Average Error: 34.7 → 28.8
Time: 20.2s
Precision: binary64
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \leq -1.1949065200363964 \cdot 10^{-175}:\\ \;\;\;\;\sqrt{\left(\left(n \cdot 2\right) \cdot U\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + \left(n \cdot \frac{\ell}{Om}\right) \cdot \left(U* - U\right)\right)\right)}\\ \mathbf{elif}\;n \leq -3.389110959089207 \cdot 10^{-284}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + \left(n \cdot \frac{\ell}{Om}\right) \cdot \left(U* - U\right)\right)\right)\right)}\\ \mathbf{elif}\;n \leq 3.85280491822023 \cdot 10^{-86}:\\ \;\;\;\;\sqrt{\left(\left(n \cdot 2\right) \cdot U\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + \left(\frac{\ell}{Om} \cdot \left(n \cdot U*\right) - \frac{\ell}{Om} \cdot \left(n \cdot U\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{U \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + \left(n \cdot \frac{\ell}{Om}\right) \cdot \left(U* - U\right)\right)\right)}\\ \end{array}\]
\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}\;n \leq -1.1949065200363964 \cdot 10^{-175}:\\
\;\;\;\;\sqrt{\left(\left(n \cdot 2\right) \cdot U\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + \left(n \cdot \frac{\ell}{Om}\right) \cdot \left(U* - U\right)\right)\right)}\\

\mathbf{elif}\;n \leq -3.389110959089207 \cdot 10^{-284}:\\
\;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + \left(n \cdot \frac{\ell}{Om}\right) \cdot \left(U* - U\right)\right)\right)\right)}\\

\mathbf{elif}\;n \leq 3.85280491822023 \cdot 10^{-86}:\\
\;\;\;\;\sqrt{\left(\left(n \cdot 2\right) \cdot U\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + \left(\frac{\ell}{Om} \cdot \left(n \cdot U*\right) - \frac{\ell}{Om} \cdot \left(n \cdot U\right)\right)\right)\right)}\\

\mathbf{else}:\\
\;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{U \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + \left(n \cdot \frac{\ell}{Om}\right) \cdot \left(U* - U\right)\right)\right)}\\

\end{array}
(FPCore (n U t l Om U*)
 :precision binary64
 (sqrt
  (*
   (* (* 2.0 n) U)
   (- (- t (* 2.0 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2.0)) (- U U*))))))
↓
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= n -1.1949065200363964e-175)
   (sqrt
    (*
     (* (* n 2.0) U)
     (+ t (* (/ l Om) (+ (* l -2.0) (* (* n (/ l Om)) (- U* U)))))))
   (if (<= n -3.389110959089207e-284)
     (sqrt
      (*
       (* n 2.0)
       (* U (+ t (* (/ l Om) (+ (* l -2.0) (* (* n (/ l Om)) (- U* U))))))))
     (if (<= n 3.85280491822023e-86)
       (sqrt
        (*
         (* (* n 2.0) U)
         (+
          t
          (*
           (/ l Om)
           (+ (* l -2.0) (- (* (/ l Om) (* n U*)) (* (/ l Om) (* n U))))))))
       (*
        (sqrt (* n 2.0))
        (sqrt
         (*
          U
          (+ t (* (/ l Om) (+ (* l -2.0) (* (* n (/ l Om)) (- U* U))))))))))))
double code(double n, double U, double t, double l, double Om, double U_42_) {
	return sqrt(((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * pow((l / Om), 2.0)) * (U - U_42_))));
}
↓
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (n <= -1.1949065200363964e-175) {
		tmp = sqrt(((n * 2.0) * U) * (t + ((l / Om) * ((l * -2.0) + ((n * (l / Om)) * (U_42_ - U))))));
	} else if (n <= -3.389110959089207e-284) {
		tmp = sqrt((n * 2.0) * (U * (t + ((l / Om) * ((l * -2.0) + ((n * (l / Om)) * (U_42_ - U)))))));
	} else if (n <= 3.85280491822023e-86) {
		tmp = sqrt(((n * 2.0) * U) * (t + ((l / Om) * ((l * -2.0) + (((l / Om) * (n * U_42_)) - ((l / Om) * (n * U)))))));
	} else {
		tmp = sqrt(n * 2.0) * sqrt(U * (t + ((l / Om) * ((l * -2.0) + ((n * (l / Om)) * (U_42_ - U))))));
	}
	return tmp;
}

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. Split input into 4 regimes

  2. if n < -1.1949065200363964e-175

    1. Initial program 32.6

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

    2. Simplified31.6

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

    4. Applied associate-*r*_binary64_1828.7

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

    5. Simplified28.7

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

    if -1.1949065200363964e-175 < n < -3.38911095908920693e-284

    1. Initial program 40.7

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

    2. Simplified36.2

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

    4. Applied associate-*r*_binary64_1836.3

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

    5. Simplified36.3

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

    7. Applied associate-*l*_binary64_1933.9

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

    if -3.38911095908920693e-284 < n < 3.85280491822023023e-86

    1. Initial program 38.2

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

    2. Simplified33.9

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

    4. Applied sub-neg_binary64_7133.9

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

    5. Applied distribute-rgt-in_binary64_2833.9

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

    6. Applied distribute-rgt-in_binary64_2833.9

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

    7. Simplified33.9

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

    8. Simplified33.9

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

    if 3.85280491822023023e-86 < n

    1. Initial program 31.7

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

    2. Simplified31.4

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

    4. Applied associate-*r*_binary64_1827.5

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

    5. Simplified27.5

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

    7. Applied associate-*l*_binary64_1928.3

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

    9. Applied sqrt-prod_binary64_9422.3

      \[\leadsto \color{blue}{\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + \left(n \cdot \frac{\ell}{Om}\right) \cdot \left(U* - U\right)\right)\right)}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification28.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.1949065200363964 \cdot 10^{-175}:\\ \;\;\;\;\sqrt{\left(\left(n \cdot 2\right) \cdot U\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + \left(n \cdot \frac{\ell}{Om}\right) \cdot \left(U* - U\right)\right)\right)}\\ \mathbf{elif}\;n \leq -3.389110959089207 \cdot 10^{-284}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + \left(n \cdot \frac{\ell}{Om}\right) \cdot \left(U* - U\right)\right)\right)\right)}\\ \mathbf{elif}\;n \leq 3.85280491822023 \cdot 10^{-86}:\\ \;\;\;\;\sqrt{\left(\left(n \cdot 2\right) \cdot U\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + \left(\frac{\ell}{Om} \cdot \left(n \cdot U*\right) - \frac{\ell}{Om} \cdot \left(n \cdot U\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{U \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + \left(n \cdot \frac{\ell}{Om}\right) \cdot \left(U* - U\right)\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020352 
(FPCore (n U t l Om U*)
  :name "Toniolo and Linder, Equation (13)"
  :precision binary64
  (sqrt (* (* (* 2.0 n) U) (- (- t (* 2.0 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2.0)) (- U U*))))))