Average Error: 34.8 → 27.9
Time: 18.1s
Precision: binary64
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \leq -7.081305533044869 \cdot 10^{-212}:\\ \;\;\;\;\sqrt{\left(\left(n \cdot 2\right) \cdot U\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + \left(n \cdot \frac{\ell}{Om}\right) \cdot \left(U* - U\right)\right)\right)}\\ \mathbf{elif}\;n \leq 4.937289606001945 \cdot 10^{-287}:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right) - 4 \cdot \frac{U \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{U \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + \left(n \cdot \frac{\ell}{Om}\right) \cdot \left(U* - U\right)\right)\right)}\\ \end{array}\]
\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 -7.081305533044869 \cdot 10^{-212}:\\
\;\;\;\;\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 4.937289606001945 \cdot 10^{-287}:\\
\;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right) - 4 \cdot \frac{U \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)}{Om}}\\

\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 -7.081305533044869e-212)
   (sqrt
    (*
     (* (* n 2.0) U)
     (+ t (* (/ l Om) (+ (* l -2.0) (* (* n (/ l Om)) (- U* U)))))))
   (if (<= n 4.937289606001945e-287)
     (sqrt (- (* 2.0 (* t (* n U))) (* 4.0 (/ (* U (* n (* l l))) Om))))
     (*
      (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 <= -7.081305533044869e-212) {
		tmp = sqrt(((n * 2.0) * U) * (t + ((l / Om) * ((l * -2.0) + ((n * (l / Om)) * (U_42_ - U))))));
	} else if (n <= 4.937289606001945e-287) {
		tmp = sqrt((2.0 * (t * (n * U))) - (4.0 * ((U * (n * (l * l))) / Om)));
	} 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 3 regimes

  2. if n < -7.08130553304486937e-212

    1. Initial program 33.4

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

    2. 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*_binary6429.5

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

    5. Simplified29.5

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

    7. Applied pow1_binary6429.5

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{{\left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + \left(n \cdot \frac{\ell}{Om}\right) \cdot \left(U* - U\right)\right)\right)}^{1}}}\]

    8. Applied pow1_binary6429.5

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot \color{blue}{{U}^{1}}\right) \cdot {\left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + \left(n \cdot \frac{\ell}{Om}\right) \cdot \left(U* - U\right)\right)\right)}^{1}}\]

    9. Applied pow1_binary6429.5

      \[\leadsto \sqrt{\left(\left(2 \cdot \color{blue}{{n}^{1}}\right) \cdot {U}^{1}\right) \cdot {\left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + \left(n \cdot \frac{\ell}{Om}\right) \cdot \left(U* - U\right)\right)\right)}^{1}}\]

    10. Applied pow1_binary6429.5

      \[\leadsto \sqrt{\left(\left(\color{blue}{{2}^{1}} \cdot {n}^{1}\right) \cdot {U}^{1}\right) \cdot {\left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + \left(n \cdot \frac{\ell}{Om}\right) \cdot \left(U* - U\right)\right)\right)}^{1}}\]

    11. Applied pow-prod-down_binary6429.5

      \[\leadsto \sqrt{\left(\color{blue}{{\left(2 \cdot n\right)}^{1}} \cdot {U}^{1}\right) \cdot {\left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + \left(n \cdot \frac{\ell}{Om}\right) \cdot \left(U* - U\right)\right)\right)}^{1}}\]

    12. Applied pow-prod-down_binary6429.5

      \[\leadsto \sqrt{\color{blue}{{\left(\left(2 \cdot n\right) \cdot U\right)}^{1}} \cdot {\left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + \left(n \cdot \frac{\ell}{Om}\right) \cdot \left(U* - U\right)\right)\right)}^{1}}\]

    13. Applied pow-prod-down_binary6429.5

      \[\leadsto \sqrt{\color{blue}{{\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + \left(n \cdot \frac{\ell}{Om}\right) \cdot \left(U* - U\right)\right)\right)\right)}^{1}}}\]

    if -7.08130553304486937e-212 < n < 4.9372896060019451e-287

    1. Initial program 39.3

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

    2. Simplified36.4

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

    3. Taylor expanded around inf 37.4

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(t \cdot \left(U \cdot n\right)\right) - 4 \cdot \frac{U \cdot \left(n \cdot {\ell}^{2}\right)}{Om}}}\]

    4. Simplified37.4

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(t \cdot \left(U \cdot n\right)\right) - 4 \cdot \frac{U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)}{Om}}}\]

    if 4.9372896060019451e-287 < n

    1. Initial program 34.7

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

    2. 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*_binary6430.3

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

    5. Simplified30.3

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

    7. Applied associate-*l*_binary6430.7

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

    8. Simplified30.8

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

    10. Applied sqrt-prod_binary6424.0

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

    11. Simplified23.9

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

  4. Final simplification27.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -7.081305533044869 \cdot 10^{-212}:\\ \;\;\;\;\sqrt{\left(\left(n \cdot 2\right) \cdot U\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + \left(n \cdot \frac{\ell}{Om}\right) \cdot \left(U* - U\right)\right)\right)}\\ \mathbf{elif}\;n \leq 4.937289606001945 \cdot 10^{-287}:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right) - 4 \cdot \frac{U \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{U \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + \left(n \cdot \frac{\ell}{Om}\right) \cdot \left(U* - U\right)\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2021175 
(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*))))))