Average Error: 43.2 → 9.9
Time: 28.4s
Precision: 64
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -1.45871046711995538 \cdot 10^{125}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(2 \cdot \left(\frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \frac{t}{\sqrt{2} \cdot x}\right) - 2 \cdot \frac{t}{\sqrt{2} \cdot {x}^{2}}\right) - \sqrt{2} \cdot t}\\ \mathbf{elif}\;t \le -9.22209258437025154 \cdot 10^{-219}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \left(t \cdot t + \frac{\left|\ell\right|}{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \frac{\left|\ell\right|}{\sqrt[3]{x}}\right) + 4 \cdot \frac{{t}^{2}}{x}}}\\ \mathbf{elif}\;t \le -1.6506396057830864 \cdot 10^{-267}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(2 \cdot \left(\frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \frac{t}{\sqrt{2} \cdot x}\right) - 2 \cdot \frac{t}{\sqrt{2} \cdot {x}^{2}}\right) - \sqrt{2} \cdot t}\\ \mathbf{elif}\;t \le 1.1078784155040597 \cdot 10^{-245}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\sqrt{2 \cdot \left(t \cdot t + \frac{{\ell}^{2}}{x}\right) + 4 \cdot \frac{{t}^{2}}{x}}} \cdot \sqrt{\sqrt{2 \cdot \left(t \cdot t + \frac{{\ell}^{2}}{x}\right) + 4 \cdot \frac{{t}^{2}}{x}}}}\\ \mathbf{elif}\;t \le 5.50990658974305019 \cdot 10^{-162} \lor \neg \left(t \le 9.0463317445752689 \cdot 10^{25}\right):\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(2 \cdot \frac{t}{\sqrt{2} \cdot x} + \left(\sqrt{2} \cdot t + 2 \cdot \frac{t}{\sqrt{2} \cdot {x}^{2}}\right)\right) - 2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \left(t \cdot t + \frac{\ell}{\frac{x}{\ell}}\right) + 4 \cdot \frac{{t}^{2}}{x}}}\\ \end{array}\]
\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 -1.45871046711995538 \cdot 10^{125}:\\
\;\;\;\;\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}\right) - 2 \cdot \frac{t}{\sqrt{2} \cdot {x}^{2}}\right) - \sqrt{2} \cdot t}\\

\mathbf{elif}\;t \le -9.22209258437025154 \cdot 10^{-219}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \left(t \cdot t + \frac{\left|\ell\right|}{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \frac{\left|\ell\right|}{\sqrt[3]{x}}\right) + 4 \cdot \frac{{t}^{2}}{x}}}\\

\mathbf{elif}\;t \le -1.6506396057830864 \cdot 10^{-267}:\\
\;\;\;\;\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}\right) - 2 \cdot \frac{t}{\sqrt{2} \cdot {x}^{2}}\right) - \sqrt{2} \cdot t}\\

\mathbf{elif}\;t \le 1.1078784155040597 \cdot 10^{-245}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\sqrt{2 \cdot \left(t \cdot t + \frac{{\ell}^{2}}{x}\right) + 4 \cdot \frac{{t}^{2}}{x}}} \cdot \sqrt{\sqrt{2 \cdot \left(t \cdot t + \frac{{\ell}^{2}}{x}\right) + 4 \cdot \frac{{t}^{2}}{x}}}}\\

\mathbf{elif}\;t \le 5.50990658974305019 \cdot 10^{-162} \lor \neg \left(t \le 9.0463317445752689 \cdot 10^{25}\right):\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\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}}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \left(t \cdot t + \frac{\ell}{\frac{x}{\ell}}\right) + 4 \cdot \frac{{t}^{2}}{x}}}\\

