Maksimov and Kolovsky, Equation (3)

Percentage Accurate: 73.1% → 99.7%

Time: 20.5s

Alternatives: 11

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: 73.1% 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: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := J \cdot t_0\\ t_2 := \left(\left(-2 \cdot J\right) \cdot t_0\right) \cdot \sqrt{1 + {\left(\frac{U}{t_0 \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;-2 \cdot \left(U \cdot 0.5\right)\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{+306}:\\ \;\;\;\;-2 \cdot \left(t_1 \cdot \mathsf{hypot}\left(1, \frac{U}{2 \cdot t_1}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5\right)\\ \end{array} \end{array} \]

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 (* J t_0))
        (t_2
         (*
          (* (* -2.0 J) t_0)
          (sqrt (+ 1.0 (pow (/ U (* t_0 (* J 2.0))) 2.0))))))
   (if (<= t_2 (- INFINITY))
     (* -2.0 (* U 0.5))
     (if (<= t_2 5e+306)
       (* -2.0 (* t_1 (hypot 1.0 (/ U (* 2.0 t_1)))))
       (* -2.0 (* U -0.5))))))

C

U = abs(U);
double code(double J, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = J * t_0;
	double t_2 = ((-2.0 * J) * t_0) * sqrt((1.0 + pow((U / (t_0 * (J * 2.0))), 2.0)));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = -2.0 * (U * 0.5);
	} else if (t_2 <= 5e+306) {
		tmp = -2.0 * (t_1 * hypot(1.0, (U / (2.0 * t_1))));
	} else {
		tmp = -2.0 * (U * -0.5);
	}
	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 = J * t_0;
	double t_2 = ((-2.0 * J) * t_0) * Math.sqrt((1.0 + Math.pow((U / (t_0 * (J * 2.0))), 2.0)));
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = -2.0 * (U * 0.5);
	} else if (t_2 <= 5e+306) {
		tmp = -2.0 * (t_1 * Math.hypot(1.0, (U / (2.0 * t_1))));
	} else {
		tmp = -2.0 * (U * -0.5);
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	t_0 = math.cos((K / 2.0))
	t_1 = J * t_0
	t_2 = ((-2.0 * J) * t_0) * math.sqrt((1.0 + math.pow((U / (t_0 * (J * 2.0))), 2.0)))
	tmp = 0
	if t_2 <= -math.inf:
		tmp = -2.0 * (U * 0.5)
	elif t_2 <= 5e+306:
		tmp = -2.0 * (t_1 * math.hypot(1.0, (U / (2.0 * t_1))))
	else:
		tmp = -2.0 * (U * -0.5)
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(J * t_0)
	t_2 = Float64(Float64(Float64(-2.0 * J) * t_0) * sqrt(Float64(1.0 + (Float64(U / Float64(t_0 * Float64(J * 2.0))) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(-2.0 * Float64(U * 0.5));
	elseif (t_2 <= 5e+306)
		tmp = Float64(-2.0 * Float64(t_1 * hypot(1.0, Float64(U / Float64(2.0 * t_1)))));
	else
		tmp = Float64(-2.0 * Float64(U * -0.5));
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	t_0 = cos((K / 2.0));
	t_1 = J * t_0;
	t_2 = ((-2.0 * J) * t_0) * sqrt((1.0 + ((U / (t_0 * (J * 2.0))) ^ 2.0)));
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = -2.0 * (U * 0.5);
	elseif (t_2 <= 5e+306)
		tmp = -2.0 * (t_1 * hypot(1.0, (U / (2.0 * t_1))));
	else
		tmp = -2.0 * (U * -0.5);
	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[(J * t$95$0), $MachinePrecision]}, Block[{t$95$2 = 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]}, If[LessEqual[t$95$2, (-Infinity)], N[(-2.0 * N[(U * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e+306], N[(-2.0 * N[(t$95$1 * N[Sqrt[1.0 ^ 2 + N[(U / N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(U * -0.5), $MachinePrecision]), $MachinePrecision]]]]]]

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 6.0%

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

   3. Simplified 64.7%

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

   4. Taylor expanded in J around 0 55.9%

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

   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)))) < 4.99999999999999993e306

   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 4.99999999999999993e306 < (*.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%

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

   3. Simplified 59.7%

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

   4. Taylor expanded in U around -inf 54.9%

      \[\leadsto -2 \cdot \color{blue}{\left(-0.5 \cdot U\right)} \]
   5. Step-by-step derivation

      1. *-commutative 54.9%

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

   6. Simplified 54.9%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -\infty:\\ \;\;\;\;-2 \cdot \left(U \cdot 0.5\right)\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 5 \cdot 10^{+306}:\\ \;\;\;\;-2 \cdot \left(\left(J \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5\right)\\ \end{array} \]

Alternative 2: 69.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := -2 \cdot \left(J \cdot t_0\right)\\ \mathbf{if}\;t_0 \leq 0.34:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq 0.5:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5\right)\\ \mathbf{elif}\;t_0 \leq 0.66:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(J \cdot \mathsf{hypot}\left(1, \frac{U \cdot 0.5}{J}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))) (t_1 (* -2.0 (* J t_0))))
   (if (<= t_0 0.34)
     t_1
     (if (<= t_0 0.5)
       (* -2.0 (* U -0.5))
       (if (<= t_0 0.66) t_1 (* -2.0 (* J (hypot 1.0 (/ (* U 0.5) J)))))))))

