Average Error: 42.8 → 9.2
Time: 22.9s
Precision: 64
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -5.835726090767108954912351025030367187301 \cdot 10^{135}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(2 \cdot \left(\frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \frac{t}{\sqrt{2} \cdot {x}^{2}}\right) - \sqrt{2} \cdot t\right) - 2 \cdot \frac{t}{\sqrt{2} \cdot x}}\\ \mathbf{elif}\;t \le 5.259716730497140175693065871416647553685 \cdot 10^{65}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left(\sqrt{{t}^{2} + \frac{{\ell}^{1}}{\frac{x}{\ell}}} \cdot \sqrt{{t}^{2} + \frac{{\ell}^{1}}{\frac{x}{\ell}}}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \left(\frac{t}{\sqrt{2} \cdot {x}^{2}} + \frac{t}{\sqrt{2} \cdot x}\right) + \left(\sqrt{2} \cdot t - 2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}\right)}\\ \end{array}\]
\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 -5.835726090767108954912351025030367187301 \cdot 10^{135}:\\
\;\;\;\;\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 5.259716730497140175693065871416647553685 \cdot 10^{65}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left(\sqrt{{t}^{2} + \frac{{\ell}^{1}}{\frac{x}{\ell}}} \cdot \sqrt{{t}^{2} + \frac{{\ell}^{1}}{\frac{x}{\ell}}}\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 r43202 = 2.0;
        double r43203 = sqrt(r43202);
        double r43204 = t;
        double r43205 = r43203 * r43204;
        double r43206 = x;
        double r43207 = 1.0;
        double r43208 = r43206 + r43207;
        double r43209 = r43206 - r43207;
        double r43210 = r43208 / r43209;
        double r43211 = l;
        double r43212 = r43211 * r43211;
        double r43213 = r43204 * r43204;
        double r43214 = r43202 * r43213;
        double r43215 = r43212 + r43214;
        double r43216 = r43210 * r43215;
        double r43217 = r43216 - r43212;
        double r43218 = sqrt(r43217);
        double r43219 = r43205 / r43218;
        return r43219;
}
↓
double f(double x, double l, double t) {
        double r43220 = t;
        double r43221 = -5.835726090767109e+135;
        bool r43222 = r43220 <= r43221;
        double r43223 = 2.0;
        double r43224 = sqrt(r43223);
        double r43225 = r43224 * r43220;
        double r43226 = 3.0;
        double r43227 = pow(r43224, r43226);
        double r43228 = x;
        double r43229 = 2.0;
        double r43230 = pow(r43228, r43229);
        double r43231 = r43227 * r43230;
        double r43232 = r43220 / r43231;
        double r43233 = r43224 * r43230;
        double r43234 = r43220 / r43233;
        double r43235 = r43232 - r43234;
        double r43236 = r43223 * r43235;
        double r43237 = r43236 - r43225;
        double r43238 = r43224 * r43228;
        double r43239 = r43220 / r43238;
        double r43240 = r43223 * r43239;
        double r43241 = r43237 - r43240;
        double r43242 = r43225 / r43241;
        double r43243 = 5.25971673049714e+65;
        bool r43244 = r43220 <= r43243;
        double r43245 = 4.0;
        double r43246 = pow(r43220, r43229);
        double r43247 = r43246 / r43228;
        double r43248 = r43245 * r43247;
        double r43249 = l;
        double r43250 = 1.0;
        double r43251 = pow(r43249, r43250);
        double r43252 = r43228 / r43249;
        double r43253 = r43251 / r43252;
        double r43254 = r43246 + r43253;
        double r43255 = sqrt(r43254);
        double r43256 = r43255 * r43255;
        double r43257 = r43223 * r43256;
        double r43258 = r43248 + r43257;
        double r43259 = sqrt(r43258);
        double r43260 = r43225 / r43259;
        double r43261 = r43234 + r43239;
        double r43262 = r43223 * r43261;
        double r43263 = r43223 * r43232;
        double r43264 = r43225 - r43263;
        double r43265 = r43262 + r43264;
        double r43266 = r43225 / r43265;
        double r43267 = r43244 ? r43260 : r43266;
        double r43268 = r43222 ? r43242 : r43267;
        return r43268;
}

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 < -5.835726090767109e+135

    1. Initial program 57.6

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

    2. Taylor expanded around -inf 2.3

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

    3. Simplified2.3

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

    if -5.835726090767109e+135 < t < 5.25971673049714e+65

    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 17.4

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

    3. Simplified17.4

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

    5. Applied sqr-pow17.4

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

    6. Applied associate-/l*13.3

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

    7. Simplified13.3

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

    9. Applied add-sqr-sqrt13.3

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

    10. Simplified13.3

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

    11. Simplified13.3

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

    if 5.25971673049714e+65 < t

    1. Initial program 46.7

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

    2. Taylor expanded around inf 3.6

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

    3. Simplified3.6

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

  4. Final simplification9.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -5.835726090767108954912351025030367187301 \cdot 10^{135}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(2 \cdot \left(\frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \frac{t}{\sqrt{2} \cdot {x}^{2}}\right) - \sqrt{2} \cdot t\right) - 2 \cdot \frac{t}{\sqrt{2} \cdot x}}\\ \mathbf{elif}\;t \le 5.259716730497140175693065871416647553685 \cdot 10^{65}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left(\sqrt{{t}^{2} + \frac{{\ell}^{1}}{\frac{x}{\ell}}} \cdot \sqrt{{t}^{2} + \frac{{\ell}^{1}}{\frac{x}{\ell}}}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \left(\frac{t}{\sqrt{2} \cdot {x}^{2}} + \frac{t}{\sqrt{2} \cdot x}\right) + \left(\sqrt{2} \cdot t - 2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019297 
(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)))))