\end{array}
double f(double x, double l, double t) {
        double r47236 = 2.0;
        double r47237 = sqrt(r47236);
        double r47238 = t;
        double r47239 = r47237 * r47238;
        double r47240 = x;
        double r47241 = 1.0;
        double r47242 = r47240 + r47241;
        double r47243 = r47240 - r47241;
        double r47244 = r47242 / r47243;
        double r47245 = l;
        double r47246 = r47245 * r47245;
        double r47247 = r47238 * r47238;
        double r47248 = r47236 * r47247;
        double r47249 = r47246 + r47248;
        double r47250 = r47244 * r47249;
        double r47251 = r47250 - r47246;
        double r47252 = sqrt(r47251);
        double r47253 = r47239 / r47252;
        return r47253;
}
↓
double f(double x, double l, double t) {
        double r47254 = t;
        double r47255 = -1.4587104671199554e+125;
        bool r47256 = r47254 <= r47255;
        double r47257 = 2.0;
        double r47258 = sqrt(r47257);
        double r47259 = r47258 * r47254;
        double r47260 = 3.0;
        double r47261 = pow(r47258, r47260);
        double r47262 = x;
        double r47263 = 2.0;
        double r47264 = pow(r47262, r47263);
        double r47265 = r47261 * r47264;
        double r47266 = r47254 / r47265;
        double r47267 = r47258 * r47262;
        double r47268 = r47254 / r47267;
        double r47269 = r47266 - r47268;
        double r47270 = r47257 * r47269;
        double r47271 = r47258 * r47264;
        double r47272 = r47254 / r47271;
        double r47273 = r47257 * r47272;
        double r47274 = r47270 - r47273;
        double r47275 = r47274 - r47259;
        double r47276 = r47259 / r47275;
        double r47277 = -9.222092584370252e-219;
        bool r47278 = r47254 <= r47277;
        double r47279 = r47254 * r47254;
        double r47280 = l;
        double r47281 = fabs(r47280);
        double r47282 = cbrt(r47262);
        double r47283 = r47282 * r47282;
        double r47284 = r47281 / r47283;
        double r47285 = r47281 / r47282;
        double r47286 = r47284 * r47285;
        double r47287 = r47279 + r47286;
        double r47288 = r47257 * r47287;
        double r47289 = 4.0;
        double r47290 = pow(r47254, r47263);
        double r47291 = r47290 / r47262;
        double r47292 = r47289 * r47291;
        double r47293 = r47288 + r47292;
        double r47294 = sqrt(r47293);
        double r47295 = r47259 / r47294;
        double r47296 = -1.6506396057830864e-267;
        bool r47297 = r47254 <= r47296;
        double r47298 = 1.1078784155040597e-245;
        bool r47299 = r47254 <= r47298;
        double r47300 = pow(r47280, r47263);
        double r47301 = r47300 / r47262;
        double r47302 = r47279 + r47301;
        double r47303 = r47257 * r47302;
        double r47304 = r47303 + r47292;
        double r47305 = sqrt(r47304);
        double r47306 = sqrt(r47305);
        double r47307 = r47306 * r47306;
        double r47308 = r47259 / r47307;
        double r47309 = 5.50990658974305e-162;
        bool r47310 = r47254 <= r47309;
        double r47311 = 9.046331744575269e+25;
        bool r47312 = r47254 <= r47311;
        double r47313 = !r47312;
        bool r47314 = r47310 || r47313;
        double r47315 = r47257 * r47268;
        double r47316 = r47259 + r47273;
        double r47317 = r47315 + r47316;
        double r47318 = r47257 * r47266;
        double r47319 = r47317 - r47318;
        double r47320 = r47259 / r47319;
        double r47321 = r47262 / r47280;
        double r47322 = r47280 / r47321;
        double r47323 = r47279 + r47322;
        double r47324 = r47257 * r47323;
        double r47325 = r47324 + r47292;
        double r47326 = sqrt(r47325);
        double r47327 = r47259 / r47326;
        double r47328 = r47314 ? r47320 : r47327;
        double r47329 = r47299 ? r47308 : r47328;
        double r47330 = r47297 ? r47276 : r47329;
        double r47331 = r47278 ? r47295 : r47330;
        double r47332 = r47256 ? r47276 : r47331;
        return r47332;
}

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 < -1.4587104671199554e+125 or -9.222092584370252e-219 < t < -1.6506396057830864e-267

    1. Initial program 55.9

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

    2. Taylor expanded around -inf 7.8

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

    3. Simplified7.8

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

    if -1.4587104671199554e+125 < t < -9.222092584370252e-219

    1. Initial program 31.5

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

    2. Taylor expanded around inf 14.2

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

    3. Simplified14.2

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

    5. Applied add-cube-cbrt14.3

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

    6. Applied add-sqr-sqrt14.3

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

    7. Applied times-frac14.3

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

    8. Simplified14.3

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

    9. Simplified9.2

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

    if -1.6506396057830864e-267 < t < 1.1078784155040597e-245

    1. Initial program 63.2

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

    2. Taylor expanded around inf 31.1

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

    3. Simplified31.1

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

    5. Applied add-sqr-sqrt31.1

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

    6. Applied sqrt-prod31.1

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

    if 1.1078784155040597e-245 < t < 5.50990658974305e-162 or 9.046331744575269e+25 < t

    1. Initial program 47.1

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

    2. Taylor expanded around inf 10.1

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

    if 5.50990658974305e-162 < t < 9.046331744575269e+25

    1. Initial program 28.1

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

    2. Taylor expanded around inf 9.3

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

    3. Simplified9.3

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

    5. Applied unpow29.3

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

    6. Applied associate-/l*4.5

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \left(t \cdot t + \color{blue}{\frac{\ell}{\frac{x}{\ell}}}\right) + 4 \cdot \frac{{t}^{2}}{x}}}\]
  3. Recombined 5 regimes into one program.

  4. Final simplification9.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.45871046711995538 \cdot 10^{125}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(2 \cdot \left(\frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \frac{t}{\sqrt{2} \cdot x}\right) - 2 \cdot \frac{t}{\sqrt{2} \cdot {x}^{2}}\right) - \sqrt{2} \cdot t}\\ \mathbf{elif}\;t \le -9.22209258437025154 \cdot 10^{-219}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \left(t \cdot t + \frac{\left|\ell\right|}{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \frac{\left|\ell\right|}{\sqrt[3]{x}}\right) + 4 \cdot \frac{{t}^{2}}{x}}}\\ \mathbf{elif}\;t \le -1.6506396057830864 \cdot 10^{-267}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(2 \cdot \left(\frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \frac{t}{\sqrt{2} \cdot x}\right) - 2 \cdot \frac{t}{\sqrt{2} \cdot {x}^{2}}\right) - \sqrt{2} \cdot t}\\ \mathbf{elif}\;t \le 1.1078784155040597 \cdot 10^{-245}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\sqrt{2 \cdot \left(t \cdot t + \frac{{\ell}^{2}}{x}\right) + 4 \cdot \frac{{t}^{2}}{x}}} \cdot \sqrt{\sqrt{2 \cdot \left(t \cdot t + \frac{{\ell}^{2}}{x}\right) + 4 \cdot \frac{{t}^{2}}{x}}}}\\ \mathbf{elif}\;t \le 5.50990658974305019 \cdot 10^{-162} \lor \neg \left(t \le 9.0463317445752689 \cdot 10^{25}\right):\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(2 \cdot \frac{t}{\sqrt{2} \cdot x} + \left(\sqrt{2} \cdot t + 2 \cdot \frac{t}{\sqrt{2} \cdot {x}^{2}}\right)\right) - 2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \left(t \cdot t + \frac{\ell}{\frac{x}{\ell}}\right) + 4 \cdot \frac{{t}^{2}}{x}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019195 
(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)))))