Average Error: 43.3 → 9.1
Time: 8.7s
Precision: 64
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -1.829047846176416821165387134663461020861 \cdot 10^{148}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(2 \cdot \left(\frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \frac{t}{\sqrt{2} \cdot {x}^{2}}\right) - \sqrt{2} \cdot t\right) - 2 \cdot \frac{t}{\sqrt{2} \cdot x}}\\ \mathbf{elif}\;t \le -2.203593120725198032992656866890125604787 \cdot 10^{-171}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \left|\ell\right| \cdot \frac{\left|\ell\right|}{x}\right)}}\\ \mathbf{elif}\;t \le -3.303971389439813148225044693459689801897 \cdot 10^{-212}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(2 \cdot \left(\frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \frac{t}{\sqrt{2} \cdot {x}^{2}}\right) - \sqrt{2} \cdot t\right) - 2 \cdot \frac{t}{\sqrt{2} \cdot x}}\\ \mathbf{elif}\;t \le 2.11300280733476473087748313915491345477 \cdot 10^{52}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \left|\ell\right| \cdot \frac{\left|\ell\right|}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \left(\frac{t}{\sqrt{2} \cdot {x}^{2}} + \frac{t}{\sqrt{2} \cdot x}\right) + \left(\sqrt{2} \cdot t - 2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}\right)}\\ \end{array}\]
\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 -1.829047846176416821165387134663461020861 \cdot 10^{148}:\\
\;\;\;\;\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 -2.203593120725198032992656866890125604787 \cdot 10^{-171}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \left|\ell\right| \cdot \frac{\left|\ell\right|}{x}\right)}}\\

\mathbf{elif}\;t \le -3.303971389439813148225044693459689801897 \cdot 10^{-212}:\\
\;\;\;\;\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 2.11300280733476473087748313915491345477 \cdot 10^{52}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({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 r34905 = 2.0;
        double r34906 = sqrt(r34905);
        double r34907 = t;
        double r34908 = r34906 * r34907;
        double r34909 = x;
        double r34910 = 1.0;
        double r34911 = r34909 + r34910;
        double r34912 = r34909 - r34910;
        double r34913 = r34911 / r34912;
        double r34914 = l;
        double r34915 = r34914 * r34914;
        double r34916 = r34907 * r34907;
        double r34917 = r34905 * r34916;
        double r34918 = r34915 + r34917;
        double r34919 = r34913 * r34918;
        double r34920 = r34919 - r34915;
        double r34921 = sqrt(r34920);
        double r34922 = r34908 / r34921;
        return r34922;
}
↓
double f(double x, double l, double t) {
        double r34923 = t;
        double r34924 = -1.8290478461764168e+148;
        bool r34925 = r34923 <= r34924;
        double r34926 = 2.0;
        double r34927 = sqrt(r34926);
        double r34928 = r34927 * r34923;
        double r34929 = 3.0;
        double r34930 = pow(r34927, r34929);
        double r34931 = x;
        double r34932 = 2.0;
        double r34933 = pow(r34931, r34932);
        double r34934 = r34930 * r34933;
        double r34935 = r34923 / r34934;
        double r34936 = r34927 * r34933;
        double r34937 = r34923 / r34936;
        double r34938 = r34935 - r34937;
        double r34939 = r34926 * r34938;
        double r34940 = r34939 - r34928;
        double r34941 = r34927 * r34931;
        double r34942 = r34923 / r34941;
        double r34943 = r34926 * r34942;
        double r34944 = r34940 - r34943;
        double r34945 = r34928 / r34944;
        double r34946 = -2.203593120725198e-171;
        bool r34947 = r34923 <= r34946;
        double r34948 = 4.0;
        double r34949 = pow(r34923, r34932);
        double r34950 = r34949 / r34931;
        double r34951 = r34948 * r34950;
        double r34952 = l;
        double r34953 = fabs(r34952);
        double r34954 = r34953 / r34931;
        double r34955 = r34953 * r34954;
        double r34956 = r34949 + r34955;
        double r34957 = r34926 * r34956;
        double r34958 = r34951 + r34957;
        double r34959 = sqrt(r34958);
        double r34960 = r34928 / r34959;
        double r34961 = -3.303971389439813e-212;
        bool r34962 = r34923 <= r34961;
        double r34963 = 2.1130028073347647e+52;
        bool r34964 = r34923 <= r34963;
        double r34965 = r34937 + r34942;
        double r34966 = r34926 * r34965;
        double r34967 = r34926 * r34935;
        double r34968 = r34928 - r34967;
        double r34969 = r34966 + r34968;
        double r34970 = r34928 / r34969;
        double r34971 = r34964 ? r34960 : r34970;
        double r34972 = r34962 ? r34945 : r34971;
        double r34973 = r34947 ? r34960 : r34972;
        double r34974 = r34925 ? r34945 : r34973;
        return r34974;
}

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 < -1.8290478461764168e+148 or -2.203593120725198e-171 < t < -3.303971389439813e-212

    1. Initial program 61.4

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

    2. Taylor expanded around -inf 6.8

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

    3. Simplified6.8

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

    if -1.8290478461764168e+148 < t < -2.203593120725198e-171 or -3.303971389439813e-212 < t < 2.1130028073347647e+52

    1. Initial program 36.7

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

    2. Taylor expanded around inf 16.4

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

    3. Simplified16.4

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

    5. Applied *-un-lft-identity16.4

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

    6. Applied add-sqr-sqrt16.4

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

    7. Applied times-frac16.4

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

    8. Simplified16.4

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

    9. Simplified12.0

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

    if 2.1130028073347647e+52 < t

    1. Initial program 44.7

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

    2. Taylor expanded around inf 3.9

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

    3. Simplified3.9

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

  4. Final simplification9.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.829047846176416821165387134663461020861 \cdot 10^{148}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(2 \cdot \left(\frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \frac{t}{\sqrt{2} \cdot {x}^{2}}\right) - \sqrt{2} \cdot t\right) - 2 \cdot \frac{t}{\sqrt{2} \cdot x}}\\ \mathbf{elif}\;t \le -2.203593120725198032992656866890125604787 \cdot 10^{-171}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \left|\ell\right| \cdot \frac{\left|\ell\right|}{x}\right)}}\\ \mathbf{elif}\;t \le -3.303971389439813148225044693459689801897 \cdot 10^{-212}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(2 \cdot \left(\frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \frac{t}{\sqrt{2} \cdot {x}^{2}}\right) - \sqrt{2} \cdot t\right) - 2 \cdot \frac{t}{\sqrt{2} \cdot x}}\\ \mathbf{elif}\;t \le 2.11300280733476473087748313915491345477 \cdot 10^{52}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \left|\ell\right| \cdot \frac{\left|\ell\right|}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \left(\frac{t}{\sqrt{2} \cdot {x}^{2}} + \frac{t}{\sqrt{2} \cdot x}\right) + \left(\sqrt{2} \cdot t - 2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 
(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)))))