Toniolo and Linder, Equation (13)

Average Error: 35.0 → 28.2

Time: 26.7s

Precision: binary64

Cost: 15491

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\ell \leq -8.843543244185774 \cdot 10^{+189}:\\ \;\;\;\;-\sqrt{2} \cdot \left(\ell \cdot \sqrt{-2 \cdot \frac{n \cdot U}{Om}}\right)\\ \mathbf{elif}\;\ell \leq -1.0210835637414531 \cdot 10^{-297}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + U* \cdot \left(n \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.1815522068552908 \cdot 10^{+108}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + \left(n \cdot \frac{\ell}{Om}\right) \cdot \left(U* - U\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot U\right) \cdot \left(\frac{n \cdot U*}{Om \cdot Om} - \left(\frac{n \cdot U}{Om \cdot Om} + \frac{2}{Om}\right)\right)} \cdot \left(\ell \cdot \sqrt{2}\right)\\ \end{array}\]

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 (<= l -8.843543244185774e+189)
   (- (* (sqrt 2.0) (* l (sqrt (* -2.0 (/ (* n U) Om))))))
   (if (<= l -1.0210835637414531e-297)
     (sqrt
      (*
       (* 2.0 n)
       (* U (+ t (* (/ l Om) (+ (* l -2.0) (* U* (* n (/ l Om)))))))))
     (if (<= l 1.1815522068552908e+108)
       (sqrt
        (*
         (* U (* 2.0 n))
         (+ t (* (/ l Om) (+ (* l -2.0) (* (* n (/ l Om)) (- U* U)))))))
       (*
        (sqrt
         (*
          (* n U)
          (- (/ (* n U*) (* Om Om)) (+ (/ (* n U) (* Om Om)) (/ 2.0 Om)))))
        (* l (sqrt 2.0)))))))

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 (l <= -8.843543244185774e+189) {
		tmp = -(sqrt(2.0) * (l * sqrt(-2.0 * ((n * U) / Om))));
	} else if (l <= -1.0210835637414531e-297) {
		tmp = sqrt((2.0 * n) * (U * (t + ((l / Om) * ((l * -2.0) + (U_42_ * (n * (l / Om))))))));
	} else if (l <= 1.1815522068552908e+108) {
		tmp = sqrt((U * (2.0 * n)) * (t + ((l / Om) * ((l * -2.0) + ((n * (l / Om)) * (U_42_ - U))))));
	} else {
		tmp = sqrt((n * U) * (((n * U_42_) / (Om * Om)) - (((n * U) / (Om * Om)) + (2.0 / Om)))) * (l * sqrt(2.0));
	}
	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*

Alternatives

Alternative 1
Error	33.5
Cost	43328

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

Alternative 2
Error	35.4
Cost	41408

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

Alternative 3
Error	58.8
Cost	33856

\[\left(\frac{\ell}{Om} \cdot \left(n \cdot \sqrt{2}\right)\right) \cdot \sqrt{U \cdot U*} + \sqrt{\frac{U}{U*}} \cdot \left(0.5 \cdot \frac{t \cdot \left(Om \cdot \sqrt{2}\right)}{\ell} - \ell \cdot \sqrt{2}\right)\]

Alternative 4
Error	33.3
Cost	30528

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

Alternative 5
Error	53.0
Cost	30016

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

Alternative 6
Error	53.9
Cost	30016

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

Alternative 7
Error	33.3
Cost	29760

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

Alternative 8
Error	33.1
Cost	28864

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

Alternative 9
Error	54.4
Cost	28096

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

Alternative 10
Error	60.0
Cost	27840

\[\sqrt{2} \cdot \sqrt{-2 \cdot \frac{U \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)}{Om}} + 0.5 \cdot \frac{t \cdot \left(U \cdot \left(n \cdot \sqrt{2}\right)\right)}{\sqrt{-2 \cdot \frac{U \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)}{Om}}}\]

