Average Error: 42.9 → 12.9
Time: 13.4s
Precision: 64
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -6.9713348016436594 \cdot 10^{-265}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - 2 \cdot \left(\frac{t}{\sqrt{2} \cdot {x}^{2}} + \frac{t}{\sqrt{2} \cdot x}\right)\right) - \sqrt{2} \cdot t}\\ \mathbf{elif}\;t \le 5.6007075831503198 \cdot 10^{56}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot {t}^{2} + \left(2 \cdot \frac{{\ell}^{2}}{x} + 4 \cdot \frac{{t}^{2}}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(2 \cdot \left(\frac{t}{\sqrt{2} \cdot {x}^{2}} + \frac{t}{\sqrt{2} \cdot x}\right) + \sqrt{2} \cdot t\right) - 2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}}\\ \end{array}\]
\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 -6.9713348016436594 \cdot 10^{-265}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\left(2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - 2 \cdot \left(\frac{t}{\sqrt{2} \cdot {x}^{2}} + \frac{t}{\sqrt{2} \cdot x}\right)\right) - \sqrt{2} \cdot t}\\

\mathbf{elif}\;t \le 5.6007075831503198 \cdot 10^{56}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot {t}^{2} + \left(2 \cdot \frac{{\ell}^{2}}{x} + 4 \cdot \frac{{t}^{2}}{x}\right)}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\left(2 \cdot \left(\frac{t}{\sqrt{2} \cdot {x}^{2}} + \frac{t}{\sqrt{2} \cdot x}\right) + \sqrt{2} \cdot t\right) - 2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}}\\

\end{array}
double f(double x, double l, double t) {
        double r40378 = 2.0;
        double r40379 = sqrt(r40378);
        double r40380 = t;
        double r40381 = r40379 * r40380;
        double r40382 = x;
        double r40383 = 1.0;
        double r40384 = r40382 + r40383;
        double r40385 = r40382 - r40383;
        double r40386 = r40384 / r40385;
        double r40387 = l;
        double r40388 = r40387 * r40387;
        double r40389 = r40380 * r40380;
        double r40390 = r40378 * r40389;
        double r40391 = r40388 + r40390;
        double r40392 = r40386 * r40391;
        double r40393 = r40392 - r40388;
        double r40394 = sqrt(r40393);
        double r40395 = r40381 / r40394;
        return r40395;
}
↓
double f(double x, double l, double t) {
        double r40396 = t;
        double r40397 = -6.971334801643659e-265;
        bool r40398 = r40396 <= r40397;
        double r40399 = 2.0;
        double r40400 = sqrt(r40399);
        double r40401 = r40400 * r40396;
        double r40402 = 3.0;
        double r40403 = pow(r40400, r40402);
        double r40404 = x;
        double r40405 = 2.0;
        double r40406 = pow(r40404, r40405);
        double r40407 = r40403 * r40406;
        double r40408 = r40396 / r40407;
        double r40409 = r40399 * r40408;
        double r40410 = r40400 * r40406;
        double r40411 = r40396 / r40410;
        double r40412 = r40400 * r40404;
        double r40413 = r40396 / r40412;
        double r40414 = r40411 + r40413;
        double r40415 = r40399 * r40414;
        double r40416 = r40409 - r40415;
        double r40417 = r40416 - r40401;
        double r40418 = r40401 / r40417;
        double r40419 = 5.60070758315032e+56;
        bool r40420 = r40396 <= r40419;
        double r40421 = pow(r40396, r40405);
        double r40422 = r40399 * r40421;
        double r40423 = l;
        double r40424 = pow(r40423, r40405);
        double r40425 = r40424 / r40404;
        double r40426 = r40399 * r40425;
        double r40427 = 4.0;
        double r40428 = r40421 / r40404;
        double r40429 = r40427 * r40428;
        double r40430 = r40426 + r40429;
        double r40431 = r40422 + r40430;
        double r40432 = sqrt(r40431);
        double r40433 = r40401 / r40432;
        double r40434 = r40415 + r40401;
        double r40435 = r40434 - r40409;
        double r40436 = r40401 / r40435;
        double r40437 = r40420 ? r40433 : r40436;
        double r40438 = r40398 ? r40418 : r40437;
        return r40438;
}

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 < -6.971334801643659e-265

    1. Initial program 41.5

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

    2. Taylor expanded around -inf 14.0

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

    3. Simplified14.0

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

    if -6.971334801643659e-265 < t < 5.60070758315032e+56

    1. Initial program 42.7

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

    2. Taylor expanded around inf 19.0

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

    if 5.60070758315032e+56 < t

    1. Initial program 46.0

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

    2. Taylor expanded around inf 3.6

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

    3. Simplified3.6

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

  4. Final simplification12.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -6.9713348016436594 \cdot 10^{-265}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - 2 \cdot \left(\frac{t}{\sqrt{2} \cdot {x}^{2}} + \frac{t}{\sqrt{2} \cdot x}\right)\right) - \sqrt{2} \cdot t}\\ \mathbf{elif}\;t \le 5.6007075831503198 \cdot 10^{56}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot {t}^{2} + \left(2 \cdot \frac{{\ell}^{2}}{x} + 4 \cdot \frac{{t}^{2}}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(2 \cdot \left(\frac{t}{\sqrt{2} \cdot {x}^{2}} + \frac{t}{\sqrt{2} \cdot x}\right) + \sqrt{2} \cdot t\right) - 2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 
(FPCore (x l t)
  :name "Toniolo and Linder, Equation (7)"
  :precision binary64
  (/ (* (sqrt 2) t) (sqrt (- (* (/ (+ x 1) (- x 1)) (+ (* l l) (* 2 (* t t)))) (* l l)))))