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);
	double tmp;
	if (t_0 <= 0.34) {
		tmp = t_1;
	} else if (t_0 <= 0.5) {
		tmp = -2.0 * (U * -0.5);
	} else if (t_0 <= 0.66) {
		tmp = t_1;
	} 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 t_0 = Math.cos((K / 2.0));
	double t_1 = -2.0 * (J * t_0);
	double tmp;
	if (t_0 <= 0.34) {
		tmp = t_1;
	} else if (t_0 <= 0.5) {
		tmp = -2.0 * (U * -0.5);
	} else if (t_0 <= 0.66) {
		tmp = t_1;
	} else {
		tmp = -2.0 * (J * Math.hypot(1.0, ((U * 0.5) / J)));
	}
	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)
	tmp = 0
	if t_0 <= 0.34:
		tmp = t_1
	elif t_0 <= 0.5:
		tmp = -2.0 * (U * -0.5)
	elif t_0 <= 0.66:
		tmp = t_1
	else:
		tmp = -2.0 * (J * math.hypot(1.0, ((U * 0.5) / J)))
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(-2.0 * Float64(J * t_0))
	tmp = 0.0
	if (t_0 <= 0.34)
		tmp = t_1;
	elseif (t_0 <= 0.5)
		tmp = Float64(-2.0 * Float64(U * -0.5));
	elseif (t_0 <= 0.66)
		tmp = t_1;
	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)
	t_0 = cos((K / 2.0));
	t_1 = -2.0 * (J * t_0);
	tmp = 0.0;
	if (t_0 <= 0.34)
		tmp = t_1;
	elseif (t_0 <= 0.5)
		tmp = -2.0 * (U * -0.5);
	elseif (t_0 <= 0.66)
		tmp = t_1;
	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_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(-2.0 * N[(J * t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.34], t$95$1, If[LessEqual[t$95$0, 0.5], N[(-2.0 * N[(U * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.66], t$95$1, 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 3 regimes

2. if (cos.f64 (/.f64 K 2)) < 0.340000000000000024 or 0.5 < (cos.f64 (/.f64 K 2)) < 0.660000000000000031

   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. Step-by-step derivation

      1. associate-*l* 75.1%

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

      2. associate-*l* 75.1%

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

      3. *-commutative 75.1%

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

      4. unpow2 75.1%

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

      5. sqr-neg 75.1%

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

      6. distribute-frac-neg 75.1%

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

      7. distribute-frac-neg 75.1%

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

      8. unpow2 75.1%

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

   3. Simplified 92.5%

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

   4. Taylor expanded in J around inf 56.1%

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

   if 0.340000000000000024 < (cos.f64 (/.f64 K 2)) < 0.5

   1. Initial program 46.1%

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

   3. Simplified 59.4%

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

   4. Taylor expanded in U around -inf 44.1%

      \[\leadsto -2 \cdot \color{blue}{\left(-0.5 \cdot U\right)} \]
   5. Step-by-step derivation

      1. *-commutative 44.1%

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

   6. Simplified 44.1%

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

   if 0.660000000000000031 < (cos.f64 (/.f64 K 2))

   1. Initial program 69.9%

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

   3. Simplified 87.4%

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

   4. Taylor expanded in K around 0 46.2%

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

      1. unpow2 46.2%

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

      2. unpow2 46.2%

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

   6. Simplified 46.2%

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

      1. add-sqr-sqrt 46.2%

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

      2. hypot-1-def 46.2%

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

      3. sqrt-prod 46.2%

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

      4. metadata-eval 46.2%

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

      5. times-frac 64.2%

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

      6. sqrt-prod 38.2%

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

      7. add-sqr-sqrt 82.3%

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

      8. associate-*r/ 82.3%

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

   8. Applied egg-rr 82.3%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.34:\\ \;\;\;\;-2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)\\ \mathbf{elif}\;\cos \left(\frac{K}{2}\right) \leq 0.5:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5\right)\\ \mathbf{elif}\;\cos \left(\frac{K}{2}\right) \leq 0.66:\\ \;\;\;\;-2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(J \cdot \mathsf{hypot}\left(1, \frac{U \cdot 0.5}{J}\right)\right)\\ \end{array} \]

Alternative 3: 87.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;U \leq 1.55 \cdot 10^{+270}:\\ \;\;\;\;-2 \cdot \left(J \cdot \left(t_0 \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot t_0\right)}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(U \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= U 1.55e+270)
     (* -2.0 (* J (* t_0 (hypot 1.0 (/ U (* J (* 2.0 t_0)))))))
     (* -2.0 (* U 0.5)))))

C

U = abs(U);
double code(double J, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (U <= 1.55e+270) {
		tmp = -2.0 * (J * (t_0 * hypot(1.0, (U / (J * (2.0 * t_0))))));
	} else {
		tmp = -2.0 * (U * 0.5);
	}
	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 (U <= 1.55e+270) {
		tmp = -2.0 * (J * (t_0 * Math.hypot(1.0, (U / (J * (2.0 * t_0))))));
	} else {
		tmp = -2.0 * (U * 0.5);
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if U <= 1.55e+270:
		tmp = -2.0 * (J * (t_0 * math.hypot(1.0, (U / (J * (2.0 * t_0))))))
	else:
		tmp = -2.0 * (U * 0.5)
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (U <= 1.55e+270)
		tmp = Float64(-2.0 * Float64(J * Float64(t_0 * hypot(1.0, Float64(U / Float64(J * Float64(2.0 * t_0)))))));
	else
		tmp = Float64(-2.0 * Float64(U * 0.5));
	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 (U <= 1.55e+270)
		tmp = -2.0 * (J * (t_0 * hypot(1.0, (U / (J * (2.0 * t_0))))));
	else
		tmp = -2.0 * (U * 0.5);
	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[LessEqual[U, 1.55e+270], N[(-2.0 * N[(J * N[(t$95$0 * N[Sqrt[1.0 ^ 2 + N[(U / N[(J * N[(2.0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(U * 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if U < 1.55e270

   1. Initial program 72.4%

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

   3. Simplified 89.9%

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

   if 1.55e270 < U

   1. Initial program 31.8%

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

   3. Simplified 43.3%

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

   4. Taylor expanded in J around 0 87.5%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 1.55 \cdot 10^{+270}:\\ \;\;\;\;-2 \cdot \left(J \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(U \cdot 0.5\right)\\ \end{array} \]

Alternative 4: 87.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;U \leq 1.65 \cdot 10^{+270}:\\ \;\;\;\;-2 \cdot \left(t_0 \cdot \left(J \cdot \mathsf{hypot}\left(1, 0.5 \cdot \frac{U}{J \cdot t_0}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(U \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= U 1.65e+270)
     (* -2.0 (* t_0 (* J (hypot 1.0 (* 0.5 (/ U (* J t_0)))))))
     (* -2.0 (* U 0.5)))))

