Average Error: 42.8 → 9.5
Time: 41.1s
Precision: 64
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -8.830678604357355 \cdot 10^{+59}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{1}{\sqrt{2}} - \frac{2}{\sqrt{2}}\right) \cdot \frac{t}{x \cdot x} - \left(\frac{2}{x \cdot \sqrt{2}} \cdot t + \sqrt{2} \cdot t\right)}\\ \mathbf{elif}\;t \le 5.314962770796106 \cdot 10^{-253}:\\ \;\;\;\;\frac{\left(t \cdot \sqrt{\sqrt{2}}\right) \cdot \sqrt{\sqrt{2}}}{\sqrt{2 \cdot \left(t \cdot t + \frac{\ell}{\frac{x}{\ell}}\right) + \frac{t}{\frac{x}{t}} \cdot 4}}\\ \mathbf{elif}\;t \le 6.060995535621362 \cdot 10^{-188}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{2}{x \cdot \sqrt{2}} \cdot t + \sqrt{2} \cdot t\right) - \left(\frac{1}{\sqrt{2}} - \frac{2}{\sqrt{2}}\right) \cdot \frac{t}{x \cdot x}}\\ \mathbf{elif}\;t \le 3.525018208516232 \cdot 10^{+71}:\\ \;\;\;\;\frac{\left(\sqrt{\sqrt[3]{\sqrt{2}}} \cdot \left(t \cdot \sqrt{\sqrt{2}}\right)\right) \cdot \sqrt{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}}{\sqrt{2 \cdot \left(t \cdot t + \frac{\ell}{\frac{x}{\ell}}\right) + \frac{t}{\frac{x}{t}} \cdot 4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{2}{x \cdot \sqrt{2}} \cdot t + \sqrt{2} \cdot t\right) - \left(\frac{1}{\sqrt{2}} - \frac{2}{\sqrt{2}}\right) \cdot \frac{t}{x \cdot x}}\\ \end{array}\]
\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 -8.830678604357355 \cdot 10^{+59}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{1}{\sqrt{2}} - \frac{2}{\sqrt{2}}\right) \cdot \frac{t}{x \cdot x} - \left(\frac{2}{x \cdot \sqrt{2}} \cdot t + \sqrt{2} \cdot t\right)}\\

\mathbf{elif}\;t \le 5.314962770796106 \cdot 10^{-253}:\\
\;\;\;\;\frac{\left(t \cdot \sqrt{\sqrt{2}}\right) \cdot \sqrt{\sqrt{2}}}{\sqrt{2 \cdot \left(t \cdot t + \frac{\ell}{\frac{x}{\ell}}\right) + \frac{t}{\frac{x}{t}} \cdot 4}}\\

\mathbf{elif}\;t \le 6.060995535621362 \cdot 10^{-188}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{2}{x \cdot \sqrt{2}} \cdot t + \sqrt{2} \cdot t\right) - \left(\frac{1}{\sqrt{2}} - \frac{2}{\sqrt{2}}\right) \cdot \frac{t}{x \cdot x}}\\

\mathbf{elif}\;t \le 3.525018208516232 \cdot 10^{+71}:\\
\;\;\;\;\frac{\left(\sqrt{\sqrt[3]{\sqrt{2}}} \cdot \left(t \cdot \sqrt{\sqrt{2}}\right)\right) \cdot \sqrt{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}}{\sqrt{2 \cdot \left(t \cdot t + \frac{\ell}{\frac{x}{\ell}}\right) + \frac{t}{\frac{x}{t}} \cdot 4}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{2}{x \cdot \sqrt{2}} \cdot t + \sqrt{2} \cdot t\right) - \left(\frac{1}{\sqrt{2}} - \frac{2}{\sqrt{2}}\right) \cdot \frac{t}{x \cdot x}}\\

