Average Error: 42.6 → 9.4
Time: 24.1s
Precision: 64
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -2.723485209640235 \cdot 10^{+72}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \left(\frac{\frac{t}{2 \cdot \sqrt{2}}}{x \cdot x} - \frac{\frac{t}{\sqrt{2}}}{x \cdot x}\right) - \left(\sqrt{2} \cdot t + \frac{2 \cdot t}{\sqrt{2} \cdot x}\right)}\\ \mathbf{elif}\;t \le -5.64615525414727 \cdot 10^{-160}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \left(\frac{\ell}{\frac{x}{\ell}} + t \cdot t\right) + 4 \cdot \frac{t \cdot t}{x}}}\\ \mathbf{elif}\;t \le -5.330429831411802 \cdot 10^{-184}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \left(\frac{\frac{t}{2 \cdot \sqrt{2}}}{x \cdot x} - \frac{\frac{t}{\sqrt{2}}}{x \cdot x}\right) - \left(\sqrt{2} \cdot t + \frac{2 \cdot t}{\sqrt{2} \cdot x}\right)}\\ \mathbf{elif}\;t \le 9.234874397993673 \cdot 10^{+103}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \left(\frac{\ell}{\frac{x}{\ell}} + t \cdot t\right) + 4 \cdot \frac{t \cdot t}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\sqrt{2} \cdot t + \frac{\frac{t}{\sqrt{2}}}{x \cdot x} \cdot 2\right) + \left(\frac{2 \cdot t}{\sqrt{2} \cdot x} - \frac{2}{x \cdot x} \cdot \frac{t}{2 \cdot \sqrt{2}}\right)}\\ \end{array}\]
\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 -2.723485209640235 \cdot 10^{+72}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \left(\frac{\frac{t}{2 \cdot \sqrt{2}}}{x \cdot x} - \frac{\frac{t}{\sqrt{2}}}{x \cdot x}\right) - \left(\sqrt{2} \cdot t + \frac{2 \cdot t}{\sqrt{2} \cdot x}\right)}\\

\mathbf{elif}\;t \le -5.64615525414727 \cdot 10^{-160}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \left(\frac{\ell}{\frac{x}{\ell}} + t \cdot t\right) + 4 \cdot \frac{t \cdot t}{x}}}\\

\mathbf{elif}\;t \le -5.330429831411802 \cdot 10^{-184}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \left(\frac{\frac{t}{2 \cdot \sqrt{2}}}{x \cdot x} - \frac{\frac{t}{\sqrt{2}}}{x \cdot x}\right) - \left(\sqrt{2} \cdot t + \frac{2 \cdot t}{\sqrt{2} \cdot x}\right)}\\

\mathbf{elif}\;t \le 9.234874397993673 \cdot 10^{+103}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \left(\frac{\ell}{\frac{x}{\ell}} + t \cdot t\right) + 4 \cdot \frac{t \cdot t}{x}}}\\

