Average Error: 34.6 → 26.5
Time: 18.4s
Precision: binary64
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
↓
\[\begin{array}{l} t_1 := \left(2 \cdot n\right) \cdot U\\ t_2 := t_1 \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;t_2 \leq 1.530135305633502 \cdot 10^{-299}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(\ell, -2, \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right), t\right)\right)}\\ \mathbf{elif}\;t_2 \leq \infty:\\ \;\;\;\;\sqrt{t_1 \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \left(U* - U\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \mathsf{fma}\left(2, U \cdot t, \left(\ell \cdot \frac{U \cdot \ell}{Om}\right) \cdot -4\right)}\\ \end{array} \]
\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}
t_1 := \left(2 \cdot n\right) \cdot U\\
t_2 := t_1 \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)\\
\mathbf{if}\;t_2 \leq 1.530135305633502 \cdot 10^{-299}:\\
\;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(\ell, -2, \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right), t\right)\right)}\\

\mathbf{elif}\;t_2 \leq \infty:\\
\;\;\;\;\sqrt{t_1 \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \left(U* - U\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right)\right)}\\

\mathbf{else}:\\
\;\;\;\;\sqrt{n \cdot \mathsf{fma}\left(2, U \cdot t, \left(\ell \cdot \frac{U \cdot \ell}{Om}\right) \cdot -4\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
 (let* ((t_1 (* (* 2.0 n) U))
        (t_2
         (*
          t_1
          (-
           (- t (* 2.0 (/ (* l l) Om)))
           (* (* n (pow (/ l Om) 2.0)) (- U U*))))))
   (if (<= t_2 1.530135305633502e-299)
     (sqrt
      (* (* 2.0 n) (* U (fma (/ l Om) (fma l -2.0 (/ (* n (* l U*)) Om)) t))))
     (if (<= t_2 INFINITY)
       (sqrt
        (* t_1 (+ t (* (/ l Om) (fma l -2.0 (* (- U* U) (* n (/ l Om))))))))
       (sqrt (* n (fma 2.0 (* U t) (* (* l (/ (* U l) Om)) -4.0))))))))
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 = (2.0 * n) * U;
	double t_2 = t_1 * ((t - (2.0 * ((l * l) / Om))) - ((n * pow((l / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_2 <= 1.530135305633502e-299) {
		tmp = sqrt((2.0 * n) * (U * fma((l / Om), fma(l, -2.0, ((n * (l * U_42_)) / Om)), t)));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt(t_1 * (t + ((l / Om) * fma(l, -2.0, ((U_42_ - U) * (n * (l / Om)))))));
	} else {
		tmp = sqrt(n * fma(2.0, (U * t), ((l * ((U * l) / Om)) * -4.0)));
	}
	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*

Derivation

  1. Split input into 3 regimes

  2. if (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*)))) < 1.53013530563350188e-299

    1. Initial program 55.1

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

    2. Simplified48.8

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

    4. Applied associate-*l*_binary6435.3

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

    5. Simplified35.9

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

    6. Taylor expanded in U* around inf 37.5

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

    if 1.53013530563350188e-299 < (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*)))) < +inf.0

    1. Initial program 24.3

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

    2. Simplified20.2

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

    if +inf.0 < (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))

    1. Initial program 64.0

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

    2. Simplified63.5

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

    3. Taylor expanded in n around 0 59.1

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

    4. Simplified61.9

      \[\leadsto \color{blue}{\sqrt{n \cdot \mathsf{fma}\left(2, t \cdot U, \left(\frac{\ell \cdot \ell}{Om} \cdot U\right) \cdot -4\right)}} \]
    5. Using strategy rm

    6. Applied *-un-lft-identity_binary6461.9

      \[\leadsto \sqrt{n \cdot \mathsf{fma}\left(2, t \cdot U, \left(\frac{\ell \cdot \ell}{\color{blue}{1 \cdot Om}} \cdot U\right) \cdot -4\right)} \]

    7. Applied times-frac_binary6457.2

      \[\leadsto \sqrt{n \cdot \mathsf{fma}\left(2, t \cdot U, \left(\color{blue}{\left(\frac{\ell}{1} \cdot \frac{\ell}{Om}\right)} \cdot U\right) \cdot -4\right)} \]

    8. Applied associate-*l*_binary6447.4

      \[\leadsto \sqrt{n \cdot \mathsf{fma}\left(2, t \cdot U, \color{blue}{\left(\frac{\ell}{1} \cdot \left(\frac{\ell}{Om} \cdot U\right)\right)} \cdot -4\right)} \]

    9. Simplified47.4

      \[\leadsto \sqrt{n \cdot \mathsf{fma}\left(2, t \cdot U, \left(\frac{\ell}{1} \cdot \color{blue}{\left(U \cdot \frac{\ell}{Om}\right)}\right) \cdot -4\right)} \]
    10. Using strategy rm

    11. Applied div-inv_binary6447.4

      \[\leadsto \sqrt{n \cdot \mathsf{fma}\left(2, t \cdot U, \left(\color{blue}{\left(\ell \cdot \frac{1}{1}\right)} \cdot \left(U \cdot \frac{\ell}{Om}\right)\right) \cdot -4\right)} \]

    12. Applied associate-*l*_binary6447.4

      \[\leadsto \sqrt{n \cdot \mathsf{fma}\left(2, t \cdot U, \color{blue}{\left(\ell \cdot \left(\frac{1}{1} \cdot \left(U \cdot \frac{\ell}{Om}\right)\right)\right)} \cdot -4\right)} \]

    13. Simplified46.2

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

  4. Final simplification26.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq 1.530135305633502 \cdot 10^{-299}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(\ell, -2, \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right), t\right)\right)}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq \infty:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \left(U* - U\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \mathsf{fma}\left(2, U \cdot t, \left(\ell \cdot \frac{U \cdot \ell}{Om}\right) \cdot -4\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2021212 
(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*))))))