Maksimov and Kolovsky, Equation (3)

Percentage Accurate: 73.3% → 99.5%

Time: 23.7s

Alternatives: 10

Speedup: 1.9×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \left(\left(-2 \cdot J\right) \cdot t_0\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot t_0}\right)}^{2}} \end{array} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (* (* (* -2.0 J) t_0) (sqrt (+ 1.0 (pow (/ U (* (* 2.0 J) t_0)) 2.0))))))

C

double code(double J, double K, double U) {
	double t_0 = cos((K / 2.0));
	return ((-2.0 * J) * t_0) * sqrt((1.0 + pow((U / ((2.0 * J) * t_0)), 2.0)));
}

Fortran

real(8) function code(j, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    t_0 = cos((k / 2.0d0))
    code = (((-2.0d0) * j) * t_0) * sqrt((1.0d0 + ((u / ((2.0d0 * j) * t_0)) ** 2.0d0)))
end function

Java

public static double code(double J, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	return ((-2.0 * J) * t_0) * Math.sqrt((1.0 + Math.pow((U / ((2.0 * J) * t_0)), 2.0)));
}

Python

def code(J, K, U):
	t_0 = math.cos((K / 2.0))
	return ((-2.0 * J) * t_0) * math.sqrt((1.0 + math.pow((U / ((2.0 * J) * t_0)), 2.0)))

Julia

function code(J, K, U)
	t_0 = cos(Float64(K / 2.0))
	return Float64(Float64(Float64(-2.0 * J) * t_0) * sqrt(Float64(1.0 + (Float64(U / Float64(Float64(2.0 * J) * t_0)) ^ 2.0))))
end

MATLAB

function tmp = code(J, K, U)
	t_0 = cos((K / 2.0));
	tmp = ((-2.0 * J) * t_0) * sqrt((1.0 + ((U / ((2.0 * J) * t_0)) ^ 2.0)));
end

Wolfram

code[J_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U / N[(N[(2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Initial Program: 73.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \left(\left(-2 \cdot J\right) \cdot t_0\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot t_0}\right)}^{2}} \end{array} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (* (* (* -2.0 J) t_0) (sqrt (+ 1.0 (pow (/ U (* (* 2.0 J) t_0)) 2.0))))))

C

double code(double J, double K, double U) {
	double t_0 = cos((K / 2.0));
	return ((-2.0 * J) * t_0) * sqrt((1.0 + pow((U / ((2.0 * J) * t_0)), 2.0)));
}

Fortran

real(8) function code(j, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    t_0 = cos((k / 2.0d0))
    code = (((-2.0d0) * j) * t_0) * sqrt((1.0d0 + ((u / ((2.0d0 * j) * t_0)) ** 2.0d0)))
end function

Java

public static double code(double J, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	return ((-2.0 * J) * t_0) * Math.sqrt((1.0 + Math.pow((U / ((2.0 * J) * t_0)), 2.0)));
}

Python

def code(J, K, U):
	t_0 = math.cos((K / 2.0))
	return ((-2.0 * J) * t_0) * math.sqrt((1.0 + math.pow((U / ((2.0 * J) * t_0)), 2.0)))

Julia

function code(J, K, U)
	t_0 = cos(Float64(K / 2.0))
	return Float64(Float64(Float64(-2.0 * J) * t_0) * sqrt(Float64(1.0 + (Float64(U / Float64(Float64(2.0 * J) * t_0)) ^ 2.0))))
end

MATLAB

function tmp = code(J, K, U)
	t_0 = cos((K / 2.0));
	tmp = ((-2.0 * J) * t_0) * sqrt((1.0 + ((U / ((2.0 * J) * t_0)) ^ 2.0)));
end

Wolfram

code[J_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U / N[(N[(2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Alternative 1: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\right) \cdot t_0\right) \cdot \sqrt{1 + {\left(\frac{U}{t_0 \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ t_2 := J \cdot t_0\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;t_1 \leq 10^{+303}:\\ \;\;\;\;-2 \cdot \left(t_2 \cdot \mathsf{hypot}\left(1, \frac{U}{2 \cdot t_2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* (* -2.0 J) t_0)
          (sqrt (+ 1.0 (pow (/ U (* t_0 (* J 2.0))) 2.0)))))
        (t_2 (* J t_0)))
   (if (<= t_1 (- INFINITY))
     (- U)
     (if (<= t_1 1e+303) (* -2.0 (* t_2 (hypot 1.0 (/ U (* 2.0 t_2))))) U))))

C

U = abs(U);
double code(double J, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * J) * t_0) * sqrt((1.0 + pow((U / (t_0 * (J * 2.0))), 2.0)));
	double t_2 = J * t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -U;
	} else if (t_1 <= 1e+303) {
		tmp = -2.0 * (t_2 * hypot(1.0, (U / (2.0 * t_2))));
	} else {
		tmp = U;
	}
	return tmp;
}

Java

U = Math.abs(U);
public static double code(double J, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = ((-2.0 * J) * t_0) * Math.sqrt((1.0 + Math.pow((U / (t_0 * (J * 2.0))), 2.0)));
	double t_2 = J * t_0;
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = -U;
	} else if (t_1 <= 1e+303) {
		tmp = -2.0 * (t_2 * Math.hypot(1.0, (U / (2.0 * t_2))));
	} else {
		tmp = U;
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	t_0 = math.cos((K / 2.0))
	t_1 = ((-2.0 * J) * t_0) * math.sqrt((1.0 + math.pow((U / (t_0 * (J * 2.0))), 2.0)))
	t_2 = J * t_0
	tmp = 0
	if t_1 <= -math.inf:
		tmp = -U
	elif t_1 <= 1e+303:
		tmp = -2.0 * (t_2 * math.hypot(1.0, (U / (2.0 * t_2))))
	else:
		tmp = U
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * J) * t_0) * sqrt(Float64(1.0 + (Float64(U / Float64(t_0 * Float64(J * 2.0))) ^ 2.0))))
	t_2 = Float64(J * t_0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-U);
	elseif (t_1 <= 1e+303)
		tmp = Float64(-2.0 * Float64(t_2 * hypot(1.0, Float64(U / Float64(2.0 * t_2)))));
	else
		tmp = U;
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	t_0 = cos((K / 2.0));
	t_1 = ((-2.0 * J) * t_0) * sqrt((1.0 + ((U / (t_0 * (J * 2.0))) ^ 2.0)));
	t_2 = J * t_0;
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = -U;
	elseif (t_1 <= 1e+303)
		tmp = -2.0 * (t_2 * hypot(1.0, (U / (2.0 * t_2))));
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U / N[(t$95$0 * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(J * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], (-U), If[LessEqual[t$95$1, 1e+303], N[(-2.0 * N[(t$95$2 * N[Sqrt[1.0 ^ 2 + N[(U / N[(2.0 * t$95$2), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], U]]]]]