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

\end{array}
double f(double x, double l, double t) {
        double r577461 = 2.0;
        double r577462 = sqrt(r577461);
        double r577463 = t;
        double r577464 = r577462 * r577463;
        double r577465 = x;
        double r577466 = 1.0;
        double r577467 = r577465 + r577466;
        double r577468 = r577465 - r577466;
        double r577469 = r577467 / r577468;
        double r577470 = l;
        double r577471 = r577470 * r577470;
        double r577472 = r577463 * r577463;
        double r577473 = r577461 * r577472;
        double r577474 = r577471 + r577473;
        double r577475 = r577469 * r577474;
        double r577476 = r577475 - r577471;
        double r577477 = sqrt(r577476);
        double r577478 = r577464 / r577477;
        return r577478;
}
↓
double f(double x, double l, double t) {
        double r577479 = t;
        double r577480 = -2.723485209640235e+72;
        bool r577481 = r577479 <= r577480;
        double r577482 = 2.0;
        double r577483 = sqrt(r577482);
        double r577484 = r577483 * r577479;
        double r577485 = r577482 * r577483;
        double r577486 = r577479 / r577485;
        double r577487 = x;
        double r577488 = r577487 * r577487;
        double r577489 = r577486 / r577488;
        double r577490 = r577479 / r577483;
        double r577491 = r577490 / r577488;
        double r577492 = r577489 - r577491;
        double r577493 = r577482 * r577492;
        double r577494 = r577482 * r577479;
        double r577495 = r577483 * r577487;
        double r577496 = r577494 / r577495;
        double r577497 = r577484 + r577496;
        double r577498 = r577493 - r577497;
        double r577499 = r577484 / r577498;
        double r577500 = -5.64615525414727e-160;
        bool r577501 = r577479 <= r577500;
        double r577502 = l;
        double r577503 = r577487 / r577502;
        double r577504 = r577502 / r577503;
        double r577505 = r577479 * r577479;
        double r577506 = r577504 + r577505;
        double r577507 = r577482 * r577506;
        double r577508 = 4.0;
        double r577509 = r577505 / r577487;
        double r577510 = r577508 * r577509;
        double r577511 = r577507 + r577510;
        double r577512 = sqrt(r577511);
        double r577513 = r577484 / r577512;
        double r577514 = -5.330429831411802e-184;
        bool r577515 = r577479 <= r577514;
        double r577516 = 9.234874397993673e+103;
        bool r577517 = r577479 <= r577516;
        double r577518 = r577491 * r577482;
        double r577519 = r577484 + r577518;
        double r577520 = r577482 / r577488;
        double r577521 = r577520 * r577486;
        double r577522 = r577496 - r577521;
        double r577523 = r577519 + r577522;
        double r577524 = r577484 / r577523;
        double r577525 = r577517 ? r577513 : r577524;
        double r577526 = r577515 ? r577499 : r577525;
        double r577527 = r577501 ? r577513 : r577526;
        double r577528 = r577481 ? r577499 : r577527;
        return r577528;
}

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 < -2.723485209640235e+72 or -5.64615525414727e-160 < t < -5.330429831411802e-184

    1. Initial program 48.3

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

    2. Taylor expanded around -inf 5.2

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

    3. Simplified5.3

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

    if -2.723485209640235e+72 < t < -5.64615525414727e-160 or -5.330429831411802e-184 < t < 9.234874397993673e+103

    1. Initial program 37.4

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

    2. Taylor expanded around inf 17.5

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

    3. Simplified17.5

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

    5. Applied associate-/l*13.4

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

    if 9.234874397993673e+103 < t

    1. Initial program 50.2

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

    2. Taylor expanded around inf 3.0

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

    3. Simplified3.0

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

  4. Final simplification9.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -2.723485209640235 \cdot 10^{+72}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \left(\frac{\frac{t}{2 \cdot \sqrt{2}}}{x \cdot x} - \frac{\frac{t}{\sqrt{2}}}{x \cdot x}\right) - \left(\sqrt{2} \cdot t + \frac{2 \cdot t}{\sqrt{2} \cdot x}\right)}\\ \mathbf{elif}\;t \le -5.64615525414727 \cdot 10^{-160}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \left(\frac{\ell}{\frac{x}{\ell}} + t \cdot t\right) + 4 \cdot \frac{t \cdot t}{x}}}\\ \mathbf{elif}\;t \le -5.330429831411802 \cdot 10^{-184}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \left(\frac{\frac{t}{2 \cdot \sqrt{2}}}{x \cdot x} - \frac{\frac{t}{\sqrt{2}}}{x \cdot x}\right) - \left(\sqrt{2} \cdot t + \frac{2 \cdot t}{\sqrt{2} \cdot x}\right)}\\ \mathbf{elif}\;t \le 9.234874397993673 \cdot 10^{+103}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \left(\frac{\ell}{\frac{x}{\ell}} + t \cdot t\right) + 4 \cdot \frac{t \cdot t}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\sqrt{2} \cdot t + \frac{\frac{t}{\sqrt{2}}}{x \cdot x} \cdot 2\right) + \left(\frac{2 \cdot t}{\sqrt{2} \cdot x} - \frac{2}{x \cdot x} \cdot \frac{t}{2 \cdot \sqrt{2}}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019156 
(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)))))