Average Error: 42.8 → 9.3
Time: 26.8s
Precision: 64
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -6.172764092322424109949396121339735582564 \cdot 10^{59}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{2}{x \cdot \left(2 \cdot \sqrt{2}\right)} \cdot \frac{t}{x} - \left(\sqrt{2} \cdot t + \frac{2}{x} \cdot \frac{t}{\sqrt{2}}\right)\right) - \frac{\frac{t}{\sqrt{2}}}{x} \cdot \frac{2}{x}}\\ \mathbf{elif}\;t \le 1.623710652061729557401163312786003220273 \cdot 10^{60}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{t \cdot t}{x} \cdot 4 + 2 \cdot \left(\sqrt{\ell \cdot \frac{\ell}{x} + t \cdot t} \cdot \sqrt{\ell \cdot \frac{\ell}{x} + t \cdot t}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{\frac{t}{\sqrt{2}}}{x} \cdot \frac{2}{x} - \frac{2}{x \cdot \left(2 \cdot \sqrt{2}\right)} \cdot \frac{t}{x}\right) + \left(\sqrt{2} \cdot t + \frac{2}{x} \cdot \frac{t}{\sqrt{2}}\right)}\\ \end{array}\]
\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 -6.172764092322424109949396121339735582564 \cdot 10^{59}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{2}{x \cdot \left(2 \cdot \sqrt{2}\right)} \cdot \frac{t}{x} - \left(\sqrt{2} \cdot t + \frac{2}{x} \cdot \frac{t}{\sqrt{2}}\right)\right) - \frac{\frac{t}{\sqrt{2}}}{x} \cdot \frac{2}{x}}\\

\mathbf{elif}\;t \le 1.623710652061729557401163312786003220273 \cdot 10^{60}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{t \cdot t}{x} \cdot 4 + 2 \cdot \left(\sqrt{\ell \cdot \frac{\ell}{x} + t \cdot t} \cdot \sqrt{\ell \cdot \frac{\ell}{x} + t \cdot t}\right)}}\\

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

\end{array}
double f(double x, double l, double t) {
        double r1517388 = 2.0;
        double r1517389 = sqrt(r1517388);
        double r1517390 = t;
        double r1517391 = r1517389 * r1517390;
        double r1517392 = x;
        double r1517393 = 1.0;
        double r1517394 = r1517392 + r1517393;
        double r1517395 = r1517392 - r1517393;
        double r1517396 = r1517394 / r1517395;
        double r1517397 = l;
        double r1517398 = r1517397 * r1517397;
        double r1517399 = r1517390 * r1517390;
        double r1517400 = r1517388 * r1517399;
        double r1517401 = r1517398 + r1517400;
        double r1517402 = r1517396 * r1517401;
        double r1517403 = r1517402 - r1517398;
        double r1517404 = sqrt(r1517403);
        double r1517405 = r1517391 / r1517404;
        return r1517405;
}
↓
double f(double x, double l, double t) {
        double r1517406 = t;
        double r1517407 = -6.172764092322424e+59;
        bool r1517408 = r1517406 <= r1517407;
        double r1517409 = 2.0;
        double r1517410 = sqrt(r1517409);
        double r1517411 = r1517410 * r1517406;
        double r1517412 = x;
        double r1517413 = r1517409 * r1517410;
        double r1517414 = r1517412 * r1517413;
        double r1517415 = r1517409 / r1517414;
        double r1517416 = r1517406 / r1517412;
        double r1517417 = r1517415 * r1517416;
        double r1517418 = r1517409 / r1517412;
        double r1517419 = r1517406 / r1517410;
        double r1517420 = r1517418 * r1517419;
        double r1517421 = r1517411 + r1517420;
        double r1517422 = r1517417 - r1517421;
        double r1517423 = r1517419 / r1517412;
        double r1517424 = r1517423 * r1517418;
        double r1517425 = r1517422 - r1517424;
        double r1517426 = r1517411 / r1517425;
        double r1517427 = 1.6237106520617296e+60;
        bool r1517428 = r1517406 <= r1517427;
        double r1517429 = r1517406 * r1517406;
        double r1517430 = r1517429 / r1517412;
        double r1517431 = 4.0;
        double r1517432 = r1517430 * r1517431;
        double r1517433 = l;
        double r1517434 = r1517433 / r1517412;
        double r1517435 = r1517433 * r1517434;
        double r1517436 = r1517435 + r1517429;
        double r1517437 = sqrt(r1517436);
        double r1517438 = r1517437 * r1517437;
        double r1517439 = r1517409 * r1517438;
        double r1517440 = r1517432 + r1517439;
        double r1517441 = sqrt(r1517440);
        double r1517442 = r1517411 / r1517441;
        double r1517443 = r1517424 - r1517417;
        double r1517444 = r1517443 + r1517421;
        double r1517445 = r1517411 / r1517444;
        double r1517446 = r1517428 ? r1517442 : r1517445;
        double r1517447 = r1517408 ? r1517426 : r1517446;
        return r1517447;
}

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 < -6.172764092322424e+59

    1. Initial program 45.1

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

    2. Taylor expanded around -inf 3.5

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

    3. Simplified3.5

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

    if -6.172764092322424e+59 < t < 1.6237106520617296e+60

    1. Initial program 40.3

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

    2. Taylor expanded around inf 17.9

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

    3. Simplified17.9

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

    5. Applied *-un-lft-identity17.9

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{t \cdot t}{x} + 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{1 \cdot x}} + t \cdot t\right)}}\]

    6. Applied times-frac14.5

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{t \cdot t}{x} + 2 \cdot \left(\color{blue}{\frac{\ell}{1} \cdot \frac{\ell}{x}} + t \cdot t\right)}}\]

    7. Simplified14.5

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

    9. Applied add-sqr-sqrt14.5

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

    if 1.6237106520617296e+60 < t

    1. Initial program 45.8

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

    2. Taylor expanded around inf 3.8

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

    3. Simplified3.8

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

  4. Final simplification9.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -6.172764092322424109949396121339735582564 \cdot 10^{59}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{2}{x \cdot \left(2 \cdot \sqrt{2}\right)} \cdot \frac{t}{x} - \left(\sqrt{2} \cdot t + \frac{2}{x} \cdot \frac{t}{\sqrt{2}}\right)\right) - \frac{\frac{t}{\sqrt{2}}}{x} \cdot \frac{2}{x}}\\ \mathbf{elif}\;t \le 1.623710652061729557401163312786003220273 \cdot 10^{60}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{t \cdot t}{x} \cdot 4 + 2 \cdot \left(\sqrt{\ell \cdot \frac{\ell}{x} + t \cdot t} \cdot \sqrt{\ell \cdot \frac{\ell}{x} + t \cdot t}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{\frac{t}{\sqrt{2}}}{x} \cdot \frac{2}{x} - \frac{2}{x \cdot \left(2 \cdot \sqrt{2}\right)} \cdot \frac{t}{x}\right) + \left(\sqrt{2} \cdot t + \frac{2}{x} \cdot \frac{t}{\sqrt{2}}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 
(FPCore (x l t)
  :name "Toniolo and Linder, Equation (7)"
  (/ (* (sqrt 2.0) t) (sqrt (- (* (/ (+ x 1.0) (- x 1.0)) (+ (* l l) (* 2.0 (* t t)))) (* l l)))))