Average Error: 43.8 → 9.8
Time: 14.3s
Precision: 64
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -4.4134163461362922 \cdot 10^{69}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \left(\frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \frac{t}{\sqrt{2} \cdot {x}^{2}}\right) - \left(2 \cdot \frac{t}{\sqrt{2} \cdot x} + t \cdot \sqrt{2}\right)}\\ \mathbf{elif}\;t \le -2.10932377839678249 \cdot 10^{-178}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot t\right)}{\sqrt{\left(2 \cdot \frac{{\ell}^{\left(\frac{2}{2}\right)}}{\frac{x}{\ell}} + 4 \cdot \frac{{t}^{2}}{x}\right) + 2 \cdot {t}^{2}}}\\ \mathbf{elif}\;t \le -5.0733881302293798 \cdot 10^{-287}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \left(\frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \frac{t}{\sqrt{2} \cdot {x}^{2}}\right) - \left(2 \cdot \frac{t}{\sqrt{2} \cdot x} + t \cdot \sqrt{2}\right)}\\ \mathbf{elif}\;t \le 6.76918902262626632 \cdot 10^{24}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot t\right)}{\sqrt{\left(2 \cdot \frac{{\ell}^{\left(\frac{2}{2}\right)}}{\frac{x}{\ell}} + 4 \cdot \frac{{t}^{2}}{x}\right) + 2 \cdot {t}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{t \cdot \sqrt{2} + 2 \cdot \frac{t}{\sqrt{2} \cdot x}}\\ \end{array}\]
\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 -4.4134163461362922 \cdot 10^{69}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \left(\frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \frac{t}{\sqrt{2} \cdot {x}^{2}}\right) - \left(2 \cdot \frac{t}{\sqrt{2} \cdot x} + t \cdot \sqrt{2}\right)}\\

\mathbf{elif}\;t \le -2.10932377839678249 \cdot 10^{-178}:\\
\;\;\;\;\frac{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot t\right)}{\sqrt{\left(2 \cdot \frac{{\ell}^{\left(\frac{2}{2}\right)}}{\frac{x}{\ell}} + 4 \cdot \frac{{t}^{2}}{x}\right) + 2 \cdot {t}^{2}}}\\

\mathbf{elif}\;t \le -5.0733881302293798 \cdot 10^{-287}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \left(\frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \frac{t}{\sqrt{2} \cdot {x}^{2}}\right) - \left(2 \cdot \frac{t}{\sqrt{2} \cdot x} + t \cdot \sqrt{2}\right)}\\

\mathbf{elif}\;t \le 6.76918902262626632 \cdot 10^{24}:\\
\;\;\;\;\frac{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot t\right)}{\sqrt{\left(2 \cdot \frac{{\ell}^{\left(\frac{2}{2}\right)}}{\frac{x}{\ell}} + 4 \cdot \frac{{t}^{2}}{x}\right) + 2 \cdot {t}^{2}}}\\

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

\end{array}
double f(double x, double l, double t) {
        double r39991 = 2.0;
        double r39992 = sqrt(r39991);
        double r39993 = t;
        double r39994 = r39992 * r39993;
        double r39995 = x;
        double r39996 = 1.0;
        double r39997 = r39995 + r39996;
        double r39998 = r39995 - r39996;
        double r39999 = r39997 / r39998;
        double r40000 = l;
        double r40001 = r40000 * r40000;
        double r40002 = r39993 * r39993;
        double r40003 = r39991 * r40002;
        double r40004 = r40001 + r40003;
        double r40005 = r39999 * r40004;
        double r40006 = r40005 - r40001;
        double r40007 = sqrt(r40006);
        double r40008 = r39994 / r40007;
        return r40008;
}
↓
double f(double x, double l, double t) {
        double r40009 = t;
        double r40010 = -4.413416346136292e+69;
        bool r40011 = r40009 <= r40010;
        double r40012 = 2.0;
        double r40013 = sqrt(r40012);
        double r40014 = r40013 * r40009;
        double r40015 = 3.0;
        double r40016 = pow(r40013, r40015);
        double r40017 = x;
        double r40018 = 2.0;
        double r40019 = pow(r40017, r40018);
        double r40020 = r40016 * r40019;
        double r40021 = r40009 / r40020;
        double r40022 = r40013 * r40019;
        double r40023 = r40009 / r40022;
        double r40024 = r40021 - r40023;
        double r40025 = r40012 * r40024;
        double r40026 = r40013 * r40017;
        double r40027 = r40009 / r40026;
        double r40028 = r40012 * r40027;
        double r40029 = r40009 * r40013;
        double r40030 = r40028 + r40029;
        double r40031 = r40025 - r40030;
        double r40032 = r40014 / r40031;
        double r40033 = -2.1093237783967825e-178;
        bool r40034 = r40009 <= r40033;
        double r40035 = cbrt(r40013);
        double r40036 = r40035 * r40035;
        double r40037 = r40035 * r40009;
        double r40038 = r40036 * r40037;
        double r40039 = l;
        double r40040 = r40018 / r40018;
        double r40041 = pow(r40039, r40040);
        double r40042 = r40017 / r40039;
        double r40043 = r40041 / r40042;
        double r40044 = r40012 * r40043;
        double r40045 = 4.0;
        double r40046 = pow(r40009, r40018);
        double r40047 = r40046 / r40017;
        double r40048 = r40045 * r40047;
        double r40049 = r40044 + r40048;
        double r40050 = r40012 * r40046;
        double r40051 = r40049 + r40050;
        double r40052 = sqrt(r40051);
        double r40053 = r40038 / r40052;
        double r40054 = -5.07338813022938e-287;
        bool r40055 = r40009 <= r40054;
        double r40056 = 6.769189022626266e+24;
        bool r40057 = r40009 <= r40056;
        double r40058 = r40029 + r40028;
        double r40059 = r40014 / r40058;
        double r40060 = r40057 ? r40053 : r40059;
        double r40061 = r40055 ? r40032 : r40060;
        double r40062 = r40034 ? r40053 : r40061;
        double r40063 = r40011 ? r40032 : r40062;
        return r40063;
}

