Average Error: 42.8 → 9.3
Time: 14.6s
Precision: 64
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -5.6277721314429949 \cdot 10^{125}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \left(\frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \frac{t}{\sqrt{2} \cdot {x}^{2}}\right) - \left(2 \cdot \frac{t}{\sqrt{2} \cdot x} + t \cdot \sqrt{2}\right)}\\ \mathbf{elif}\;t \le -9.4314582946777516 \cdot 10^{-250}:\\ \;\;\;\;\frac{\sqrt{\sqrt{2}} \cdot \left(\sqrt{\sqrt{2}} \cdot t\right)}{\sqrt{\left(\left(\ell \cdot \frac{\ell}{x}\right) \cdot 2 + 4 \cdot \frac{{t}^{2}}{x}\right) + 2 \cdot {t}^{2}}}\\ \mathbf{elif}\;t \le -6.3448789190600404 \cdot 10^{-289}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{-\left(t \cdot \sqrt{2} + 2 \cdot \frac{t}{\sqrt{2} \cdot x}\right)}\\ \mathbf{elif}\;t \le 3.34251302474556348 \cdot 10^{119}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\left(\left(\ell \cdot \frac{\ell}{x}\right) \cdot 2 + 4 \cdot \frac{{t}^{2}}{x}\right) + 2 \cdot {t}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \left(\frac{t}{\sqrt{2} \cdot {x}^{2}} + \frac{t}{\sqrt{2} \cdot x}\right) + \left(\sqrt{2} \cdot t - 2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}\right)}\\ \end{array}\]
\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 -5.6277721314429949 \cdot 10^{125}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \left(\frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \frac{t}{\sqrt{2} \cdot {x}^{2}}\right) - \left(2 \cdot \frac{t}{\sqrt{2} \cdot x} + t \cdot \sqrt{2}\right)}\\

\mathbf{elif}\;t \le -9.4314582946777516 \cdot 10^{-250}:\\
\;\;\;\;\frac{\sqrt{\sqrt{2}} \cdot \left(\sqrt{\sqrt{2}} \cdot t\right)}{\sqrt{\left(\left(\ell \cdot \frac{\ell}{x}\right) \cdot 2 + 4 \cdot \frac{{t}^{2}}{x}\right) + 2 \cdot {t}^{2}}}\\

\mathbf{elif}\;t \le -6.3448789190600404 \cdot 10^{-289}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{-\left(t \cdot \sqrt{2} + 2 \cdot \frac{t}{\sqrt{2} \cdot x}\right)}\\

\mathbf{elif}\;t \le 3.34251302474556348 \cdot 10^{119}:\\
\;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\left(\left(\ell \cdot \frac{\ell}{x}\right) \cdot 2 + 4 \cdot \frac{{t}^{2}}{x}\right) + 2 \cdot {t}^{2}}}\\

\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 r40131 = 2.0;
        double r40132 = sqrt(r40131);
        double r40133 = t;
        double r40134 = r40132 * r40133;
        double r40135 = x;
        double r40136 = 1.0;
        double r40137 = r40135 + r40136;
        double r40138 = r40135 - r40136;
        double r40139 = r40137 / r40138;
        double r40140 = l;
        double r40141 = r40140 * r40140;
        double r40142 = r40133 * r40133;
        double r40143 = r40131 * r40142;
        double r40144 = r40141 + r40143;
        double r40145 = r40139 * r40144;
        double r40146 = r40145 - r40141;
        double r40147 = sqrt(r40146);
        double r40148 = r40134 / r40147;
        return r40148;
}
↓
double f(double x, double l, double t) {
        double r40149 = t;
        double r40150 = -5.627772131442995e+125;
        bool r40151 = r40149 <= r40150;
        double r40152 = 2.0;
        double r40153 = sqrt(r40152);
        double r40154 = r40153 * r40149;
        double r40155 = 3.0;
        double r40156 = pow(r40153, r40155);
        double r40157 = x;
        double r40158 = 2.0;
        double r40159 = pow(r40157, r40158);
        double r40160 = r40156 * r40159;
        double r40161 = r40149 / r40160;
        double r40162 = r40153 * r40159;
        double r40163 = r40149 / r40162;
        double r40164 = r40161 - r40163;
        double r40165 = r40152 * r40164;
        double r40166 = r40153 * r40157;
        double r40167 = r40149 / r40166;
        double r40168 = r40152 * r40167;
        double r40169 = r40149 * r40153;
        double r40170 = r40168 + r40169;
        double r40171 = r40165 - r40170;
        double r40172 = r40154 / r40171;
        double r40173 = -9.431458294677752e-250;
        bool r40174 = r40149 <= r40173;
        double r40175 = sqrt(r40153);
        double r40176 = r40175 * r40149;
        double r40177 = r40175 * r40176;
        double r40178 = l;
        double r40179 = r40178 / r40157;
        double r40180 = r40178 * r40179;
        double r40181 = r40180 * r40152;
        double r40182 = 4.0;
        double r40183 = pow(r40149, r40158);
        double r40184 = r40183 / r40157;
        double r40185 = r40182 * r40184;
        double r40186 = r40181 + r40185;
        double r40187 = r40152 * r40183;
        double r40188 = r40186 + r40187;
        double r40189 = sqrt(r40188);
        double r40190 = r40177 / r40189;
        double r40191 = -6.34487891906004e-289;
        bool r40192 = r40149 <= r40191;
        double r40193 = r40169 + r40168;
        double r40194 = -r40193;
        double r40195 = r40154 / r40194;
        double r40196 = 3.3425130247455635e+119;
        bool r40197 = r40149 <= r40196;
        double r40198 = r40169 / r40189;
        double r40199 = r40163 + r40167;
        double r40200 = r40152 * r40199;
        double r40201 = r40152 * r40161;
        double r40202 = r40154 - r40201;
        double r40203 = r40200 + r40202;
        double r40204 = r40154 / r40203;
        double r40205 = r40197 ? r40198 : r40204;
        double r40206 = r40192 ? r40195 : r40205;
        double r40207 = r40174 ? r40190 : r40206;
        double r40208 = r40151 ? r40172 : r40207;
        return r40208;
}

