Toniolo and Linder, Equation (10+)

Average Error: 32.5 → 10.2

Time: 22.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -5.02981601344431446 \cdot 10^{-47} \lor \neg \left(t \le 3.061555101649697 \cdot 10^{-53}\right):\\ \;\;\;\;\frac{2}{\frac{\left(\left({\left(\sqrt[3]{t}\right)}^{3} \cdot \left(\frac{\sqrt[3]{{\left(\sqrt[3]{t}\right)}^{3}} \cdot \sqrt[3]{{\left(\sqrt[3]{t}\right)}^{3}}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \left(\frac{\sqrt[3]{{\left(\sqrt[3]{t}\right)}^{3}}}{\sqrt[3]{\ell}} \cdot \sin k\right)\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(2, \frac{{t}^{2} \cdot {\left(\sin k\right)}^{2}}{\cos k \cdot \ell}, \frac{{k}^{2} \cdot {\left(\sin k\right)}^{2}}{\cos k \cdot \ell}\right)}{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3}}}}\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r139450 = 2.0;
        double r139451 = t;
        double r139452 = 3.0;
        double r139453 = pow(r139451, r139452);
        double r139454 = l;
        double r139455 = r139454 * r139454;
        double r139456 = r139453 / r139455;
        double r139457 = k;
        double r139458 = sin(r139457);
        double r139459 = r139456 * r139458;
        double r139460 = tan(r139457);
        double r139461 = r139459 * r139460;
        double r139462 = 1.0;
        double r139463 = r139457 / r139451;
        double r139464 = pow(r139463, r139450);
        double r139465 = r139462 + r139464;
        double r139466 = r139465 + r139462;
        double r139467 = r139461 * r139466;
        double r139468 = r139450 / r139467;
        return r139468;
}

↓

double f(double t, double l, double k) {
        double r139469 = t;
        double r139470 = -5.029816013444314e-47;
        bool r139471 = r139469 <= r139470;
        double r139472 = 3.0615551016496975e-53;
        bool r139473 = r139469 <= r139472;
        double r139474 = !r139473;
        bool r139475 = r139471 || r139474;
        double r139476 = 2.0;
        double r139477 = cbrt(r139469);
        double r139478 = 3.0;
        double r139479 = pow(r139477, r139478);
        double r139480 = cbrt(r139479);
        double r139481 = r139480 * r139480;
        double r139482 = l;
        double r139483 = cbrt(r139482);
        double r139484 = r139483 * r139483;
        double r139485 = r139481 / r139484;
        double r139486 = r139480 / r139483;
        double r139487 = k;
        double r139488 = sin(r139487);
        double r139489 = r139486 * r139488;
        double r139490 = r139485 * r139489;
        double r139491 = r139479 * r139490;
        double r139492 = tan(r139487);
        double r139493 = r139491 * r139492;
        double r139494 = 1.0;
        double r139495 = r139487 / r139469;
        double r139496 = pow(r139495, r139476);
        double r139497 = r139494 + r139496;
        double r139498 = r139497 + r139494;
        double r139499 = r139493 * r139498;
        double r139500 = r139482 / r139479;
        double r139501 = r139499 / r139500;
        double r139502 = r139476 / r139501;
        double r139503 = 2.0;
        double r139504 = pow(r139469, r139503);
        double r139505 = pow(r139488, r139503);
        double r139506 = r139504 * r139505;
        double r139507 = cos(r139487);
        double r139508 = r139507 * r139482;
        double r139509 = r139506 / r139508;
        double r139510 = pow(r139487, r139503);
        double r139511 = r139510 * r139505;
        double r139512 = r139511 / r139508;
        double r139513 = fma(r139476, r139509, r139512);
        double r139514 = r139513 / r139500;
        double r139515 = r139476 / r139514;
        double r139516 = r139475 ? r139502 : r139515;
        return r139516;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 2 regimes

2. if t < -5.029816013444314e-47 or 3.0615551016496975e-53 < t

   1. Initial program 22.8

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

   3. Applied add-cube-cbrt 23.0

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

   4. Applied unpow-prod-down 23.0

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

   5. Applied times-frac 16.3

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

   6. Applied associate-*l* 14.1

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}}{\ell} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
   7. Using strategy rm

   8. Applied unpow-prod-down 14.1

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

   9. Applied associate-/l* 8.9

      \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3}}}} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
   10. Using strategy rm

   11. Applied associate-*l/ 7.7

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

   12. Applied associate-*l/ 5.4

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

   13. Applied associate-*l/ 4.9

       \[\leadsto \frac{2}{\color{blue}{\frac{\left(\left({\left(\sqrt[3]{t}\right)}^{3} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3}}}}}\]
   14. Using strategy rm

   15. Applied add-cube-cbrt 4.9

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

   16. Applied add-cube-cbrt 4.9

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

   17. Applied times-frac 4.9

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

   18. Applied associate-*l* 4.8

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

   if -5.029816013444314e-47 < t < 3.0615551016496975e-53

   1. Initial program 54.4

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

   3. Applied add-cube-cbrt 54.5

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

   4. Applied unpow-prod-down 54.5

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

   5. Applied times-frac 46.3

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

   6. Applied associate-*l* 45.7

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}}{\ell} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
   7. Using strategy rm

   8. Applied unpow-prod-down 45.7

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

   9. Applied associate-/l* 39.7

      \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3}}}} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
   10. Using strategy rm

   11. Applied associate-*l/ 39.7

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

   12. Applied associate-*l/ 41.0

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

   13. Applied associate-*l/ 37.5

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

   14. Taylor expanded around inf 22.2

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

   15. Simplified 22.2

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

4. Final simplification 10.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -5.02981601344431446 \cdot 10^{-47} \lor \neg \left(t \le 3.061555101649697 \cdot 10^{-53}\right):\\ \;\;\;\;\frac{2}{\frac{\left(\left({\left(\sqrt[3]{t}\right)}^{3} \cdot \left(\frac{\sqrt[3]{{\left(\sqrt[3]{t}\right)}^{3}} \cdot \sqrt[3]{{\left(\sqrt[3]{t}\right)}^{3}}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \left(\frac{\sqrt[3]{{\left(\sqrt[3]{t}\right)}^{3}}}{\sqrt[3]{\ell}} \cdot \sin k\right)\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(2, \frac{{t}^{2} \cdot {\left(\sin k\right)}^{2}}{\cos k \cdot \ell}, \frac{{k}^{2} \cdot {\left(\sin k\right)}^{2}}{\cos k \cdot \ell}\right)}{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3}}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020057 +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))))