Average Error: 42.0 → 9.6
Time: 1.2m
Precision: 64
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -1.9272954767798406 \cdot 10^{+17}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\frac{t}{x \cdot x} \cdot \frac{1}{\sqrt{2}} - \left(\frac{2}{\sqrt{2}} \cdot \left(\frac{t}{x} + \frac{t}{x \cdot x}\right) + \sqrt{2} \cdot t\right)}\\ \mathbf{elif}\;t \le 6.398010735699759 \cdot 10^{+134}:\\ \;\;\;\;\frac{\sqrt{\sqrt{2}} \cdot \left(\sqrt{\sqrt{2}} \cdot t\right)}{\sqrt{\left(2 \cdot \ell\right) \cdot \frac{\ell}{x} + \left(t \cdot t\right) \cdot \left(2 + \frac{4}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\sqrt{2} + \frac{2}{x \cdot \sqrt{2}}\right) \cdot t - \frac{\frac{\frac{t}{x}}{\sqrt{2}}}{x}}\\ \end{array}\]
\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}
↓
\begin{array}{l}
\mathbf{if}\;t \le -1.9272954767798406 \cdot 10^{+17}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\frac{t}{x \cdot x} \cdot \frac{1}{\sqrt{2}} - \left(\frac{2}{\sqrt{2}} \cdot \left(\frac{t}{x} + \frac{t}{x \cdot x}\right) + \sqrt{2} \cdot t\right)}\\

\mathbf{elif}\;t \le 6.398010735699759 \cdot 10^{+134}:\\
\;\;\;\;\frac{\sqrt{\sqrt{2}} \cdot \left(\sqrt{\sqrt{2}} \cdot t\right)}{\sqrt{\left(2 \cdot \ell\right) \cdot \frac{\ell}{x} + \left(t \cdot t\right) \cdot \left(2 + \frac{4}{x}\right)}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\sqrt{2} + \frac{2}{x \cdot \sqrt{2}}\right) \cdot t - \frac{\frac{\frac{t}{x}}{\sqrt{2}}}{x}}\\

\end{array}
double f(double x, double l, double t) {
        double r4239438 = 2.0;
        double r4239439 = sqrt(r4239438);
        double r4239440 = t;
        double r4239441 = r4239439 * r4239440;
        double r4239442 = x;
        double r4239443 = 1.0;
        double r4239444 = r4239442 + r4239443;
        double r4239445 = r4239442 - r4239443;
        double r4239446 = r4239444 / r4239445;
        double r4239447 = l;
        double r4239448 = r4239447 * r4239447;
        double r4239449 = r4239440 * r4239440;
        double r4239450 = r4239438 * r4239449;
        double r4239451 = r4239448 + r4239450;
        double r4239452 = r4239446 * r4239451;
        double r4239453 = r4239452 - r4239448;
        double r4239454 = sqrt(r4239453);
        double r4239455 = r4239441 / r4239454;
        return r4239455;
}
↓
double f(double x, double l, double t) {
        double r4239456 = t;
        double r4239457 = -1.9272954767798406e+17;
        bool r4239458 = r4239456 <= r4239457;
        double r4239459 = 2.0;
        double r4239460 = sqrt(r4239459);
        double r4239461 = r4239460 * r4239456;
        double r4239462 = x;
        double r4239463 = r4239462 * r4239462;
        double r4239464 = r4239456 / r4239463;
        double r4239465 = 1.0;
        double r4239466 = r4239465 / r4239460;
        double r4239467 = r4239464 * r4239466;
        double r4239468 = r4239459 / r4239460;
        double r4239469 = r4239456 / r4239462;
        double r4239470 = r4239469 + r4239464;
        double r4239471 = r4239468 * r4239470;
        double r4239472 = r4239471 + r4239461;
        double r4239473 = r4239467 - r4239472;
        double r4239474 = r4239461 / r4239473;
        double r4239475 = 6.398010735699759e+134;
        bool r4239476 = r4239456 <= r4239475;
        double r4239477 = sqrt(r4239460);
        double r4239478 = r4239477 * r4239456;
        double r4239479 = r4239477 * r4239478;
        double r4239480 = l;
        double r4239481 = r4239459 * r4239480;
        double r4239482 = r4239480 / r4239462;
        double r4239483 = r4239481 * r4239482;
        double r4239484 = r4239456 * r4239456;
        double r4239485 = 4.0;
        double r4239486 = r4239485 / r4239462;
        double r4239487 = r4239459 + r4239486;
        double r4239488 = r4239484 * r4239487;
        double r4239489 = r4239483 + r4239488;
        double r4239490 = sqrt(r4239489);
        double r4239491 = r4239479 / r4239490;
        double r4239492 = r4239462 * r4239460;
        double r4239493 = r4239459 / r4239492;
        double r4239494 = r4239460 + r4239493;
        double r4239495 = r4239494 * r4239456;
        double r4239496 = r4239469 / r4239460;
        double r4239497 = r4239496 / r4239462;
        double r4239498 = r4239495 - r4239497;
        double r4239499 = r4239461 / r4239498;
        double r4239500 = r4239476 ? r4239491 : r4239499;
        double r4239501 = r4239458 ? r4239474 : r4239500;
        return r4239501;
}