C

U = abs(U);
double code(double J, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (U <= 1.65e+270) {
		tmp = -2.0 * (t_0 * (J * hypot(1.0, (0.5 * (U / (J * t_0))))));
	} else {
		tmp = -2.0 * (U * 0.5);
	}
	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 (U <= 1.65e+270) {
		tmp = -2.0 * (t_0 * (J * Math.hypot(1.0, (0.5 * (U / (J * t_0))))));
	} else {
		tmp = -2.0 * (U * 0.5);
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if U <= 1.65e+270:
		tmp = -2.0 * (t_0 * (J * math.hypot(1.0, (0.5 * (U / (J * t_0))))))
	else:
		tmp = -2.0 * (U * 0.5)
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (U <= 1.65e+270)
		tmp = Float64(-2.0 * Float64(t_0 * Float64(J * hypot(1.0, Float64(0.5 * Float64(U / Float64(J * t_0)))))));
	else
		tmp = Float64(-2.0 * Float64(U * 0.5));
	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 (U <= 1.65e+270)
		tmp = -2.0 * (t_0 * (J * hypot(1.0, (0.5 * (U / (J * t_0))))));
	else
		tmp = -2.0 * (U * 0.5);
	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[LessEqual[U, 1.65e+270], N[(-2.0 * N[(t$95$0 * N[(J * N[Sqrt[1.0 ^ 2 + N[(0.5 * N[(U / N[(J * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(U * 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if U < 1.64999999999999996e270

   1. Initial program 72.4%

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

      1. associate-*l* 72.4%

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

      2. associate-*l* 72.4%

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

      3. *-commutative 72.4%

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

      4. unpow2 72.4%

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

      5. sqr-neg 72.4%

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

      6. distribute-frac-neg 72.4%

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

      7. distribute-frac-neg 72.4%

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

      8. unpow2 72.4%

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

   3. Simplified 89.9%

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

   if 1.64999999999999996e270 < U

   1. Initial program 31.8%

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

   3. Simplified 43.3%

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

   4. Taylor expanded in J around 0 87.5%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 1.65 \cdot 10^{+270}:\\ \;\;\;\;-2 \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \mathsf{hypot}\left(1, 0.5 \cdot \frac{U}{J \cdot \cos \left(\frac{K}{2}\right)}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(U \cdot 0.5\right)\\ \end{array} \]

Alternative 5: 78.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} t_0 := -2 \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \mathsf{hypot}\left(1, 0.5 \cdot \frac{U}{J}\right)\right)\right)\\ \mathbf{if}\;J \leq -2.65 \cdot 10^{-146}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq -4 \cdot 10^{-310}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5\right)\\ \mathbf{elif}\;J \leq 3.6 \cdot 10^{-107}:\\ \;\;\;\;-2 \cdot \left(U \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (* -2.0 (* (cos (/ K 2.0)) (* J (hypot 1.0 (* 0.5 (/ U J))))))))
   (if (<= J -2.65e-146)
     t_0
     (if (<= J -4e-310)
       (* -2.0 (* U -0.5))
       (if (<= J 3.6e-107) (* -2.0 (* U 0.5)) t_0)))))

C

U = abs(U);
double code(double J, double K, double U) {
	double t_0 = -2.0 * (cos((K / 2.0)) * (J * hypot(1.0, (0.5 * (U / J)))));
	double tmp;
	if (J <= -2.65e-146) {
		tmp = t_0;
	} else if (J <= -4e-310) {
		tmp = -2.0 * (U * -0.5);
	} else if (J <= 3.6e-107) {
		tmp = -2.0 * (U * 0.5);
	} 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 * (Math.cos((K / 2.0)) * (J * Math.hypot(1.0, (0.5 * (U / J)))));
	double tmp;
	if (J <= -2.65e-146) {
		tmp = t_0;
	} else if (J <= -4e-310) {
		tmp = -2.0 * (U * -0.5);
	} else if (J <= 3.6e-107) {
		tmp = -2.0 * (U * 0.5);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	t_0 = -2.0 * (math.cos((K / 2.0)) * (J * math.hypot(1.0, (0.5 * (U / J)))))
	tmp = 0
	if J <= -2.65e-146:
		tmp = t_0
	elif J <= -4e-310:
		tmp = -2.0 * (U * -0.5)
	elif J <= 3.6e-107:
		tmp = -2.0 * (U * 0.5)
	else:
		tmp = t_0
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	t_0 = Float64(-2.0 * Float64(cos(Float64(K / 2.0)) * Float64(J * hypot(1.0, Float64(0.5 * Float64(U / J))))))
	tmp = 0.0
	if (J <= -2.65e-146)
		tmp = t_0;
	elseif (J <= -4e-310)
		tmp = Float64(-2.0 * Float64(U * -0.5));
	elseif (J <= 3.6e-107)
		tmp = Float64(-2.0 * Float64(U * 0.5));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	t_0 = -2.0 * (cos((K / 2.0)) * (J * hypot(1.0, (0.5 * (U / J)))));
	tmp = 0.0;
	if (J <= -2.65e-146)
		tmp = t_0;
	elseif (J <= -4e-310)
		tmp = -2.0 * (U * -0.5);
	elseif (J <= 3.6e-107)
		tmp = -2.0 * (U * 0.5);
	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[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(J * N[Sqrt[1.0 ^ 2 + N[(0.5 * N[(U / J), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[J, -2.65e-146], t$95$0, If[LessEqual[J, -4e-310], N[(-2.0 * N[(U * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[J, 3.6e-107], N[(-2.0 * N[(U * 0.5), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if J < -2.64999999999999991e-146 or 3.59999999999999976e-107 < J

   1. Initial program 85.0%

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

      1. associate-*l* 85.0%

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

      2. associate-*l* 85.0%

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

      3. *-commutative 85.0%

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

      4. unpow2 85.0%

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

      5. sqr-neg 85.0%

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

      6. distribute-frac-neg 85.0%

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

      7. distribute-frac-neg 85.0%

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

      8. unpow2 85.0%

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

   3. Simplified 97.7%

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

   4. Taylor expanded in K around 0 82.1%

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

   if -2.64999999999999991e-146 < J < -3.999999999999988e-310

   1. Initial program 40.8%

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

   3. Simplified 64.1%

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

   4. Taylor expanded in U around -inf 42.1%

      \[\leadsto -2 \cdot \color{blue}{\left(-0.5 \cdot U\right)} \]
   5. Step-by-step derivation

