Maksimov and Kolovsky, Equation (3)

Average Error: 18.1 → 9.5

Time: 11.4s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := J \cdot \cos \left(\frac{K}{2}\right)\\ t_1 := -2 \cdot \left(t_0 \cdot \mathsf{hypot}\left(1, \frac{U}{2 \cdot t_0}\right)\right)\\ t_2 := -2 \cdot \left(U \cdot -0.5\right)\\ \mathbf{if}\;J \leq -8.148302528701179 \cdot 10^{-170}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;J \leq -2.151444545834551 \cdot 10^{-266}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;J \leq -1.3034423682823207 \cdot 10^{-306}:\\ \;\;\;\;-2 \cdot \left(U \cdot 0.5\right)\\ \mathbf{elif}\;J \leq 1.2148140507579032 \cdot 10^{-235}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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))))))
        (t_2 (* -2.0 (* U -0.5))))
   (if (<= J -8.148302528701179e-170)
     t_1
     (if (<= J -2.151444545834551e-266)
       t_2
       (if (<= J -1.3034423682823207e-306)
         (* -2.0 (* U 0.5))
         (if (<= J 1.2148140507579032e-235) t_2 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 t_2 = -2.0 * (U * -0.5);
	double tmp;
	if (J <= -8.148302528701179e-170) {
		tmp = t_1;
	} else if (J <= -2.151444545834551e-266) {
		tmp = t_2;
	} else if (J <= -1.3034423682823207e-306) {
		tmp = -2.0 * (U * 0.5);
	} else if (J <= 1.2148140507579032e-235) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

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

↓

public static double code(double J, double K, double U) {
	double t_0 = J * Math.cos((K / 2.0));
	double t_1 = -2.0 * (t_0 * Math.hypot(1.0, (U / (2.0 * t_0))));
	double t_2 = -2.0 * (U * -0.5);
	double tmp;
	if (J <= -8.148302528701179e-170) {
		tmp = t_1;
	} else if (J <= -2.151444545834551e-266) {
		tmp = t_2;
	} else if (J <= -1.3034423682823207e-306) {
		tmp = -2.0 * (U * 0.5);
	} else if (J <= 1.2148140507579032e-235) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(J, K, U):
	return ((-2.0 * J) * math.cos((K / 2.0))) * math.sqrt((1.0 + math.pow((U / ((2.0 * J) * math.cos((K / 2.0)))), 2.0)))

↓

def code(J, K, U):
	t_0 = J * math.cos((K / 2.0))
	t_1 = -2.0 * (t_0 * math.hypot(1.0, (U / (2.0 * t_0))))
	t_2 = -2.0 * (U * -0.5)
	tmp = 0
	if J <= -8.148302528701179e-170:
		tmp = t_1
	elif J <= -2.151444545834551e-266:
		tmp = t_2
	elif J <= -1.3034423682823207e-306:
		tmp = -2.0 * (U * 0.5)
	elif J <= 1.2148140507579032e-235:
		tmp = t_2
	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)))))
	t_2 = Float64(-2.0 * Float64(U * -0.5))
	tmp = 0.0
	if (J <= -8.148302528701179e-170)
		tmp = t_1;
	elseif (J <= -2.151444545834551e-266)
		tmp = t_2;
	elseif (J <= -1.3034423682823207e-306)
		tmp = Float64(-2.0 * Float64(U * 0.5));
	elseif (J <= 1.2148140507579032e-235)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp = code(J, K, U)
	tmp = ((-2.0 * J) * cos((K / 2.0))) * sqrt((1.0 + ((U / ((2.0 * J) * cos((K / 2.0)))) ^ 2.0)));
end

↓

function tmp_2 = code(J, K, U)
	t_0 = J * cos((K / 2.0));
	t_1 = -2.0 * (t_0 * hypot(1.0, (U / (2.0 * t_0))));
	t_2 = -2.0 * (U * -0.5);
	tmp = 0.0;
	if (J <= -8.148302528701179e-170)
		tmp = t_1;
	elseif (J <= -2.151444545834551e-266)
		tmp = t_2;
	elseif (J <= -1.3034423682823207e-306)
		tmp = -2.0 * (U * 0.5);
	elseif (J <= 1.2148140507579032e-235)
		tmp = t_2;
	else
		tmp = t_1;
	end
	tmp_2 = 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]}, Block[{t$95$2 = N[(-2.0 * N[(U * -0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[J, -8.148302528701179e-170], t$95$1, If[LessEqual[J, -2.151444545834551e-266], t$95$2, If[LessEqual[J, -1.3034423682823207e-306], N[(-2.0 * N[(U * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[J, 1.2148140507579032e-235], t$95$2, t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

2. if J < -8.14830252870117906e-170 or 1.21481405075790322e-235 < J

   1. Initial program 13.2

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

   2. Simplified 4.7

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

   if -8.14830252870117906e-170 < J < -2.1514445458345509e-266 or -1.30344236828232073e-306 < J < 1.21481405075790322e-235

   1. Initial program 42.5

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

   2. Simplified 25.1

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

   3. Taylor expanded in U around -inf 34.3

      \[\leadsto -2 \cdot \color{blue}{\left(-0.5 \cdot U\right)} \]

   if -2.1514445458345509e-266 < J < -1.30344236828232073e-306

   1. Initial program 45.4

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

   2. Simplified 29.2

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

   3. Taylor expanded in U around inf 31.8

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

4. Final simplification 9.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -8.148302528701179 \cdot 10^{-170}:\\ \;\;\;\;-2 \cdot \left(\left(J \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)}\right)\right)\\ \mathbf{elif}\;J \leq -2.151444545834551 \cdot 10^{-266}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5\right)\\ \mathbf{elif}\;J \leq -1.3034423682823207 \cdot 10^{-306}:\\ \;\;\;\;-2 \cdot \left(U \cdot 0.5\right)\\ \mathbf{elif}\;J \leq 1.2148140507579032 \cdot 10^{-235}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\left(J \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)}\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022185 
(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)))))