Average Error: 43.2 → 9.6
Time: 11.2s
Precision: 64
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -1.35488772505874831 \cdot 10^{150}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(2 \cdot \left(\frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \frac{t}{\sqrt{2} \cdot {x}^{2}}\right) - \sqrt{2} \cdot t\right) - 2 \cdot \frac{t}{\sqrt{2} \cdot x}}\\ \mathbf{elif}\;t \le 5.4123788187210028 \cdot 10^{86}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left(\sqrt{{t}^{2} + \left|\ell\right| \cdot \frac{\left|\ell\right|}{x}} \cdot \sqrt{{t}^{2} + \left|\ell\right| \cdot \frac{\left|\ell\right|}{x}}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \left(\frac{t}{\sqrt{2} \cdot {x}^{2}} + \frac{t}{\sqrt{2} \cdot x}\right) + \left(\sqrt{2} \cdot t - 2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}\right)}\\ \end{array}\]
\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 -1.35488772505874831 \cdot 10^{150}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\left(2 \cdot \left(\frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \frac{t}{\sqrt{2} \cdot {x}^{2}}\right) - \sqrt{2} \cdot t\right) - 2 \cdot \frac{t}{\sqrt{2} \cdot x}}\\

\mathbf{elif}\;t \le 5.4123788187210028 \cdot 10^{86}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left(\sqrt{{t}^{2} + \left|\ell\right| \cdot \frac{\left|\ell\right|}{x}} \cdot \sqrt{{t}^{2} + \left|\ell\right| \cdot \frac{\left|\ell\right|}{x}}\right)}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \left(\frac{t}{\sqrt{2} \cdot {x}^{2}} + \frac{t}{\sqrt{2} \cdot x}\right) + \left(\sqrt{2} \cdot t - 2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}\right)}\\

\end{array}
double f(double x, double l, double t) {
        double r33726 = 2.0;
        double r33727 = sqrt(r33726);
        double r33728 = t;
        double r33729 = r33727 * r33728;
        double r33730 = x;
        double r33731 = 1.0;
        double r33732 = r33730 + r33731;
        double r33733 = r33730 - r33731;
        double r33734 = r33732 / r33733;
        double r33735 = l;
        double r33736 = r33735 * r33735;
        double r33737 = r33728 * r33728;
        double r33738 = r33726 * r33737;
        double r33739 = r33736 + r33738;
        double r33740 = r33734 * r33739;
        double r33741 = r33740 - r33736;
        double r33742 = sqrt(r33741);
        double r33743 = r33729 / r33742;
        return r33743;
}
↓
double f(double x, double l, double t) {
        double r33744 = t;
        double r33745 = -1.3548877250587483e+150;
        bool r33746 = r33744 <= r33745;
        double r33747 = 2.0;
        double r33748 = sqrt(r33747);
        double r33749 = r33748 * r33744;
        double r33750 = 3.0;
        double r33751 = pow(r33748, r33750);
        double r33752 = x;
        double r33753 = 2.0;
        double r33754 = pow(r33752, r33753);
        double r33755 = r33751 * r33754;
        double r33756 = r33744 / r33755;
        double r33757 = r33748 * r33754;
        double r33758 = r33744 / r33757;
        double r33759 = r33756 - r33758;
        double r33760 = r33747 * r33759;
        double r33761 = r33760 - r33749;
        double r33762 = r33748 * r33752;
        double r33763 = r33744 / r33762;
        double r33764 = r33747 * r33763;
        double r33765 = r33761 - r33764;
        double r33766 = r33749 / r33765;
        double r33767 = 5.412378818721003e+86;
        bool r33768 = r33744 <= r33767;
        double r33769 = 4.0;
        double r33770 = pow(r33744, r33753);
        double r33771 = r33770 / r33752;
        double r33772 = r33769 * r33771;
        double r33773 = l;
        double r33774 = fabs(r33773);
        double r33775 = r33774 / r33752;
        double r33776 = r33774 * r33775;
        double r33777 = r33770 + r33776;
        double r33778 = sqrt(r33777);
        double r33779 = r33778 * r33778;
        double r33780 = r33747 * r33779;
        double r33781 = r33772 + r33780;
        double r33782 = sqrt(r33781);
        double r33783 = r33749 / r33782;
        double r33784 = r33758 + r33763;
        double r33785 = r33747 * r33784;
        double r33786 = r33747 * r33756;
        double r33787 = r33749 - r33786;
        double r33788 = r33785 + r33787;
        double r33789 = r33749 / r33788;
        double r33790 = r33768 ? r33783 : r33789;
        double r33791 = r33746 ? r33766 : r33790;
        return r33791;
}

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 < -1.3548877250587483e+150

    1. Initial program 61.2

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

    2. Taylor expanded around -inf 2.0

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

    3. Simplified2.0

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

    if -1.3548877250587483e+150 < t < 5.412378818721003e+86

    1. Initial program 36.5

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

    2. Taylor expanded around inf 17.7

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

    3. Simplified17.7

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

    5. Applied *-un-lft-identity17.7

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \frac{{\ell}^{2}}{\color{blue}{1 \cdot x}}\right)}}\]

    6. Applied add-sqr-sqrt17.7

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \frac{\color{blue}{\sqrt{{\ell}^{2}} \cdot \sqrt{{\ell}^{2}}}}{1 \cdot x}\right)}}\]

    7. Applied times-frac17.7

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \color{blue}{\frac{\sqrt{{\ell}^{2}}}{1} \cdot \frac{\sqrt{{\ell}^{2}}}{x}}\right)}}\]

    8. Simplified17.7

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

    9. Simplified13.6

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

    11. Applied add-sqr-sqrt13.7

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

    if 5.412378818721003e+86 < t

    1. Initial program 49.2

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

    2. Taylor expanded around inf 3.5

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

    3. Simplified3.5

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

  4. Final simplification9.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.35488772505874831 \cdot 10^{150}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(2 \cdot \left(\frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \frac{t}{\sqrt{2} \cdot {x}^{2}}\right) - \sqrt{2} \cdot t\right) - 2 \cdot \frac{t}{\sqrt{2} \cdot x}}\\ \mathbf{elif}\;t \le 5.4123788187210028 \cdot 10^{86}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left(\sqrt{{t}^{2} + \left|\ell\right| \cdot \frac{\left|\ell\right|}{x}} \cdot \sqrt{{t}^{2} + \left|\ell\right| \cdot \frac{\left|\ell\right|}{x}}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \left(\frac{t}{\sqrt{2} \cdot {x}^{2}} + \frac{t}{\sqrt{2} \cdot x}\right) + \left(\sqrt{2} \cdot t - 2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020036 
(FPCore (x l t)
  :name "Toniolo and Linder, Equation (7)"
  :precision binary64
  (/ (* (sqrt 2) t) (sqrt (- (* (/ (+ x 1) (- x 1)) (+ (* l l) (* 2 (* t t)))) (* l l)))))