      1. *-commutative 42.1%

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

   6. Simplified 42.1%

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

   if -3.999999999999988e-310 < J < 3.59999999999999976e-107

   1. Initial program 33.3%

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

   3. Simplified 67.1%

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

   4. Taylor expanded in J around 0 44.5%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -2.65 \cdot 10^{-146}:\\ \;\;\;\;-2 \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \mathsf{hypot}\left(1, 0.5 \cdot \frac{U}{J}\right)\right)\right)\\ \mathbf{elif}\;J \leq -4 \cdot 10^{-310}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5\right)\\ \mathbf{elif}\;J \leq 3.6 \cdot 10^{-107}:\\ \;\;\;\;-2 \cdot \left(U \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \mathsf{hypot}\left(1, 0.5 \cdot \frac{U}{J}\right)\right)\right)\\ \end{array} \]

Alternative 6: 65.2% accurate, 3.5× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} t_0 := -2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)\\ \mathbf{if}\;J \leq -2.35 \cdot 10^{+30}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq -5.7 \cdot 10^{-12}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5 - \frac{J}{\frac{U}{J}}\right)\\ \mathbf{elif}\;J \leq -2.3 \cdot 10^{-91}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq -4 \cdot 10^{-310}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5\right)\\ \mathbf{elif}\;J \leq 1.5 \cdot 10^{-98} \lor \neg \left(J \leq 5.2 \cdot 10^{+26}\right) \land J \leq 5.8 \cdot 10^{+63}:\\ \;\;\;\;-2 \cdot \left(U \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

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))))))
   (if (<= J -2.35e+30)
     t_0
     (if (<= J -5.7e-12)
       (* -2.0 (- (* U -0.5) (/ J (/ U J))))
       (if (<= J -2.3e-91)
         t_0
         (if (<= J -4e-310)
           (* -2.0 (* U -0.5))
           (if (or (<= J 1.5e-98) (and (not (<= J 5.2e+26)) (<= J 5.8e+63)))
             (* -2.0 (* U 0.5))
             t_0)))))))

C

U = abs(U);
double code(double J, double K, double U) {
	double t_0 = -2.0 * (J * cos((K / 2.0)));
	double tmp;
	if (J <= -2.35e+30) {
		tmp = t_0;
	} else if (J <= -5.7e-12) {
		tmp = -2.0 * ((U * -0.5) - (J / (U / J)));
	} else if (J <= -2.3e-91) {
		tmp = t_0;
	} else if (J <= -4e-310) {
		tmp = -2.0 * (U * -0.5);
	} else if ((J <= 1.5e-98) || (!(J <= 5.2e+26) && (J <= 5.8e+63))) {
		tmp = -2.0 * (U * 0.5);
	} 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 = (-2.0d0) * (j * cos((k / 2.0d0)))
    if (j <= (-2.35d+30)) then
        tmp = t_0
    else if (j <= (-5.7d-12)) then
        tmp = (-2.0d0) * ((u * (-0.5d0)) - (j / (u / j)))
    else if (j <= (-2.3d-91)) then
        tmp = t_0
    else if (j <= (-4d-310)) then
        tmp = (-2.0d0) * (u * (-0.5d0))
    else if ((j <= 1.5d-98) .or. (.not. (j <= 5.2d+26)) .and. (j <= 5.8d+63)) then
        tmp = (-2.0d0) * (u * 0.5d0)
    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 = -2.0 * (J * Math.cos((K / 2.0)));
	double tmp;
	if (J <= -2.35e+30) {
		tmp = t_0;
	} else if (J <= -5.7e-12) {
		tmp = -2.0 * ((U * -0.5) - (J / (U / J)));
	} else if (J <= -2.3e-91) {
		tmp = t_0;
	} else if (J <= -4e-310) {
		tmp = -2.0 * (U * -0.5);
	} else if ((J <= 1.5e-98) || (!(J <= 5.2e+26) && (J <= 5.8e+63))) {
		tmp = -2.0 * (U * 0.5);
	} 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)))
	tmp = 0
	if J <= -2.35e+30:
		tmp = t_0
	elif J <= -5.7e-12:
		tmp = -2.0 * ((U * -0.5) - (J / (U / J)))
	elif J <= -2.3e-91:
		tmp = t_0
	elif J <= -4e-310:
		tmp = -2.0 * (U * -0.5)
	elif (J <= 1.5e-98) or (not (J <= 5.2e+26) and (J <= 5.8e+63)):
		tmp = -2.0 * (U * 0.5)
	else:
		tmp = t_0
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	t_0 = Float64(-2.0 * Float64(J * cos(Float64(K / 2.0))))
	tmp = 0.0
	if (J <= -2.35e+30)
		tmp = t_0;
	elseif (J <= -5.7e-12)
		tmp = Float64(-2.0 * Float64(Float64(U * -0.5) - Float64(J / Float64(U / J))));
	elseif (J <= -2.3e-91)
		tmp = t_0;
	elseif (J <= -4e-310)
		tmp = Float64(-2.0 * Float64(U * -0.5));
	elseif ((J <= 1.5e-98) || (!(J <= 5.2e+26) && (J <= 5.8e+63)))
		tmp = Float64(-2.0 * Float64(U * 0.5));
	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)));
	tmp = 0.0;
	if (J <= -2.35e+30)
		tmp = t_0;
	elseif (J <= -5.7e-12)
		tmp = -2.0 * ((U * -0.5) - (J / (U / J)));
	elseif (J <= -2.3e-91)
		tmp = t_0;
	elseif (J <= -4e-310)
		tmp = -2.0 * (U * -0.5);
	elseif ((J <= 1.5e-98) || (~((J <= 5.2e+26)) && (J <= 5.8e+63)))
		tmp = -2.0 * (U * 0.5);
	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[(J * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[J, -2.35e+30], t$95$0, If[LessEqual[J, -5.7e-12], N[(-2.0 * N[(N[(U * -0.5), $MachinePrecision] - N[(J / N[(U / J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[J, -2.3e-91], t$95$0, If[LessEqual[J, -4e-310], N[(-2.0 * N[(U * -0.5), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[J, 1.5e-98], And[N[Not[LessEqual[J, 5.2e+26]], $MachinePrecision], LessEqual[J, 5.8e+63]]], N[(-2.0 * N[(U * 0.5), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]