Derivation

1. Split input into 3 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 6.3%

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

   2. Taylor expanded in J around 0 41.5%

      \[\leadsto \color{blue}{-1 \cdot U} \]
   3. Step-by-step derivation

      1. neg-mul-1 41.5%

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

   4. Simplified 41.5%

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

   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)))) < 1e303

   1. Initial program 99.8%

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

      1. associate-*l* 99.8%

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

      2. associate-*l* 99.8%

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

      3. unpow2 99.8%

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

      4. sqr-neg 99.8%

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

      5. distribute-frac-neg 99.8%

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

      6. distribute-frac-neg 99.8%

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

      7. unpow2 99.8%

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

   3. Simplified 99.8%

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

   if 1e303 < (*.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 5.1%

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

   2. Taylor expanded in U around -inf 50.4%

      \[\leadsto \color{blue}{U} \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 10^{+303}:\\ \;\;\;\;-2 \cdot \left(\left(J \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternative 2: 88.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;J \leq -2.85 \cdot 10^{-218} \lor \neg \left(J \leq -6.8 \cdot 10^{-304}\right):\\ \;\;\;\;-2 \cdot \left(t_0 \cdot \left(J \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot t_0} \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (or (<= J -2.85e-218) (not (<= J -6.8e-304)))
     (* -2.0 (* t_0 (* J (hypot 1.0 (* (/ U (* J t_0)) 0.5)))))
     U)))

C

U = abs(U);
double code(double J, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if ((J <= -2.85e-218) || !(J <= -6.8e-304)) {
		tmp = -2.0 * (t_0 * (J * hypot(1.0, ((U / (J * t_0)) * 0.5))));
	} else {
		tmp = U;
	}
	return tmp;
}

Java

U = Math.abs(U);
public static double code(double J, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double tmp;
	if ((J <= -2.85e-218) || !(J <= -6.8e-304)) {
		tmp = -2.0 * (t_0 * (J * Math.hypot(1.0, ((U / (J * t_0)) * 0.5))));
	} else {
		tmp = U;
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if (J <= -2.85e-218) or not (J <= -6.8e-304):
		tmp = -2.0 * (t_0 * (J * math.hypot(1.0, ((U / (J * t_0)) * 0.5))))
	else:
		tmp = U
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if ((J <= -2.85e-218) || !(J <= -6.8e-304))
		tmp = Float64(-2.0 * Float64(t_0 * Float64(J * hypot(1.0, Float64(Float64(U / Float64(J * t_0)) * 0.5)))));
	else
		tmp = U;
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	t_0 = cos((K / 2.0));
	tmp = 0.0;
	if ((J <= -2.85e-218) || ~((J <= -6.8e-304)))
		tmp = -2.0 * (t_0 * (J * hypot(1.0, ((U / (J * t_0)) * 0.5))));
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[J, -2.85e-218], N[Not[LessEqual[J, -6.8e-304]], $MachinePrecision]], N[(-2.0 * N[(t$95$0 * N[(J * N[Sqrt[1.0 ^ 2 + N[(N[(U / N[(J * t$95$0), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], U]]

Derivation

1. Split input into 2 regimes

2. if J < -2.8499999999999999e-218 or -6.7999999999999997e-304 < J

   1. Initial program 71.6%

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

      1. associate-*l* 71.6%

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

      2. associate-*l* 71.6%

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

      3. *-commutative 71.6%

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

      4. unpow2 71.6%

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

      5. sqr-neg 71.6%

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

      6. distribute-frac-neg 71.6%

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

      7. distribute-frac-neg 71.6%

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

      8. unpow2 71.6%

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

   3. Simplified 90.3%

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

   if -2.8499999999999999e-218 < J < -6.7999999999999997e-304

   1. Initial program 24.0%

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

   2. Taylor expanded in U around -inf 45.7%

      \[\leadsto \color{blue}{U} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -2.85 \cdot 10^{-218} \lor \neg \left(J \leq -6.8 \cdot 10^{-304}\right):\\ \;\;\;\;-2 \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot \cos \left(\frac{K}{2}\right)} \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternative 3: 77.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} t_0 := -2 \cdot \left(\left(J \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, U \cdot \frac{0.5}{J}\right)\right)\\ \mathbf{if}\;J \leq -9 \cdot 10^{-209}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq -1 \cdot 10^{-310}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5 - J \cdot \frac{J}{U}\right)\\ \mathbf{elif}\;J \leq 4.2 \cdot 10^{-155} \lor \neg \left(J \leq 3.8 \cdot 10^{-107}\right) \land J \leq 2.6 \cdot 10^{-71}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (* -2.0 (* (* J (cos (/ K 2.0))) (hypot 1.0 (* U (/ 0.5 J)))))))
   (if (<= J -9e-209)
     t_0
     (if (<= J -1e-310)
       (* -2.0 (- (* U -0.5) (* J (/ J U))))
       (if (or (<= J 4.2e-155) (and (not (<= J 3.8e-107)) (<= J 2.6e-71)))
         (- U)
         t_0)))))

C

U = abs(U);
double code(double J, double K, double U) {
	double t_0 = -2.0 * ((J * cos((K / 2.0))) * hypot(1.0, (U * (0.5 / J))));
	double tmp;
	if (J <= -9e-209) {
		tmp = t_0;
	} else if (J <= -1e-310) {
		tmp = -2.0 * ((U * -0.5) - (J * (J / U)));
	} else if ((J <= 4.2e-155) || (!(J <= 3.8e-107) && (J <= 2.6e-71))) {
		tmp = -U;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

U = Math.abs(U);
public static double code(double J, double K, double U) {
	double t_0 = -2.0 * ((J * Math.cos((K / 2.0))) * Math.hypot(1.0, (U * (0.5 / J))));
	double tmp;
	if (J <= -9e-209) {
		tmp = t_0;
	} else if (J <= -1e-310) {
		tmp = -2.0 * ((U * -0.5) - (J * (J / U)));
	} else if ((J <= 4.2e-155) || (!(J <= 3.8e-107) && (J <= 2.6e-71))) {
		tmp = -U;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	t_0 = -2.0 * ((J * math.cos((K / 2.0))) * math.hypot(1.0, (U * (0.5 / J))))
	tmp = 0
	if J <= -9e-209:
		tmp = t_0
	elif J <= -1e-310:
		tmp = -2.0 * ((U * -0.5) - (J * (J / U)))
	elif (J <= 4.2e-155) or (not (J <= 3.8e-107) and (J <= 2.6e-71)):
		tmp = -U
	else:
		tmp = t_0
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	t_0 = Float64(-2.0 * Float64(Float64(J * cos(Float64(K / 2.0))) * hypot(1.0, Float64(U * Float64(0.5 / J)))))
	tmp = 0.0
	if (J <= -9e-209)
		tmp = t_0;
	elseif (J <= -1e-310)
		tmp = Float64(-2.0 * Float64(Float64(U * -0.5) - Float64(J * Float64(J / U))));
	elseif ((J <= 4.2e-155) || (!(J <= 3.8e-107) && (J <= 2.6e-71)))
		tmp = Float64(-U);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	t_0 = -2.0 * ((J * cos((K / 2.0))) * hypot(1.0, (U * (0.5 / J))));
	tmp = 0.0;
	if (J <= -9e-209)
		tmp = t_0;
	elseif (J <= -1e-310)
		tmp = -2.0 * ((U * -0.5) - (J * (J / U)));
	elseif ((J <= 4.2e-155) || (~((J <= 3.8e-107)) && (J <= 2.6e-71)))
		tmp = -U;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := Block[{t$95$0 = N[(-2.0 * N[(N[(J * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[(U * N[(0.5 / J), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[J, -9e-209], t$95$0, If[LessEqual[J, -1e-310], N[(-2.0 * N[(N[(U * -0.5), $MachinePrecision] - N[(J * N[(J / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[J, 4.2e-155], And[N[Not[LessEqual[J, 3.8e-107]], $MachinePrecision], LessEqual[J, 2.6e-71]]], (-U), t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if J < -8.9999999999999996e-209 or 4.2000000000000003e-155 < J < 3.8000000000000002e-107 or 2.5999999999999999e-71 < J

