Average Error: 43.2 → 9.5
Time: 35.9s
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 -1.940005881918408859869096974602770049363 \cdot 10^{94}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{2 \cdot t}{\left(x \cdot x\right) \cdot \left(2 \cdot \sqrt{2}\right)} - \left(\sqrt{2} \cdot t + \frac{2 \cdot t}{\sqrt{2} \cdot x}\right)\right) - \frac{\frac{2 \cdot t}{\sqrt{2}}}{x \cdot x}}\\ \mathbf{elif}\;t \le 1.839658841198905283015739574372348749582 \cdot 10^{-241}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\sqrt{\sqrt[3]{2 \cdot \left(\frac{\ell}{\frac{x}{\ell}} + t \cdot t\right) + \frac{4}{\frac{x}{t \cdot t}}} \cdot \sqrt[3]{2 \cdot \left(\frac{\ell}{\frac{x}{\ell}} + t \cdot t\right) + \frac{4}{\frac{x}{t \cdot t}}}} \cdot \sqrt{\sqrt[3]{2 \cdot \left(\frac{\ell}{\frac{x}{\ell}} + t \cdot t\right) + \frac{4}{\frac{x}{t \cdot t}}}}} \cdot \sqrt{\sqrt{2 \cdot \left(\frac{\ell}{\frac{x}{\ell}} + t \cdot t\right) + \frac{4}{\frac{x}{t \cdot t}}}}}\\ \mathbf{elif}\;t \le 3.247980644380315344262592988201111260205 \cdot 10^{-157}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\frac{-2 \cdot t}{\left(x \cdot x\right) \cdot \left(2 \cdot \sqrt{2}\right)} + \left(\frac{2 \cdot t}{\sqrt{2} \cdot x} + \left(\frac{\frac{2 \cdot t}{\sqrt{2}}}{x \cdot x} + \sqrt{2} \cdot t\right)\right)}\\ \mathbf{elif}\;t \le 4.214549585149478093549971775733778272653 \cdot 10^{55}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\sqrt{\sqrt[3]{2 \cdot \left(\frac{\ell}{\frac{x}{\ell}} + t \cdot t\right) + \frac{4}{\frac{x}{t \cdot t}}} \cdot \sqrt[3]{2 \cdot \left(\frac{\ell}{\frac{x}{\ell}} + t \cdot t\right) + \frac{4}{\frac{x}{t \cdot t}}}} \cdot \sqrt{\sqrt[3]{2 \cdot \left(\frac{\ell}{\frac{x}{\ell}} + t \cdot t\right) + \frac{4}{\frac{x}{t \cdot t}}}}} \cdot \sqrt{\sqrt{2 \cdot \left(\frac{\ell}{\frac{x}{\ell}} + t \cdot t\right) + \frac{4}{\frac{x}{t \cdot t}}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\frac{-2 \cdot t}{\left(x \cdot x\right) \cdot \left(2 \cdot \sqrt{2}\right)} + \left(\frac{2 \cdot t}{\sqrt{2} \cdot x} + \left(\frac{\frac{2 \cdot t}{\sqrt{2}}}{x \cdot x} + \sqrt{2} \cdot t\right)\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 -1.940005881918408859869096974602770049363 \cdot 10^{94}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{2 \cdot t}{\left(x \cdot x\right) \cdot \left(2 \cdot \sqrt{2}\right)} - \left(\sqrt{2} \cdot t + \frac{2 \cdot t}{\sqrt{2} \cdot x}\right)\right) - \frac{\frac{2 \cdot t}{\sqrt{2}}}{x \cdot x}}\\

\mathbf{elif}\;t \le 1.839658841198905283015739574372348749582 \cdot 10^{-241}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\sqrt{\sqrt[3]{2 \cdot \left(\frac{\ell}{\frac{x}{\ell}} + t \cdot t\right) + \frac{4}{\frac{x}{t \cdot t}}} \cdot \sqrt[3]{2 \cdot \left(\frac{\ell}{\frac{x}{\ell}} + t \cdot t\right) + \frac{4}{\frac{x}{t \cdot t}}}} \cdot \sqrt{\sqrt[3]{2 \cdot \left(\frac{\ell}{\frac{x}{\ell}} + t \cdot t\right) + \frac{4}{\frac{x}{t \cdot t}}}}} \cdot \sqrt{\sqrt{2 \cdot \left(\frac{\ell}{\frac{x}{\ell}} + t \cdot t\right) + \frac{4}{\frac{x}{t \cdot t}}}}}\\

