Average Error: 32.0 → 2.2
Time: 41.7s
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\sqrt{2} \cdot \frac{\sqrt{2}}{\mathsf{fma}\left(\frac{\sin k \cdot \sin k}{\cos k}, \frac{t}{\frac{\ell}{k}} \cdot \frac{1}{\frac{\ell}{k}}, 2 \cdot \frac{\sqrt[3]{t} \cdot \left(\left(\sqrt[3]{t} \cdot \frac{\sin k \cdot t}{\ell}\right) \cdot \left(\sqrt[3]{t} \cdot \frac{\sin k \cdot t}{\ell}\right)\right)}{\cos k}\right)}\]
\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)}
↓
\sqrt{2} \cdot \frac{\sqrt{2}}{\mathsf{fma}\left(\frac{\sin k \cdot \sin k}{\cos k}, \frac{t}{\frac{\ell}{k}} \cdot \frac{1}{\frac{\ell}{k}}, 2 \cdot \frac{\sqrt[3]{t} \cdot \left(\left(\sqrt[3]{t} \cdot \frac{\sin k \cdot t}{\ell}\right) \cdot \left(\sqrt[3]{t} \cdot \frac{\sin k \cdot t}{\ell}\right)\right)}{\cos k}\right)}
double f(double t, double l, double k) {
        double r3761454 = 2.0;
        double r3761455 = t;
        double r3761456 = 3.0;
        double r3761457 = pow(r3761455, r3761456);
        double r3761458 = l;
        double r3761459 = r3761458 * r3761458;
        double r3761460 = r3761457 / r3761459;
        double r3761461 = k;
        double r3761462 = sin(r3761461);
        double r3761463 = r3761460 * r3761462;
        double r3761464 = tan(r3761461);
        double r3761465 = r3761463 * r3761464;
        double r3761466 = 1.0;
        double r3761467 = r3761461 / r3761455;
        double r3761468 = pow(r3761467, r3761454);
        double r3761469 = r3761466 + r3761468;
        double r3761470 = r3761469 + r3761466;
        double r3761471 = r3761465 * r3761470;
        double r3761472 = r3761454 / r3761471;
        return r3761472;
}
↓
double f(double t, double l, double k) {
        double r3761473 = 2.0;
        double r3761474 = sqrt(r3761473);
        double r3761475 = k;
        double r3761476 = sin(r3761475);
        double r3761477 = r3761476 * r3761476;
        double r3761478 = cos(r3761475);
        double r3761479 = r3761477 / r3761478;
        double r3761480 = t;
        double r3761481 = l;
        double r3761482 = r3761481 / r3761475;
        double r3761483 = r3761480 / r3761482;
        double r3761484 = 1.0;
        double r3761485 = r3761484 / r3761482;
        double r3761486 = r3761483 * r3761485;
        double r3761487 = cbrt(r3761480);
        double r3761488 = r3761476 * r3761480;
        double r3761489 = r3761488 / r3761481;
        double r3761490 = r3761487 * r3761489;
        double r3761491 = r3761490 * r3761490;
        double r3761492 = r3761487 * r3761491;
        double r3761493 = r3761492 / r3761478;
        double r3761494 = r3761473 * r3761493;
        double r3761495 = fma(r3761479, r3761486, r3761494);
        double r3761496 = r3761474 / r3761495;
        double r3761497 = r3761474 * r3761496;
        return r3761497;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

  1. Initial program 32.0

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

  2. Taylor expanded around inf 30.8

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

  3. Simplified19.9

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

  5. Applied *-un-lft-identity19.9

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

  6. Applied add-sqr-sqrt20.0

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

  7. Applied times-frac20.0

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

  8. Simplified20.0

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

  9. Simplified7.3

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

  11. Applied *-un-lft-identity7.3

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

  12. Applied times-frac4.2

    \[\leadsto \sqrt{2} \cdot \frac{\sqrt{2}}{\mathsf{fma}\left(\frac{\sin k \cdot \sin k}{\cos k}, \color{blue}{\frac{1}{\frac{\ell}{k}} \cdot \frac{t}{\frac{\ell}{k}}}, \frac{\left(\left(t \cdot \frac{\sin k}{\ell}\right) \cdot \left(t \cdot \frac{\sin k}{\ell}\right)\right) \cdot t}{\cos k} \cdot 2\right)}\]
  13. Using strategy rm

  14. Applied add-cube-cbrt4.3

    \[\leadsto \sqrt{2} \cdot \frac{\sqrt{2}}{\mathsf{fma}\left(\frac{\sin k \cdot \sin k}{\cos k}, \frac{1}{\frac{\ell}{k}} \cdot \frac{t}{\frac{\ell}{k}}, \frac{\left(\left(t \cdot \frac{\sin k}{\ell}\right) \cdot \left(t \cdot \frac{\sin k}{\ell}\right)\right) \cdot \color{blue}{\left(\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}\right)}}{\cos k} \cdot 2\right)}\]

  15. Applied associate-*r*4.3

    \[\leadsto \sqrt{2} \cdot \frac{\sqrt{2}}{\mathsf{fma}\left(\frac{\sin k \cdot \sin k}{\cos k}, \frac{1}{\frac{\ell}{k}} \cdot \frac{t}{\frac{\ell}{k}}, \frac{\color{blue}{\left(\left(\left(t \cdot \frac{\sin k}{\ell}\right) \cdot \left(t \cdot \frac{\sin k}{\ell}\right)\right) \cdot \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)\right) \cdot \sqrt[3]{t}}}{\cos k} \cdot 2\right)}\]

  16. Simplified2.2

    \[\leadsto \sqrt{2} \cdot \frac{\sqrt{2}}{\mathsf{fma}\left(\frac{\sin k \cdot \sin k}{\cos k}, \frac{1}{\frac{\ell}{k}} \cdot \frac{t}{\frac{\ell}{k}}, \frac{\color{blue}{\left(\left(\sqrt[3]{t} \cdot \frac{\sin k \cdot t}{\ell}\right) \cdot \left(\sqrt[3]{t} \cdot \frac{\sin k \cdot t}{\ell}\right)\right)} \cdot \sqrt[3]{t}}{\cos k} \cdot 2\right)}\]

  17. Final simplification2.2

    \[\leadsto \sqrt{2} \cdot \frac{\sqrt{2}}{\mathsf{fma}\left(\frac{\sin k \cdot \sin k}{\cos k}, \frac{t}{\frac{\ell}{k}} \cdot \frac{1}{\frac{\ell}{k}}, 2 \cdot \frac{\sqrt[3]{t} \cdot \left(\left(\sqrt[3]{t} \cdot \frac{\sin k \cdot t}{\ell}\right) \cdot \left(\sqrt[3]{t} \cdot \frac{\sin k \cdot t}{\ell}\right)\right)}{\cos k}\right)}\]

Reproduce

herbie shell --seed 2019169 +o rules:numerics
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10+)"
  (/ 2.0 (* (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k)) (+ (+ 1.0 (pow (/ k t) 2.0)) 1.0))))