   1. Initial program 83.3%

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

      1. associate-*l* 83.3%

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

      2. associate-*l* 83.3%

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

      3. unpow2 83.3%

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

      4. sqr-neg 83.3%

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

      5. distribute-frac-neg 83.3%

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

      6. distribute-frac-neg 83.3%

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

      7. unpow2 83.3%

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

   3. Simplified 96.2%

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

   4. Taylor expanded in K around 0 84.3%

      \[\leadsto -2 \cdot \left(\mathsf{hypot}\left(1, \color{blue}{0.5 \cdot \frac{U}{J}}\right) \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)\right) \]
   5. Step-by-step derivation

      1. associate-*r/ 84.3%

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

      2. *-commutative 84.3%

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

      3. associate-*r/ 84.2%

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

   6. Simplified 84.2%

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

   if -8.9999999999999996e-209 < J < -9.999999999999969e-311

   1. Initial program 26.4%

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

   2. Taylor expanded in K around 0 3.7%

      \[\leadsto \color{blue}{-2 \cdot \left(\sqrt{1 + 0.25 \cdot \frac{{U}^{2}}{{J}^{2}}} \cdot J\right)} \]
   3. Step-by-step derivation

      1. *-commutative 3.7%

         \[\leadsto -2 \cdot \color{blue}{\left(J \cdot \sqrt{1 + 0.25 \cdot \frac{{U}^{2}}{{J}^{2}}}\right)} \]

      2. unpow2 3.7%

         \[\leadsto -2 \cdot \left(J \cdot \sqrt{1 + 0.25 \cdot \frac{\color{blue}{U \cdot U}}{{J}^{2}}}\right) \]

      3. unpow2 3.7%

         \[\leadsto -2 \cdot \left(J \cdot \sqrt{1 + 0.25 \cdot \frac{U \cdot U}{\color{blue}{J \cdot J}}}\right) \]

   4. Simplified 3.7%

      \[\leadsto \color{blue}{-2 \cdot \left(J \cdot \sqrt{1 + 0.25 \cdot \frac{U \cdot U}{J \cdot J}}\right)} \]

   5. Taylor expanded in U around -inf 25.0%

      \[\leadsto -2 \cdot \left(J \cdot \color{blue}{\left(-0.5 \cdot \frac{U}{J} + -1 \cdot \frac{J}{U}\right)}\right) \]

   6. Taylor expanded in J around 0 46.6%

      \[\leadsto -2 \cdot \color{blue}{\left(-1 \cdot \frac{{J}^{2}}{U} + -0.5 \cdot U\right)} \]
   7. Step-by-step derivation

      1. +-commutative 46.6%

         \[\leadsto -2 \cdot \color{blue}{\left(-0.5 \cdot U + -1 \cdot \frac{{J}^{2}}{U}\right)} \]

      2. mul-1-neg 46.6%

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

      3. unsub-neg 46.6%

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

      4. *-commutative 46.6%

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

      5. unpow2 46.6%

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

      6. associate-*r/ 46.6%

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

   8. Simplified 46.6%

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

   if -9.999999999999969e-311 < J < 4.2000000000000003e-155 or 3.8000000000000002e-107 < J < 2.5999999999999999e-71

   1. Initial program 33.2%

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

   2. Taylor expanded in J around 0 32.7%

      \[\leadsto \color{blue}{-1 \cdot U} \]
   3. Step-by-step derivation

      1. neg-mul-1 32.7%

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

   4. Simplified 32.7%

      \[\leadsto \color{blue}{-U} \]
3. Recombined 3 regimes into one program.

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -9 \cdot 10^{-209}:\\ \;\;\;\;-2 \cdot \left(\left(J \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, U \cdot \frac{0.5}{J}\right)\right)\\ \mathbf{elif}\;J \leq -1 \cdot 10^{-310}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5 - J \cdot \frac{J}{U}\right)\\ \mathbf{elif}\;J \leq 4.2 \cdot 10^{-155} \lor \neg \left(J \leq 3.8 \cdot 10^{-107}\right) \land J \leq 2.6 \cdot 10^{-71}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\left(J \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, U \cdot \frac{0.5}{J}\right)\right)\\ \end{array} \]

Alternative 4: 77.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\right) \cdot t_0\right) \cdot \mathsf{hypot}\left(1, 0.5 \cdot \frac{U}{J}\right)\\ \mathbf{if}\;J \leq -1.05 \cdot 10^{-208}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;J \leq -1 \cdot 10^{-310}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5 - J \cdot \frac{J}{U}\right)\\ \mathbf{elif}\;J \leq 9 \cdot 10^{-155}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 9.5 \cdot 10^{-108}:\\ \;\;\;\;-2 \cdot \left(\left(J \cdot t_0\right) \cdot \mathsf{hypot}\left(1, U \cdot \frac{0.5}{J}\right)\right)\\ \mathbf{elif}\;J \leq 4.7 \cdot 10^{-73}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1 (* (* (* -2.0 J) t_0) (hypot 1.0 (* 0.5 (/ U J))))))
   (if (<= J -1.05e-208)
     t_1
     (if (<= J -1e-310)
       (* -2.0 (- (* U -0.5) (* J (/ J U))))
       (if (<= J 9e-155)
         (- U)
         (if (<= J 9.5e-108)
           (* -2.0 (* (* J t_0) (hypot 1.0 (* U (/ 0.5 J)))))
           (if (<= J 4.7e-73) (- U) t_1)))))))

C

U = abs(U);
double code(double J, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * J) * t_0) * hypot(1.0, (0.5 * (U / J)));
	double tmp;
	if (J <= -1.05e-208) {
		tmp = t_1;
	} else if (J <= -1e-310) {
		tmp = -2.0 * ((U * -0.5) - (J * (J / U)));
	} else if (J <= 9e-155) {
		tmp = -U;
	} else if (J <= 9.5e-108) {
		tmp = -2.0 * ((J * t_0) * hypot(1.0, (U * (0.5 / J))));
	} else if (J <= 4.7e-73) {
		tmp = -U;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