\mathbf{elif}\;t \le 3.247980644380315344262592988201111260205 \cdot 10^{-157}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\frac{-2 \cdot t}{\left(x \cdot x\right) \cdot \left(2 \cdot \sqrt{2}\right)} + \left(\frac{2 \cdot t}{\sqrt{2} \cdot x} + \left(\frac{\frac{2 \cdot t}{\sqrt{2}}}{x \cdot x} + \sqrt{2} \cdot t\right)\right)}\\

\mathbf{elif}\;t \le 4.214549585149478093549971775733778272653 \cdot 10^{55}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\sqrt{\sqrt[3]{2 \cdot \left(\frac{\ell}{\frac{x}{\ell}} + t \cdot t\right) + \frac{4}{\frac{x}{t \cdot t}}} \cdot \sqrt[3]{2 \cdot \left(\frac{\ell}{\frac{x}{\ell}} + t \cdot t\right) + \frac{4}{\frac{x}{t \cdot t}}}} \cdot \sqrt{\sqrt[3]{2 \cdot \left(\frac{\ell}{\frac{x}{\ell}} + t \cdot t\right) + \frac{4}{\frac{x}{t \cdot t}}}}} \cdot \sqrt{\sqrt{2 \cdot \left(\frac{\ell}{\frac{x}{\ell}} + t \cdot t\right) + \frac{4}{\frac{x}{t \cdot t}}}}}\\

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

\end{array}
double f(double x, double l, double t) {
        double r1517494 = 2.0;
        double r1517495 = sqrt(r1517494);
        double r1517496 = t;
        double r1517497 = r1517495 * r1517496;
        double r1517498 = x;
        double r1517499 = 1.0;
        double r1517500 = r1517498 + r1517499;
        double r1517501 = r1517498 - r1517499;
        double r1517502 = r1517500 / r1517501;
        double r1517503 = l;
        double r1517504 = r1517503 * r1517503;
        double r1517505 = r1517496 * r1517496;
        double r1517506 = r1517494 * r1517505;
        double r1517507 = r1517504 + r1517506;
        double r1517508 = r1517502 * r1517507;
        double r1517509 = r1517508 - r1517504;
        double r1517510 = sqrt(r1517509);
        double r1517511 = r1517497 / r1517510;
        return r1517511;
}
↓
double f(double x, double l, double t) {
        double r1517512 = t;
        double r1517513 = -1.940005881918409e+94;
        bool r1517514 = r1517512 <= r1517513;
        double r1517515 = 2.0;
        double r1517516 = sqrt(r1517515);
        double r1517517 = r1517516 * r1517512;
        double r1517518 = r1517515 * r1517512;
        double r1517519 = x;
        double r1517520 = r1517519 * r1517519;
        double r1517521 = r1517515 * r1517516;
        double r1517522 = r1517520 * r1517521;
        double r1517523 = r1517518 / r1517522;
        double r1517524 = r1517516 * r1517519;
        double r1517525 = r1517518 / r1517524;
        double r1517526 = r1517517 + r1517525;
        double r1517527 = r1517523 - r1517526;
        double r1517528 = r1517518 / r1517516;
        double r1517529 = r1517528 / r1517520;
        double r1517530 = r1517527 - r1517529;
        double r1517531 = r1517517 / r1517530;
        double r1517532 = 1.8396588411989053e-241;
        bool r1517533 = r1517512 <= r1517532;
        double r1517534 = l;
        double r1517535 = r1517519 / r1517534;
        double r1517536 = r1517534 / r1517535;
        double r1517537 = r1517512 * r1517512;
        double r1517538 = r1517536 + r1517537;
        double r1517539 = r1517515 * r1517538;
        double r1517540 = 4.0;
        double r1517541 = r1517519 / r1517537;
        double r1517542 = r1517540 / r1517541;
        double r1517543 = r1517539 + r1517542;
        double r1517544 = cbrt(r1517543);
        double r1517545 = r1517544 * r1517544;
        double r1517546 = sqrt(r1517545);
        double r1517547 = sqrt(r1517544);
        double r1517548 = r1517546 * r1517547;
        double r1517549 = sqrt(r1517548);
        double r1517550 = sqrt(r1517543);
        double r1517551 = sqrt(r1517550);
        double r1517552 = r1517549 * r1517551;
        double r1517553 = r1517517 / r1517552;
        double r1517554 = 3.2479806443803153e-157;
        bool r1517555 = r1517512 <= r1517554;
        double r1517556 = -r1517518;
        double r1517557 = r1517556 / r1517522;
        double r1517558 = r1517529 + r1517517;
        double r1517559 = r1517525 + r1517558;
        double r1517560 = r1517557 + r1517559;
        double r1517561 = r1517517 / r1517560;
        double r1517562 = 4.214549585149478e+55;
        bool r1517563 = r1517512 <= r1517562;
        double r1517564 = r1517563 ? r1517553 : r1517561;
        double r1517565 = r1517555 ? r1517561 : r1517564;
        double r1517566 = r1517533 ? r1517553 : r1517565;
        double r1517567 = r1517514 ? r1517531 : r1517566;
        return r1517567;
}

