Average Error: 42.8 → 9.3
Time: 27.6s
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 r1517386 = 2.0;
        double r1517387 = sqrt(r1517386);
        double r1517388 = t;
        double r1517389 = r1517387 * r1517388;
        double r1517390 = x;
        double r1517391 = 1.0;
        double r1517392 = r1517390 + r1517391;
        double r1517393 = r1517390 - r1517391;
        double r1517394 = r1517392 / r1517393;
        double r1517395 = l;
        double r1517396 = r1517395 * r1517395;
        double r1517397 = r1517388 * r1517388;
        double r1517398 = r1517386 * r1517397;
        double r1517399 = r1517396 + r1517398;
        double r1517400 = r1517394 * r1517399;
        double r1517401 = r1517400 - r1517396;
        double r1517402 = sqrt(r1517401);
        double r1517403 = r1517389 / r1517402;
        return r1517403;
}
↓
double f(double x, double l, double t) {
        double r1517404 = t;
        double r1517405 = -6.172764092322424e+59;
        bool r1517406 = r1517404 <= r1517405;
        double r1517407 = 2.0;
        double r1517408 = sqrt(r1517407);
        double r1517409 = r1517408 * r1517404;
        double r1517410 = x;
        double r1517411 = r1517407 * r1517408;
        double r1517412 = r1517410 * r1517411;
        double r1517413 = r1517407 / r1517412;
        double r1517414 = r1517404 / r1517410;
        double r1517415 = r1517413 * r1517414;
        double r1517416 = r1517407 / r1517410;
        double r1517417 = r1517404 / r1517408;
        double r1517418 = r1517416 * r1517417;
        double r1517419 = r1517409 + r1517418;
        double r1517420 = r1517415 - r1517419;
        double r1517421 = r1517417 / r1517410;
        double r1517422 = r1517421 * r1517416;
        double r1517423 = r1517420 - r1517422;
        double r1517424 = r1517409 / r1517423;
        double r1517425 = 1.6237106520617296e+60;
        bool r1517426 = r1517404 <= r1517425;
        double r1517427 = r1517404 * r1517404;
        double r1517428 = r1517427 / r1517410;
        double r1517429 = 4.0;
        double r1517430 = r1517428 * r1517429;
        double r1517431 = l;
        double r1517432 = r1517431 / r1517410;
        double r1517433 = r1517431 * r1517432;
        double r1517434 = r1517433 + r1517427;
        double r1517435 = sqrt(r1517434);
        double r1517436 = r1517435 * r1517435;
        double r1517437 = r1517407 * r1517436;
        double r1517438 = r1517430 + r1517437;
        double r1517439 = sqrt(r1517438);
        double r1517440 = r1517409 / r1517439;
        double r1517441 = r1517422 - r1517415;
        double r1517442 = r1517441 + r1517419;
        double r1517443 = r1517409 / r1517442;
        double r1517444 = r1517426 ? r1517440 : r1517443;
        double r1517445 = r1517406 ? r1517424 : r1517444;
        return r1517445;
}

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)))))