Average Error: 42.3 → 9.4
Time: 13.5s
Precision: 64
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -6.06158033024633525 \cdot 10^{80}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \left(\frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \frac{t}{\sqrt{2} \cdot {x}^{2}}\right) - \left(2 \cdot \frac{t}{\sqrt{2} \cdot x} + t \cdot \sqrt{2}\right)}\\ \mathbf{elif}\;t \le -3.4742680211531412 \cdot 10^{-205}:\\ \;\;\;\;\frac{\sqrt{1} \cdot \left(t \cdot \sqrt{2}\right)}{\sqrt{\left(2 \cdot \frac{{\ell}^{\left(\frac{2}{2}\right)}}{\frac{x}{\ell}} + 4 \cdot \frac{{t}^{2}}{x}\right) + 2 \cdot {t}^{2}}}\\ \mathbf{elif}\;t \le -4.9375663824827687 \cdot 10^{-283}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \left(\frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \frac{t}{\sqrt{2} \cdot {x}^{2}}\right) - \left(2 \cdot \frac{t}{\sqrt{2} \cdot x} + t \cdot \sqrt{2}\right)}\\ \mathbf{elif}\;t \le 2.41471964083382225 \cdot 10^{115}:\\ \;\;\;\;\frac{\sqrt{1} \cdot \left(t \cdot \sqrt{2}\right)}{\sqrt{\left(2 \cdot \frac{{\ell}^{\left(\frac{2}{2}\right)}}{\frac{x}{\ell}} + 4 \cdot \frac{{t}^{2}}{x}\right) + 2 \cdot {t}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \left(\frac{t}{\sqrt{2} \cdot {x}^{2}} + \frac{t}{\sqrt{2} \cdot x}\right) + \left(\sqrt{2} \cdot t - 2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}\right)}\\ \end{array}\]
\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 -6.06158033024633525 \cdot 10^{80}:\\
\;\;\;\;\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 -3.4742680211531412 \cdot 10^{-205}:\\
\;\;\;\;\frac{\sqrt{1} \cdot \left(t \cdot \sqrt{2}\right)}{\sqrt{\left(2 \cdot \frac{{\ell}^{\left(\frac{2}{2}\right)}}{\frac{x}{\ell}} + 4 \cdot \frac{{t}^{2}}{x}\right) + 2 \cdot {t}^{2}}}\\

\mathbf{elif}\;t \le -4.9375663824827687 \cdot 10^{-283}:\\
\;\;\;\;\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 2.41471964083382225 \cdot 10^{115}:\\
\;\;\;\;\frac{\sqrt{1} \cdot \left(t \cdot \sqrt{2}\right)}{\sqrt{\left(2 \cdot \frac{{\ell}^{\left(\frac{2}{2}\right)}}{\frac{x}{\ell}} + 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 r40340 = 2.0;
        double r40341 = sqrt(r40340);
        double r40342 = t;
        double r40343 = r40341 * r40342;
        double r40344 = x;
        double r40345 = 1.0;
        double r40346 = r40344 + r40345;
        double r40347 = r40344 - r40345;
        double r40348 = r40346 / r40347;
        double r40349 = l;
        double r40350 = r40349 * r40349;
        double r40351 = r40342 * r40342;
        double r40352 = r40340 * r40351;
        double r40353 = r40350 + r40352;
        double r40354 = r40348 * r40353;
        double r40355 = r40354 - r40350;
        double r40356 = sqrt(r40355);
        double r40357 = r40343 / r40356;
        return r40357;
}
↓
double f(double x, double l, double t) {
        double r40358 = t;
        double r40359 = -6.061580330246335e+80;
        bool r40360 = r40358 <= r40359;
        double r40361 = 2.0;
        double r40362 = sqrt(r40361);
        double r40363 = r40362 * r40358;
        double r40364 = 3.0;
        double r40365 = pow(r40362, r40364);
        double r40366 = x;
        double r40367 = 2.0;
        double r40368 = pow(r40366, r40367);
        double r40369 = r40365 * r40368;
        double r40370 = r40358 / r40369;
        double r40371 = r40362 * r40368;
        double r40372 = r40358 / r40371;
        double r40373 = r40370 - r40372;
        double r40374 = r40361 * r40373;
        double r40375 = r40362 * r40366;
        double r40376 = r40358 / r40375;
        double r40377 = r40361 * r40376;
        double r40378 = r40358 * r40362;
        double r40379 = r40377 + r40378;
        double r40380 = r40374 - r40379;
        double r40381 = r40363 / r40380;
        double r40382 = -3.4742680211531412e-205;
        bool r40383 = r40358 <= r40382;
        double r40384 = 1.0;
        double r40385 = sqrt(r40384);
        double r40386 = r40385 * r40378;
        double r40387 = l;
        double r40388 = r40367 / r40367;
        double r40389 = pow(r40387, r40388);
        double r40390 = r40366 / r40387;
        double r40391 = r40389 / r40390;
        double r40392 = r40361 * r40391;
        double r40393 = 4.0;
        double r40394 = pow(r40358, r40367);
        double r40395 = r40394 / r40366;
        double r40396 = r40393 * r40395;
        double r40397 = r40392 + r40396;
        double r40398 = r40361 * r40394;
        double r40399 = r40397 + r40398;
        double r40400 = sqrt(r40399);
        double r40401 = r40386 / r40400;
        double r40402 = -4.937566382482769e-283;
        bool r40403 = r40358 <= r40402;
        double r40404 = 2.4147196408338222e+115;
        bool r40405 = r40358 <= r40404;
        double r40406 = r40372 + r40376;
        double r40407 = r40361 * r40406;
        double r40408 = r40361 * r40370;
        double r40409 = r40363 - r40408;
        double r40410 = r40407 + r40409;
        double r40411 = r40363 / r40410;
        double r40412 = r40405 ? r40401 : r40411;
        double r40413 = r40403 ? r40381 : r40412;
        double r40414 = r40383 ? r40401 : r40413;
        double r40415 = r40360 ? r40381 : r40414;
        return r40415;
}

