Average Error: 34.6 → 26.9
Time: 18.4s
Precision: binary64
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
↓
\[\begin{array}{l} t_1 := U \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + n \cdot \left(\frac{\ell}{Om} \cdot \left(U* - U\right)\right)\right)\right)\\ \mathbf{if}\;n \leq 2.5655838064577735 \cdot 10^{-306}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot t_1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{t_1}\\ \end{array} \]
\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}
t_1 := U \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + n \cdot \left(\frac{\ell}{Om} \cdot \left(U* - U\right)\right)\right)\right)\\
\mathbf{if}\;n \leq 2.5655838064577735 \cdot 10^{-306}:\\
\;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot t_1}\\

\mathbf{else}:\\
\;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{t_1}\\


\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
 (let* ((t_1
         (* U (+ t (* (/ l Om) (+ (* l -2.0) (* n (* (/ l Om) (- U* U)))))))))
   (if (<= n 2.5655838064577735e-306)
     (sqrt (* (* n 2.0) t_1))
     (* (sqrt (* n 2.0)) (sqrt t_1)))))
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 t_1 = U * (t + ((l / Om) * ((l * -2.0) + (n * ((l / Om) * (U_42_ - U))))));
	double tmp;
	if (n <= 2.5655838064577735e-306) {
		tmp = sqrt((n * 2.0) * t_1);
	} else {
		tmp = sqrt(n * 2.0) * sqrt(t_1);
	}
	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 2 regimes

  2. if n < 2.5655838064577735e-306

    1. Initial program 35.0

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

    2. Simplified33.0

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

    4. Applied associate-*r*_binary6431.0

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

    5. Simplified31.0

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

    7. Applied associate-*l*_binary6430.1

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

    8. Simplified30.5

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

    10. Applied add-sqr-sqrt_binary6430.7

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

    11. Simplified30.7

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

    12. Simplified30.7

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

    13. Taylor expanded in l around 0 32.4

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

    14. Simplified30.5

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

    if 2.5655838064577735e-306 < n

    1. Initial program 34.1

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

    2. Simplified32.3

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

    4. Applied associate-*r*_binary6430.0

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

    5. Simplified30.0

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

    7. Applied associate-*l*_binary6430.1

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

    8. Simplified30.6

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

    10. Applied sqrt-prod_binary6423.2

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

    11. Simplified23.2

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

  4. Final simplification26.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 2.5655838064577735 \cdot 10^{-306}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + n \cdot \left(\frac{\ell}{Om} \cdot \left(U* - U\right)\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{U \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + n \cdot \left(\frac{\ell}{Om} \cdot \left(U* - U\right)\right)\right)\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2021206 
(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*))))))