Alternative 11
Error	33.0
Cost	27712

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

Alternative 12
Error	59.7
Cost	27584

\[\frac{t \cdot \left(U \cdot \left(n \cdot \sqrt{2}\right)\right)}{\ell \cdot \sqrt{-2 \cdot \frac{n \cdot U}{Om}}} \cdot -0.5 - \sqrt{2} \cdot \left(\ell \cdot \sqrt{-2 \cdot \frac{n \cdot U}{Om}}\right)\]

Alternative 13
Error	59.7
Cost	27584

\[\sqrt{2} \cdot \left(\ell \cdot \sqrt{-2 \cdot \frac{n \cdot U}{Om}}\right) + \frac{t \cdot \left(U \cdot \left(n \cdot \sqrt{2}\right)\right)}{\ell \cdot \sqrt{-2 \cdot \frac{n \cdot U}{Om}}} \cdot 0.5\]

Alternative 14
Error	43.0
Cost	26944

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

Alternative 15
Error	58.2
Cost	26816

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

Alternative 16
Error	58.0
Cost	21120

\[\frac{\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t \cdot t - \left(\ell \cdot 4\right) \cdot \frac{{\ell}^{3}}{Om \cdot Om}\right)\right)\right)}}{\sqrt{t + 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)}}\]

Alternative 17
Error	40.2
Cost	20864

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

Alternative 18
Error	35.0
Cost	20800

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

Alternative 19
Error	36.8
Cost	20160

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

Alternative 20
Error	56.9
Cost	14592

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

Alternative 21
Error	41.3
Cost	14528

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

Alternative 22
Error	56.8
Cost	14528

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

Alternative 23
Error	35.0
Cost	14464

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

Alternative 24
Error	61.1
Cost	14400

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

Alternative 25
Error	43.7
Cost	14400

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

Alternative 26
Error	45.0
Cost	14400

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

Alternative 27
Error	43.8
Cost	14272

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

Alternative 28
Error	56.9
Cost	14080

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

Alternative 29
Error	56.8
Cost	14016

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

Alternative 30
Error	60.3
Cost	13888

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

Alternative 31
Error	62.3
Cost	13824

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

Alternative 32
Error	35.0
Cost	13760

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

Alternative 33
Error	58.1
Cost	13568

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

Alternative 34
Error	57.8
Cost	13568

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

Alternative 35
Error	57.7
Cost	13504

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

Alternative 36
Error	58.0
Cost	13504

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

Alternative 37
Error	40.6
Cost	13248

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

Alternative 38
Error	40.9
Cost	13248

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

Alternative 39
Error	47.2
Cost	10560

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

Alternative 40
Error	48.3
Cost	10560

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

Alternative 41
Error	47.3
Cost	10176

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

Alternative 42
Error	47.6
Cost	9664

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

Alternative 43
Error	40.5
Cost	9664

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

Alternative 44
Error	59.4
Cost	9280

\[\sqrt{\frac{n \cdot \left(U \cdot \left(\ell \cdot \ell\right)\right)}{Om} \cdot -4 + 2 \cdot \left(\frac{U \cdot \left(\left(n \cdot n\right) \cdot \left(\left(\ell \cdot \ell\right) \cdot U*\right)\right)}{Om \cdot Om} - \frac{\left(U \cdot U\right) \cdot \left(\left(n \cdot n\right) \cdot \left(\ell \cdot \ell\right)\right)}{Om \cdot Om}\right)}\]

Alternative 45
Error	44.0
Cost	8512

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

Alternative 46
Error	43.0
Cost	8384

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

Alternative 47
Error	57.3
Cost	8256

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

Alternative 48
Error	41.2
Cost	8128

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

Alternative 49
Error	61.1
Cost	8064

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

Alternative 50
Error	35.9
Cost	8000

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

Alternative 51
Error	33.0
Cost	8000

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

Alternative 52
Error	31.0
Cost	8000

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

Alternative 53
Error	33.5
Cost	8000

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

Alternative 54
Error	31.3
Cost	7872

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

Alternative 55
Error	31.2
Cost	7872

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

Alternative 56
Error	36.1
Cost	7872

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

Alternative 57
Error	37.5
Cost	7872

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

Alternative 58
Error	33.1
Cost	7872

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

Alternative 59
Error	35.2
Cost	7872

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

Alternative 60
Error	57.5
Cost	7744

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

Alternative 61
Error	38.6
Cost	7744

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

Alternative 62
Error	38.5
Cost	7616

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

Alternative 63
Error	60.8
Cost	7616

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

Alternative 64
Error	62.4
Cost	7488

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

Alternative 65
Error	38.3
Cost	7488

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

Alternative 66
Error	37.8
Cost	7488

\[\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{1}{Om}\right)\right)\right)\right)}\]

