Average Error: 42.8 → 9.3
Time: 41.5s
Precision: 64
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -4.290053800826923246197267651223841654495 \cdot 10^{123}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{t}{\left(\sqrt{2} \cdot 2\right) \cdot \left(x \cdot x\right)} - \left(\frac{t}{\left(x \cdot \sqrt{2}\right) \cdot x} + \frac{\frac{t}{\sqrt{2}}}{x}\right)\right) \cdot 2 - \sqrt{2} \cdot t}\\ \mathbf{elif}\;t \le 2.818948520043459281423123944713297721074 \cdot 10^{72}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{4}{\frac{x}{t \cdot t}} + \left(\frac{\frac{\ell}{\sqrt[3]{x}}}{\sqrt[3]{\sqrt[3]{x}}} \cdot \frac{\frac{\ell}{\sqrt[3]{x}}}{\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}} + t \cdot t\right) \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{\frac{t}{\sqrt{2}} \cdot 2}{x \cdot x} + \sqrt{2} \cdot t\right) + 2 \cdot \left(\frac{\frac{t}{\sqrt{2}}}{x} - \frac{t}{\left(\sqrt{2} \cdot 2\right) \cdot \left(x \cdot x\right)}\right)}\\ \end{array}\]
\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 -4.290053800826923246197267651223841654495 \cdot 10^{123}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{t}{\left(\sqrt{2} \cdot 2\right) \cdot \left(x \cdot x\right)} - \left(\frac{t}{\left(x \cdot \sqrt{2}\right) \cdot x} + \frac{\frac{t}{\sqrt{2}}}{x}\right)\right) \cdot 2 - \sqrt{2} \cdot t}\\

\mathbf{elif}\;t \le 2.818948520043459281423123944713297721074 \cdot 10^{72}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{4}{\frac{x}{t \cdot t}} + \left(\frac{\frac{\ell}{\sqrt[3]{x}}}{\sqrt[3]{\sqrt[3]{x}}} \cdot \frac{\frac{\ell}{\sqrt[3]{x}}}{\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}} + t \cdot t\right) \cdot 2}}\\

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

\end{array}
double f(double x, double l, double t) {
        double r1709562 = 2.0;
        double r1709563 = sqrt(r1709562);
        double r1709564 = t;
        double r1709565 = r1709563 * r1709564;
        double r1709566 = x;
        double r1709567 = 1.0;
        double r1709568 = r1709566 + r1709567;
        double r1709569 = r1709566 - r1709567;
        double r1709570 = r1709568 / r1709569;
        double r1709571 = l;
        double r1709572 = r1709571 * r1709571;
        double r1709573 = r1709564 * r1709564;
        double r1709574 = r1709562 * r1709573;
        double r1709575 = r1709572 + r1709574;
        double r1709576 = r1709570 * r1709575;
        double r1709577 = r1709576 - r1709572;
        double r1709578 = sqrt(r1709577);
        double r1709579 = r1709565 / r1709578;
        return r1709579;
}
↓
double f(double x, double l, double t) {
        double r1709580 = t;
        double r1709581 = -4.290053800826923e+123;
        bool r1709582 = r1709580 <= r1709581;
        double r1709583 = 2.0;
        double r1709584 = sqrt(r1709583);
        double r1709585 = r1709584 * r1709580;
        double r1709586 = r1709584 * r1709583;
        double r1709587 = x;
        double r1709588 = r1709587 * r1709587;
        double r1709589 = r1709586 * r1709588;
        double r1709590 = r1709580 / r1709589;
        double r1709591 = r1709587 * r1709584;
        double r1709592 = r1709591 * r1709587;
        double r1709593 = r1709580 / r1709592;
        double r1709594 = r1709580 / r1709584;
        double r1709595 = r1709594 / r1709587;
        double r1709596 = r1709593 + r1709595;
        double r1709597 = r1709590 - r1709596;
        double r1709598 = r1709597 * r1709583;
        double r1709599 = r1709598 - r1709585;
        double r1709600 = r1709585 / r1709599;
        double r1709601 = 2.8189485200434593e+72;
        bool r1709602 = r1709580 <= r1709601;
        double r1709603 = 4.0;
        double r1709604 = r1709580 * r1709580;
        double r1709605 = r1709587 / r1709604;
        double r1709606 = r1709603 / r1709605;
        double r1709607 = l;
        double r1709608 = cbrt(r1709587);
        double r1709609 = r1709607 / r1709608;
        double r1709610 = cbrt(r1709608);
        double r1709611 = r1709609 / r1709610;
        double r1709612 = r1709608 * r1709608;
        double r1709613 = cbrt(r1709612);
        double r1709614 = r1709609 / r1709613;
        double r1709615 = r1709611 * r1709614;
        double r1709616 = r1709615 + r1709604;
        double r1709617 = r1709616 * r1709583;
        double r1709618 = r1709606 + r1709617;
        double r1709619 = sqrt(r1709618);
        double r1709620 = r1709585 / r1709619;
        double r1709621 = r1709594 * r1709583;
        double r1709622 = r1709621 / r1709588;
        double r1709623 = r1709622 + r1709585;
        double r1709624 = r1709595 - r1709590;
        double r1709625 = r1709583 * r1709624;
        double r1709626 = r1709623 + r1709625;
        double r1709627 = r1709585 / r1709626;
        double r1709628 = r1709602 ? r1709620 : r1709627;
        double r1709629 = r1709582 ? r1709600 : r1709628;
        return r1709629;
}

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 < -4.290053800826923e+123

    1. Initial program 54.8

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

    2. Taylor expanded around -inf 2.8

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

    3. Simplified2.8

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

    if -4.290053800826923e+123 < t < 2.8189485200434593e+72

    1. Initial program 37.9

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

    2. Taylor expanded around inf 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

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

    5. Applied add-cube-cbrt17.6

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

    6. Applied associate-/r*17.6

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

    7. Simplified14.5

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

    9. Applied add-cube-cbrt14.5

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

    10. Applied cbrt-prod14.5

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

    11. Applied times-frac13.3

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

    if 2.8189485200434593e+72 < t

    1. Initial program 46.4

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

    2. Taylor expanded around inf 3.7

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

    3. Simplified3.7

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

  4. Final simplification9.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -4.290053800826923246197267651223841654495 \cdot 10^{123}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{t}{\left(\sqrt{2} \cdot 2\right) \cdot \left(x \cdot x\right)} - \left(\frac{t}{\left(x \cdot \sqrt{2}\right) \cdot x} + \frac{\frac{t}{\sqrt{2}}}{x}\right)\right) \cdot 2 - \sqrt{2} \cdot t}\\ \mathbf{elif}\;t \le 2.818948520043459281423123944713297721074 \cdot 10^{72}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{4}{\frac{x}{t \cdot t}} + \left(\frac{\frac{\ell}{\sqrt[3]{x}}}{\sqrt[3]{\sqrt[3]{x}}} \cdot \frac{\frac{\ell}{\sqrt[3]{x}}}{\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}} + t \cdot t\right) \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{\frac{t}{\sqrt{2}} \cdot 2}{x \cdot x} + \sqrt{2} \cdot t\right) + 2 \cdot \left(\frac{\frac{t}{\sqrt{2}}}{x} - \frac{t}{\left(\sqrt{2} \cdot 2\right) \cdot \left(x \cdot x\right)}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019200 
(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)))))