Error

Bits error versus x

Bits error versus l

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if t < -1.9272954767798406e+17

    1. Initial program 41.7

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]

    2. Taylor expanded around -inf 5.1

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \left(2 \cdot \frac{t}{\sqrt{2} \cdot {x}^{2}} + \left(t \cdot \sqrt{2} + 2 \cdot \frac{t}{\sqrt{2} \cdot x}\right)\right)}}\]

    3. Simplified5.1

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\frac{t}{x \cdot x} \cdot \frac{1}{\sqrt{2}} - \left(\sqrt{2} \cdot t + \frac{2}{\sqrt{2}} \cdot \left(\frac{t}{x} + \frac{t}{x \cdot x}\right)\right)}}\]

    if -1.9272954767798406e+17 < t < 6.398010735699759e+134

    1. Initial program 37.5

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]

    2. Taylor expanded around -inf 17.7

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{2 \cdot {t}^{2} + \left(2 \cdot \frac{{\ell}^{2}}{x} + 4 \cdot \frac{{t}^{2}}{x}\right)}}}\]

    3. Simplified13.9

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\left(t \cdot t\right) \cdot \left(\frac{4}{x} + 2\right) + \left(\ell \cdot 2\right) \cdot \frac{\ell}{x}}}}\]
    4. Using strategy rm

    5. Applied add-sqr-sqrt13.9

      \[\leadsto \frac{\sqrt{\color{blue}{\sqrt{2} \cdot \sqrt{2}}} \cdot t}{\sqrt{\left(t \cdot t\right) \cdot \left(\frac{4}{x} + 2\right) + \left(\ell \cdot 2\right) \cdot \frac{\ell}{x}}}\]

    6. Applied sqrt-prod14.0

      \[\leadsto \frac{\color{blue}{\left(\sqrt{\sqrt{2}} \cdot \sqrt{\sqrt{2}}\right)} \cdot t}{\sqrt{\left(t \cdot t\right) \cdot \left(\frac{4}{x} + 2\right) + \left(\ell \cdot 2\right) \cdot \frac{\ell}{x}}}\]

    7. Applied associate-*l*13.9

      \[\leadsto \frac{\color{blue}{\sqrt{\sqrt{2}} \cdot \left(\sqrt{\sqrt{2}} \cdot t\right)}}{\sqrt{\left(t \cdot t\right) \cdot \left(\frac{4}{x} + 2\right) + \left(\ell \cdot 2\right) \cdot \frac{\ell}{x}}}\]

    if 6.398010735699759e+134 < t

    1. Initial program 57.0

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]

    2. Taylor expanded around -inf 56.8

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{2 \cdot {t}^{2} + \left(2 \cdot \frac{{\ell}^{2}}{x} + 4 \cdot \frac{{t}^{2}}{x}\right)}}}\]

    3. Simplified56.0

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\left(t \cdot t\right) \cdot \left(\frac{4}{x} + 2\right) + \left(\ell \cdot 2\right) \cdot \frac{\ell}{x}}}}\]

    4. Taylor expanded around inf 2.3

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\left(t \cdot \sqrt{2} + 2 \cdot \frac{t}{\sqrt{2} \cdot x}\right) - 2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}}}\]

    5. Simplified2.3

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{t \cdot \left(\sqrt{2} + \frac{2}{x \cdot \sqrt{2}}\right) - \frac{\frac{\frac{t}{x}}{\sqrt{2}}}{x}}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification9.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.9272954767798406 \cdot 10^{+17}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\frac{t}{x \cdot x} \cdot \frac{1}{\sqrt{2}} - \left(\frac{2}{\sqrt{2}} \cdot \left(\frac{t}{x} + \frac{t}{x \cdot x}\right) + \sqrt{2} \cdot t\right)}\\ \mathbf{elif}\;t \le 6.398010735699759 \cdot 10^{+134}:\\ \;\;\;\;\frac{\sqrt{\sqrt{2}} \cdot \left(\sqrt{\sqrt{2}} \cdot t\right)}{\sqrt{\left(2 \cdot \ell\right) \cdot \frac{\ell}{x} + \left(t \cdot t\right) \cdot \left(2 + \frac{4}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\sqrt{2} + \frac{2}{x \cdot \sqrt{2}}\right) \cdot t - \frac{\frac{\frac{t}{x}}{\sqrt{2}}}{x}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019107 
(FPCore (x l t)
  :name "Toniolo and Linder, Equation (7)"
  (/ (* (sqrt 2) t) (sqrt (- (* (/ (+ x 1) (- x 1)) (+ (* l l) (* 2 (* t t)))) (* l l)))))