Average Error: 43.3 → 9.4
Time: 9.4s
Precision: 64
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -3.09314035729689678 \cdot 10^{118}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(2 \cdot \left(\frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \frac{t}{\sqrt{2} \cdot {x}^{2}}\right) - \sqrt{2} \cdot t\right) - 2 \cdot \frac{t}{\sqrt{2} \cdot x}}\\ \mathbf{elif}\;t \le -7.0895915203531395 \cdot 10^{-211}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \frac{\frac{\left|\ell\right|}{\sqrt[3]{x}}}{\sqrt[3]{x}} \cdot \frac{\left|\ell\right|}{\sqrt[3]{x}}\right)}}\\ \mathbf{elif}\;t \le -9.1577416971198005 \cdot 10^{-244}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(2 \cdot \left(\frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \frac{t}{\sqrt{2} \cdot {x}^{2}}\right) - \sqrt{2} \cdot t\right) - 2 \cdot \frac{t}{\sqrt{2} \cdot x}}\\ \mathbf{elif}\;t \le 4.8080161767920681 \cdot 10^{61}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \frac{\frac{\left|\ell\right|}{\sqrt[3]{x}}}{\sqrt[3]{x}} \cdot \frac{\left|\ell\right|}{\sqrt[3]{x}}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(2 \cdot \frac{t}{\sqrt{2} \cdot x} + t \cdot \sqrt{2}\right) - 2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}}\\ \end{array}\]
\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 -3.09314035729689678 \cdot 10^{118}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\left(2 \cdot \left(\frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \frac{t}{\sqrt{2} \cdot {x}^{2}}\right) - \sqrt{2} \cdot t\right) - 2 \cdot \frac{t}{\sqrt{2} \cdot x}}\\

\mathbf{elif}\;t \le -7.0895915203531395 \cdot 10^{-211}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \frac{\frac{\left|\ell\right|}{\sqrt[3]{x}}}{\sqrt[3]{x}} \cdot \frac{\left|\ell\right|}{\sqrt[3]{x}}\right)}}\\

\mathbf{elif}\;t \le -9.1577416971198005 \cdot 10^{-244}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\left(2 \cdot \left(\frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \frac{t}{\sqrt{2} \cdot {x}^{2}}\right) - \sqrt{2} \cdot t\right) - 2 \cdot \frac{t}{\sqrt{2} \cdot x}}\\

\mathbf{elif}\;t \le 4.8080161767920681 \cdot 10^{61}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \frac{\frac{\left|\ell\right|}{\sqrt[3]{x}}}{\sqrt[3]{x}} \cdot \frac{\left|\ell\right|}{\sqrt[3]{x}}\right)}}\\

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

\end{array}
double f(double x, double l, double t) {
        double r38390 = 2.0;
        double r38391 = sqrt(r38390);
        double r38392 = t;
        double r38393 = r38391 * r38392;
        double r38394 = x;
        double r38395 = 1.0;
        double r38396 = r38394 + r38395;
        double r38397 = r38394 - r38395;
        double r38398 = r38396 / r38397;
        double r38399 = l;
        double r38400 = r38399 * r38399;
        double r38401 = r38392 * r38392;
        double r38402 = r38390 * r38401;
        double r38403 = r38400 + r38402;
        double r38404 = r38398 * r38403;
        double r38405 = r38404 - r38400;
        double r38406 = sqrt(r38405);
        double r38407 = r38393 / r38406;
        return r38407;
}
↓
double f(double x, double l, double t) {
        double r38408 = t;
        double r38409 = -3.093140357296897e+118;
        bool r38410 = r38408 <= r38409;
        double r38411 = 2.0;
        double r38412 = sqrt(r38411);
        double r38413 = r38412 * r38408;
        double r38414 = 3.0;
        double r38415 = pow(r38412, r38414);
        double r38416 = x;
        double r38417 = 2.0;
        double r38418 = pow(r38416, r38417);
        double r38419 = r38415 * r38418;
        double r38420 = r38408 / r38419;
        double r38421 = r38412 * r38418;
        double r38422 = r38408 / r38421;
        double r38423 = r38420 - r38422;
        double r38424 = r38411 * r38423;
        double r38425 = r38424 - r38413;
        double r38426 = r38412 * r38416;
        double r38427 = r38408 / r38426;
        double r38428 = r38411 * r38427;
        double r38429 = r38425 - r38428;
        double r38430 = r38413 / r38429;
        double r38431 = -7.0895915203531395e-211;
        bool r38432 = r38408 <= r38431;
        double r38433 = 4.0;
        double r38434 = pow(r38408, r38417);
        double r38435 = r38434 / r38416;
        double r38436 = r38433 * r38435;
        double r38437 = l;
        double r38438 = fabs(r38437);
        double r38439 = cbrt(r38416);
        double r38440 = r38438 / r38439;
        double r38441 = r38440 / r38439;
        double r38442 = r38441 * r38440;
        double r38443 = r38434 + r38442;
        double r38444 = r38411 * r38443;
        double r38445 = r38436 + r38444;
        double r38446 = sqrt(r38445);
        double r38447 = r38413 / r38446;
        double r38448 = -9.1577416971198e-244;
        bool r38449 = r38408 <= r38448;
        double r38450 = 4.808016176792068e+61;
        bool r38451 = r38408 <= r38450;
        double r38452 = r38408 * r38412;
        double r38453 = r38428 + r38452;
        double r38454 = r38411 * r38420;
        double r38455 = r38453 - r38454;
        double r38456 = r38413 / r38455;
        double r38457 = r38451 ? r38447 : r38456;
        double r38458 = r38449 ? r38430 : r38457;
        double r38459 = r38432 ? r38447 : r38458;
        double r38460 = r38410 ? r38430 : r38459;
        return r38460;
}

