Average Error: 42.8 → 9.1
Time: 34.1s
Precision: 64
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -6.377624199137871 \cdot 10^{+80}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\frac{t}{x \cdot \sqrt{2}} \cdot -2 - \sqrt{2} \cdot t}\\ \mathbf{elif}\;t \le 3.3304656260583313 \cdot 10^{+124}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{t \cdot t}{x} + \left(t \cdot t + \frac{\ell}{\sqrt[3]{x}} \cdot \frac{\frac{\ell}{\sqrt[3]{x}}}{\sqrt[3]{x}}\right) \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\frac{\frac{2}{x} \cdot t}{\sqrt{2}} + \sqrt{2} \cdot t}\\ \end{array}\]
\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}{\frac{t}{x \cdot \sqrt{2}} \cdot -2 - \sqrt{2} \cdot t}\\

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

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

\end{array}
double f(double x, double l, double t) {
        double r1471461 = 2.0;
        double r1471462 = sqrt(r1471461);
        double r1471463 = t;
        double r1471464 = r1471462 * r1471463;
        double r1471465 = x;
        double r1471466 = 1.0;
        double r1471467 = r1471465 + r1471466;
        double r1471468 = r1471465 - r1471466;
        double r1471469 = r1471467 / r1471468;
        double r1471470 = l;
        double r1471471 = r1471470 * r1471470;
        double r1471472 = r1471463 * r1471463;
        double r1471473 = r1471461 * r1471472;
        double r1471474 = r1471471 + r1471473;
        double r1471475 = r1471469 * r1471474;
        double r1471476 = r1471475 - r1471471;
        double r1471477 = sqrt(r1471476);
        double r1471478 = r1471464 / r1471477;
        return r1471478;
}
↓
double f(double x, double l, double t) {
        double r1471479 = t;
        double r1471480 = -6.377624199137871e+80;
        bool r1471481 = r1471479 <= r1471480;
        double r1471482 = 2.0;
        double r1471483 = sqrt(r1471482);
        double r1471484 = r1471483 * r1471479;
        double r1471485 = x;
        double r1471486 = r1471485 * r1471483;
        double r1471487 = r1471479 / r1471486;
        double r1471488 = -2.0;
        double r1471489 = r1471487 * r1471488;
        double r1471490 = r1471489 - r1471484;
        double r1471491 = r1471484 / r1471490;
        double r1471492 = 3.3304656260583313e+124;
        bool r1471493 = r1471479 <= r1471492;
        double r1471494 = 4.0;
        double r1471495 = r1471479 * r1471479;
        double r1471496 = r1471495 / r1471485;
        double r1471497 = r1471494 * r1471496;
        double r1471498 = l;
        double r1471499 = cbrt(r1471485);
        double r1471500 = r1471498 / r1471499;
        double r1471501 = r1471500 / r1471499;
        double r1471502 = r1471500 * r1471501;
        double r1471503 = r1471495 + r1471502;
        double r1471504 = r1471503 * r1471482;
        double r1471505 = r1471497 + r1471504;
        double r1471506 = sqrt(r1471505);
        double r1471507 = r1471484 / r1471506;
        double r1471508 = r1471482 / r1471485;
        double r1471509 = r1471508 * r1471479;
        double r1471510 = r1471509 / r1471483;
        double r1471511 = r1471510 + r1471484;
        double r1471512 = r1471484 / r1471511;
        double r1471513 = r1471493 ? r1471507 : r1471512;
        double r1471514 = r1471481 ? r1471491 : r1471513;
        return r1471514;
}

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. Taylor expanded around -inf 3.0

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

    5. Simplified3.0

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

    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. Taylor expanded around inf 2.5

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

    5. Simplified2.5

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

  4. Final simplification9.1

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

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)))))