Toniolo and Linder, Equation (10-)

Average Error: 47.9 → 5.5

Time: 54.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le 1.602726285194281 \cdot 10^{+155}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{\sin k} \cdot \frac{\frac{\ell}{k \cdot t} \cdot \frac{\ell}{k}}{\sin k}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \left(\frac{\ell}{k} \cdot \frac{\frac{\ell}{t}}{k}\right)}{\sin k \cdot \sin k}\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r1460506 = 2.0;
        double r1460507 = t;
        double r1460508 = 3.0;
        double r1460509 = pow(r1460507, r1460508);
        double r1460510 = l;
        double r1460511 = r1460510 * r1460510;
        double r1460512 = r1460509 / r1460511;
        double r1460513 = k;
        double r1460514 = sin(r1460513);
        double r1460515 = r1460512 * r1460514;
        double r1460516 = tan(r1460513);
        double r1460517 = r1460515 * r1460516;
        double r1460518 = 1.0;
        double r1460519 = r1460513 / r1460507;
        double r1460520 = pow(r1460519, r1460506);
        double r1460521 = r1460518 + r1460520;
        double r1460522 = r1460521 - r1460518;
        double r1460523 = r1460517 * r1460522;
        double r1460524 = r1460506 / r1460523;
        return r1460524;
}

↓

double f(double t, double l, double k) {
        double r1460525 = t;
        double r1460526 = 1.602726285194281e+155;
        bool r1460527 = r1460525 <= r1460526;
        double r1460528 = 2.0;
        double r1460529 = k;
        double r1460530 = cos(r1460529);
        double r1460531 = sin(r1460529);
        double r1460532 = r1460530 / r1460531;
        double r1460533 = l;
        double r1460534 = r1460529 * r1460525;
        double r1460535 = r1460533 / r1460534;
        double r1460536 = r1460533 / r1460529;
        double r1460537 = r1460535 * r1460536;
        double r1460538 = r1460537 / r1460531;
        double r1460539 = r1460532 * r1460538;
        double r1460540 = r1460528 * r1460539;
        double r1460541 = r1460533 / r1460525;
        double r1460542 = r1460541 / r1460529;
        double r1460543 = r1460536 * r1460542;
        double r1460544 = r1460530 * r1460543;
        double r1460545 = r1460531 * r1460531;
        double r1460546 = r1460544 / r1460545;
        double r1460547 = r1460528 * r1460546;
        double r1460548 = r1460527 ? r1460540 : r1460547;
        return r1460548;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 2 regimes

2. if t < 1.602726285194281e+155

   1. Initial program 46.6

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

   2. Simplified 31.1

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

   3. Taylor expanded around inf 23.5

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

   4. Simplified 21.6

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

   6. Applied times-frac 14.9

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

   8. Applied *-un-lft-identity 14.9

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

   9. Applied times-frac 11.2

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

   10. Applied associate-*r* 7.8

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

   11. Simplified 7.8

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

   13. Applied *-un-lft-identity 7.8

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

   14. Applied times-frac 7.8

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

   15. Applied associate-*r* 7.1

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

   16. Simplified 5.0

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

   if 1.602726285194281e+155 < t

   1. Initial program 54.8

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

   2. Simplified 32.1

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

   3. Taylor expanded around inf 23.1

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

   4. Simplified 19.9

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

   6. Applied times-frac 12.5

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

   8. Applied *-un-lft-identity 12.5

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

   9. Applied times-frac 8.9

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

   10. Applied associate-*r* 8.7

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

   11. Simplified 8.7

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

   13. Applied associate-*r/ 8.7

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

4. Final simplification 5.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le 1.602726285194281 \cdot 10^{+155}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{\sin k} \cdot \frac{\frac{\ell}{k \cdot t} \cdot \frac{\ell}{k}}{\sin k}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \left(\frac{\ell}{k} \cdot \frac{\frac{\ell}{t}}{k}\right)}{\sin k \cdot \sin k}\\ \end{array}\]

Reproduce

herbie shell --seed 2019154 
(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))))