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 < -6.061580330246335e+80 or -3.4742680211531412e-205 < t < -4.937566382482769e-283

    1. Initial program 49.9

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

    2. Taylor expanded around -inf 9.8

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

    3. Simplified9.8

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

    if -6.061580330246335e+80 < t < -3.4742680211531412e-205 or -4.937566382482769e-283 < t < 2.4147196408338222e+115

    1. Initial program 35.3

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

    2. Taylor expanded around inf 15.8

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

    4. Applied sqr-pow15.8

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

    5. Applied associate-/l*11.6

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

    6. Simplified11.6

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

    8. Applied *-un-lft-identity11.6

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

    9. Applied sqrt-prod11.6

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

    10. Applied associate-*l*11.6

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

    11. Simplified11.6

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

    if 2.4147196408338222e+115 < t

    1. Initial program 52.4

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

    2. Taylor expanded around inf 2.4

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

    3. Simplified2.4

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

  4. Final simplification9.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -6.06158033024633525 \cdot 10^{80}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \left(\frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \frac{t}{\sqrt{2} \cdot {x}^{2}}\right) - \left(2 \cdot \frac{t}{\sqrt{2} \cdot x} + t \cdot \sqrt{2}\right)}\\ \mathbf{elif}\;t \le -3.4742680211531412 \cdot 10^{-205}:\\ \;\;\;\;\frac{\sqrt{1} \cdot \left(t \cdot \sqrt{2}\right)}{\sqrt{\left(2 \cdot \frac{{\ell}^{\left(\frac{2}{2}\right)}}{\frac{x}{\ell}} + 4 \cdot \frac{{t}^{2}}{x}\right) + 2 \cdot {t}^{2}}}\\ \mathbf{elif}\;t \le -4.9375663824827687 \cdot 10^{-283}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \left(\frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \frac{t}{\sqrt{2} \cdot {x}^{2}}\right) - \left(2 \cdot \frac{t}{\sqrt{2} \cdot x} + t \cdot \sqrt{2}\right)}\\ \mathbf{elif}\;t \le 2.41471964083382225 \cdot 10^{115}:\\ \;\;\;\;\frac{\sqrt{1} \cdot \left(t \cdot \sqrt{2}\right)}{\sqrt{\left(2 \cdot \frac{{\ell}^{\left(\frac{2}{2}\right)}}{\frac{x}{\ell}} + 4 \cdot \frac{{t}^{2}}{x}\right) + 2 \cdot {t}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \left(\frac{t}{\sqrt{2} \cdot {x}^{2}} + \frac{t}{\sqrt{2} \cdot x}\right) + \left(\sqrt{2} \cdot t - 2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 
(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)))))