Average Error: 34.6 → 30.7
Time: 1.3m
Precision: 64
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;U \le -2.8641562817760440835220634548776257285 \cdot 10^{-277}:\\ \;\;\;\;\sqrt{\left(t - \mathsf{fma}\left(\frac{\ell}{Om}, \ell \cdot 2, \sqrt[3]{U - U*} \cdot \left(\left(\sqrt[3]{U - U*} \cdot \sqrt[3]{U - U*}\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\ \mathbf{elif}\;U \le 2.387210800340655478652547601628847274612 \cdot 10^{-169}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\left(\left(t - \mathsf{fma}\left(\ell \cdot 2, \frac{\ell}{Om}, \left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)\right) \cdot U\right) \cdot \left(2 \cdot n\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(t - \mathsf{fma}\left(\frac{\ell}{Om}, \ell \cdot 2, \sqrt[3]{U - U*} \cdot \left(\left(\sqrt[3]{U - U*} \cdot \sqrt[3]{U - U*}\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\ \end{array}\]
\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}
↓
\begin{array}{l}
\mathbf{if}\;U \le -2.8641562817760440835220634548776257285 \cdot 10^{-277}:\\
\;\;\;\;\sqrt{\left(t - \mathsf{fma}\left(\frac{\ell}{Om}, \ell \cdot 2, \sqrt[3]{U - U*} \cdot \left(\left(\sqrt[3]{U - U*} \cdot \sqrt[3]{U - U*}\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\

\mathbf{elif}\;U \le 2.387210800340655478652547601628847274612 \cdot 10^{-169}:\\
\;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\left(\left(t - \mathsf{fma}\left(\ell \cdot 2, \frac{\ell}{Om}, \left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)\right) \cdot U\right) \cdot \left(2 \cdot n\right)}\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\sqrt{\left(t - \mathsf{fma}\left(\frac{\ell}{Om}, \ell \cdot 2, \sqrt[3]{U - U*} \cdot \left(\left(\sqrt[3]{U - U*} \cdot \sqrt[3]{U - U*}\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\

\end{array}
double f(double n, double U, double t, double l, double Om, double U_) {
        double r3282372 = 2.0;
        double r3282373 = n;
        double r3282374 = r3282372 * r3282373;
        double r3282375 = U;
        double r3282376 = r3282374 * r3282375;
        double r3282377 = t;
        double r3282378 = l;
        double r3282379 = r3282378 * r3282378;
        double r3282380 = Om;
        double r3282381 = r3282379 / r3282380;
        double r3282382 = r3282372 * r3282381;
        double r3282383 = r3282377 - r3282382;
        double r3282384 = r3282378 / r3282380;
        double r3282385 = pow(r3282384, r3282372);
        double r3282386 = r3282373 * r3282385;
        double r3282387 = U_;
        double r3282388 = r3282375 - r3282387;
        double r3282389 = r3282386 * r3282388;
        double r3282390 = r3282383 - r3282389;
        double r3282391 = r3282376 * r3282390;
        double r3282392 = sqrt(r3282391);
        return r3282392;
}
↓
double f(double n, double U, double t, double l, double Om, double U_) {
        double r3282393 = U;
        double r3282394 = -2.864156281776044e-277;
        bool r3282395 = r3282393 <= r3282394;
        double r3282396 = t;
        double r3282397 = l;
        double r3282398 = Om;
        double r3282399 = r3282397 / r3282398;
        double r3282400 = 2.0;
        double r3282401 = r3282397 * r3282400;
        double r3282402 = U_;
        double r3282403 = r3282393 - r3282402;
        double r3282404 = cbrt(r3282403);
        double r3282405 = r3282404 * r3282404;
        double r3282406 = n;
        double r3282407 = pow(r3282399, r3282400);
        double r3282408 = r3282406 * r3282407;
        double r3282409 = r3282405 * r3282408;
        double r3282410 = r3282404 * r3282409;
        double r3282411 = fma(r3282399, r3282401, r3282410);
        double r3282412 = r3282396 - r3282411;
        double r3282413 = r3282400 * r3282406;
        double r3282414 = r3282413 * r3282393;
        double r3282415 = r3282412 * r3282414;
        double r3282416 = sqrt(r3282415);
        double r3282417 = 2.3872108003406555e-169;
        bool r3282418 = r3282393 <= r3282417;
        double r3282419 = r3282403 * r3282408;
        double r3282420 = fma(r3282401, r3282399, r3282419);
        double r3282421 = r3282396 - r3282420;
        double r3282422 = r3282421 * r3282393;
        double r3282423 = r3282422 * r3282413;
        double r3282424 = sqrt(r3282423);
        double r3282425 = log1p(r3282424);
        double r3282426 = expm1(r3282425);
        double r3282427 = r3282418 ? r3282426 : r3282416;
        double r3282428 = r3282395 ? r3282416 : r3282427;
        return r3282428;
}