Derivation

1. Split input into 4 regimes

2. if J < -2.34999999999999995e30 or -5.7000000000000003e-12 < J < -2.29999999999999996e-91 or 1.5e-98 < J < 5.20000000000000004e26 or 5.7999999999999999e63 < J

   1. Initial program 89.4%

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

      1. associate-*l* 89.4%

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

      2. associate-*l* 89.4%

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

      3. *-commutative 89.4%

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

      4. unpow2 89.4%

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

      5. sqr-neg 89.4%

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

      6. distribute-frac-neg 89.4%

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

      7. distribute-frac-neg 89.4%

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

      8. unpow2 89.4%

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

   3. Simplified 97.9%

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

   4. Taylor expanded in J around inf 71.6%

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

   if -2.34999999999999995e30 < J < -5.7000000000000003e-12

   1. Initial program 62.6%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in K around 0 54.1%

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

      1. unpow2 54.1%

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

      2. unpow2 54.1%

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

   6. Simplified 54.1%

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

   7. Taylor expanded in U around -inf 51.3%

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

      1. +-commutative 51.3%

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

      2. mul-1-neg 51.3%

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

      3. unsub-neg 51.3%

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

      4. *-commutative 51.3%

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

      5. unpow2 51.3%

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

      6. associate-/l* 51.3%

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

   9. Simplified 51.3%

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

   if -2.29999999999999996e-91 < J < -3.999999999999988e-310

   1. Initial program 47.3%

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

   3. Simplified 69.7%

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

   4. Taylor expanded in U around -inf 46.3%

      \[\leadsto -2 \cdot \color{blue}{\left(-0.5 \cdot U\right)} \]
   5. Step-by-step derivation

      1. *-commutative 46.3%

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

   6. Simplified 46.3%

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

   if -3.999999999999988e-310 < J < 1.5e-98 or 5.20000000000000004e26 < J < 5.7999999999999999e63

   1. Initial program 38.7%

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

   3. Simplified 74.5%

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

   4. Taylor expanded in J around 0 42.9%

      \[\leadsto -2 \cdot \color{blue}{\left(0.5 \cdot U\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -2.35 \cdot 10^{+30}:\\ \;\;\;\;-2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)\\ \mathbf{elif}\;J \leq -5.7 \cdot 10^{-12}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5 - \frac{J}{\frac{U}{J}}\right)\\ \mathbf{elif}\;J \leq -2.3 \cdot 10^{-91}:\\ \;\;\;\;-2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)\\ \mathbf{elif}\;J \leq -4 \cdot 10^{-310}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5\right)\\ \mathbf{elif}\;J \leq 1.5 \cdot 10^{-98} \lor \neg \left(J \leq 5.2 \cdot 10^{+26}\right) \land J \leq 5.8 \cdot 10^{+63}:\\ \;\;\;\;-2 \cdot \left(U \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)\\ \end{array} \]

Alternative 7: 48.1% accurate, 19.7× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} t_0 := -2 \cdot \left(J + -0.125 \cdot \left(J \cdot \left(K \cdot K\right)\right)\right)\\ t_1 := -2 \cdot \left(U \cdot -0.5\right)\\ \mathbf{if}\;J \leq -2.9 \cdot 10^{+151}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;J \leq -4.5 \cdot 10^{-45}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;J \leq -2.1 \cdot 10^{-91}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq -4 \cdot 10^{-310}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;J \leq 1.1 \cdot 10^{+64}:\\ \;\;\;\;-2 \cdot \left(U \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (* -2.0 (+ J (* -0.125 (* J (* K K))))))
        (t_1 (* -2.0 (* U -0.5))))
   (if (<= J -2.9e+151)
     (* -2.0 J)
     (if (<= J -4.5e-45)
       t_1
       (if (<= J -2.1e-91)
         t_0
         (if (<= J -4e-310)
           t_1
           (if (<= J 1.1e+64) (* -2.0 (* U 0.5)) t_0)))))))

C

U = abs(U);
double code(double J, double K, double U) {
	double t_0 = -2.0 * (J + (-0.125 * (J * (K * K))));
	double t_1 = -2.0 * (U * -0.5);
	double tmp;
	if (J <= -2.9e+151) {
		tmp = -2.0 * J;
	} else if (J <= -4.5e-45) {
		tmp = t_1;
	} else if (J <= -2.1e-91) {
		tmp = t_0;
	} else if (J <= -4e-310) {
		tmp = t_1;
	} else if (J <= 1.1e+64) {
		tmp = -2.0 * (U * 0.5);
	} 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) :: t_1
    real(8) :: tmp
    t_0 = (-2.0d0) * (j + ((-0.125d0) * (j * (k * k))))
    t_1 = (-2.0d0) * (u * (-0.5d0))
    if (j <= (-2.9d+151)) then
        tmp = (-2.0d0) * j
    else if (j <= (-4.5d-45)) then
        tmp = t_1
    else if (j <= (-2.1d-91)) then
        tmp = t_0
    else if (j <= (-4d-310)) then
        tmp = t_1
    else if (j <= 1.1d+64) then
        tmp = (-2.0d0) * (u * 0.5d0)
    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 = -2.0 * (J + (-0.125 * (J * (K * K))));
	double t_1 = -2.0 * (U * -0.5);
	double tmp;
	if (J <= -2.9e+151) {
		tmp = -2.0 * J;
	} else if (J <= -4.5e-45) {
		tmp = t_1;
	} else if (J <= -2.1e-91) {
		tmp = t_0;
	} else if (J <= -4e-310) {
		tmp = t_1;
	} else if (J <= 1.1e+64) {
		tmp = -2.0 * (U * 0.5);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	t_0 = -2.0 * (J + (-0.125 * (J * (K * K))))
	t_1 = -2.0 * (U * -0.5)
	tmp = 0
	if J <= -2.9e+151:
		tmp = -2.0 * J
	elif J <= -4.5e-45:
		tmp = t_1
	elif J <= -2.1e-91:
		tmp = t_0
	elif J <= -4e-310:
		tmp = t_1
	elif J <= 1.1e+64:
		tmp = -2.0 * (U * 0.5)
	else:
		tmp = t_0
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	t_0 = Float64(-2.0 * Float64(J + Float64(-0.125 * Float64(J * Float64(K * K)))))
	t_1 = Float64(-2.0 * Float64(U * -0.5))
	tmp = 0.0
	if (J <= -2.9e+151)
		tmp = Float64(-2.0 * J);
	elseif (J <= -4.5e-45)
		tmp = t_1;
	elseif (J <= -2.1e-91)
		tmp = t_0;
	elseif (J <= -4e-310)
		tmp = t_1;
	elseif (J <= 1.1e+64)
		tmp = Float64(-2.0 * Float64(U * 0.5));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	t_0 = -2.0 * (J + (-0.125 * (J * (K * K))));
	t_1 = -2.0 * (U * -0.5);
	tmp = 0.0;
	if (J <= -2.9e+151)
		tmp = -2.0 * J;
	elseif (J <= -4.5e-45)
		tmp = t_1;
	elseif (J <= -2.1e-91)
		tmp = t_0;
	elseif (J <= -4e-310)
		tmp = t_1;
	elseif (J <= 1.1e+64)
		tmp = -2.0 * (U * 0.5);
	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[(J + N[(-0.125 * N[(J * N[(K * K), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-2.0 * N[(U * -0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[J, -2.9e+151], N[(-2.0 * J), $MachinePrecision], If[LessEqual[J, -4.5e-45], t$95$1, If[LessEqual[J, -2.1e-91], t$95$0, If[LessEqual[J, -4e-310], t$95$1, If[LessEqual[J, 1.1e+64], N[(-2.0 * N[(U * 0.5), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]]

