Maksimov and Kolovsky, Equation (3)

Average Error: 18.0 → 8.2

Time: 11.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;U \leq -2.1573275483081895 \cdot 10^{+221}:\\ \;\;\;\;-2 \cdot \left(U \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\mathsf{hypot}\left(1, \frac{U}{\cos \left(0.5 \cdot K\right) \cdot \left(2 \cdot J\right)}\right) \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)\right)\\ \end{array} \]

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
 (if (<= U -2.1573275483081895e+221)
   (* -2.0 (* U 0.5))
   (*
    -2.0
    (*
     (hypot 1.0 (/ U (* (cos (* 0.5 K)) (* 2.0 J))))
     (* J (cos (/ K 2.0)))))))

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 tmp;
	if (U <= -2.1573275483081895e+221) {
		tmp = -2.0 * (U * 0.5);
	} else {
		tmp = -2.0 * (hypot(1.0, (U / (cos((0.5 * K)) * (2.0 * J)))) * (J * cos((K / 2.0))));
	}
	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 tmp;
	if (U <= -2.1573275483081895e+221) {
		tmp = -2.0 * (U * 0.5);
	} else {
		tmp = -2.0 * (Math.hypot(1.0, (U / (Math.cos((0.5 * K)) * (2.0 * J)))) * (J * Math.cos((K / 2.0))));
	}
	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):
	tmp = 0
	if U <= -2.1573275483081895e+221:
		tmp = -2.0 * (U * 0.5)
	else:
		tmp = -2.0 * (math.hypot(1.0, (U / (math.cos((0.5 * K)) * (2.0 * J)))) * (J * math.cos((K / 2.0))))
	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)
	tmp = 0.0
	if (U <= -2.1573275483081895e+221)
		tmp = Float64(-2.0 * Float64(U * 0.5));
	else
		tmp = Float64(-2.0 * Float64(hypot(1.0, Float64(U / Float64(cos(Float64(0.5 * K)) * Float64(2.0 * J)))) * Float64(J * cos(Float64(K / 2.0)))));
	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)
	tmp = 0.0;
	if (U <= -2.1573275483081895e+221)
		tmp = -2.0 * (U * 0.5);
	else
		tmp = -2.0 * (hypot(1.0, (U / (cos((0.5 * K)) * (2.0 * J)))) * (J * cos((K / 2.0))));
	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_] := If[LessEqual[U, -2.1573275483081895e+221], N[(-2.0 * N[(U * 0.5), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(N[Sqrt[1.0 ^ 2 + N[(U / N[(N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] * N[(2.0 * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] * N[(J * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Error

Bits error versus J

Bits error versus K

Bits error versus U

Derivation

1. Split input into 2 regimes

2. if U < -2.1573275483081895e221

   1. Initial program 41.4

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

   2. Simplified 25.6

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

   3. Taylor expanded in U around inf 32.7

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

   if -2.1573275483081895e221 < U

   1. Initial program 16.2

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

   2. Simplified 6.3

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

   3. Applied egg-rr 6.3

      \[\leadsto -2 \cdot \left(\mathsf{hypot}\left(1, \color{blue}{1 \cdot \frac{U}{\cos \left(K \cdot 0.5\right) \cdot \left(2 \cdot J\right)}}\right) \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 8.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq -2.1573275483081895 \cdot 10^{+221}:\\ \;\;\;\;-2 \cdot \left(U \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\mathsf{hypot}\left(1, \frac{U}{\cos \left(0.5 \cdot K\right) \cdot \left(2 \cdot J\right)}\right) \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022153 
(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)))))