Average Error: 35.2 → 28.9
Time: 33.0s
Precision: binary64
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \leq 9.437245162500533 \cdot 10^{-300}:\\ \;\;\;\;\sqrt{\left(\left(n \cdot 2\right) \cdot U\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + \left(n \cdot \frac{\ell}{Om}\right) \cdot \left(U* - U\right)\right)\right)}\\ \mathbf{elif}\;n \leq 1.3211102329035494 \cdot 10^{-190}:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{U \cdot \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)}\\ \mathbf{elif}\;n \leq 7.872510036920366 \cdot 10^{+64}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{U \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + n \cdot \frac{\ell \cdot \left(U* - U\right)}{Om}\right)\right)}\\ \end{array}\]
\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 9.437245162500533 \cdot 10^{-300}:\\
\;\;\;\;\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 1.3211102329035494 \cdot 10^{-190}:\\
\;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{U \cdot \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)}\\

\mathbf{elif}\;n \leq 7.872510036920366 \cdot 10^{+64}:\\
\;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \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)\right)}\\

\mathbf{else}:\\
\;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{U \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + n \cdot \frac{\ell \cdot \left(U* - U\right)}{Om}\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 9.437245162500533e-300)
   (sqrt
    (*
     (* (* n 2.0) U)
     (+ t (* (/ l Om) (+ (* l -2.0) (* (* n (/ l Om)) (- U* U)))))))
   (if (<= n 1.3211102329035494e-190)
     (* (sqrt (* n 2.0)) (sqrt (* U (- t (* 2.0 (/ (* l l) Om))))))
     (if (<= n 7.872510036920366e+64)
       (sqrt
        (*
         (* n 2.0)
         (* U (+ t (* (/ l Om) (+ (* l -2.0) (* (/ l Om) (* n (- U* U)))))))))
       (*
        (sqrt (* n 2.0))
        (sqrt
         (*
          U
          (+ t (* (/ l Om) (+ (* l -2.0) (* n (/ (* l (- U* U)) Om))))))))))))
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 <= 9.437245162500533e-300) {
		tmp = sqrt(((n * 2.0) * U) * (t + ((l / Om) * ((l * -2.0) + ((n * (l / Om)) * (U_42_ - U))))));
	} else if (n <= 1.3211102329035494e-190) {
		tmp = sqrt(n * 2.0) * sqrt(U * (t - (2.0 * ((l * l) / Om))));
	} else if (n <= 7.872510036920366e+64) {
		tmp = sqrt((n * 2.0) * (U * (t + ((l / Om) * ((l * -2.0) + ((l / Om) * (n * (U_42_ - U))))))));
	} else {
		tmp = sqrt(n * 2.0) * sqrt(U * (t + ((l / Om) * ((l * -2.0) + (n * ((l * (U_42_ - U)) / Om))))));
	}
	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 < 9.4372451625005325e-300

    1. Initial program 35.7

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

    2. Simplified33.5

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

    4. Applied associate-*r*_binary64_35931.5

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

    if 9.4372451625005325e-300 < n < 1.32111023290354939e-190

    1. Initial program 38.1

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

    2. Simplified35.4

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

    4. Applied associate-*l*_binary64_36033.4

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

    5. Simplified36.3

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

    6. Taylor expanded around 0 37.2

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

    7. Simplified37.2

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

    9. Applied sqrt-prod_binary64_43528.2

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

    if 1.32111023290354939e-190 < n < 7.872510036920366e64

    1. Initial program 32.2

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

    2. Simplified28.0

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

    4. Applied associate-*l*_binary64_36027.2

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

    if 7.872510036920366e64 < n

    1. Initial program 35.4

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

    2. Simplified39.5

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

    4. Applied associate-*l*_binary64_36041.8

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

    5. Simplified34.1

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

    7. Applied sqrt-prod_binary64_43522.6

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

  4. Final simplification28.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 9.437245162500533 \cdot 10^{-300}:\\ \;\;\;\;\sqrt{\left(\left(n \cdot 2\right) \cdot U\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + \left(n \cdot \frac{\ell}{Om}\right) \cdot \left(U* - U\right)\right)\right)}\\ \mathbf{elif}\;n \leq 1.3211102329035494 \cdot 10^{-190}:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{U \cdot \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)}\\ \mathbf{elif}\;n \leq 7.872510036920366 \cdot 10^{+64}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{U \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + n \cdot \frac{\ell \cdot \left(U* - U\right)}{Om}\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2021025 
(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*))))))