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 5 regimes

  2. if t < -5.627772131442995e+125

    1. Initial program 54.5

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

    2. Taylor expanded around -inf 2.1

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

    3. Simplified2.1

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

    if -5.627772131442995e+125 < t < -9.431458294677752e-250

    1. Initial program 33.4

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

    2. Taylor expanded around inf 16.0

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

    4. Applied *-un-lft-identity16.0

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

    5. Applied sqr-pow16.0

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

    6. Applied times-frac11.3

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

    7. Simplified11.3

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

    8. Simplified11.3

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

    10. Applied add-sqr-sqrt11.3

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

    11. Applied sqrt-prod11.5

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

    12. Applied associate-*l*11.4

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

    if -9.431458294677752e-250 < t < -6.34487891906004e-289

    1. Initial program 63.1

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

    2. Taylor expanded around inf 28.6

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

    3. Taylor expanded around -inf 42.7

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

    if -6.34487891906004e-289 < t < 3.3425130247455635e+119

    1. Initial program 37.1

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

    2. Taylor expanded around inf 17.3

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

    4. Applied *-un-lft-identity17.3

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

    5. Applied sqr-pow17.3

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

    6. Applied times-frac13.3

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

    7. Simplified13.3

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

    8. Simplified13.3

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

    if 3.3425130247455635e+119 < t

    1. Initial program 54.0

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

    2. Taylor expanded around inf 2.3

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

    3. Simplified2.3

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{2 \cdot \left(\frac{t}{\sqrt{2} \cdot {x}^{2}} + \frac{t}{\sqrt{2} \cdot x}\right) + \left(\sqrt{2} \cdot t - 2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}\right)}}\]
  3. Recombined 5 regimes into one program.

  4. Final simplification9.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -5.6277721314429949 \cdot 10^{125}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \left(\frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \frac{t}{\sqrt{2} \cdot {x}^{2}}\right) - \left(2 \cdot \frac{t}{\sqrt{2} \cdot x} + t \cdot \sqrt{2}\right)}\\ \mathbf{elif}\;t \le -9.4314582946777516 \cdot 10^{-250}:\\ \;\;\;\;\frac{\sqrt{\sqrt{2}} \cdot \left(\sqrt{\sqrt{2}} \cdot t\right)}{\sqrt{\left(\left(\ell \cdot \frac{\ell}{x}\right) \cdot 2 + 4 \cdot \frac{{t}^{2}}{x}\right) + 2 \cdot {t}^{2}}}\\ \mathbf{elif}\;t \le -6.3448789190600404 \cdot 10^{-289}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{-\left(t \cdot \sqrt{2} + 2 \cdot \frac{t}{\sqrt{2} \cdot x}\right)}\\ \mathbf{elif}\;t \le 3.34251302474556348 \cdot 10^{119}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\left(\left(\ell \cdot \frac{\ell}{x}\right) \cdot 2 + 4 \cdot \frac{{t}^{2}}{x}\right) + 2 \cdot {t}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \left(\frac{t}{\sqrt{2} \cdot {x}^{2}} + \frac{t}{\sqrt{2} \cdot x}\right) + \left(\sqrt{2} \cdot t - 2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 
(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)))))