Error

Bits error versus n

Bits error versus U

Bits error versus t

Bits error versus l

Bits error versus Om

Bits error versus U*

Derivation

  1. Split input into 2 regimes

  2. if U < -2.864156281776044e-277 or 2.3872108003406555e-169 < U

    1. Initial program 32.8

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]

    2. Simplified29.7

      \[\leadsto \color{blue}{\sqrt{\left(t - \mathsf{fma}\left(\frac{\ell}{Om}, \ell \cdot 2, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) \cdot \left(\left(n \cdot 2\right) \cdot U\right)}}\]
    3. Using strategy rm

    4. Applied add-cube-cbrt29.7

      \[\leadsto \sqrt{\left(t - \mathsf{fma}\left(\frac{\ell}{Om}, \ell \cdot 2, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \color{blue}{\left(\left(\sqrt[3]{U - U*} \cdot \sqrt[3]{U - U*}\right) \cdot \sqrt[3]{U - U*}\right)}\right)\right) \cdot \left(\left(n \cdot 2\right) \cdot U\right)}\]

    5. Applied associate-*r*29.7

      \[\leadsto \sqrt{\left(t - \mathsf{fma}\left(\frac{\ell}{Om}, \ell \cdot 2, \color{blue}{\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(\sqrt[3]{U - U*} \cdot \sqrt[3]{U - U*}\right)\right) \cdot \sqrt[3]{U - U*}}\right)\right) \cdot \left(\left(n \cdot 2\right) \cdot U\right)}\]

    if -2.864156281776044e-277 < U < 2.3872108003406555e-169

    1. Initial program 42.8

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]

    2. Simplified40.7

      \[\leadsto \color{blue}{\sqrt{\left(t - \mathsf{fma}\left(\frac{\ell}{Om}, \ell \cdot 2, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) \cdot \left(\left(n \cdot 2\right) \cdot U\right)}}\]
    3. Using strategy rm

    4. Applied add-cube-cbrt40.9

      \[\leadsto \sqrt{\color{blue}{\left(\sqrt[3]{\left(t - \mathsf{fma}\left(\frac{\ell}{Om}, \ell \cdot 2, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) \cdot \left(\left(n \cdot 2\right) \cdot U\right)} \cdot \sqrt[3]{\left(t - \mathsf{fma}\left(\frac{\ell}{Om}, \ell \cdot 2, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) \cdot \left(\left(n \cdot 2\right) \cdot U\right)}\right) \cdot \sqrt[3]{\left(t - \mathsf{fma}\left(\frac{\ell}{Om}, \ell \cdot 2, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) \cdot \left(\left(n \cdot 2\right) \cdot U\right)}}}\]
    5. Using strategy rm

    6. Applied expm1-log1p-u41.4

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\left(\sqrt[3]{\left(t - \mathsf{fma}\left(\frac{\ell}{Om}, \ell \cdot 2, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) \cdot \left(\left(n \cdot 2\right) \cdot U\right)} \cdot \sqrt[3]{\left(t - \mathsf{fma}\left(\frac{\ell}{Om}, \ell \cdot 2, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) \cdot \left(\left(n \cdot 2\right) \cdot U\right)}\right) \cdot \sqrt[3]{\left(t - \mathsf{fma}\left(\frac{\ell}{Om}, \ell \cdot 2, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) \cdot \left(\left(n \cdot 2\right) \cdot U\right)}}\right)\right)}\]

    7. Simplified35.6

      \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, \left(U - U*\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)\right)\right)\right)}\right)}\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification30.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;U \le -2.8641562817760440835220634548776257285 \cdot 10^{-277}:\\ \;\;\;\;\sqrt{\left(t - \mathsf{fma}\left(\frac{\ell}{Om}, \ell \cdot 2, \sqrt[3]{U - U*} \cdot \left(\left(\sqrt[3]{U - U*} \cdot \sqrt[3]{U - U*}\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\ \mathbf{elif}\;U \le 2.387210800340655478652547601628847274612 \cdot 10^{-169}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\left(\left(t - \mathsf{fma}\left(\ell \cdot 2, \frac{\ell}{Om}, \left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)\right) \cdot U\right) \cdot \left(2 \cdot n\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(t - \mathsf{fma}\left(\frac{\ell}{Om}, \ell \cdot 2, \sqrt[3]{U - U*} \cdot \left(\left(\sqrt[3]{U - U*} \cdot \sqrt[3]{U - U*}\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(FPCore (n U t l Om U*)
  :name "Toniolo and Linder, Equation (13)"
  (sqrt (* (* (* 2.0 n) U) (- (- t (* 2.0 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2.0)) (- U U*))))))