Average Error: 42.9 → 10.2
Time: 16.0s
Precision: 64
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -2404699594045954599540100621504872448:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \left(\frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \frac{t}{\sqrt{2} \cdot x}\right) - t \cdot \sqrt{2}}\\ \mathbf{elif}\;t \le -6.492099203484521869436081722227008598355 \cdot 10^{-157}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left(t \cdot t + \frac{\ell}{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \frac{\ell}{\sqrt[3]{x}}\right)}}\\ \mathbf{elif}\;t \le -7.231141217156239276359457821248495233406 \cdot 10^{-225}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \left(\frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \frac{t}{\sqrt{2} \cdot x}\right) - t \cdot \sqrt{2}}\\ \mathbf{elif}\;t \le 1.53909661729405029639481098041358476011 \cdot 10^{-299} \lor \neg \left(t \le 1.464783658109499293496367809888962941264 \cdot 10^{-233} \lor \neg \left(t \le 7.020103372014630694165473293132646183048 \cdot 10^{147}\right)\right):\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left(t \cdot t + \left|\ell\right| \cdot \frac{\left|\ell\right|}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(2 \cdot \frac{t}{\sqrt{2} \cdot x} + t \cdot \sqrt{2}\right) - \frac{t}{{x}^{2}} \cdot \left(\frac{2}{\sqrt{2} \cdot 2} - \frac{2}{\sqrt{2}}\right)}\\ \end{array}\]
\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 -2404699594045954599540100621504872448:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \left(\frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \frac{t}{\sqrt{2} \cdot x}\right) - t \cdot \sqrt{2}}\\

\mathbf{elif}\;t \le -6.492099203484521869436081722227008598355 \cdot 10^{-157}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left(t \cdot t + \frac{\ell}{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \frac{\ell}{\sqrt[3]{x}}\right)}}\\

\mathbf{elif}\;t \le -7.231141217156239276359457821248495233406 \cdot 10^{-225}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \left(\frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \frac{t}{\sqrt{2} \cdot x}\right) - t \cdot \sqrt{2}}\\

\mathbf{elif}\;t \le 1.53909661729405029639481098041358476011 \cdot 10^{-299} \lor \neg \left(t \le 1.464783658109499293496367809888962941264 \cdot 10^{-233} \lor \neg \left(t \le 7.020103372014630694165473293132646183048 \cdot 10^{147}\right)\right):\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left(t \cdot t + \left|\ell\right| \cdot \frac{\left|\ell\right|}{x}\right)}}\\

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

\end{array}
double f(double x, double l, double t) {
        double r36924 = 2.0;
        double r36925 = sqrt(r36924);
        double r36926 = t;
        double r36927 = r36925 * r36926;
        double r36928 = x;
        double r36929 = 1.0;
        double r36930 = r36928 + r36929;
        double r36931 = r36928 - r36929;
        double r36932 = r36930 / r36931;
        double r36933 = l;
        double r36934 = r36933 * r36933;
        double r36935 = r36926 * r36926;
        double r36936 = r36924 * r36935;
        double r36937 = r36934 + r36936;
        double r36938 = r36932 * r36937;
        double r36939 = r36938 - r36934;
        double r36940 = sqrt(r36939);
        double r36941 = r36927 / r36940;
        return r36941;
}
↓
double f(double x, double l, double t) {
        double r36942 = t;
        double r36943 = -2.4046995940459546e+36;
        bool r36944 = r36942 <= r36943;
        double r36945 = 2.0;
        double r36946 = sqrt(r36945);
        double r36947 = r36946 * r36942;
        double r36948 = 3.0;
        double r36949 = pow(r36946, r36948);
        double r36950 = x;
        double r36951 = 2.0;
        double r36952 = pow(r36950, r36951);
        double r36953 = r36949 * r36952;
        double r36954 = r36942 / r36953;
        double r36955 = r36946 * r36950;
        double r36956 = r36942 / r36955;
        double r36957 = r36954 - r36956;
        double r36958 = r36945 * r36957;
        double r36959 = r36942 * r36946;
        double r36960 = r36958 - r36959;
        double r36961 = r36947 / r36960;
        double r36962 = -6.492099203484522e-157;
        bool r36963 = r36942 <= r36962;
        double r36964 = 4.0;
        double r36965 = pow(r36942, r36951);
        double r36966 = r36965 / r36950;
        double r36967 = r36964 * r36966;
        double r36968 = r36942 * r36942;
        double r36969 = l;
        double r36970 = cbrt(r36950);
        double r36971 = r36970 * r36970;
        double r36972 = r36969 / r36971;
        double r36973 = r36969 / r36970;
        double r36974 = r36972 * r36973;
        double r36975 = r36968 + r36974;
        double r36976 = r36945 * r36975;
        double r36977 = r36967 + r36976;
        double r36978 = sqrt(r36977);
        double r36979 = r36947 / r36978;
        double r36980 = -7.231141217156239e-225;
        bool r36981 = r36942 <= r36980;
        double r36982 = 1.5390966172940503e-299;
        bool r36983 = r36942 <= r36982;
        double r36984 = 1.4647836581094993e-233;
        bool r36985 = r36942 <= r36984;
        double r36986 = 7.02010337201463e+147;
        bool r36987 = r36942 <= r36986;
        double r36988 = !r36987;
        bool r36989 = r36985 || r36988;
        double r36990 = !r36989;
        bool r36991 = r36983 || r36990;
        double r36992 = fabs(r36969);
        double r36993 = r36992 / r36950;
        double r36994 = r36992 * r36993;
        double r36995 = r36968 + r36994;
        double r36996 = r36945 * r36995;
        double r36997 = r36967 + r36996;
        double r36998 = sqrt(r36997);
        double r36999 = r36947 / r36998;
        double r37000 = r36945 * r36956;
        double r37001 = r37000 + r36959;
        double r37002 = r36942 / r36952;
        double r37003 = r36946 * r36945;
        double r37004 = r36945 / r37003;
        double r37005 = r36945 / r36946;
        double r37006 = r37004 - r37005;
        double r37007 = r37002 * r37006;
        double r37008 = r37001 - r37007;
        double r37009 = r36947 / r37008;
        double r37010 = r36991 ? r36999 : r37009;
        double r37011 = r36981 ? r36961 : r37010;
        double r37012 = r36963 ? r36979 : r37011;
        double r37013 = r36944 ? r36961 : r37012;
        return r37013;
}