Alternative 67
Error	61.3
Cost	7488

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

Alternative 68
Error	62.3
Cost	7424

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

Alternative 69
Error	34.9
Cost	7360

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

Alternative 70
Error	35.2
Cost	7360

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

Alternative 71
Error	55.6
Cost	7232

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

Alternative 72
Error	58.1
Cost	7232

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

Alternative 73
Error	40.5
Cost	6848

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

Alternative 74
Error	64.0
Cost	6848

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

Alternative 75
Error	40.8
Cost	6848

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

Alternative 76
Error	64.0
Cost	1600

\[\mathsf{NaN} \cdot \left(n \cdot \left(\left(t + \frac{n \cdot \left(\left(\ell \cdot \ell\right) \cdot U*\right)}{Om \cdot Om}\right) - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\]

Alternative 77
Error	64.0
Cost	832

\[\mathsf{NaN} \cdot \left(n \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\]

Alternative 78
Error	64.0
Cost	320

\[\left(n \cdot U\right) \cdot \mathsf{NaN}\]

Alternative 79
Error	60.1
Cost	64

\[1\]

Alternative 80
Error	61.3
Cost	64

\[0\]

Alternative 81
Error	63.0
Cost	64

\[-1\]

Derivation

1. Split input into 4 regimes

2. if l < -8.8435432441857741e189

   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 53.0

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

   4. Applied associate-*r*_binary64_344 52.9

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

   5. Simplified 52.9

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

   6. Taylor expanded around inf 57.5

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

   7. Simplified 52.9

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

   8. Taylor expanded around 0 64.0

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

   9. Simplified 52.1

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

   10. Taylor expanded around -inf 36.8

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

   11. Simplified 36.8

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

   12. Simplified 36.8

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

   if -8.8435432441857741e189 < l < -1.02108356374145313e-297

   1. Initial program 30.9

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

   2. Simplified 30.3

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

   4. Applied associate-*r*_binary64_344 28.1

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

   5. Simplified 28.1

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

   6. Taylor expanded around inf 30.8

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

   7. Simplified 28.3

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

   9. Applied associate-*l*_binary64_345 27.7

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

   10. Simplified 27.7

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

   if -1.02108356374145313e-297 < l < 1.18155220685529079e108

   1. Initial program 26.8

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

   2. Simplified 27.8

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

   4. Applied associate-*r*_binary64_344 25.3

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

   5. Simplified 25.3

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

   6. Simplified 25.3

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

   if 1.18155220685529079e108 < l

   1. Initial program 57.5

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

   2. Simplified 46.3

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

   3. Taylor expanded around inf 33.6

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

   4. Simplified 33.6

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

   5. Simplified 33.6

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

4. Final simplification 28.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -8.843543244185774 \cdot 10^{+189}:\\ \;\;\;\;-\sqrt{2} \cdot \left(\ell \cdot \sqrt{-2 \cdot \frac{n \cdot U}{Om}}\right)\\ \mathbf{elif}\;\ell \leq -1.0210835637414531 \cdot 10^{-297}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + U* \cdot \left(n \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.1815522068552908 \cdot 10^{+108}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + \left(n \cdot \frac{\ell}{Om}\right) \cdot \left(U* - U\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot U\right) \cdot \left(\frac{n \cdot U*}{Om \cdot Om} - \left(\frac{n \cdot U}{Om \cdot Om} + \frac{2}{Om}\right)\right)} \cdot \left(\ell \cdot \sqrt{2}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021042 
(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*))))))