U = Math.abs(U);
public static double code(double J, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = ((-2.0 * J) * t_0) * Math.hypot(1.0, (0.5 * (U / J)));
	double tmp;
	if (J <= -1.05e-208) {
		tmp = t_1;
	} else if (J <= -1e-310) {
		tmp = -2.0 * ((U * -0.5) - (J * (J / U)));
	} else if (J <= 9e-155) {
		tmp = -U;
	} else if (J <= 9.5e-108) {
		tmp = -2.0 * ((J * t_0) * Math.hypot(1.0, (U * (0.5 / J))));
	} else if (J <= 4.7e-73) {
		tmp = -U;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	t_0 = math.cos((K / 2.0))
	t_1 = ((-2.0 * J) * t_0) * math.hypot(1.0, (0.5 * (U / J)))
	tmp = 0
	if J <= -1.05e-208:
		tmp = t_1
	elif J <= -1e-310:
		tmp = -2.0 * ((U * -0.5) - (J * (J / U)))
	elif J <= 9e-155:
		tmp = -U
	elif J <= 9.5e-108:
		tmp = -2.0 * ((J * t_0) * math.hypot(1.0, (U * (0.5 / J))))
	elif J <= 4.7e-73:
		tmp = -U
	else:
		tmp = t_1
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * J) * t_0) * hypot(1.0, Float64(0.5 * Float64(U / J))))
	tmp = 0.0
	if (J <= -1.05e-208)
		tmp = t_1;
	elseif (J <= -1e-310)
		tmp = Float64(-2.0 * Float64(Float64(U * -0.5) - Float64(J * Float64(J / U))));
	elseif (J <= 9e-155)
		tmp = Float64(-U);
	elseif (J <= 9.5e-108)
		tmp = Float64(-2.0 * Float64(Float64(J * t_0) * hypot(1.0, Float64(U * Float64(0.5 / J)))));
	elseif (J <= 4.7e-73)
		tmp = Float64(-U);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	t_0 = cos((K / 2.0));
	t_1 = ((-2.0 * J) * t_0) * hypot(1.0, (0.5 * (U / J)));
	tmp = 0.0;
	if (J <= -1.05e-208)
		tmp = t_1;
	elseif (J <= -1e-310)
		tmp = -2.0 * ((U * -0.5) - (J * (J / U)));
	elseif (J <= 9e-155)
		tmp = -U;
	elseif (J <= 9.5e-108)
		tmp = -2.0 * ((J * t_0) * hypot(1.0, (U * (0.5 / J))));
	elseif (J <= 4.7e-73)
		tmp = -U;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[(0.5 * N[(U / J), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[J, -1.05e-208], t$95$1, If[LessEqual[J, -1e-310], N[(-2.0 * N[(N[(U * -0.5), $MachinePrecision] - N[(J * N[(J / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[J, 9e-155], (-U), If[LessEqual[J, 9.5e-108], N[(-2.0 * N[(N[(J * t$95$0), $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[(U * N[(0.5 / J), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[J, 4.7e-73], (-U), t$95$1]]]]]]]

Derivation

1. Split input into 4 regimes

2. if J < -1.05000000000000006e-208 or 4.69999999999999994e-73 < J

   1. Initial program 83.6%

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

      1. metadata-eval 83.6%

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

      2. associate-*r* 83.6%

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

      3. pow2 83.6%

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

      4. hypot-udef 97.0%

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

      5. *-un-lft-identity 97.0%

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

      6. times-frac 97.0%

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

      7. metadata-eval 97.0%

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

      8. *-commutative 97.0%

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

      9. associate-/r* 97.0%

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

      10. div-inv 97.0%

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

      11. metadata-eval 97.0%

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

   3. Applied egg-rr 97.0%

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

   4. Taylor expanded in K around 0 85.0%

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

   if -1.05000000000000006e-208 < J < -9.999999999999969e-311

   1. Initial program 26.4%

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

   2. Taylor expanded in K around 0 3.7%

      \[\leadsto \color{blue}{-2 \cdot \left(\sqrt{1 + 0.25 \cdot \frac{{U}^{2}}{{J}^{2}}} \cdot J\right)} \]
   3. Step-by-step derivation

      1. *-commutative 3.7%

         \[\leadsto -2 \cdot \color{blue}{\left(J \cdot \sqrt{1 + 0.25 \cdot \frac{{U}^{2}}{{J}^{2}}}\right)} \]

      2. unpow2 3.7%

         \[\leadsto -2 \cdot \left(J \cdot \sqrt{1 + 0.25 \cdot \frac{\color{blue}{U \cdot U}}{{J}^{2}}}\right) \]

      3. unpow2 3.7%

         \[\leadsto -2 \cdot \left(J \cdot \sqrt{1 + 0.25 \cdot \frac{U \cdot U}{\color{blue}{J \cdot J}}}\right) \]

   4. Simplified 3.7%

      \[\leadsto \color{blue}{-2 \cdot \left(J \cdot \sqrt{1 + 0.25 \cdot \frac{U \cdot U}{J \cdot J}}\right)} \]

   5. Taylor expanded in U around -inf 25.0%

      \[\leadsto -2 \cdot \left(J \cdot \color{blue}{\left(-0.5 \cdot \frac{U}{J} + -1 \cdot \frac{J}{U}\right)}\right) \]

   6. Taylor expanded in J around 0 46.6%

      \[\leadsto -2 \cdot \color{blue}{\left(-1 \cdot \frac{{J}^{2}}{U} + -0.5 \cdot U\right)} \]
   7. Step-by-step derivation

      1. +-commutative 46.6%

         \[\leadsto -2 \cdot \color{blue}{\left(-0.5 \cdot U + -1 \cdot \frac{{J}^{2}}{U}\right)} \]

      2. mul-1-neg 46.6%

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

      3. unsub-neg 46.6%

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

      4. *-commutative 46.6%

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

      5. unpow2 46.6%

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

      6. associate-*r/ 46.6%

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

   8. Simplified 46.6%

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

   if -9.999999999999969e-311 < J < 9.0000000000000007e-155 or 9.5000000000000005e-108 < J < 4.69999999999999994e-73

   1. Initial program 33.2%

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

   2. Taylor expanded in J around 0 32.7%

      \[\leadsto \color{blue}{-1 \cdot U} \]
   3. Step-by-step derivation

      1. neg-mul-1 32.7%

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

   4. Simplified 32.7%

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

   if 9.0000000000000007e-155 < J < 9.5000000000000005e-108

   1. Initial program 78.7%

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

      1. associate-*l* 78.7%

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

      2. associate-*l* 78.7%

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

      3. unpow2 78.7%

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

      4. sqr-neg 78.7%

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

      5. distribute-frac-neg 78.7%

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

      6. distribute-frac-neg 78.7%

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

      7. unpow2 78.7%

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

   3. Simplified 86.0%

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

   4. Taylor expanded in K around 0 75.1%

      \[\leadsto -2 \cdot \left(\mathsf{hypot}\left(1, \color{blue}{0.5 \cdot \frac{U}{J}}\right) \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)\right) \]
   5. Step-by-step derivation

      1. associate-*r/ 75.1%

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

      2. *-commutative 75.1%

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

      3. associate-*r/ 75.1%

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

   6. Simplified 75.1%

      \[\leadsto -2 \cdot \left(\mathsf{hypot}\left(1, \color{blue}{U \cdot \frac{0.5}{J}}\right) \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -1.05 \cdot 10^{-208}:\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, 0.5 \cdot \frac{U}{J}\right)\\ \mathbf{elif}\;J \leq -1 \cdot 10^{-310}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5 - J \cdot \frac{J}{U}\right)\\ \mathbf{elif}\;J \leq 9 \cdot 10^{-155}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 9.5 \cdot 10^{-108}:\\ \;\;\;\;-2 \cdot \left(\left(J \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, U \cdot \frac{0.5}{J}\right)\right)\\ \mathbf{elif}\;J \leq 4.7 \cdot 10^{-73}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, 0.5 \cdot \frac{U}{J}\right)\\ \end{array} \]

Alternative 5: 70.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.7:\\ \;\;\;\;J \cdot \left(-2 \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(J \cdot \mathsf{hypot}\left(1, \frac{U \cdot 0.5}{J}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.7)
   (* J (* -2.0 (cos (* K 0.5))))
   (* -2.0 (* J (hypot 1.0 (/ (* U 0.5) J))))))