Derivation

1. Split input into 4 regimes

2. if J < -2.90000000000000018e151

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in K around 0 35.8%

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

      1. unpow2 35.8%

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

      2. unpow2 35.8%

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

   6. Simplified 35.8%

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

   7. Taylor expanded in J around inf 50.4%

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

   if -2.90000000000000018e151 < J < -4.4999999999999999e-45 or -2.0999999999999999e-91 < J < -3.999999999999988e-310

   1. Initial program 61.1%

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

   3. Simplified 81.6%

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

   4. Taylor expanded in U around -inf 39.9%

      \[\leadsto -2 \cdot \color{blue}{\left(-0.5 \cdot U\right)} \]
   5. Step-by-step derivation

      1. *-commutative 39.9%

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

   6. Simplified 39.9%

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

   if -4.4999999999999999e-45 < J < -2.0999999999999999e-91 or 1.10000000000000001e64 < J

   1. Initial program 92.3%

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

   3. Simplified 99.7%

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

      1. expm1-log1p-u 85.9%

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

      2. expm1-udef 85.8%

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

   5. Applied egg-rr 85.8%

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

   6. Taylor expanded in U around 0 75.0%

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

      1. *-commutative 75.0%

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

   8. Simplified 75.0%

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

   9. Taylor expanded in K around 0 45.0%

      \[\leadsto -2 \cdot \color{blue}{\left(J + -0.125 \cdot \left(J \cdot {K}^{2}\right)\right)} \]
   10. Step-by-step derivation

       1. unpow2 45.0%

          \[\leadsto -2 \cdot \left(J + -0.125 \cdot \left(J \cdot \color{blue}{\left(K \cdot K\right)}\right)\right) \]

   11. Simplified 45.0%

       \[\leadsto -2 \cdot \color{blue}{\left(J + -0.125 \cdot \left(J \cdot \left(K \cdot K\right)\right)\right)} \]

   if -3.999999999999988e-310 < J < 1.10000000000000001e64

   1. Initial program 54.4%

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

   3. Simplified 82.3%

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

   4. Taylor expanded in J around 0 39.3%

      \[\leadsto -2 \cdot \color{blue}{\left(0.5 \cdot U\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 42.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -2.9 \cdot 10^{+151}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;J \leq -4.5 \cdot 10^{-45}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5\right)\\ \mathbf{elif}\;J \leq -2.1 \cdot 10^{-91}:\\ \;\;\;\;-2 \cdot \left(J + -0.125 \cdot \left(J \cdot \left(K \cdot K\right)\right)\right)\\ \mathbf{elif}\;J \leq -4 \cdot 10^{-310}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5\right)\\ \mathbf{elif}\;J \leq 1.1 \cdot 10^{+64}:\\ \;\;\;\;-2 \cdot \left(U \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(J + -0.125 \cdot \left(J \cdot \left(K \cdot K\right)\right)\right)\\ \end{array} \]

Alternative 8: 48.1% accurate, 19.7× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} t_0 := -2 \cdot \left(J + -0.125 \cdot \left(J \cdot \left(K \cdot K\right)\right)\right)\\ \mathbf{if}\;J \leq -4 \cdot 10^{+151}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;J \leq -4.5 \cdot 10^{-45}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5 - \frac{J}{\frac{U}{J}}\right)\\ \mathbf{elif}\;J \leq -2.3 \cdot 10^{-91}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq -4 \cdot 10^{-310}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5\right)\\ \mathbf{elif}\;J \leq 1.4 \cdot 10^{+64}:\\ \;\;\;\;-2 \cdot \left(U \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (* -2.0 (+ J (* -0.125 (* J (* K K)))))))
   (if (<= J -4e+151)
     (* -2.0 J)
     (if (<= J -4.5e-45)
       (* -2.0 (- (* U -0.5) (/ J (/ U J))))
       (if (<= J -2.3e-91)
         t_0
         (if (<= J -4e-310)
           (* -2.0 (* U -0.5))
           (if (<= J 1.4e+64) (* -2.0 (* U 0.5)) t_0)))))))

