Average Error: 43.0 → 9.0
Time: 26.5s
Precision: 64
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -6.758930516841978210246323496861710927032 \cdot 10^{97}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{2}{x \cdot \left(2 \cdot \sqrt{2}\right)} \cdot \frac{t}{x} - \left(\sqrt{2} \cdot t + \frac{2}{x} \cdot \frac{t}{\sqrt{2}}\right)\right) - \frac{\frac{t}{\sqrt{2}}}{x} \cdot \frac{2}{x}}\\ \mathbf{elif}\;t \le 4.225315466536892026145009010018028395833 \cdot 10^{92}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\left(\ell \cdot \frac{\ell}{x} + t \cdot t\right) \cdot 2 + \frac{t \cdot t}{x} \cdot 4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\sqrt{2} \cdot t + \frac{2}{x} \cdot \frac{t}{\sqrt{2}}\right) + \left(\frac{\frac{t}{\sqrt{2}}}{x} \cdot \frac{2}{x} - \frac{2}{x \cdot \left(2 \cdot \sqrt{2}\right)} \cdot \frac{t}{x}\right)}\\ \end{array}\]
\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 -6.758930516841978210246323496861710927032 \cdot 10^{97}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{2}{x \cdot \left(2 \cdot \sqrt{2}\right)} \cdot \frac{t}{x} - \left(\sqrt{2} \cdot t + \frac{2}{x} \cdot \frac{t}{\sqrt{2}}\right)\right) - \frac{\frac{t}{\sqrt{2}}}{x} \cdot \frac{2}{x}}\\

\mathbf{elif}\;t \le 4.225315466536892026145009010018028395833 \cdot 10^{92}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\left(\ell \cdot \frac{\ell}{x} + t \cdot t\right) \cdot 2 + \frac{t \cdot t}{x} \cdot 4}}\\

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

\end{array}
double f(double x, double l, double t) {
        double r1452094 = 2.0;
        double r1452095 = sqrt(r1452094);
        double r1452096 = t;
        double r1452097 = r1452095 * r1452096;
        double r1452098 = x;
        double r1452099 = 1.0;
        double r1452100 = r1452098 + r1452099;
        double r1452101 = r1452098 - r1452099;
        double r1452102 = r1452100 / r1452101;
        double r1452103 = l;
        double r1452104 = r1452103 * r1452103;
        double r1452105 = r1452096 * r1452096;
        double r1452106 = r1452094 * r1452105;
        double r1452107 = r1452104 + r1452106;
        double r1452108 = r1452102 * r1452107;
        double r1452109 = r1452108 - r1452104;
        double r1452110 = sqrt(r1452109);
        double r1452111 = r1452097 / r1452110;
        return r1452111;
}
↓
double f(double x, double l, double t) {
        double r1452112 = t;
        double r1452113 = -6.758930516841978e+97;
        bool r1452114 = r1452112 <= r1452113;
        double r1452115 = 2.0;
        double r1452116 = sqrt(r1452115);
        double r1452117 = r1452116 * r1452112;
        double r1452118 = x;
        double r1452119 = r1452115 * r1452116;
        double r1452120 = r1452118 * r1452119;
        double r1452121 = r1452115 / r1452120;
        double r1452122 = r1452112 / r1452118;
        double r1452123 = r1452121 * r1452122;
        double r1452124 = r1452115 / r1452118;
        double r1452125 = r1452112 / r1452116;
        double r1452126 = r1452124 * r1452125;
        double r1452127 = r1452117 + r1452126;
        double r1452128 = r1452123 - r1452127;
        double r1452129 = r1452125 / r1452118;
        double r1452130 = r1452129 * r1452124;
        double r1452131 = r1452128 - r1452130;
        double r1452132 = r1452117 / r1452131;
        double r1452133 = 4.225315466536892e+92;
        bool r1452134 = r1452112 <= r1452133;
        double r1452135 = l;
        double r1452136 = r1452135 / r1452118;
        double r1452137 = r1452135 * r1452136;
        double r1452138 = r1452112 * r1452112;
        double r1452139 = r1452137 + r1452138;
        double r1452140 = r1452139 * r1452115;
        double r1452141 = r1452138 / r1452118;
        double r1452142 = 4.0;
        double r1452143 = r1452141 * r1452142;
        double r1452144 = r1452140 + r1452143;
        double r1452145 = sqrt(r1452144);
        double r1452146 = r1452117 / r1452145;
        double r1452147 = r1452130 - r1452123;
        double r1452148 = r1452127 + r1452147;
        double r1452149 = r1452117 / r1452148;
        double r1452150 = r1452134 ? r1452146 : r1452149;
        double r1452151 = r1452114 ? r1452132 : r1452150;
        return r1452151;
}

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 < -6.758930516841978e+97

    1. Initial program 49.7

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

    2. Taylor expanded around -inf 3.2

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

    3. Simplified3.2

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

    if -6.758930516841978e+97 < t < 4.225315466536892e+92

    1. Initial program 38.5

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

    2. Taylor expanded around inf 17.8

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

    3. Simplified17.8

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

    5. Applied *-un-lft-identity17.8

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

    6. Applied times-frac13.2

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

    7. Simplified13.2

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

    if 4.225315466536892e+92 < t

    1. Initial program 49.3

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

    2. Taylor expanded around inf 2.8

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

    3. Simplified2.8

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

  4. Final simplification9.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -6.758930516841978210246323496861710927032 \cdot 10^{97}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{2}{x \cdot \left(2 \cdot \sqrt{2}\right)} \cdot \frac{t}{x} - \left(\sqrt{2} \cdot t + \frac{2}{x} \cdot \frac{t}{\sqrt{2}}\right)\right) - \frac{\frac{t}{\sqrt{2}}}{x} \cdot \frac{2}{x}}\\ \mathbf{elif}\;t \le 4.225315466536892026145009010018028395833 \cdot 10^{92}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\left(\ell \cdot \frac{\ell}{x} + t \cdot t\right) \cdot 2 + \frac{t \cdot t}{x} \cdot 4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\sqrt{2} \cdot t + \frac{2}{x} \cdot \frac{t}{\sqrt{2}}\right) + \left(\frac{\frac{t}{\sqrt{2}}}{x} \cdot \frac{2}{x} - \frac{2}{x \cdot \left(2 \cdot \sqrt{2}\right)} \cdot \frac{t}{x}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 
(FPCore (x l t)
  :name "Toniolo and Linder, Equation (7)"
  (/ (* (sqrt 2.0) t) (sqrt (- (* (/ (+ x 1.0) (- x 1.0)) (+ (* l l) (* 2.0 (* t t)))) (* l l)))))