C

U = abs(U);
double code(double J, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= 0.7) {
		tmp = J * (-2.0 * cos((K * 0.5)));
	} else {
		tmp = -2.0 * (J * hypot(1.0, ((U * 0.5) / J)));
	}
	return tmp;
}

Java

U = Math.abs(U);
public static double code(double J, double K, double U) {
	double tmp;
	if (Math.cos((K / 2.0)) <= 0.7) {
		tmp = J * (-2.0 * Math.cos((K * 0.5)));
	} else {
		tmp = -2.0 * (J * Math.hypot(1.0, ((U * 0.5) / J)));
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	tmp = 0
	if math.cos((K / 2.0)) <= 0.7:
		tmp = J * (-2.0 * math.cos((K * 0.5)))
	else:
		tmp = -2.0 * (J * math.hypot(1.0, ((U * 0.5) / J)))
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= 0.7)
		tmp = Float64(J * Float64(-2.0 * cos(Float64(K * 0.5))));
	else
		tmp = Float64(-2.0 * Float64(J * hypot(1.0, Float64(Float64(U * 0.5) / J))));
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	tmp = 0.0;
	if (cos((K / 2.0)) <= 0.7)
		tmp = J * (-2.0 * cos((K * 0.5)));
	else
		tmp = -2.0 * (J * hypot(1.0, ((U * 0.5) / J)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], 0.7], N[(J * N[(-2.0 * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(J * N[Sqrt[1.0 ^ 2 + N[(N[(U * 0.5), $MachinePrecision] / J), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K 2)) < 0.69999999999999996

   1. Initial program 70.3%

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

   2. Taylor expanded in J around inf 53.0%

      \[\leadsto \color{blue}{-2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot J\right)} \]
   3. Step-by-step derivation

      1. associate-*r* 53.0%

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

      2. metadata-eval 53.0%

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

      3. *-commutative 53.0%

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

      4. distribute-lft-neg-in 53.0%

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

      5. distribute-lft-neg-in 53.0%

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

      6. *-commutative 53.0%

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

      7. distribute-rgt-neg-in 53.0%

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

      8. distribute-lft-neg-in 53.0%

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

      9. metadata-eval 53.0%

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

      10. *-commutative 53.0%

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

   4. Simplified 53.0%

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

   if 0.69999999999999996 < (cos.f64 (/.f64 K 2))

   1. Initial program 66.5%

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

   2. Taylor expanded in K around 0 41.6%

      \[\leadsto \color{blue}{-2 \cdot \left(\sqrt{1 + 0.25 \cdot \frac{{U}^{2}}{{J}^{2}}} \cdot J\right)} \]
   3. Step-by-step derivation

      1. *-commutative 41.6%

         \[\leadsto -2 \cdot \color{blue}{\left(J \cdot \sqrt{1 + 0.25 \cdot \frac{{U}^{2}}{{J}^{2}}}\right)} \]

      2. unpow2 41.6%

         \[\leadsto -2 \cdot \left(J \cdot \sqrt{1 + 0.25 \cdot \frac{\color{blue}{U \cdot U}}{{J}^{2}}}\right) \]

      3. unpow2 41.6%

         \[\leadsto -2 \cdot \left(J \cdot \sqrt{1 + 0.25 \cdot \frac{U \cdot U}{\color{blue}{J \cdot J}}}\right) \]

   4. Simplified 41.6%

      \[\leadsto \color{blue}{-2 \cdot \left(J \cdot \sqrt{1 + 0.25 \cdot \frac{U \cdot U}{J \cdot J}}\right)} \]
   5. Step-by-step derivation

      1. expm1-log1p-u 41.2%

         \[\leadsto -2 \cdot \left(J \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 + 0.25 \cdot \frac{U \cdot U}{J \cdot J}}\right)\right)}\right) \]

      2. expm1-udef 41.2%

         \[\leadsto -2 \cdot \left(J \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{1 + 0.25 \cdot \frac{U \cdot U}{J \cdot J}}\right)} - 1\right)}\right) \]

      3. add-sqr-sqrt 41.2%

         \[\leadsto -2 \cdot \left(J \cdot \left(e^{\mathsf{log1p}\left(\sqrt{1 + \color{blue}{\sqrt{0.25 \cdot \frac{U \cdot U}{J \cdot J}} \cdot \sqrt{0.25 \cdot \frac{U \cdot U}{J \cdot J}}}}\right)} - 1\right)\right) \]

      4. hypot-1-def 41.2%

         \[\leadsto -2 \cdot \left(J \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\mathsf{hypot}\left(1, \sqrt{0.25 \cdot \frac{U \cdot U}{J \cdot J}}\right)}\right)} - 1\right)\right) \]

      5. *-commutative 41.2%

         \[\leadsto -2 \cdot \left(J \cdot \left(e^{\mathsf{log1p}\left(\mathsf{hypot}\left(1, \sqrt{\color{blue}{\frac{U \cdot U}{J \cdot J} \cdot 0.25}}\right)\right)} - 1\right)\right) \]

      6. sqrt-prod 41.2%

         \[\leadsto -2 \cdot \left(J \cdot \left(e^{\mathsf{log1p}\left(\mathsf{hypot}\left(1, \color{blue}{\sqrt{\frac{U \cdot U}{J \cdot J}} \cdot \sqrt{0.25}}\right)\right)} - 1\right)\right) \]

      7. times-frac 58.4%

         \[\leadsto -2 \cdot \left(J \cdot \left(e^{\mathsf{log1p}\left(\mathsf{hypot}\left(1, \sqrt{\color{blue}{\frac{U}{J} \cdot \frac{U}{J}}} \cdot \sqrt{0.25}\right)\right)} - 1\right)\right) \]

      8. sqrt-prod 40.7%

         \[\leadsto -2 \cdot \left(J \cdot \left(e^{\mathsf{log1p}\left(\mathsf{hypot}\left(1, \color{blue}{\left(\sqrt{\frac{U}{J}} \cdot \sqrt{\frac{U}{J}}\right)} \cdot \sqrt{0.25}\right)\right)} - 1\right)\right) \]

      9. add-sqr-sqrt 74.8%

         \[\leadsto -2 \cdot \left(J \cdot \left(e^{\mathsf{log1p}\left(\mathsf{hypot}\left(1, \color{blue}{\frac{U}{J}} \cdot \sqrt{0.25}\right)\right)} - 1\right)\right) \]

      10. metadata-eval 74.8%

          \[\leadsto -2 \cdot \left(J \cdot \left(e^{\mathsf{log1p}\left(\mathsf{hypot}\left(1, \frac{U}{J} \cdot \color{blue}{0.5}\right)\right)} - 1\right)\right) \]

   6. Applied egg-rr 74.8%

      \[\leadsto -2 \cdot \left(J \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\mathsf{hypot}\left(1, \frac{U}{J} \cdot 0.5\right)\right)} - 1\right)}\right) \]
   7. Step-by-step derivation

      1. expm1-def 74.8%

         \[\leadsto -2 \cdot \left(J \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(1, \frac{U}{J} \cdot 0.5\right)\right)\right)}\right) \]

