = l;
        double r89519 = r89518 * r89518;
        double r89520 = r89517 / r89519;
        double r89521 = k;
        double r89522 = sin(r89521);
        double r89523 = r89520 * r89522;
        double r89524 = tan(r89521);
        double r89525 = r89523 * r89524;
        double r89526 = 1.0;
        double r89527 = r89521 / r89515;
        double r89528 = pow(r89527, r89514);
        double r89529 = r89526 + r89528;
        double r89530 = r89529 - r89526;
        double r89531 = r89525 * r89530;
        double r89532 = r89514 / r89531;
        return r89532;
}

↓

double f(double t, double l, double k) {
        double r89533 = k;
        double r89534 = -7.116427264207922e+154;
        bool r89535 = r89533 <= r89534;
        double r89536 = 1.5960164217416833e+147;
        bool r89537 = r89533 <= r89536;
        double r89538 = !r89537;
        bool r89539 = r89535 || r89538;
        double r89540 = 2.0;
        double r89541 = l;
        double r89542 = r89540 * r89541;
        double r89543 = 2.0;
        double r89544 = r89540 / r89543;
        double r89545 = pow(r89533, r89544);
        double r89546 = t;
        double r89547 = 1.0;
        double r89548 = pow(r89546, r89547);
        double r89549 = r89545 * r89548;
        double r89550 = r89545 * r89549;
        double r89551 = pow(r89550, r89547);
        double r89552 = r89542 / r89551;
        double r89553 = sin(r89533);
        double r89554 = pow(r89553, r89543);
        double r89555 = r89541 / r89554;
        double r89556 = r89552 * r89555;
        double r89557 = cos(r89533);
        double r89558 = r89556 * r89557;
        double r89559 = pow(r89533, r89540);
        double r89560 = pow(r89559, r89547);
        double r89561 = r89542 / r89560;
        double r89562 = pow(r89548, r89547);
        double r89563 = r89561 / r89562;
        double r89564 = r89563 * r89555;
        double r89565 = r89564 * r89557;
        double r89566 = r89539 ? r89558 : r89565;
        return r89566;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 2 regimes

2. if k < -7.116427264207922e+154 or 1.5960164217416833e+147 < k

   1. Initial program 40.0

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

   2. Simplified 34.6

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

   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* 19.2

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

   8. Applied associate-*r/ 19.2

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

   9. Applied associate-*l/ 19.2

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

   10. Applied associate-/r/ 19.2

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

   11. Simplified 22.2

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

   13. Applied sqr-pow 22.2

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

   14. Applied associate-*l* 14.7

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

   if -7.116427264207922e+154 < k < 1.5960164217416833e+147

   1. Initial program 54.0

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

   2. Simplified 44.3

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

   3. Taylor expanded around inf 21.6

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

   5. Applied sqr-pow 21.6

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

   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 10.6

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

   13. Applied unpow-prod-down 10.6

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

   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 9.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \le -7.116427264207922396198059745161573311828 \cdot 10^{154} \lor \neg \left(k \le 1.596016421741683330355613183588958719568 \cdot 10^{147}\right):\\ \;\;\;\;\left(\frac{2 \cdot \ell}{{\left({k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)\right)}^{1}} \cdot \frac{\ell}{{\left(\sin k\right)}^{2}}\right) \cdot \cos k\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{2 \cdot \ell}{{\left({k}^{2}\right)}^{1}}}{{\left({t}^{1}\right)}^{1}} \cdot \frac{\ell}{{\left(\sin k\right)}^{2}}\right) \cdot \cos k\\ \end{array}\]

Reproduce

herbie shell --seed 2019347 +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))))