Average Error: 42.3 → 8.9
Time: 27.6s
Precision: 64
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -1.178101575424767 \cdot 10^{+99}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(2 \cdot \left(\frac{t}{\left(x \cdot x\right) \cdot \left(2 \cdot \sqrt{2}\right)} - \frac{\frac{t}{\sqrt{2}}}{x \cdot x}\right) - \frac{t}{\sqrt{2}} \cdot \frac{2}{x}\right) - \sqrt{2} \cdot t}\\ \mathbf{elif}\;t \le 1.4235245726759946 \cdot 10^{+99}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{t \cdot t}{x} \cdot 4 + \left(\frac{\ell}{\sqrt[3]{x}} \cdot \frac{\frac{\ell}{\sqrt[3]{x}}}{\sqrt[3]{x}} + t \cdot t\right) \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{t}{\sqrt{2}} \cdot \frac{2}{x} + \sqrt{2} \cdot t\right) - 2 \cdot \left(\frac{t}{\left(x \cdot x\right) \cdot \left(2 \cdot \sqrt{2}\right)} - \frac{\frac{t}{\sqrt{2}}}{x \cdot x}\right)}\\ \end{array}\]
\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 -1.178101575424767 \cdot 10^{+99}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\left(2 \cdot \left(\frac{t}{\left(x \cdot x\right) \cdot \left(2 \cdot \sqrt{2}\right)} - \frac{\frac{t}{\sqrt{2}}}{x \cdot x}\right) - \frac{t}{\sqrt{2}} \cdot \frac{2}{x}\right) - \sqrt{2} \cdot t}\\

\mathbf{elif}\;t \le 1.4235245726759946 \cdot 10^{+99}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{t \cdot t}{x} \cdot 4 + \left(\frac{\ell}{\sqrt[3]{x}} \cdot \frac{\frac{\ell}{\sqrt[3]{x}}}{\sqrt[3]{x}} + t \cdot t\right) \cdot 2}}\\

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

\end{array}
double f(double x, double l, double t) {
        double r1576369 = 2.0;
        double r1576370 = sqrt(r1576369);
        double r1576371 = t;
        double r1576372 = r1576370 * r1576371;
        double r1576373 = x;
        double r1576374 = 1.0;
        double r1576375 = r1576373 + r1576374;
        double r1576376 = r1576373 - r1576374;
        double r1576377 = r1576375 / r1576376;
        double r1576378 = l;
        double r1576379 = r1576378 * r1576378;
        double r1576380 = r1576371 * r1576371;
        double r1576381 = r1576369 * r1576380;
        double r1576382 = r1576379 + r1576381;
        double r1576383 = r1576377 * r1576382;
        double r1576384 = r1576383 - r1576379;
        double r1576385 = sqrt(r1576384);
        double r1576386 = r1576372 / r1576385;
        return r1576386;
}
↓
double f(double x, double l, double t) {
        double r1576387 = t;
        double r1576388 = -1.178101575424767e+99;
        bool r1576389 = r1576387 <= r1576388;
        double r1576390 = 2.0;
        double r1576391 = sqrt(r1576390);
        double r1576392 = r1576391 * r1576387;
        double r1576393 = x;
        double r1576394 = r1576393 * r1576393;
        double r1576395 = r1576390 * r1576391;
        double r1576396 = r1576394 * r1576395;
        double r1576397 = r1576387 / r1576396;
        double r1576398 = r1576387 / r1576391;
        double r1576399 = r1576398 / r1576394;
        double r1576400 = r1576397 - r1576399;
        double r1576401 = r1576390 * r1576400;
        double r1576402 = r1576390 / r1576393;
        double r1576403 = r1576398 * r1576402;
        double r1576404 = r1576401 - r1576403;
        double r1576405 = r1576404 - r1576392;
        double r1576406 = r1576392 / r1576405;
        double r1576407 = 1.4235245726759946e+99;
        bool r1576408 = r1576387 <= r1576407;
        double r1576409 = r1576387 * r1576387;
        double r1576410 = r1576409 / r1576393;
        double r1576411 = 4.0;
        double r1576412 = r1576410 * r1576411;
        double r1576413 = l;
        double r1576414 = cbrt(r1576393);
        double r1576415 = r1576413 / r1576414;
        double r1576416 = r1576415 / r1576414;
        double r1576417 = r1576415 * r1576416;
        double r1576418 = r1576417 + r1576409;
        double r1576419 = r1576418 * r1576390;
        double r1576420 = r1576412 + r1576419;
        double r1576421 = sqrt(r1576420);
        double r1576422 = r1576392 / r1576421;
        double r1576423 = r1576403 + r1576392;
        double r1576424 = r1576423 - r1576401;
        double r1576425 = r1576392 / r1576424;
        double r1576426 = r1576408 ? r1576422 : r1576425;
        double r1576427 = r1576389 ? r1576406 : r1576426;
        return r1576427;
}

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 < -1.178101575424767e+99

    1. Initial program 48.9

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

    2. Taylor expanded around -inf 2.5

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

    3. Simplified2.5

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

    if -1.178101575424767e+99 < t < 1.4235245726759946e+99

    1. Initial program 37.8

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

    2. Taylor expanded around inf 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-frac12.9

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

    7. Simplified12.9

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

    if 1.4235245726759946e+99 < t

    1. Initial program 49.1

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

    2. Taylor expanded around inf 3.3

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

    3. Simplified3.3

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

  4. Final simplification8.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.178101575424767 \cdot 10^{+99}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(2 \cdot \left(\frac{t}{\left(x \cdot x\right) \cdot \left(2 \cdot \sqrt{2}\right)} - \frac{\frac{t}{\sqrt{2}}}{x \cdot x}\right) - \frac{t}{\sqrt{2}} \cdot \frac{2}{x}\right) - \sqrt{2} \cdot t}\\ \mathbf{elif}\;t \le 1.4235245726759946 \cdot 10^{+99}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{t \cdot t}{x} \cdot 4 + \left(\frac{\ell}{\sqrt[3]{x}} \cdot \frac{\frac{\ell}{\sqrt[3]{x}}}{\sqrt[3]{x}} + t \cdot t\right) \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{t}{\sqrt{2}} \cdot \frac{2}{x} + \sqrt{2} \cdot t\right) - 2 \cdot \left(\frac{t}{\left(x \cdot x\right) \cdot \left(2 \cdot \sqrt{2}\right)} - \frac{\frac{t}{\sqrt{2}}}{x \cdot x}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 
(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)))))