      2. expm1-log1p 78.2%

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

      3. associate-*l/ 78.2%

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

   8. Simplified 78.2%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.7:\\ \;\;\;\;J \cdot \left(-2 \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(J \cdot \mathsf{hypot}\left(1, \frac{U \cdot 0.5}{J}\right)\right)\\ \end{array} \]

Alternative 6: 64.8% accurate, 3.7× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} t_0 := J \cdot \left(-2 \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{if}\;J \leq -4.5 \cdot 10^{-58}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq -1 \cdot 10^{-310}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5 - J \cdot \frac{J}{U}\right)\\ \mathbf{elif}\;J \leq 7 \cdot 10^{+93}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (* J (* -2.0 (cos (* K 0.5))))))
   (if (<= J -4.5e-58)
     t_0
     (if (<= J -1e-310)
       (* -2.0 (- (* U -0.5) (* J (/ J U))))
       (if (<= J 7e+93) (- U) t_0)))))

C

U = abs(U);
double code(double J, double K, double U) {
	double t_0 = J * (-2.0 * cos((K * 0.5)));
	double tmp;
	if (J <= -4.5e-58) {
		tmp = t_0;
	} else if (J <= -1e-310) {
		tmp = -2.0 * ((U * -0.5) - (J * (J / U)));
	} else if (J <= 7e+93) {
		tmp = -U;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

NOTE: U should be positive before calling this function
real(8) function code(j, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = j * ((-2.0d0) * cos((k * 0.5d0)))
    if (j <= (-4.5d-58)) then
        tmp = t_0
    else if (j <= (-1d-310)) then
        tmp = (-2.0d0) * ((u * (-0.5d0)) - (j * (j / u)))
    else if (j <= 7d+93) then
        tmp = -u
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

U = Math.abs(U);
public static double code(double J, double K, double U) {
	double t_0 = J * (-2.0 * Math.cos((K * 0.5)));
	double tmp;
	if (J <= -4.5e-58) {
		tmp = t_0;
	} else if (J <= -1e-310) {
		tmp = -2.0 * ((U * -0.5) - (J * (J / U)));
	} else if (J <= 7e+93) {
		tmp = -U;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	t_0 = J * (-2.0 * math.cos((K * 0.5)))
	tmp = 0
	if J <= -4.5e-58:
		tmp = t_0
	elif J <= -1e-310:
		tmp = -2.0 * ((U * -0.5) - (J * (J / U)))
	elif J <= 7e+93:
		tmp = -U
	else:
		tmp = t_0
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	t_0 = Float64(J * Float64(-2.0 * cos(Float64(K * 0.5))))
	tmp = 0.0
	if (J <= -4.5e-58)
		tmp = t_0;
	elseif (J <= -1e-310)
		tmp = Float64(-2.0 * Float64(Float64(U * -0.5) - Float64(J * Float64(J / U))));
	elseif (J <= 7e+93)
		tmp = Float64(-U);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	t_0 = J * (-2.0 * cos((K * 0.5)));
	tmp = 0.0;
	if (J <= -4.5e-58)
		tmp = t_0;
	elseif (J <= -1e-310)
		tmp = -2.0 * ((U * -0.5) - (J * (J / U)));
	elseif (J <= 7e+93)
		tmp = -U;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := Block[{t$95$0 = N[(J * N[(-2.0 * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[J, -4.5e-58], t$95$0, If[LessEqual[J, -1e-310], N[(-2.0 * N[(N[(U * -0.5), $MachinePrecision] - N[(J * N[(J / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[J, 7e+93], (-U), t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if J < -4.5000000000000003e-58 or 6.99999999999999996e93 < J

   1. Initial program 95.0%

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

   2. Taylor expanded in J around inf 78.6%

      \[\leadsto \color{blue}{-2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot J\right)} \]
   3. Step-by-step derivation

      1. associate-*r* 78.6%

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

      2. metadata-eval 78.6%

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

      3. *-commutative 78.6%

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

      4. distribute-lft-neg-in 78.6%

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

      5. distribute-lft-neg-in 78.6%

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

      6. *-commutative 78.6%

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

      7. distribute-rgt-neg-in 78.6%

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

      8. distribute-lft-neg-in 78.6%

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

      9. metadata-eval 78.6%

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

      10. *-commutative 78.6%

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

   4. Simplified 78.6%

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

   if -4.5000000000000003e-58 < J < -9.999999999999969e-311

   1. Initial program 42.3%

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

   2. Taylor expanded in K around 0 17.5%

      \[\leadsto \color{blue}{-2 \cdot \left(\sqrt{1 + 0.25 \cdot \frac{{U}^{2}}{{J}^{2}}} \cdot J\right)} \]
   3. Step-by-step derivation

      1. *-commutative 17.5%

         \[\leadsto -2 \cdot \color{blue}{\left(J \cdot \sqrt{1 + 0.25 \cdot \frac{{U}^{2}}{{J}^{2}}}\right)} \]

      2. unpow2 17.5%

         \[\leadsto -2 \cdot \left(J \cdot \sqrt{1 + 0.25 \cdot \frac{\color{blue}{U \cdot U}}{{J}^{2}}}\right) \]

      3. unpow2 17.5%

         \[\leadsto -2 \cdot \left(J \cdot \sqrt{1 + 0.25 \cdot \frac{U \cdot U}{\color{blue}{J \cdot J}}}\right) \]

   4. Simplified 17.5%

      \[\leadsto \color{blue}{-2 \cdot \left(J \cdot \sqrt{1 + 0.25 \cdot \frac{U \cdot U}{J \cdot J}}\right)} \]

   5. Taylor expanded in U around -inf 30.4%

      \[\leadsto -2 \cdot \left(J \cdot \color{blue}{\left(-0.5 \cdot \frac{U}{J} + -1 \cdot \frac{J}{U}\right)}\right) \]

   6. Taylor expanded in J around 0 46.0%

      \[\leadsto -2 \cdot \color{blue}{\left(-1 \cdot \frac{{J}^{2}}{U} + -0.5 \cdot U\right)} \]
   7. Step-by-step derivation

      1. +-commutative 46.0%

         \[\leadsto -2 \cdot \color{blue}{\left(-0.5 \cdot U + -1 \cdot \frac{{J}^{2}}{U}\right)} \]

      2. mul-1-neg 46.0%

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

      3. unsub-neg 46.0%

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

      4. *-commutative 46.0%

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

      5. unpow2 46.0%

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

      6. associate-*r/ 45.9%

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

   8. Simplified 45.9%

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

   if -9.999999999999969e-311 < J < 6.99999999999999996e93

   1. Initial program 47.7%

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

   2. Taylor expanded in J around 0 34.8%

      \[\leadsto \color{blue}{-1 \cdot U} \]
   3. Step-by-step derivation

      1. neg-mul-1 34.8%

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

   4. Simplified 34.8%

      \[\leadsto \color{blue}{-U} \]
3. Recombined 3 regimes into one program.

