Average Error: 32.2 → 9.3
Time: 43.8s
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -7.451781871449174 \cdot 10^{-25}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \left(\frac{\frac{t \cdot \sin k}{\sqrt[3]{\frac{\ell}{t}} \cdot \sqrt[3]{\frac{\ell}{t}}}}{\sqrt[3]{\frac{\ell}{t}}} \cdot \tan k\right)}{\frac{\ell}{t}}}\\ \mathbf{elif}\;t \le 2.030695630825502 \cdot 10^{-34}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(2, \frac{t \cdot t}{\frac{\ell \cdot \cos k}{\sin k \cdot \sin k}}, \frac{\sin k \cdot \sin k}{\frac{\ell \cdot \cos k}{k \cdot k}}\right)}{\frac{\ell}{t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \left(\frac{\frac{t \cdot \sin k}{\sqrt[3]{\frac{\ell}{t}} \cdot \sqrt[3]{\frac{\ell}{t}}}}{\sqrt[3]{\frac{\ell}{t}}} \cdot \tan k\right)}{\frac{\ell}{t}}}\\ \end{array}\]
\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 -7.451781871449174 \cdot 10^{-25}:\\
\;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \left(\frac{\frac{t \cdot \sin k}{\sqrt[3]{\frac{\ell}{t}} \cdot \sqrt[3]{\frac{\ell}{t}}}}{\sqrt[3]{\frac{\ell}{t}}} \cdot \tan k\right)}{\frac{\ell}{t}}}\\

\mathbf{elif}\;t \le 2.030695630825502 \cdot 10^{-34}:\\
\;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(2, \frac{t \cdot t}{\frac{\ell \cdot \cos k}{\sin k \cdot \sin k}}, \frac{\sin k \cdot \sin k}{\frac{\ell \cdot \cos k}{k \cdot k}}\right)}{\frac{\ell}{t}}}\\

\mathbf{else}:\\
\;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \left(\frac{\frac{t \cdot \sin k}{\sqrt[3]{\frac{\ell}{t}} \cdot \sqrt[3]{\frac{\ell}{t}}}}{\sqrt[3]{\frac{\ell}{t}}} \cdot \tan k\right)}{\frac{\ell}{t}}}\\

\end{array}
double f(double t, double l, double k) {
        double r1284474 = 2.0;
        double r1284475 = t;
        double r1284476 = 3.0;
        double r1284477 = pow(r1284475, r1284476);
        double r1284478 = l;
        double r1284479 = r1284478 * r1284478;
        double r1284480 = r1284477 / r1284479;
        double r1284481 = k;
        double r1284482 = sin(r1284481);
        double r1284483 = r1284480 * r1284482;
        double r1284484 = tan(r1284481);
        double r1284485 = r1284483 * r1284484;
        double r1284486 = 1.0;
        double r1284487 = r1284481 / r1284475;
        double r1284488 = pow(r1284487, r1284474);
        double r1284489 = r1284486 + r1284488;
        double r1284490 = r1284489 + r1284486;
        double r1284491 = r1284485 * r1284490;
        double r1284492 = r1284474 / r1284491;
        return r1284492;
}
↓
double f(double t, double l, double k) {
        double r1284493 = t;
        double r1284494 = -7.451781871449174e-25;
        bool r1284495 = r1284493 <= r1284494;
        double r1284496 = 2.0;
        double r1284497 = k;
        double r1284498 = r1284497 / r1284493;
        double r1284499 = fma(r1284498, r1284498, r1284496);
        double r1284500 = sin(r1284497);
        double r1284501 = r1284493 * r1284500;
        double r1284502 = l;
        double r1284503 = r1284502 / r1284493;
        double r1284504 = cbrt(r1284503);
        double r1284505 = r1284504 * r1284504;
        double r1284506 = r1284501 / r1284505;
        double r1284507 = r1284506 / r1284504;
        double r1284508 = tan(r1284497);
        double r1284509 = r1284507 * r1284508;
        double r1284510 = r1284499 * r1284509;
        double r1284511 = r1284510 / r1284503;
        double r1284512 = r1284496 / r1284511;
        double r1284513 = 2.030695630825502e-34;
        bool r1284514 = r1284493 <= r1284513;
        double r1284515 = r1284493 * r1284493;
        double r1284516 = cos(r1284497);
        double r1284517 = r1284502 * r1284516;
        double r1284518 = r1284500 * r1284500;
        double r1284519 = r1284517 / r1284518;
        double r1284520 = r1284515 / r1284519;
        double r1284521 = r1284497 * r1284497;
        double r1284522 = r1284517 / r1284521;
        double r1284523 = r1284518 / r1284522;
        double r1284524 = fma(r1284496, r1284520, r1284523);
        double r1284525 = r1284524 / r1284503;
        double r1284526 = r1284496 / r1284525;
        double r1284527 = r1284514 ? r1284526 : r1284512;
        double r1284528 = r1284495 ? r1284512 : r1284527;
        return r1284528;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

  1. Split input into 2 regimes

  2. if t < -7.451781871449174e-25 or 2.030695630825502e-34 < t

    1. Initial program 22.0

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

    2. Simplified12.1

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

    4. Applied *-un-lft-identity12.1

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

    5. Applied times-frac11.4

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

    6. Applied associate-*l*8.4

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

    8. Applied associate-*l/7.0

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

    9. Applied associate-*r/7.0

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

    10. Applied associate-*l/4.1

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

    11. Applied associate-*l/3.7

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

    12. Simplified3.5

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

    14. Applied add-cube-cbrt3.8

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

    15. Applied associate-/r*3.8

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

    if -7.451781871449174e-25 < t < 2.030695630825502e-34

    1. Initial program 52.1

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

    2. Simplified39.1

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

    4. Applied *-un-lft-identity39.1

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

    5. Applied times-frac38.3

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

    6. Applied associate-*l*37.2

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

    8. Applied associate-*l/37.8

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

    9. Applied associate-*r/37.8

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

    10. Applied associate-*l/38.7

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

    11. Applied associate-*l/35.0

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

    12. Simplified35.0

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

    13. Taylor expanded around -inf 22.8

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

    14. Simplified20.0

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

  4. Final simplification9.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -7.451781871449174 \cdot 10^{-25}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \left(\frac{\frac{t \cdot \sin k}{\sqrt[3]{\frac{\ell}{t}} \cdot \sqrt[3]{\frac{\ell}{t}}}}{\sqrt[3]{\frac{\ell}{t}}} \cdot \tan k\right)}{\frac{\ell}{t}}}\\ \mathbf{elif}\;t \le 2.030695630825502 \cdot 10^{-34}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(2, \frac{t \cdot t}{\frac{\ell \cdot \cos k}{\sin k \cdot \sin k}}, \frac{\sin k \cdot \sin k}{\frac{\ell \cdot \cos k}{k \cdot k}}\right)}{\frac{\ell}{t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \left(\frac{\frac{t \cdot \sin k}{\sqrt[3]{\frac{\ell}{t}} \cdot \sqrt[3]{\frac{\ell}{t}}}}{\sqrt[3]{\frac{\ell}{t}}} \cdot \tan k\right)}{\frac{\ell}{t}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019153 +o rules:numerics
(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))))