Average Error: 18.4 → 8.3
Time: 17.1s
Precision: binary64
\[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
↓
\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \frac{U}{t_0 \cdot \left(J \cdot 2\right)}\\ t_2 := \left(-2 \cdot J\right) \cdot t_0\\ \mathbf{if}\;t_2 \cdot \sqrt{1 + {t_1}^{2}} \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \mathsf{hypot}\left(1, t_1\right)\\ \end{array} \]
(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 (cos (/ K 2.0)))
        (t_1 (/ U (* t_0 (* J 2.0))))
        (t_2 (* (* -2.0 J) t_0)))
   (if (<= (* t_2 (sqrt (+ 1.0 (pow t_1 2.0)))) (- INFINITY))
     (- U)
     (* t_2 (hypot 1.0 t_1)))))
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 = cos((K / 2.0));
	double t_1 = U / (t_0 * (J * 2.0));
	double t_2 = (-2.0 * J) * t_0;
	double tmp;
	if ((t_2 * sqrt((1.0 + pow(t_1, 2.0)))) <= -((double) INFINITY)) {
		tmp = -U;
	} else {
		tmp = t_2 * hypot(1.0, t_1);
	}
	return tmp;
}

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 = Math.cos((K / 2.0));
	double t_1 = U / (t_0 * (J * 2.0));
	double t_2 = (-2.0 * J) * t_0;
	double tmp;
	if ((t_2 * Math.sqrt((1.0 + Math.pow(t_1, 2.0)))) <= -Double.POSITIVE_INFINITY) {
		tmp = -U;
	} else {
		tmp = t_2 * Math.hypot(1.0, t_1);
	}
	return tmp;
}

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 = math.cos((K / 2.0))
	t_1 = U / (t_0 * (J * 2.0))
	t_2 = (-2.0 * J) * t_0
	tmp = 0
	if (t_2 * math.sqrt((1.0 + math.pow(t_1, 2.0)))) <= -math.inf:
		tmp = -U
	else:
		tmp = t_2 * math.hypot(1.0, t_1)
	return tmp

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 = cos(Float64(K / 2.0))
	t_1 = Float64(U / Float64(t_0 * Float64(J * 2.0)))
	t_2 = Float64(Float64(-2.0 * J) * t_0)
	tmp = 0.0
	if (Float64(t_2 * sqrt(Float64(1.0 + (t_1 ^ 2.0)))) <= Float64(-Inf))
		tmp = Float64(-U);
	else
		tmp = Float64(t_2 * hypot(1.0, t_1));
	end
	return tmp
end

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 = cos((K / 2.0));
	t_1 = U / (t_0 * (J * 2.0));
	t_2 = (-2.0 * J) * t_0;
	tmp = 0.0;
	if ((t_2 * sqrt((1.0 + (t_1 ^ 2.0)))) <= -Inf)
		tmp = -U;
	else
		tmp = t_2 * hypot(1.0, t_1);
	end
	tmp_2 = tmp;
end

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[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(U / N[(t$95$0 * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[N[(t$95$2 * N[Sqrt[N[(1.0 + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], (-Infinity)], (-U), N[(t$95$2 * N[Sqrt[1.0 ^ 2 + t$95$1 ^ 2], $MachinePrecision]), $MachinePrecision]]]]]

\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 := \cos \left(\frac{K}{2}\right)\\
t_1 := \frac{U}{t_0 \cdot \left(J \cdot 2\right)}\\
t_2 := \left(-2 \cdot J\right) \cdot t_0\\
\mathbf{if}\;t_2 \cdot \sqrt{1 + {t_1}^{2}} \leq -\infty:\\
\;\;\;\;-U\\

\mathbf{else}:\\
\;\;\;\;t_2 \cdot \mathsf{hypot}\left(1, t_1\right)\\


\end{array}

Error

Bits error versus J

Bits error versus K

Bits error versus U

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if (*.f64 (*.f64 (*.f64 -2 J) (cos.f64 (/.f64 K 2))) (sqrt.f64 (+.f64 1 (pow.f64 (/.f64 U (*.f64 (*.f64 2 J) (cos.f64 (/.f64 K 2)))) 2)))) < -inf.0

    1. Initial program 64.0

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

    2. Simplified29.5

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

    3. Taylor expanded in J around 0 30.5

      \[\leadsto \color{blue}{-1 \cdot U} \]

    4. Simplified30.5

      \[\leadsto \color{blue}{-U} \]

    if -inf.0 < (*.f64 (*.f64 (*.f64 -2 J) (cos.f64 (/.f64 K 2))) (sqrt.f64 (+.f64 1 (pow.f64 (/.f64 U (*.f64 (*.f64 2 J) (cos.f64 (/.f64 K 2)))) 2))))

    1. Initial program 10.9

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

    2. Simplified4.6

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

  4. Final simplification8.3

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

Reproduce

herbie shell --seed 2022134 
(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)))))