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 < -4.413416346136292e+69 or -2.1093237783967825e-178 < t < -5.07338813022938e-287

    1. Initial program 51.9

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

    2. Taylor expanded around -inf 11.7

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

    3. Simplified11.7

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

    if -4.413416346136292e+69 < t < -2.1093237783967825e-178 or -5.07338813022938e-287 < t < 6.769189022626266e+24

    1. Initial program 39.1

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

    2. Taylor expanded around inf 16.1

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

    4. Applied sqr-pow16.1

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

    5. Applied associate-/l*11.9

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

    6. Simplified11.9

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

    8. Applied add-cube-cbrt11.9

      \[\leadsto \frac{\color{blue}{\left(\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \sqrt[3]{\sqrt{2}}\right)} \cdot t}{\sqrt{2 \cdot {t}^{2} + \left(2 \cdot \frac{{\ell}^{\left(\frac{2}{2}\right)}}{\frac{x}{{\ell}^{1}}} + 4 \cdot \frac{{t}^{2}}{x}\right)}}\]

    9. Applied associate-*l*11.9

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot t\right)}}{\sqrt{2 \cdot {t}^{2} + \left(2 \cdot \frac{{\ell}^{\left(\frac{2}{2}\right)}}{\frac{x}{{\ell}^{1}}} + 4 \cdot \frac{{t}^{2}}{x}\right)}}\]

    if 6.769189022626266e+24 < t

    1. Initial program 41.9

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

    2. Taylor expanded around inf 39.8

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

    3. Taylor expanded around inf 4.3

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

  4. Final simplification9.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -4.4134163461362922 \cdot 10^{69}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \left(\frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \frac{t}{\sqrt{2} \cdot {x}^{2}}\right) - \left(2 \cdot \frac{t}{\sqrt{2} \cdot x} + t \cdot \sqrt{2}\right)}\\ \mathbf{elif}\;t \le -2.10932377839678249 \cdot 10^{-178}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot t\right)}{\sqrt{\left(2 \cdot \frac{{\ell}^{\left(\frac{2}{2}\right)}}{\frac{x}{\ell}} + 4 \cdot \frac{{t}^{2}}{x}\right) + 2 \cdot {t}^{2}}}\\ \mathbf{elif}\;t \le -5.0733881302293798 \cdot 10^{-287}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \left(\frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \frac{t}{\sqrt{2} \cdot {x}^{2}}\right) - \left(2 \cdot \frac{t}{\sqrt{2} \cdot x} + t \cdot \sqrt{2}\right)}\\ \mathbf{elif}\;t \le 6.76918902262626632 \cdot 10^{24}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot t\right)}{\sqrt{\left(2 \cdot \frac{{\ell}^{\left(\frac{2}{2}\right)}}{\frac{x}{\ell}} + 4 \cdot \frac{{t}^{2}}{x}\right) + 2 \cdot {t}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{t \cdot \sqrt{2} + 2 \cdot \frac{t}{\sqrt{2} \cdot x}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(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)))))