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 r34916 = 2.0;
        double r34917 = sqrt(r34916);
        double r34918 = t;
        double r34919 = r34917 * r34918;
        double r34920 = x;
        double r34921 = 1.0;
        double r34922 = r34920 + r34921;
        double r34923 = r34920 - r34921;
        double r34924 = r34922 / r34923;
        double r34925 = l;
        double r34926 = r34925 * r34925;
        double r34927 = r34918 * r34918;
        double r34928 = r34916 * r34927;
        double r34929 = r34926 + r34928;
        double r34930 = r34924 * r34929;
        double r34931 = r34930 - r34926;
        double r34932 = sqrt(r34931);
        double r34933 = r34919 / r34932;
        return r34933;
}
↓
double f(double x, double l, double t) {
        double r34934 = t;
        double r34935 = -1.8290478461764168e+148;
        bool r34936 = r34934 <= r34935;
        double r34937 = 2.0;
        double r34938 = sqrt(r34937);
        double r34939 = r34938 * r34934;
        double r34940 = 3.0;
        double r34941 = pow(r34938, r34940);
        double r34942 = x;
        double r34943 = 2.0;
        double r34944 = pow(r34942, r34943);
        double r34945 = r34941 * r34944;
        double r34946 = r34934 / r34945;
        double r34947 = r34938 * r34944;
        double r34948 = r34934 / r34947;
        double r34949 = r34946 - r34948;
        double r34950 = r34937 * r34949;
        double r34951 = r34950 - r34939;
        double r34952 = r34938 * r34942;
        double r34953 = r34934 / r34952;
        double r34954 = r34937 * r34953;
        double r34955 = r34951 - r34954;
        double r34956 = r34939 / r34955;
        double r34957 = -2.203593120725198e-171;
        bool r34958 = r34934 <= r34957;
        double r34959 = 4.0;
        double r34960 = pow(r34934, r34943);
        double r34961 = r34960 / r34942;
        double r34962 = r34959 * r34961;
        double r34963 = l;
        double r34964 = fabs(r34963);
        double r34965 = r34964 / r34942;
        double r34966 = r34964 * r34965;
        double r34967 = r34960 + r34966;
        double r34968 = r34937 * r34967;
        double r34969 = r34962 + r34968;
        double r34970 = sqrt(r34969);
        double r34971 = r34939 / r34970;
        double r34972 = -3.303971389439813e-212;
        bool r34973 = r34934 <= r34972;
        double r34974 = 2.1130028073347647e+52;
        bool r34975 = r34934 <= r34974;
        double r34976 = r34948 + r34953;
        double r34977 = r34937 * r34976;
        double r34978 = r34937 * r34946;
        double r34979 = r34939 - r34978;
        double r34980 = r34977 + r34979;
        double r34981 = r34939 / r34980;
        double r34982 = r34975 ? r34971 : r34981;
        double r34983 = r34973 ? r34956 : r34982;
        double r34984 = r34958 ? r34971 : r34983;
        double r34985 = r34936 ? r34956 : r34984;
        return r34985;
}

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)))))