Average Error: 33.1 → 29.6
Time: 40.3s
Precision: 64
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\sqrt{\sqrt{\left(\left(U \cdot n\right) \cdot 2\right) \cdot \left(t - \left(2 \cdot \ell - \left(-\left(U - U*\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)\right) \cdot \frac{\ell}{Om}\right)}} \cdot \sqrt{\sqrt{\left(\left(U \cdot n\right) \cdot 2\right) \cdot \left(t - \left(2 \cdot \ell - \left(-\left(U - U*\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)\right) \cdot \frac{\ell}{Om}\right)}}\]
\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)}
↓
\sqrt{\sqrt{\left(\left(U \cdot n\right) \cdot 2\right) \cdot \left(t - \left(2 \cdot \ell - \left(-\left(U - U*\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)\right) \cdot \frac{\ell}{Om}\right)}} \cdot \sqrt{\sqrt{\left(\left(U \cdot n\right) \cdot 2\right) \cdot \left(t - \left(2 \cdot \ell - \left(-\left(U - U*\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)\right) \cdot \frac{\ell}{Om}\right)}}
double f(double n, double U, double t, double l, double Om, double U_) {
        double r1428346 = 2.0;
        double r1428347 = n;
        double r1428348 = r1428346 * r1428347;
        double r1428349 = U;
        double r1428350 = r1428348 * r1428349;
        double r1428351 = t;
        double r1428352 = l;
        double r1428353 = r1428352 * r1428352;
        double r1428354 = Om;
        double r1428355 = r1428353 / r1428354;
        double r1428356 = r1428346 * r1428355;
        double r1428357 = r1428351 - r1428356;
        double r1428358 = r1428352 / r1428354;
        double r1428359 = pow(r1428358, r1428346);
        double r1428360 = r1428347 * r1428359;
        double r1428361 = U_;
        double r1428362 = r1428349 - r1428361;
        double r1428363 = r1428360 * r1428362;
        double r1428364 = r1428357 - r1428363;
        double r1428365 = r1428350 * r1428364;
        double r1428366 = sqrt(r1428365);
        return r1428366;
}
↓
double f(double n, double U, double t, double l, double Om, double U_) {
        double r1428367 = U;
        double r1428368 = n;
        double r1428369 = r1428367 * r1428368;
        double r1428370 = 2.0;
        double r1428371 = r1428369 * r1428370;
        double r1428372 = t;
        double r1428373 = l;
        double r1428374 = r1428370 * r1428373;
        double r1428375 = U_;
        double r1428376 = r1428367 - r1428375;
        double r1428377 = Om;
        double r1428378 = r1428373 / r1428377;
        double r1428379 = r1428378 * r1428368;
        double r1428380 = r1428376 * r1428379;
        double r1428381 = -r1428380;
        double r1428382 = r1428374 - r1428381;
        double r1428383 = r1428382 * r1428378;
        double r1428384 = r1428372 - r1428383;
        double r1428385 = r1428371 * r1428384;
        double r1428386 = sqrt(r1428385);
        double r1428387 = sqrt(r1428386);
        double r1428388 = r1428387 * r1428387;
        return r1428388;
}

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*

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 33.1

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

  2. Simplified29.5

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

  4. Applied add-sqr-sqrt29.6

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

  5. Final simplification29.6

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

Reproduce

herbie shell --seed 2019134 
(FPCore (n U t l Om U*)
  :name "Toniolo and Linder, Equation (13)"
  (sqrt (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*))))))