Toniolo and Linder, Equation (13)

Average Error: 35.0 → 26.7

Time: 5.4min

Precision: binary64

Cost: 36033

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq 0:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + \sqrt[3]{U* - U} \cdot \left(\left(n \cdot \frac{\ell}{Om}\right) \cdot \left(\sqrt[3]{U* - U} \cdot \sqrt[3]{U* - U}\right)\right)\right)\right)\right)}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq \infty:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + U* \cdot \left(n \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{n \cdot \left(U \cdot \left(\frac{n \cdot U*}{Om \cdot Om} - \left(\frac{n \cdot U}{Om \cdot Om} + \frac{2}{Om}\right)\right)\right)}\right)\\ \end{array}\]

FPCore

(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 (<=
      (*
       (* (* 2.0 n) U)
       (- (- t (* 2.0 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2.0)) (- U U*))))
      0.0)
   (sqrt
    (*
     (* 2.0 n)
     (*
      U
      (+
       t
       (*
        (/ l Om)
        (+
         (* l -2.0)
         (*
          (cbrt (- U* U))
          (* (* n (/ l Om)) (* (cbrt (- U* U)) (cbrt (- U* U)))))))))))
   (if (<=
        (*
         (* (* 2.0 n) U)
         (-
          (- t (* 2.0 (/ (* l l) Om)))
          (* (* n (pow (/ l Om) 2.0)) (- U U*))))
        INFINITY)
     (sqrt
      (*
       (* (* 2.0 n) U)
       (+ t (* (/ l Om) (+ (* l -2.0) (* U* (* n (/ l Om))))))))
     (*
      (sqrt 2.0)
      (*
       l
       (sqrt
        (*
         n
         (*
          U
          (-
           (/ (* n U*) (* Om Om))
           (+ (/ (* n U) (* Om Om)) (/ 2.0 Om)))))))))))

C

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 ((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * pow((l / Om), 2.0)) * (U - U_42_)))) <= 0.0) {
		tmp = sqrt((2.0 * n) * (U * (t + ((l / Om) * ((l * -2.0) + (cbrt(U_42_ - U) * ((n * (l / Om)) * (cbrt(U_42_ - U) * cbrt(U_42_ - U)))))))));
	} else if ((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * pow((l / Om), 2.0)) * (U - U_42_)))) <= ((double) INFINITY)) {
		tmp = sqrt(((2.0 * n) * U) * (t + ((l / Om) * ((l * -2.0) + (U_42_ * (n * (l / Om)))))));
	} else {
		tmp = sqrt(2.0) * (l * sqrt(n * (U * (((n * U_42_) / (Om * Om)) - (((n * U) / (Om * Om)) + (2.0 / 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*

Alternatives

Alternative 1
Error	26.7
Cost	31170

\[\begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq 0:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + U* \cdot \left(n \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq \infty:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + U* \cdot \left(n \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{n \cdot \left(U \cdot \left(\frac{n \cdot U*}{Om \cdot Om} - \left(\frac{n \cdot U}{Om \cdot Om} + \frac{2}{Om}\right)\right)\right)}\right)\\ \end{array}\]

Alternative 2
Error	26.7
Cost	30914

\[\begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq 0:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + U* \cdot \left(n \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq \infty:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + U* \cdot \left(n \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot U\right) \cdot \frac{\frac{n \cdot U*}{Om} - \left(2 + \frac{n \cdot U}{Om}\right)}{Om}} \cdot \left(\ell \cdot \sqrt{2}\right)\\ \end{array}\]

Alternative 3
Error	29.7
Cost	13825

\[\begin{array}{l} \mathbf{if}\;\ell \leq 9.375455422571545 \cdot 10^{+196}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + U* \cdot \left(n \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{-2 \cdot \frac{n \cdot U}{Om}}\right)\\ \end{array}\]

Alternative 4
Error	31.1
Cost	8963

\[\begin{array}{l} \mathbf{if}\;U* \leq -2.288091003434274 \cdot 10^{+174}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + U* \cdot \left(n \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;U* \leq -7.37744113393698 \cdot 10^{+27}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + U* \cdot \left(n \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{elif}\;U* \leq 1.0265641528632078 \cdot 10^{-157}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + U* \cdot \left(n \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + \left(U* - U\right) \cdot \frac{n \cdot \ell}{Om}\right)\right)}\\ \end{array}\]

Alternative 5
Error	30.9
Cost	8332

\[\begin{array}{l} \mathbf{if}\;U* \leq -4.2577670731976007 \cdot 10^{+173} \lor \neg \left(U* \leq -3.801842269998187 \cdot 10^{+27}\right) \land U* \leq 1.1066923262746806 \cdot 10^{-158}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + U* \cdot \left(n \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + U* \cdot \left(n \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \end{array}\]