4. Final simplification 56.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -4.5 \cdot 10^{-58}:\\ \;\;\;\;J \cdot \left(-2 \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{elif}\;J \leq -1 \cdot 10^{-310}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5 - J \cdot \frac{J}{U}\right)\\ \mathbf{elif}\;J \leq 7 \cdot 10^{+93}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(-2 \cdot \cos \left(K \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 7: 49.1% accurate, 27.8× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} \mathbf{if}\;J \leq -2.5 \cdot 10^{-54}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;J \leq -1 \cdot 10^{-310}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5 - J \cdot \frac{J}{U}\right)\\ \mathbf{elif}\;J \leq 1.7 \cdot 10^{+108}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot J\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (if (<= J -2.5e-54)
   (* -2.0 J)
   (if (<= J -1e-310)
     (* -2.0 (- (* U -0.5) (* J (/ J U))))
     (if (<= J 1.7e+108) (- U) (* -2.0 J)))))

C

U = abs(U);
double code(double J, double K, double U) {
	double tmp;
	if (J <= -2.5e-54) {
		tmp = -2.0 * J;
	} else if (J <= -1e-310) {
		tmp = -2.0 * ((U * -0.5) - (J * (J / U)));
	} else if (J <= 1.7e+108) {
		tmp = -U;
	} else {
		tmp = -2.0 * J;
	}
	return tmp;
}

Fortran

NOTE: U should be positive before calling this function
real(8) function code(j, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (j <= (-2.5d-54)) then
        tmp = (-2.0d0) * j
    else if (j <= (-1d-310)) then
        tmp = (-2.0d0) * ((u * (-0.5d0)) - (j * (j / u)))
    else if (j <= 1.7d+108) then
        tmp = -u
    else
        tmp = (-2.0d0) * j
    end if
    code = tmp
end function

Java

U = Math.abs(U);
public static double code(double J, double K, double U) {
	double tmp;
	if (J <= -2.5e-54) {
		tmp = -2.0 * J;
	} else if (J <= -1e-310) {
		tmp = -2.0 * ((U * -0.5) - (J * (J / U)));
	} else if (J <= 1.7e+108) {
		tmp = -U;
	} else {
		tmp = -2.0 * J;
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	tmp = 0
	if J <= -2.5e-54:
		tmp = -2.0 * J
	elif J <= -1e-310:
		tmp = -2.0 * ((U * -0.5) - (J * (J / U)))
	elif J <= 1.7e+108:
		tmp = -U
	else:
		tmp = -2.0 * J
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	tmp = 0.0
	if (J <= -2.5e-54)
		tmp = Float64(-2.0 * J);
	elseif (J <= -1e-310)
		tmp = Float64(-2.0 * Float64(Float64(U * -0.5) - Float64(J * Float64(J / U))));
	elseif (J <= 1.7e+108)
		tmp = Float64(-U);
	else
		tmp = Float64(-2.0 * J);
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	tmp = 0.0;
	if (J <= -2.5e-54)
		tmp = -2.0 * J;
	elseif (J <= -1e-310)
		tmp = -2.0 * ((U * -0.5) - (J * (J / U)));
	elseif (J <= 1.7e+108)
		tmp = -U;
	else
		tmp = -2.0 * J;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := If[LessEqual[J, -2.5e-54], N[(-2.0 * J), $MachinePrecision], If[LessEqual[J, -1e-310], N[(-2.0 * N[(N[(U * -0.5), $MachinePrecision] - N[(J * N[(J / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[J, 1.7e+108], (-U), N[(-2.0 * J), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if J < -2.50000000000000008e-54 or 1.69999999999999998e108 < J

   1. Initial program 94.9%

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

   2. Taylor expanded in K around 0 40.9%

      \[\leadsto \color{blue}{-2 \cdot \left(\sqrt{1 + 0.25 \cdot \frac{{U}^{2}}{{J}^{2}}} \cdot J\right)} \]
   3. Step-by-step derivation

      1. *-commutative 40.9%

         \[\leadsto -2 \cdot \color{blue}{\left(J \cdot \sqrt{1 + 0.25 \cdot \frac{{U}^{2}}{{J}^{2}}}\right)} \]

      2. unpow2 40.9%

         \[\leadsto -2 \cdot \left(J \cdot \sqrt{1 + 0.25 \cdot \frac{\color{blue}{U \cdot U}}{{J}^{2}}}\right) \]

      3. unpow2 40.9%

         \[\leadsto -2 \cdot \left(J \cdot \sqrt{1 + 0.25 \cdot \frac{U \cdot U}{\color{blue}{J \cdot J}}}\right) \]

   4. Simplified 40.9%

      \[\leadsto \color{blue}{-2 \cdot \left(J \cdot \sqrt{1 + 0.25 \cdot \frac{U \cdot U}{J \cdot J}}\right)} \]

   5. Taylor expanded in J around inf 42.1%

      \[\leadsto -2 \cdot \color{blue}{J} \]

   if -2.50000000000000008e-54 < J < -9.999999999999969e-311

   1. Initial program 42.3%

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

   2. Taylor expanded in K around 0 17.5%

      \[\leadsto \color{blue}{-2 \cdot \left(\sqrt{1 + 0.25 \cdot \frac{{U}^{2}}{{J}^{2}}} \cdot J\right)} \]
   3. Step-by-step derivation

      1. *-commutative 17.5%

         \[\leadsto -2 \cdot \color{blue}{\left(J \cdot \sqrt{1 + 0.25 \cdot \frac{{U}^{2}}{{J}^{2}}}\right)} \]

      2. unpow2 17.5%

         \[\leadsto -2 \cdot \left(J \cdot \sqrt{1 + 0.25 \cdot \frac{\color{blue}{U \cdot U}}{{J}^{2}}}\right) \]

      3. unpow2 17.5%

         \[\leadsto -2 \cdot \left(J \cdot \sqrt{1 + 0.25 \cdot \frac{U \cdot U}{\color{blue}{J \cdot J}}}\right) \]

   4. Simplified 17.5%

      \[\leadsto \color{blue}{-2 \cdot \left(J \cdot \sqrt{1 + 0.25 \cdot \frac{U \cdot U}{J \cdot J}}\right)} \]

   5. Taylor expanded in U around -inf 30.4%

      \[\leadsto -2 \cdot \left(J \cdot \color{blue}{\left(-0.5 \cdot \frac{U}{J} + -1 \cdot \frac{J}{U}\right)}\right) \]

   6. Taylor expanded in J around 0 46.0%

      \[\leadsto -2 \cdot \color{blue}{\left(-1 \cdot \frac{{J}^{2}}{U} + -0.5 \cdot U\right)} \]
   7. Step-by-step derivation

      1. +-commutative 46.0%

         \[\leadsto -2 \cdot \color{blue}{\left(-0.5 \cdot U + -1 \cdot \frac{{J}^{2}}{U}\right)} \]

      2. mul-1-neg 46.0%

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

      3. unsub-neg 46.0%

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

      4. *-commutative 46.0%

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

      5. unpow2 46.0%

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

      6. associate-*r/ 45.9%

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

   8. Simplified 45.9%

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

   if -9.999999999999969e-311 < J < 1.69999999999999998e108

   1. Initial program 48.7%

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

   2. Taylor expanded in J around 0 34.2%

      \[\leadsto \color{blue}{-1 \cdot U} \]
   3. Step-by-step derivation

      1. neg-mul-1 34.2%

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

   4. Simplified 34.2%

      \[\leadsto \color{blue}{-U} \]
3. Recombined 3 regimes into one program.