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 < -2.4046995940459546e+36 or -6.492099203484522e-157 < t < -7.231141217156239e-225

    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 40.8

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

    3. Simplified40.8

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

    4. Taylor expanded around -inf 9.0

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

    5. Simplified9.0

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

    if -2.4046995940459546e+36 < t < -6.492099203484522e-157

    1. Initial program 29.4

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

    2. Taylor expanded around inf 9.5

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

    3. Simplified9.5

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

    5. Applied add-cube-cbrt9.5

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

    6. Applied add-sqr-sqrt37.2

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

    7. Applied unpow-prod-down37.2

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

    8. Applied times-frac34.9

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

    9. Simplified34.9

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

    10. Simplified5.0

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

    if -7.231141217156239e-225 < t < 1.5390966172940503e-299 or 1.4647836581094993e-233 < t < 7.02010337201463e+147

    1. Initial program 35.8

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

    2. Taylor expanded around inf 17.6

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

    3. Simplified17.6

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

    5. Applied *-un-lft-identity17.6

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

    6. Applied add-sqr-sqrt17.6

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

    7. Applied times-frac17.6

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

    8. Simplified17.6

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

    9. Simplified13.4

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

    if 1.5390966172940503e-299 < t < 1.4647836581094993e-233 or 7.02010337201463e+147 < t

    1. Initial program 60.9

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

    2. Taylor expanded around inf 10.3

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

    3. Simplified10.3

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

  4. Final simplification10.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -2404699594045954599540100621504872448:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \left(\frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \frac{t}{\sqrt{2} \cdot x}\right) - t \cdot \sqrt{2}}\\ \mathbf{elif}\;t \le -6.492099203484521869436081722227008598355 \cdot 10^{-157}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left(t \cdot t + \frac{\ell}{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \frac{\ell}{\sqrt[3]{x}}\right)}}\\ \mathbf{elif}\;t \le -7.231141217156239276359457821248495233406 \cdot 10^{-225}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \left(\frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \frac{t}{\sqrt{2} \cdot x}\right) - t \cdot \sqrt{2}}\\ \mathbf{elif}\;t \le 1.53909661729405029639481098041358476011 \cdot 10^{-299} \lor \neg \left(t \le 1.464783658109499293496367809888962941264 \cdot 10^{-233} \lor \neg \left(t \le 7.020103372014630694165473293132646183048 \cdot 10^{147}\right)\right):\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left(t \cdot t + \left|\ell\right| \cdot \frac{\left|\ell\right|}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(2 \cdot \frac{t}{\sqrt{2} \cdot x} + t \cdot \sqrt{2}\right) - \frac{t}{{x}^{2}} \cdot \left(\frac{2}{\sqrt{2} \cdot 2} - \frac{2}{\sqrt{2}}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019350 
(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)))))