Alternative 6
Error	30.7
Cost	8200

\[\begin{array}{l} \mathbf{if}\;U \leq -8.939592240728657 \cdot 10^{-307} \lor \neg \left(U \leq 9.254280005184624 \cdot 10^{-198}\right):\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + U* \cdot \left(n \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \end{array}\]

Alternative 7
Error	35.2
Cost	8332

\[\begin{array}{l} \mathbf{if}\;U* \leq -8.006058640161498 \cdot 10^{+174} \lor \neg \left(U* \leq -3.198428286253744 \cdot 10^{+31}\right) \land U* \leq 4.897470277910266 \cdot 10^{-158}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{\ell}{Om} \cdot \left(n \cdot U*\right)\right)}{Om}\right)}\\ \end{array}\]

Alternative 8
Error	35.4
Cost	9028

\[\begin{array}{l} \mathbf{if}\;U* \leq -6.389588186975487 \cdot 10^{+213}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{elif}\;U* \leq -7.7637127646784 \cdot 10^{+34}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)}\\ \mathbf{elif}\;U* \leq 2.2782929962950365 \cdot 10^{-176}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{elif}\;U* \leq 2.3947496780133307 \cdot 10^{+104}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)}\\ \end{array}\]

Alternative 9
Error	32.5
Cost	7688

\[\begin{array}{l} \mathbf{if}\;U \leq -1.436021515694449 \cdot 10^{+29} \lor \neg \left(U \leq 5.523615901209126 \cdot 10^{-50}\right):\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \end{array}\]

Alternative 10
Error	34.7
Cost	7681

\[\begin{array}{l} \mathbf{if}\;U \leq 4.512649071113925 \cdot 10^{+172}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array}\]

Alternative 11
Error	41.2
Cost	8453

\[\begin{array}{l} \mathbf{if}\;t \leq -1.3874823129819738 \cdot 10^{-103}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\\ \mathbf{elif}\;t \leq -1.8628278213737705 \cdot 10^{-156}:\\ \;\;\;\;\sqrt{\frac{n \cdot \left(U \cdot \left(\ell \cdot \ell\right)\right)}{Om} \cdot -4}\\ \mathbf{elif}\;t \leq -7.593430322271435 \cdot 10^{-253}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t}\\ \mathbf{elif}\;t \leq 3.13126537796642 \cdot 10^{-141}:\\ \;\;\;\;\sqrt{\frac{n \cdot \left(U \cdot \left(\ell \cdot \ell\right)\right)}{Om} \cdot -4}\\ \mathbf{elif}\;t \leq 1.2202590128081094 \cdot 10^{+201}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\\ \end{array}\]

Alternative 12
Error	40.2
Cost	7169

\[\begin{array}{l} \mathbf{if}\;n \leq 1.8625196802125445 \cdot 10^{-172}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t}\\ \end{array}\]

Alternative 13
Error	39.6
Cost	7176

\[\begin{array}{l} \mathbf{if}\;n \leq -2.0030237459866915 \cdot 10^{-25} \lor \neg \left(n \leq -7.128042310595049 \cdot 10^{-300}\right):\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array}\]

Alternative 14
Error	40.2
Cost	6848

\[\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\]

Alternative 15
Error	60.1
Cost	64

\[1\]

Alternative 16
Error	61.4
Cost	64

\[0\]

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*)))) < 0.0

   1. Initial program 57.4

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

   2. Simplified 52.7

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

   4. Applied associate-*r*_binary64_359 51.0

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

   5. Simplified 51.0

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

   7. Applied associate-*l*_binary64_360 36.0

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

   9. Applied add-cube-cbrt_binary64_454 36.0

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

   10. Applied associate-*r*_binary64_359 36.0

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

   11. Simplified 36.0

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

   if 0.0 < (*.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*)))) < +inf.0

   1. Initial program 24.7

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

   2. Simplified 23.3

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

   4. Applied associate-*r*_binary64_359 20.4

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

   5. Simplified 20.4

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

   6. Taylor expanded around inf 23.7

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

   7. Simplified 20.6

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

   8. Simplified 20.6

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

   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. Simplified 63.6

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

   3. Taylor expanded around 0 64.0

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

   4. Simplified 64.0

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

   5. Taylor expanded around 0 49.4

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

   6. Simplified 49.4

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

   7. Simplified 49.4

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

4. Final simplification 26.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq 0:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + \sqrt[3]{U* - U} \cdot \left(\left(n \cdot \frac{\ell}{Om}\right) \cdot \left(\sqrt[3]{U* - U} \cdot \sqrt[3]{U* - U}\right)\right)\right)\right)\right)}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq \infty:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + U* \cdot \left(n \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{n \cdot \left(U \cdot \left(\frac{n \cdot U*}{Om \cdot Om} - \left(\frac{n \cdot U}{Om \cdot Om} + \frac{2}{Om}\right)\right)\right)}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021014 
(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*))))))