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 < -3.093140357296897e+118 or -7.0895915203531395e-211 < t < -9.1577416971198e-244

    1. Initial program 55.1

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

    2. Taylor expanded around -inf 6.3

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

    3. Simplified6.3

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

    if -3.093140357296897e+118 < t < -7.0895915203531395e-211 or -9.1577416971198e-244 < t < 4.808016176792068e+61

    1. Initial program 38.0

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

    2. Taylor expanded around inf 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. Simplified17.7

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

    5. Applied add-cube-cbrt17.8

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \frac{{\ell}^{2}}{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}\right)}}\]

    6. Applied add-sqr-sqrt17.8

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \frac{\color{blue}{\sqrt{{\ell}^{2}} \cdot \sqrt{{\ell}^{2}}}}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}\right)}}\]

    7. Applied times-frac17.8

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \color{blue}{\frac{\sqrt{{\ell}^{2}}}{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \frac{\sqrt{{\ell}^{2}}}{\sqrt[3]{x}}}\right)}}\]

    8. Simplified17.8

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \color{blue}{\frac{\frac{\left|\ell\right|}{\sqrt[3]{x}}}{\sqrt[3]{x}}} \cdot \frac{\sqrt{{\ell}^{2}}}{\sqrt[3]{x}}\right)}}\]

    9. Simplified12.9

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \frac{\frac{\left|\ell\right|}{\sqrt[3]{x}}}{\sqrt[3]{x}} \cdot \color{blue}{\frac{\left|\ell\right|}{\sqrt[3]{x}}}\right)}}\]

    if 4.808016176792068e+61 < t

    1. Initial program 45.9

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

    2. Taylor expanded around inf 44.8

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

    3. Simplified44.8

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

    4. Taylor expanded around inf 4.0

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

  4. Final simplification9.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -3.09314035729689678 \cdot 10^{118}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(2 \cdot \left(\frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \frac{t}{\sqrt{2} \cdot {x}^{2}}\right) - \sqrt{2} \cdot t\right) - 2 \cdot \frac{t}{\sqrt{2} \cdot x}}\\ \mathbf{elif}\;t \le -7.0895915203531395 \cdot 10^{-211}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \frac{\frac{\left|\ell\right|}{\sqrt[3]{x}}}{\sqrt[3]{x}} \cdot \frac{\left|\ell\right|}{\sqrt[3]{x}}\right)}}\\ \mathbf{elif}\;t \le -9.1577416971198005 \cdot 10^{-244}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(2 \cdot \left(\frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \frac{t}{\sqrt{2} \cdot {x}^{2}}\right) - \sqrt{2} \cdot t\right) - 2 \cdot \frac{t}{\sqrt{2} \cdot x}}\\ \mathbf{elif}\;t \le 4.8080161767920681 \cdot 10^{61}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \frac{\frac{\left|\ell\right|}{\sqrt[3]{x}}}{\sqrt[3]{x}} \cdot \frac{\left|\ell\right|}{\sqrt[3]{x}}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(2 \cdot \frac{t}{\sqrt{2} \cdot x} + t \cdot \sqrt{2}\right) - 2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020020 
(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)))))