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 < -1.940005881918409e+94

    1. Initial program 49.9

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

    2. Taylor expanded around -inf 3.3

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

    3. Simplified3.4

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

    if -1.940005881918409e+94 < t < 1.8396588411989053e-241 or 3.2479806443803153e-157 < t < 4.214549585149478e+55

    1. Initial program 37.0

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

    2. Taylor expanded around inf 15.8

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

    3. Simplified15.8

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

    5. Applied associate-/l*12.2

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

    7. Applied add-sqr-sqrt12.2

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

    8. Applied sqrt-prod12.3

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

    10. Applied add-cube-cbrt12.4

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

    11. Applied sqrt-prod12.4

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

    if 1.8396588411989053e-241 < t < 3.2479806443803153e-157 or 4.214549585149478e+55 < t

    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 8.8

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

    3. Simplified8.8

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

  4. Final simplification9.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.940005881918408859869096974602770049363 \cdot 10^{94}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{2 \cdot t}{\left(x \cdot x\right) \cdot \left(2 \cdot \sqrt{2}\right)} - \left(\sqrt{2} \cdot t + \frac{2 \cdot t}{\sqrt{2} \cdot x}\right)\right) - \frac{\frac{2 \cdot t}{\sqrt{2}}}{x \cdot x}}\\ \mathbf{elif}\;t \le 1.839658841198905283015739574372348749582 \cdot 10^{-241}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\sqrt{\sqrt[3]{2 \cdot \left(\frac{\ell}{\frac{x}{\ell}} + t \cdot t\right) + \frac{4}{\frac{x}{t \cdot t}}} \cdot \sqrt[3]{2 \cdot \left(\frac{\ell}{\frac{x}{\ell}} + t \cdot t\right) + \frac{4}{\frac{x}{t \cdot t}}}} \cdot \sqrt{\sqrt[3]{2 \cdot \left(\frac{\ell}{\frac{x}{\ell}} + t \cdot t\right) + \frac{4}{\frac{x}{t \cdot t}}}}} \cdot \sqrt{\sqrt{2 \cdot \left(\frac{\ell}{\frac{x}{\ell}} + t \cdot t\right) + \frac{4}{\frac{x}{t \cdot t}}}}}\\ \mathbf{elif}\;t \le 3.247980644380315344262592988201111260205 \cdot 10^{-157}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\frac{-2 \cdot t}{\left(x \cdot x\right) \cdot \left(2 \cdot \sqrt{2}\right)} + \left(\frac{2 \cdot t}{\sqrt{2} \cdot x} + \left(\frac{\frac{2 \cdot t}{\sqrt{2}}}{x \cdot x} + \sqrt{2} \cdot t\right)\right)}\\ \mathbf{elif}\;t \le 4.214549585149478093549971775733778272653 \cdot 10^{55}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\sqrt{\sqrt[3]{2 \cdot \left(\frac{\ell}{\frac{x}{\ell}} + t \cdot t\right) + \frac{4}{\frac{x}{t \cdot t}}} \cdot \sqrt[3]{2 \cdot \left(\frac{\ell}{\frac{x}{\ell}} + t \cdot t\right) + \frac{4}{\frac{x}{t \cdot t}}}} \cdot \sqrt{\sqrt[3]{2 \cdot \left(\frac{\ell}{\frac{x}{\ell}} + t \cdot t\right) + \frac{4}{\frac{x}{t \cdot t}}}}} \cdot \sqrt{\sqrt{2 \cdot \left(\frac{\ell}{\frac{x}{\ell}} + t \cdot t\right) + \frac{4}{\frac{x}{t \cdot t}}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\frac{-2 \cdot t}{\left(x \cdot x\right) \cdot \left(2 \cdot \sqrt{2}\right)} + \left(\frac{2 \cdot t}{\sqrt{2} \cdot x} + \left(\frac{\frac{2 \cdot t}{\sqrt{2}}}{x \cdot x} + \sqrt{2} \cdot t\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 
(FPCore (x l t)
  :name "Toniolo and Linder, Equation (7)"
  (/ (* (sqrt 2.0) t) (sqrt (- (* (/ (+ x 1.0) (- x 1.0)) (+ (* l l) (* 2.0 (* t t)))) (* l l)))))