\end{array}
double f(double x, double l, double t) {
        double r1714007 = 2.0;
        double r1714008 = sqrt(r1714007);
        double r1714009 = t;
        double r1714010 = r1714008 * r1714009;
        double r1714011 = x;
        double r1714012 = 1.0;
        double r1714013 = r1714011 + r1714012;
        double r1714014 = r1714011 - r1714012;
        double r1714015 = r1714013 / r1714014;
        double r1714016 = l;
        double r1714017 = r1714016 * r1714016;
        double r1714018 = r1714009 * r1714009;
        double r1714019 = r1714007 * r1714018;
        double r1714020 = r1714017 + r1714019;
        double r1714021 = r1714015 * r1714020;
        double r1714022 = r1714021 - r1714017;
        double r1714023 = sqrt(r1714022);
        double r1714024 = r1714010 / r1714023;
        return r1714024;
}
↓
double f(double x, double l, double t) {
        double r1714025 = t;
        double r1714026 = -8.830678604357355e+59;
        bool r1714027 = r1714025 <= r1714026;
        double r1714028 = 2.0;
        double r1714029 = sqrt(r1714028);
        double r1714030 = r1714029 * r1714025;
        double r1714031 = 1.0;
        double r1714032 = r1714031 / r1714029;
        double r1714033 = r1714028 / r1714029;
        double r1714034 = r1714032 - r1714033;
        double r1714035 = x;
        double r1714036 = r1714035 * r1714035;
        double r1714037 = r1714025 / r1714036;
        double r1714038 = r1714034 * r1714037;
        double r1714039 = r1714035 * r1714029;
        double r1714040 = r1714028 / r1714039;
        double r1714041 = r1714040 * r1714025;
        double r1714042 = r1714041 + r1714030;
        double r1714043 = r1714038 - r1714042;
        double r1714044 = r1714030 / r1714043;
        double r1714045 = 5.314962770796106e-253;
        bool r1714046 = r1714025 <= r1714045;
        double r1714047 = sqrt(r1714029);
        double r1714048 = r1714025 * r1714047;
        double r1714049 = r1714048 * r1714047;
        double r1714050 = r1714025 * r1714025;
        double r1714051 = l;
        double r1714052 = r1714035 / r1714051;
        double r1714053 = r1714051 / r1714052;
        double r1714054 = r1714050 + r1714053;
        double r1714055 = r1714028 * r1714054;
        double r1714056 = r1714035 / r1714025;
        double r1714057 = r1714025 / r1714056;
        double r1714058 = 4.0;
        double r1714059 = r1714057 * r1714058;
        double r1714060 = r1714055 + r1714059;
        double r1714061 = sqrt(r1714060);
        double r1714062 = r1714049 / r1714061;
        double r1714063 = 6.060995535621362e-188;
        bool r1714064 = r1714025 <= r1714063;
        double r1714065 = r1714042 - r1714038;
        double r1714066 = r1714030 / r1714065;
        double r1714067 = 3.525018208516232e+71;
        bool r1714068 = r1714025 <= r1714067;
        double r1714069 = cbrt(r1714029);
        double r1714070 = sqrt(r1714069);
        double r1714071 = r1714070 * r1714048;
        double r1714072 = r1714069 * r1714069;
        double r1714073 = sqrt(r1714072);
        double r1714074 = r1714071 * r1714073;
        double r1714075 = r1714074 / r1714061;
        double r1714076 = r1714068 ? r1714075 : r1714066;
        double r1714077 = r1714064 ? r1714066 : r1714076;
        double r1714078 = r1714046 ? r1714062 : r1714077;
        double r1714079 = r1714027 ? r1714044 : r1714078;
        return r1714079;
}

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 4 regimes

  2. if t < -8.830678604357355e+59

    1. Initial program 46.2

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

    2. Taylor expanded around -inf 4.0

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

    3. Simplified4.0

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

    if -8.830678604357355e+59 < t < 5.314962770796106e-253

    1. Initial program 42.2

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

    2. Taylor expanded around inf 19.6

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

    3. Simplified19.6

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

    5. Applied associate-/l*15.4

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

    7. Applied add-sqr-sqrt15.4

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

    8. Applied sqrt-prod15.5

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

    9. Applied associate-*l*15.5

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

    if 5.314962770796106e-253 < t < 6.060995535621362e-188 or 3.525018208516232e+71 < t

    1. Initial program 49.0

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

    2. Taylor expanded around inf 8.9

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

    3. Simplified8.9

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

    if 6.060995535621362e-188 < t < 3.525018208516232e+71

    1. Initial program 31.2

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

    2. Taylor expanded around inf 12.9

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

    3. Simplified12.9

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

    5. Applied associate-/l*7.9

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

    7. Applied add-sqr-sqrt7.9

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

    8. Applied sqrt-prod8.1

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

    9. Applied associate-*l*8.0

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

    11. Applied add-cube-cbrt8.0

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

    12. Applied sqrt-prod8.1

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

    13. Applied associate-*l*7.9

      \[\leadsto \frac{\color{blue}{\sqrt{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}} \cdot \left(\sqrt{\sqrt[3]{\sqrt{2}}} \cdot \left(\sqrt{\sqrt{2}} \cdot t\right)\right)}}{\sqrt{\left(t \cdot t + \frac{\ell}{\frac{x}{\ell}}\right) \cdot 2 + \frac{t}{\frac{x}{t}} \cdot 4}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification9.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -8.830678604357355 \cdot 10^{+59}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{1}{\sqrt{2}} - \frac{2}{\sqrt{2}}\right) \cdot \frac{t}{x \cdot x} - \left(\frac{2}{x \cdot \sqrt{2}} \cdot t + \sqrt{2} \cdot t\right)}\\ \mathbf{elif}\;t \le 5.314962770796106 \cdot 10^{-253}:\\ \;\;\;\;\frac{\left(t \cdot \sqrt{\sqrt{2}}\right) \cdot \sqrt{\sqrt{2}}}{\sqrt{2 \cdot \left(t \cdot t + \frac{\ell}{\frac{x}{\ell}}\right) + \frac{t}{\frac{x}{t}} \cdot 4}}\\ \mathbf{elif}\;t \le 6.060995535621362 \cdot 10^{-188}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{2}{x \cdot \sqrt{2}} \cdot t + \sqrt{2} \cdot t\right) - \left(\frac{1}{\sqrt{2}} - \frac{2}{\sqrt{2}}\right) \cdot \frac{t}{x \cdot x}}\\ \mathbf{elif}\;t \le 3.525018208516232 \cdot 10^{+71}:\\ \;\;\;\;\frac{\left(\sqrt{\sqrt[3]{\sqrt{2}}} \cdot \left(t \cdot \sqrt{\sqrt{2}}\right)\right) \cdot \sqrt{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}}{\sqrt{2 \cdot \left(t \cdot t + \frac{\ell}{\frac{x}{\ell}}\right) + \frac{t}{\frac{x}{t}} \cdot 4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{2}{x \cdot \sqrt{2}} \cdot t + \sqrt{2} \cdot t\right) - \left(\frac{1}{\sqrt{2}} - \frac{2}{\sqrt{2}}\right) \cdot \frac{t}{x \cdot x}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019139 
(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)))))