Toniolo and Linder, Equation (13)

Average Error: 34.7 → 27.2

Time: 14.0s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\\ t_2 := \left(2 \cdot n\right) \cdot U\\ t_3 := \sqrt{t_2 \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - t_1\right)}\\ t_4 := \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - t_1\\ \mathbf{if}\;t_3 \leq 0:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot U\right)\right)}\\ \mathbf{elif}\;t_3 \leq 2 \cdot 10^{+149}:\\ \;\;\;\;\sqrt{t_2 \cdot t_4}\\ \mathbf{elif}\;t_3 \leq \infty:\\ \;\;\;\;\sqrt{t_2} \cdot \sqrt{t_4}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \sqrt{\left(\left(U - U*\right) \cdot \frac{n}{Om \cdot Om} + \frac{2}{Om}\right) \cdot \left(-2 \cdot \left(U \cdot n\right)\right)}\\ \end{array} \]

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
 (let* ((t_1 (* (* n (pow (/ l Om) 2.0)) (- U U*)))
        (t_2 (* (* 2.0 n) U))
        (t_3 (sqrt (* t_2 (- (- t (* 2.0 (/ (* l l) Om))) t_1))))
        (t_4 (- (- t (* 2.0 (* l (/ l Om)))) t_1)))
   (if (<= t_3 0.0)
     (sqrt (* 2.0 (* n (* (- t (* 2.0 (/ (pow l 2.0) Om))) U))))
     (if (<= t_3 2e+149)
       (sqrt (* t_2 t_4))
       (if (<= t_3 INFINITY)
         (* (sqrt t_2) (sqrt t_4))
         (*
          l
          (sqrt
           (*
            (+ (* (- U U*) (/ n (* Om Om))) (/ 2.0 Om))
            (* -2.0 (* U n))))))))))

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 t_1 = (n * pow((l / Om), 2.0)) * (U - U_42_);
	double t_2 = (2.0 * n) * U;
	double t_3 = sqrt((t_2 * ((t - (2.0 * ((l * l) / Om))) - t_1)));
	double t_4 = (t - (2.0 * (l * (l / Om)))) - t_1;
	double tmp;
	if (t_3 <= 0.0) {
		tmp = sqrt((2.0 * (n * ((t - (2.0 * (pow(l, 2.0) / Om))) * U))));
	} else if (t_3 <= 2e+149) {
		tmp = sqrt((t_2 * t_4));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt(t_2) * sqrt(t_4);
	} else {
		tmp = l * sqrt(((((U - U_42_) * (n / (Om * Om))) + (2.0 / Om)) * (-2.0 * (U * n))));
	}
	return tmp;
}

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	return Math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * Math.pow((l / Om), 2.0)) * (U - U_42_)))));
}

↓

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (n * Math.pow((l / Om), 2.0)) * (U - U_42_);
	double t_2 = (2.0 * n) * U;
	double t_3 = Math.sqrt((t_2 * ((t - (2.0 * ((l * l) / Om))) - t_1)));
	double t_4 = (t - (2.0 * (l * (l / Om)))) - t_1;
	double tmp;
	if (t_3 <= 0.0) {
		tmp = Math.sqrt((2.0 * (n * ((t - (2.0 * (Math.pow(l, 2.0) / Om))) * U))));
	} else if (t_3 <= 2e+149) {
		tmp = Math.sqrt((t_2 * t_4));
	} else if (t_3 <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt(t_2) * Math.sqrt(t_4);
	} else {
		tmp = l * Math.sqrt(((((U - U_42_) * (n / (Om * Om))) + (2.0 / Om)) * (-2.0 * (U * n))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	return math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * math.pow((l / Om), 2.0)) * (U - U_42_)))))