C

U = abs(U);
double code(double J, double K, double U) {
	double t_0 = -2.0 * (J + (-0.125 * (J * (K * K))));
	double tmp;
	if (J <= -4e+151) {
		tmp = -2.0 * J;
	} else if (J <= -4.5e-45) {
		tmp = -2.0 * ((U * -0.5) - (J / (U / J)));
	} else if (J <= -2.3e-91) {
		tmp = t_0;
	} else if (J <= -4e-310) {
		tmp = -2.0 * (U * -0.5);
	} else if (J <= 1.4e+64) {
		tmp = -2.0 * (U * 0.5);
	} 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 = (-2.0d0) * (j + ((-0.125d0) * (j * (k * k))))
    if (j <= (-4d+151)) then
        tmp = (-2.0d0) * j
    else if (j <= (-4.5d-45)) then
        tmp = (-2.0d0) * ((u * (-0.5d0)) - (j / (u / j)))
    else if (j <= (-2.3d-91)) then
        tmp = t_0
    else if (j <= (-4d-310)) then
        tmp = (-2.0d0) * (u * (-0.5d0))
    else if (j <= 1.4d+64) then
        tmp = (-2.0d0) * (u * 0.5d0)
    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 = -2.0 * (J + (-0.125 * (J * (K * K))));
	double tmp;
	if (J <= -4e+151) {
		tmp = -2.0 * J;
	} else if (J <= -4.5e-45) {
		tmp = -2.0 * ((U * -0.5) - (J / (U / J)));
	} else if (J <= -2.3e-91) {
		tmp = t_0;
	} else if (J <= -4e-310) {
		tmp = -2.0 * (U * -0.5);
	} else if (J <= 1.4e+64) {
		tmp = -2.0 * (U * 0.5);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	t_0 = -2.0 * (J + (-0.125 * (J * (K * K))))
	tmp = 0
	if J <= -4e+151:
		tmp = -2.0 * J
	elif J <= -4.5e-45:
		tmp = -2.0 * ((U * -0.5) - (J / (U / J)))
	elif J <= -2.3e-91:
		tmp = t_0
	elif J <= -4e-310:
		tmp = -2.0 * (U * -0.5)
	elif J <= 1.4e+64:
		tmp = -2.0 * (U * 0.5)
	else:
		tmp = t_0
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	t_0 = Float64(-2.0 * Float64(J + Float64(-0.125 * Float64(J * Float64(K * K)))))
	tmp = 0.0
	if (J <= -4e+151)
		tmp = Float64(-2.0 * J);
	elseif (J <= -4.5e-45)
		tmp = Float64(-2.0 * Float64(Float64(U * -0.5) - Float64(J / Float64(U / J))));
	elseif (J <= -2.3e-91)
		tmp = t_0;
	elseif (J <= -4e-310)
		tmp = Float64(-2.0 * Float64(U * -0.5));
	elseif (J <= 1.4e+64)
		tmp = Float64(-2.0 * Float64(U * 0.5));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	t_0 = -2.0 * (J + (-0.125 * (J * (K * K))));
	tmp = 0.0;
	if (J <= -4e+151)
		tmp = -2.0 * J;
	elseif (J <= -4.5e-45)
		tmp = -2.0 * ((U * -0.5) - (J / (U / J)));
	elseif (J <= -2.3e-91)
		tmp = t_0;
	elseif (J <= -4e-310)
		tmp = -2.0 * (U * -0.5);
	elseif (J <= 1.4e+64)
		tmp = -2.0 * (U * 0.5);
	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[(J + N[(-0.125 * N[(J * N[(K * K), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[J, -4e+151], N[(-2.0 * J), $MachinePrecision], If[LessEqual[J, -4.5e-45], N[(-2.0 * N[(N[(U * -0.5), $MachinePrecision] - N[(J / N[(U / J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[J, -2.3e-91], t$95$0, If[LessEqual[J, -4e-310], N[(-2.0 * N[(U * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[J, 1.4e+64], N[(-2.0 * N[(U * 0.5), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]

Derivation

1. Split input into 5 regimes

2. if J < -4.00000000000000007e151

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in K around 0 35.8%

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

      1. unpow2 35.8%

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

      2. unpow2 35.8%

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

   6. Simplified 35.8%

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

   7. Taylor expanded in J around inf 50.4%

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

   if -4.00000000000000007e151 < J < -4.4999999999999999e-45

   1. Initial program 82.0%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in K around 0 36.0%

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

      1. unpow2 36.0%

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

      2. unpow2 36.0%

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

   6. Simplified 36.0%

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

   7. Taylor expanded in U around -inf 30.4%

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

      1. +-commutative 30.4%

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

      2. mul-1-neg 30.4%

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

      3. unsub-neg 30.4%

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

      4. *-commutative 30.4%

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

      5. unpow2 30.4%

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

      6. associate-/l* 30.4%

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

   9. Simplified 30.4%

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

   if -4.4999999999999999e-45 < J < -2.29999999999999996e-91 or 1.40000000000000012e64 < J

   1. Initial program 92.3%

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

   3. Simplified 99.7%

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

      1. expm1-log1p-u 85.9%

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

      2. expm1-udef 85.8%

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

   5. Applied egg-rr 85.8%

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

   6. Taylor expanded in U around 0 75.0%

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

      1. *-commutative 75.0%

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

   8. Simplified 75.0%

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

   9. Taylor expanded in K around 0 45.0%

      \[\leadsto -2 \cdot \color{blue}{\left(J + -0.125 \cdot \left(J \cdot {K}^{2}\right)\right)} \]
   10. Step-by-step derivation

       1. unpow2 45.0%

          \[\leadsto -2 \cdot \left(J + -0.125 \cdot \left(J \cdot \color{blue}{\left(K \cdot K\right)}\right)\right) \]

   11. Simplified 45.0%

       \[\leadsto -2 \cdot \color{blue}{\left(J + -0.125 \cdot \left(J \cdot \left(K \cdot K\right)\right)\right)} \]

   if -2.29999999999999996e-91 < J < -3.999999999999988e-310

   1. Initial program 47.3%

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

   3. Simplified 69.7%

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

   4. Taylor expanded in U around -inf 46.3%

      \[\leadsto -2 \cdot \color{blue}{\left(-0.5 \cdot U\right)} \]
   5. Step-by-step derivation

      1. *-commutative 46.3%

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

   6. Simplified 46.3%

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

   if -3.999999999999988e-310 < J < 1.40000000000000012e64

   1. Initial program 54.4%

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

   3. Simplified 82.3%

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

   4. Taylor expanded in J around 0 39.3%

      \[\leadsto -2 \cdot \color{blue}{\left(0.5 \cdot U\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 42.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -4 \cdot 10^{+151}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;J \leq -4.5 \cdot 10^{-45}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5 - \frac{J}{\frac{U}{J}}\right)\\ \mathbf{elif}\;J \leq -2.3 \cdot 10^{-91}:\\ \;\;\;\;-2 \cdot \left(J + -0.125 \cdot \left(J \cdot \left(K \cdot K\right)\right)\right)\\ \mathbf{elif}\;J \leq -4 \cdot 10^{-310}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5\right)\\ \mathbf{elif}\;J \leq 1.4 \cdot 10^{+64}:\\ \;\;\;\;-2 \cdot \left(U \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(J + -0.125 \cdot \left(J \cdot \left(K \cdot K\right)\right)\right)\\ \end{array} \]

