Average Error: 42.8 → 9.1
Time: 27.7s
Precision: 64
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -6.377624199137871 \cdot 10^{+80}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{-\left(2 \cdot \frac{t}{x \cdot \sqrt{2}} + \sqrt{2} \cdot t\right)}\\ \mathbf{elif}\;t \le 3.3304656260583313 \cdot 10^{+124}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{t \cdot t}{x} \cdot 4 + 2 \cdot \left(\frac{\ell}{\sqrt[3]{x}} \cdot \frac{\frac{\ell}{\sqrt[3]{x}}}{\sqrt[3]{x}} + t \cdot t\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{2} \cdot t + \frac{2}{x} \cdot \frac{t}{\sqrt{2}}}\\ \end{array}\]
\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 -6.377624199137871 \cdot 10^{+80}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{-\left(2 \cdot \frac{t}{x \cdot \sqrt{2}} + \sqrt{2} \cdot t\right)}\\

\mathbf{elif}\;t \le 3.3304656260583313 \cdot 10^{+124}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{t \cdot t}{x} \cdot 4 + 2 \cdot \left(\frac{\ell}{\sqrt[3]{x}} \cdot \frac{\frac{\ell}{\sqrt[3]{x}}}{\sqrt[3]{x}} + t \cdot t\right)}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{2} \cdot t + \frac{2}{x} \cdot \frac{t}{\sqrt{2}}}\\

\end{array}
double f(double x, double l, double t) {
        double r628287 = 2.0;
        double r628288 = sqrt(r628287);
        double r628289 = t;
        double r628290 = r628288 * r628289;
        double r628291 = x;
        double r628292 = 1.0;
        double r628293 = r628291 + r628292;
        double r628294 = r628291 - r628292;
        double r628295 = r628293 / r628294;
        double r628296 = l;
        double r628297 = r628296 * r628296;
        double r628298 = r628289 * r628289;
        double r628299 = r628287 * r628298;
        double r628300 = r628297 + r628299;
        double r628301 = r628295 * r628300;
        double r628302 = r628301 - r628297;
        double r628303 = sqrt(r628302);
        double r628304 = r628290 / r628303;
        return r628304;
}
↓
double f(double x, double l, double t) {
        double r628305 = t;
        double r628306 = -6.377624199137871e+80;
        bool r628307 = r628305 <= r628306;
        double r628308 = 2.0;
        double r628309 = sqrt(r628308);
        double r628310 = r628309 * r628305;
        double r628311 = x;
        double r628312 = r628311 * r628309;
        double r628313 = r628305 / r628312;
        double r628314 = r628308 * r628313;
        double r628315 = r628314 + r628310;
        double r628316 = -r628315;
        double r628317 = r628310 / r628316;
        double r628318 = 3.3304656260583313e+124;
        bool r628319 = r628305 <= r628318;
        double r628320 = r628305 * r628305;
        double r628321 = r628320 / r628311;
        double r628322 = 4.0;
        double r628323 = r628321 * r628322;
        double r628324 = l;
        double r628325 = cbrt(r628311);
        double r628326 = r628324 / r628325;
        double r628327 = r628326 / r628325;
        double r628328 = r628326 * r628327;
        double r628329 = r628328 + r628320;
        double r628330 = r628308 * r628329;
        double r628331 = r628323 + r628330;
        double r628332 = sqrt(r628331);
        double r628333 = r628310 / r628332;
        double r628334 = r628308 / r628311;
        double r628335 = r628305 / r628309;
        double r628336 = r628334 * r628335;
        double r628337 = r628310 + r628336;
        double r628338 = r628310 / r628337;
        double r628339 = r628319 ? r628333 : r628338;
        double r628340 = r628307 ? r628317 : r628339;
        return r628340;
}

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 < -6.377624199137871e+80

    1. Initial program 48.1

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

    2. Taylor expanded around inf 47.9

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

    3. Simplified47.9

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

    5. Applied *-un-lft-identity47.9

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

    6. Applied times-frac45.9

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

    7. Simplified45.9

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

    8. Taylor expanded around -inf 3.0

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

    if -6.377624199137871e+80 < t < 3.3304656260583313e+124

    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 17.3

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

    3. Simplified17.3

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

    5. Applied add-cube-cbrt17.4

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

    6. Applied times-frac13.2

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

    7. Simplified13.2

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

    if 3.3304656260583313e+124 < t

    1. Initial program 54.3

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

    2. Taylor expanded around inf 54.2

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

    3. Simplified54.2

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

    5. Applied add-cube-cbrt54.2

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

    6. Applied times-frac53.2

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

    7. Simplified53.2

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

    8. Taylor expanded around inf 2.5

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{t \cdot \sqrt{2} + 2 \cdot \frac{t}{\sqrt{2} \cdot x}}}\]

    9. Simplified2.5

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

  4. Final simplification9.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -6.377624199137871 \cdot 10^{+80}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{-\left(2 \cdot \frac{t}{x \cdot \sqrt{2}} + \sqrt{2} \cdot t\right)}\\ \mathbf{elif}\;t \le 3.3304656260583313 \cdot 10^{+124}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{t \cdot t}{x} \cdot 4 + 2 \cdot \left(\frac{\ell}{\sqrt[3]{x}} \cdot \frac{\frac{\ell}{\sqrt[3]{x}}}{\sqrt[3]{x}} + t \cdot t\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{2} \cdot t + \frac{2}{x} \cdot \frac{t}{\sqrt{2}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019152 
(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)))))