Maksimov and Kolovsky, Equation (3)

Average Error: 18.1 → 9.4

Time: 13.0s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := J \cdot \cos \left(\frac{K}{2}\right)\\ t_1 := -2 \cdot \left(t_0 \cdot \mathsf{hypot}\left(1, \frac{U}{2 \cdot t_0}\right)\right)\\ \mathbf{if}\;J \leq -6.59631123688703 \cdot 10^{-263}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;J \leq 1.815890624493559 \cdot 10^{-248}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5\right)\\ \mathbf{elif}\;J \leq 1.3800459910238586 \cdot 10^{-203}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;J \leq 5.3724426798428875 \cdot 10^{-127}:\\ \;\;\;\;-2 \cdot \left(U \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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 -6.59631123688703e-263)
     t_1
     (if (<= J 1.815890624493559e-248)
       (* -2.0 (* U -0.5))
       (if (<= J 1.3800459910238586e-203)
         t_1
         (if (<= J 5.3724426798428875e-127) (* -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 <= -6.59631123688703e-263) {
		tmp = t_1;
	} else if (J <= 1.815890624493559e-248) {
		tmp = -2.0 * (U * -0.5);
	} else if (J <= 1.3800459910238586e-203) {
		tmp = t_1;
	} else if (J <= 5.3724426798428875e-127) {
		tmp = -2.0 * (U * 0.5);
	} 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 tmp;
	if (J <= -6.59631123688703e-263) {
		tmp = t_1;
	} else if (J <= 1.815890624493559e-248) {
		tmp = -2.0 * (U * -0.5);
	} else if (J <= 1.3800459910238586e-203) {
		tmp = t_1;
	} else if (J <= 5.3724426798428875e-127) {
		tmp = -2.0 * (U * 0.5);
	} 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))))
	tmp = 0
	if J <= -6.59631123688703e-263:
		tmp = t_1
	elif J <= 1.815890624493559e-248:
		tmp = -2.0 * (U * -0.5)
	elif J <= 1.3800459910238586e-203:
		tmp = t_1
	elif J <= 5.3724426798428875e-127:
		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 <= -6.59631123688703e-263)
		tmp = t_1;
	elseif (J <= 1.815890624493559e-248)
		tmp = Float64(-2.0 * Float64(U * -0.5));
	elseif (J <= 1.3800459910238586e-203)
		tmp = t_1;
	elseif (J <= 5.3724426798428875e-127)
		tmp = Float64(-2.0 * Float64(U * 0.5));
	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))));
	tmp = 0.0;
	if (J <= -6.59631123688703e-263)
		tmp = t_1;
	elseif (J <= 1.815890624493559e-248)
		tmp = -2.0 * (U * -0.5);
	elseif (J <= 1.3800459910238586e-203)
		tmp = t_1;
	elseif (J <= 5.3724426798428875e-127)
		tmp = -2.0 * (U * 0.5);
	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]}, If[LessEqual[J, -6.59631123688703e-263], t$95$1, If[LessEqual[J, 1.815890624493559e-248], N[(-2.0 * N[(U * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[J, 1.3800459910238586e-203], t$95$1, If[LessEqual[J, 5.3724426798428875e-127], N[(-2.0 * N[(U * 0.5), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Error

Bits error versus J

Bits error versus K

Bits error versus U

Derivation

1. Split input into 3 regimes

2. if J < -6.5963112368870296e-263 or 1.81589062449355909e-248 < J < 1.38004599102385856e-203 or 5.3724426798428875e-127 < J

   1. Initial program 14.5

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

   2. Simplified 5.1

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

   if -6.5963112368870296e-263 < J < 1.81589062449355909e-248

   1. Initial program 44.9

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

   2. Simplified 29.5

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

   3. Taylor expanded in U around -inf 32.3

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

   if 1.38004599102385856e-203 < J < 5.3724426798428875e-127

   1. Initial program 31.0

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

   2. Simplified 17.4

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

   3. Taylor expanded in U around inf 38.9

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

4. Final simplification 9.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -6.59631123688703 \cdot 10^{-263}:\\ \;\;\;\;-2 \cdot \left(\left(J \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)}\right)\right)\\ \mathbf{elif}\;J \leq 1.815890624493559 \cdot 10^{-248}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5\right)\\ \mathbf{elif}\;J \leq 1.3800459910238586 \cdot 10^{-203}:\\ \;\;\;\;-2 \cdot \left(\left(J \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)}\right)\right)\\ \mathbf{elif}\;J \leq 5.3724426798428875 \cdot 10^{-127}:\\ \;\;\;\;-2 \cdot \left(U \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\left(J \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)}\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022159 
(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)))))