↓

def code(n, U, t, l, Om, U_42_):
	t_1 = (n * math.pow((l / Om), 2.0)) * (U - U_42_)
	t_2 = (2.0 * n) * U
	t_3 = math.sqrt((t_2 * ((t - (2.0 * ((l * l) / Om))) - t_1)))
	t_4 = (t - (2.0 * (l * (l / Om)))) - t_1
	tmp = 0
	if t_3 <= 0.0:
		tmp = math.sqrt((2.0 * (n * ((t - (2.0 * (math.pow(l, 2.0) / Om))) * U))))
	elif t_3 <= 2e+149:
		tmp = math.sqrt((t_2 * t_4))
	elif t_3 <= math.inf:
		tmp = math.sqrt(t_2) * math.sqrt(t_4)
	else:
		tmp = l * math.sqrt(((((U - U_42_) * (n / (Om * Om))) + (2.0 / Om)) * (-2.0 * (U * n))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	return sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) - Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U - U_42_)))))
end

↓

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U - U_42_))
	t_2 = Float64(Float64(2.0 * n) * U)
	t_3 = sqrt(Float64(t_2 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) - t_1)))
	t_4 = Float64(Float64(t - Float64(2.0 * Float64(l * Float64(l / Om)))) - t_1)
	tmp = 0.0
	if (t_3 <= 0.0)
		tmp = sqrt(Float64(2.0 * Float64(n * Float64(Float64(t - Float64(2.0 * Float64((l ^ 2.0) / Om))) * U))));
	elseif (t_3 <= 2e+149)
		tmp = sqrt(Float64(t_2 * t_4));
	elseif (t_3 <= Inf)
		tmp = Float64(sqrt(t_2) * sqrt(t_4));
	else
		tmp = Float64(l * sqrt(Float64(Float64(Float64(Float64(U - U_42_) * Float64(n / Float64(Om * Om))) + Float64(2.0 / Om)) * Float64(-2.0 * Float64(U * n)))));
	end
	return tmp
end

MATLAB

function tmp = code(n, U, t, l, Om, U_42_)
	tmp = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * ((l / Om) ^ 2.0)) * (U - U_42_)))));
end

↓

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = (n * ((l / Om) ^ 2.0)) * (U - U_42_);
	t_2 = (2.0 * n) * U;
	t_3 = sqrt((t_2 * ((t - (2.0 * ((l * l) / Om))) - t_1)));
	t_4 = (t - (2.0 * (l * (l / Om)))) - t_1;
	tmp = 0.0;
	if (t_3 <= 0.0)
		tmp = sqrt((2.0 * (n * ((t - (2.0 * ((l ^ 2.0) / Om))) * U))));
	elseif (t_3 <= 2e+149)
		tmp = sqrt((t_2 * t_4));
	elseif (t_3 <= Inf)
		tmp = sqrt(t_2) * sqrt(t_4);
	else
		tmp = l * sqrt(((((U - U_42_) * (n / (Om * Om))) + (2.0 / Om)) * (-2.0 * (U * n))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

↓

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(t$95$2 * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[(t - N[(2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]}, If[LessEqual[t$95$3, 0.0], N[Sqrt[N[(2.0 * N[(n * N[(N[(t - N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$3, 2e+149], N[Sqrt[N[(t$95$2 * t$95$4), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[Sqrt[t$95$2], $MachinePrecision] * N[Sqrt[t$95$4], $MachinePrecision]), $MachinePrecision], N[(l * N[Sqrt[N[(N[(N[(N[(U - U$42$), $MachinePrecision] * N[(n / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 / Om), $MachinePrecision]), $MachinePrecision] * N[(-2.0 * N[(U * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (sqrt.f64 (*.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 56.4

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

   2. Taylor expanded in n around 0 38.8

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

   if 0.0 < (sqrt.f64 (*.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*))))) < 2.0000000000000001e149

   1. Initial program 1.7

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

   2. Applied egg-rr 1.7

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

   if 2.0000000000000001e149 < (sqrt.f64 (*.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 63.0

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

   2. Applied egg-rr 49.8

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

   if +inf.0 < (sqrt.f64 (*.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 61.4

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

   3. Taylor expanded in l around inf 53.9

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

   4. Simplified 53.7

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

4. Final simplification 27.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \leq 0:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot U\right)\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \leq 2 \cdot 10^{+149}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \leq \infty:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \sqrt{\left(\left(U - U*\right) \cdot \frac{n}{Om \cdot Om} + \frac{2}{Om}\right) \cdot \left(-2 \cdot \left(U \cdot n\right)\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2022203 
(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*))))))