Maksimov and Kolovsky, Equation (3)

Average Error: 17.7 → 9.2

Time: 9.3s

Precision: binary64

Math

\[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

↓

\[\begin{array}{l} t_0 := J \cdot \cos \left(\frac{K}{2}\right)\\ t_1 := -2 \cdot \left(t_0 \cdot \mathsf{hypot}\left(1, \frac{U}{2 \cdot t_0}\right)\right)\\ \mathbf{if}\;J \leq -1.6428318280514014 \cdot 10^{-195}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;J \leq -1.06 \cdot 10^{-265}:\\ \;\;\;\;-2 \cdot \mathsf{fma}\left(0.5, U, J \cdot \left(\frac{J}{U} \cdot {\cos \left(K \cdot 0.5\right)}^{2}\right)\right)\\ \mathbf{elif}\;J \leq 9.087827870608433 \cdot 10^{-266}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (*
  (* (* -2.0 J) (cos (/ K 2.0)))
  (sqrt (+ 1.0 (pow (/ U (* (* 2.0 J) (cos (/ K 2.0)))) 2.0)))))

↓

(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (* J (cos (/ K 2.0))))
        (t_1 (* -2.0 (* t_0 (hypot 1.0 (/ U (* 2.0 t_0)))))))
   (if (<= J -1.6428318280514014e-195)
     t_1
     (if (<= J -1.06e-265)
       (* -2.0 (fma 0.5 U (* J (* (/ J U) (pow (cos (* K 0.5)) 2.0)))))
       (if (<= J 9.087827870608433e-266) (* -2.0 (* U -0.5)) t_1)))))

C

double code(double J, double K, double U) {
	return ((-2.0 * J) * cos((K / 2.0))) * sqrt((1.0 + pow((U / ((2.0 * J) * cos((K / 2.0)))), 2.0)));
}

↓

double code(double J, double K, double U) {
	double t_0 = J * cos((K / 2.0));
	double t_1 = -2.0 * (t_0 * hypot(1.0, (U / (2.0 * t_0))));
	double tmp;
	if (J <= -1.6428318280514014e-195) {
		tmp = t_1;
	} else if (J <= -1.06e-265) {
		tmp = -2.0 * fma(0.5, U, (J * ((J / U) * pow(cos((K * 0.5)), 2.0))));
	} else if (J <= 9.087827870608433e-266) {
		tmp = -2.0 * (U * -0.5);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(J, K, U)
	return Float64(Float64(Float64(-2.0 * J) * cos(Float64(K / 2.0))) * sqrt(Float64(1.0 + (Float64(U / Float64(Float64(2.0 * J) * cos(Float64(K / 2.0)))) ^ 2.0))))
end

↓

function code(J, K, U)
	t_0 = Float64(J * cos(Float64(K / 2.0)))
	t_1 = Float64(-2.0 * Float64(t_0 * hypot(1.0, Float64(U / Float64(2.0 * t_0)))))
	tmp = 0.0
	if (J <= -1.6428318280514014e-195)
		tmp = t_1;
	elseif (J <= -1.06e-265)
		tmp = Float64(-2.0 * fma(0.5, U, Float64(J * Float64(Float64(J / U) * (cos(Float64(K * 0.5)) ^ 2.0)))));
	elseif (J <= 9.087827870608433e-266)
		tmp = Float64(-2.0 * Float64(U * -0.5));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[J_, K_, U_] := N[(N[(N[(-2.0 * J), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U / N[(N[(2.0 * J), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

↓

code[J_, K_, U_] := Block[{t$95$0 = N[(J * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-2.0 * N[(t$95$0 * N[Sqrt[1.0 ^ 2 + N[(U / N[(2.0 * t$95$0), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[J, -1.6428318280514014e-195], t$95$1, If[LessEqual[J, -1.06e-265], N[(-2.0 * N[(0.5 * U + N[(J * N[(N[(J / U), $MachinePrecision] * N[Power[N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[J, 9.087827870608433e-266], N[(-2.0 * N[(U * -0.5), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if J < -1.64283182805140143e-195 or 9.0878278706084327e-266 < J

   1. Initial program 14.1

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   2. Simplified 5.5

      \[\leadsto \color{blue}{-2 \cdot \left(\mathsf{hypot}\left(1, \frac{U}{2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)}\right) \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)\right)} \]

   if -1.64283182805140143e-195 < J < -1.0600000000000001e-265

   1. Initial program 38.4

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   2. Simplified 23.8

      \[\leadsto \color{blue}{-2 \cdot \left(\mathsf{hypot}\left(1, \frac{U}{2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)}\right) \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)\right)} \]

   3. Applied egg-rr 23.8

      \[\leadsto -2 \cdot \color{blue}{\left(0 + J \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\cos \left(K \cdot 0.5\right) \cdot \left(2 \cdot J\right)}\right)\right)\right)} \]

   4. Taylor expanded in J around 0 36.5

      \[\leadsto -2 \cdot \left(0 + \color{blue}{\left(0.5 \cdot U + \frac{{\cos \left(0.5 \cdot K\right)}^{2} \cdot {J}^{2}}{U}\right)}\right) \]

   5. Simplified 36.4

      \[\leadsto -2 \cdot \left(0 + \color{blue}{\mathsf{fma}\left(0.5, U, \left(\frac{J}{U} \cdot {\cos \left(0.5 \cdot K\right)}^{2}\right) \cdot J\right)}\right) \]

   if -1.0600000000000001e-265 < J < 9.0878278706084327e-266

   1. Initial program 45.0

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   2. Simplified 29.7

      \[\leadsto \color{blue}{-2 \cdot \left(\mathsf{hypot}\left(1, \frac{U}{2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)}\right) \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)\right)} \]

   3. Applied egg-rr 30.5

      \[\leadsto -2 \cdot \left(\mathsf{hypot}\left(1, \frac{U}{2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)}\right) \cdot \color{blue}{{\left(\sqrt[3]{J \cdot \cos \left(K \cdot 0.5\right)}\right)}^{3}}\right) \]

   4. Applied egg-rr 30.0

      \[\leadsto -2 \cdot \left(\mathsf{hypot}\left(1, \frac{U}{2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)}\right) \cdot \color{blue}{\left({\left(\sqrt[3]{\cos \left(K \cdot 0.5\right)}\right)}^{2} \cdot \left(\sqrt[3]{\cos \left(K \cdot 0.5\right)} \cdot J\right)\right)}\right) \]

   5. Taylor expanded in U around -inf 32.7

      \[\leadsto -2 \cdot \color{blue}{\left(-0.5 \cdot U\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 9.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -1.6428318280514014 \cdot 10^{-195}:\\ \;\;\;\;-2 \cdot \left(\left(J \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)}\right)\right)\\ \mathbf{elif}\;J \leq -1.06 \cdot 10^{-265}:\\ \;\;\;\;-2 \cdot \mathsf{fma}\left(0.5, U, J \cdot \left(\frac{J}{U} \cdot {\cos \left(K \cdot 0.5\right)}^{2}\right)\right)\\ \mathbf{elif}\;J \leq 9.087827870608433 \cdot 10^{-266}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\left(J \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)}\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022209 
(FPCore (J K U)
  :name "Maksimov and Kolovsky, Equation (3)"
  :precision binary64
  (* (* (* -2.0 J) (cos (/ K 2.0))) (sqrt (+ 1.0 (pow (/ U (* (* 2.0 J) (cos (/ K 2.0)))) 2.0)))))