4. Final simplification 39.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -2.5 \cdot 10^{-54}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;J \leq -1 \cdot 10^{-310}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5 - J \cdot \frac{J}{U}\right)\\ \mathbf{elif}\;J \leq 1.7 \cdot 10^{+108}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot J\\ \end{array} \]

Alternative 8: 49.1% accurate, 45.7× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} \mathbf{if}\;J \leq -2.4 \cdot 10^{-54}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;J \leq -1 \cdot 10^{-310}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 6.8 \cdot 10^{+108}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot J\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (if (<= J -2.4e-54)
   (* -2.0 J)
   (if (<= J -1e-310) U (if (<= J 6.8e+108) (- U) (* -2.0 J)))))

C

U = abs(U);
double code(double J, double K, double U) {
	double tmp;
	if (J <= -2.4e-54) {
		tmp = -2.0 * J;
	} else if (J <= -1e-310) {
		tmp = U;
	} else if (J <= 6.8e+108) {
		tmp = -U;
	} else {
		tmp = -2.0 * J;
	}
	return tmp;
}

Fortran

NOTE: U should be positive before calling this function
real(8) function code(j, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (j <= (-2.4d-54)) then
        tmp = (-2.0d0) * j
    else if (j <= (-1d-310)) then
        tmp = u
    else if (j <= 6.8d+108) then
        tmp = -u
    else
        tmp = (-2.0d0) * j
    end if
    code = tmp
end function

Java

U = Math.abs(U);
public static double code(double J, double K, double U) {
	double tmp;
	if (J <= -2.4e-54) {
		tmp = -2.0 * J;
	} else if (J <= -1e-310) {
		tmp = U;
	} else if (J <= 6.8e+108) {
		tmp = -U;
	} else {
		tmp = -2.0 * J;
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	tmp = 0
	if J <= -2.4e-54:
		tmp = -2.0 * J
	elif J <= -1e-310:
		tmp = U
	elif J <= 6.8e+108:
		tmp = -U
	else:
		tmp = -2.0 * J
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	tmp = 0.0
	if (J <= -2.4e-54)
		tmp = Float64(-2.0 * J);
	elseif (J <= -1e-310)
		tmp = U;
	elseif (J <= 6.8e+108)
		tmp = Float64(-U);
	else
		tmp = Float64(-2.0 * J);
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	tmp = 0.0;
	if (J <= -2.4e-54)
		tmp = -2.0 * J;
	elseif (J <= -1e-310)
		tmp = U;
	elseif (J <= 6.8e+108)
		tmp = -U;
	else
		tmp = -2.0 * J;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := If[LessEqual[J, -2.4e-54], N[(-2.0 * J), $MachinePrecision], If[LessEqual[J, -1e-310], U, If[LessEqual[J, 6.8e+108], (-U), N[(-2.0 * J), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if J < -2.40000000000000013e-54 or 6.79999999999999992e108 < J

   1. Initial program 94.9%

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

   2. Taylor expanded in K around 0 40.9%

      \[\leadsto \color{blue}{-2 \cdot \left(\sqrt{1 + 0.25 \cdot \frac{{U}^{2}}{{J}^{2}}} \cdot J\right)} \]
   3. Step-by-step derivation

      1. *-commutative 40.9%

         \[\leadsto -2 \cdot \color{blue}{\left(J \cdot \sqrt{1 + 0.25 \cdot \frac{{U}^{2}}{{J}^{2}}}\right)} \]

      2. unpow2 40.9%

         \[\leadsto -2 \cdot \left(J \cdot \sqrt{1 + 0.25 \cdot \frac{\color{blue}{U \cdot U}}{{J}^{2}}}\right) \]

      3. unpow2 40.9%

         \[\leadsto -2 \cdot \left(J \cdot \sqrt{1 + 0.25 \cdot \frac{U \cdot U}{\color{blue}{J \cdot J}}}\right) \]

   4. Simplified 40.9%

      \[\leadsto \color{blue}{-2 \cdot \left(J \cdot \sqrt{1 + 0.25 \cdot \frac{U \cdot U}{J \cdot J}}\right)} \]

   5. Taylor expanded in J around inf 42.1%

      \[\leadsto -2 \cdot \color{blue}{J} \]

   if -2.40000000000000013e-54 < J < -9.999999999999969e-311

   1. Initial program 42.3%

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

   2. Taylor expanded in U around -inf 46.0%

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

   if -9.999999999999969e-311 < J < 6.79999999999999992e108

   1. Initial program 48.7%

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

   2. Taylor expanded in J around 0 34.2%

      \[\leadsto \color{blue}{-1 \cdot U} \]
   3. Step-by-step derivation

      1. neg-mul-1 34.2%

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

   4. Simplified 34.2%

      \[\leadsto \color{blue}{-U} \]
3. Recombined 3 regimes into one program.

4. Final simplification 39.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -2.4 \cdot 10^{-54}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;J \leq -1 \cdot 10^{-310}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 6.8 \cdot 10^{+108}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot J\\ \end{array} \]

Alternative 9: 39.3% accurate, 103.4× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} \mathbf{if}\;J \leq -1 \cdot 10^{-310}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U) :precision binary64 (if (<= J -1e-310) U (- U)))

C

U = abs(U);
double code(double J, double K, double U) {
	double tmp;
	if (J <= -1e-310) {
		tmp = U;
	} else {
		tmp = -U;
	}
	return tmp;
}

Fortran

NOTE: U should be positive before calling this function
real(8) function code(j, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (j <= (-1d-310)) then
        tmp = u
    else
        tmp = -u
    end if
    code = tmp
end function

Java

U = Math.abs(U);
public static double code(double J, double K, double U) {
	double tmp;
	if (J <= -1e-310) {
		tmp = U;
	} else {
		tmp = -U;
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	tmp = 0
	if J <= -1e-310:
		tmp = U
	else:
		tmp = -U
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	tmp = 0.0
	if (J <= -1e-310)
		tmp = U;
	else
		tmp = Float64(-U);
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	tmp = 0.0;
	if (J <= -1e-310)
		tmp = U;
	else
		tmp = -U;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := If[LessEqual[J, -1e-310], U, (-U)]

Derivation

1. Split input into 2 regimes

2. if J < -9.999999999999969e-311

   1. Initial program 75.1%

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

   2. Taylor expanded in U around -inf 25.6%

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

   if -9.999999999999969e-311 < J

   1. Initial program 61.5%

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

   2. Taylor expanded in J around 0 27.3%

      \[\leadsto \color{blue}{-1 \cdot U} \]
   3. Step-by-step derivation

      1. neg-mul-1 27.3%

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

   4. Simplified 27.3%

      \[\leadsto \color{blue}{-U} \]
3. Recombined 2 regimes into one program.

4. Final simplification 26.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -1 \cdot 10^{-310}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]

Alternative 10: 27.1% accurate, 420.0× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ U \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U) :precision binary64 U)

C

U = abs(U);
double code(double J, double K, double U) {
	return U;
}

Fortran

NOTE: U should be positive before calling this function
real(8) function code(j, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = u
end function

Java

U = Math.abs(U);
public static double code(double J, double K, double U) {
	return U;
}

Python

U = abs(U)
def code(J, K, U):
	return U

Julia

U = abs(U)
function code(J, K, U)
	return U
end

MATLAB

U = abs(U)
function tmp = code(J, K, U)
	tmp = U;
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := U

Derivation

1. Initial program 67.8%

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

2. Taylor expanded in U around -inf 31.1%

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

3. Final simplification 31.1%

   \[\leadsto U \]

Reproduce

herbie shell --seed 2023274 
(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)))))