Alternative 9: 48.7% accurate, 27.4× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} t_0 := -2 \cdot \left(U \cdot -0.5\right)\\ \mathbf{if}\;J \leq -4.2 \cdot 10^{+151}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;J \leq -4.6 \cdot 10^{-45}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq -2.3 \cdot 10^{-91}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;J \leq -4 \cdot 10^{-310}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq 7.6 \cdot 10^{+63}:\\ \;\;\;\;-2 \cdot \left(U \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot J\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (* -2.0 (* U -0.5))))
   (if (<= J -4.2e+151)
     (* -2.0 J)
     (if (<= J -4.6e-45)
       t_0
       (if (<= J -2.3e-91)
         (* -2.0 J)
         (if (<= J -4e-310)
           t_0
           (if (<= J 7.6e+63) (* -2.0 (* U 0.5)) (* -2.0 J))))))))

C

U = abs(U);
double code(double J, double K, double U) {
	double t_0 = -2.0 * (U * -0.5);
	double tmp;
	if (J <= -4.2e+151) {
		tmp = -2.0 * J;
	} else if (J <= -4.6e-45) {
		tmp = t_0;
	} else if (J <= -2.3e-91) {
		tmp = -2.0 * J;
	} else if (J <= -4e-310) {
		tmp = t_0;
	} else if (J <= 7.6e+63) {
		tmp = -2.0 * (U * 0.5);
	} 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) :: t_0
    real(8) :: tmp
    t_0 = (-2.0d0) * (u * (-0.5d0))
    if (j <= (-4.2d+151)) then
        tmp = (-2.0d0) * j
    else if (j <= (-4.6d-45)) then
        tmp = t_0
    else if (j <= (-2.3d-91)) then
        tmp = (-2.0d0) * j
    else if (j <= (-4d-310)) then
        tmp = t_0
    else if (j <= 7.6d+63) then
        tmp = (-2.0d0) * (u * 0.5d0)
    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 t_0 = -2.0 * (U * -0.5);
	double tmp;
	if (J <= -4.2e+151) {
		tmp = -2.0 * J;
	} else if (J <= -4.6e-45) {
		tmp = t_0;
	} else if (J <= -2.3e-91) {
		tmp = -2.0 * J;
	} else if (J <= -4e-310) {
		tmp = t_0;
	} else if (J <= 7.6e+63) {
		tmp = -2.0 * (U * 0.5);
	} else {
		tmp = -2.0 * J;
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	t_0 = -2.0 * (U * -0.5)
	tmp = 0
	if J <= -4.2e+151:
		tmp = -2.0 * J
	elif J <= -4.6e-45:
		tmp = t_0
	elif J <= -2.3e-91:
		tmp = -2.0 * J
	elif J <= -4e-310:
		tmp = t_0
	elif J <= 7.6e+63:
		tmp = -2.0 * (U * 0.5)
	else:
		tmp = -2.0 * J
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	t_0 = Float64(-2.0 * Float64(U * -0.5))
	tmp = 0.0
	if (J <= -4.2e+151)
		tmp = Float64(-2.0 * J);
	elseif (J <= -4.6e-45)
		tmp = t_0;
	elseif (J <= -2.3e-91)
		tmp = Float64(-2.0 * J);
	elseif (J <= -4e-310)
		tmp = t_0;
	elseif (J <= 7.6e+63)
		tmp = Float64(-2.0 * Float64(U * 0.5));
	else
		tmp = Float64(-2.0 * J);
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	t_0 = -2.0 * (U * -0.5);
	tmp = 0.0;
	if (J <= -4.2e+151)
		tmp = -2.0 * J;
	elseif (J <= -4.6e-45)
		tmp = t_0;
	elseif (J <= -2.3e-91)
		tmp = -2.0 * J;
	elseif (J <= -4e-310)
		tmp = t_0;
	elseif (J <= 7.6e+63)
		tmp = -2.0 * (U * 0.5);
	else
		tmp = -2.0 * J;
	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[(U * -0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[J, -4.2e+151], N[(-2.0 * J), $MachinePrecision], If[LessEqual[J, -4.6e-45], t$95$0, If[LessEqual[J, -2.3e-91], N[(-2.0 * J), $MachinePrecision], If[LessEqual[J, -4e-310], t$95$0, If[LessEqual[J, 7.6e+63], N[(-2.0 * N[(U * 0.5), $MachinePrecision]), $MachinePrecision], N[(-2.0 * J), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if J < -4.2000000000000001e151 or -4.59999999999999983e-45 < J < -2.29999999999999996e-91 or 7.6000000000000002e63 < J

   1. Initial program 94.8%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in K around 0 40.5%

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

      1. unpow2 40.5%

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

      2. unpow2 40.5%

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

   6. Simplified 40.5%

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

   7. Taylor expanded in J around inf 46.6%

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

   if -4.2000000000000001e151 < J < -4.59999999999999983e-45 or -2.29999999999999996e-91 < J < -3.999999999999988e-310

   1. Initial program 61.1%

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

   3. Simplified 81.6%

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

   4. Taylor expanded in U around -inf 39.9%

      \[\leadsto -2 \cdot \color{blue}{\left(-0.5 \cdot U\right)} \]
   5. Step-by-step derivation

      1. *-commutative 39.9%

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

   6. Simplified 39.9%

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

   if -3.999999999999988e-310 < J < 7.6000000000000002e63

   1. Initial program 54.4%

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

   3. Simplified 82.3%

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

   4. Taylor expanded in J around 0 39.3%

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

4. Final simplification 42.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -4.2 \cdot 10^{+151}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;J \leq -4.6 \cdot 10^{-45}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5\right)\\ \mathbf{elif}\;J \leq -2.3 \cdot 10^{-91}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;J \leq -4 \cdot 10^{-310}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5\right)\\ \mathbf{elif}\;J \leq 7.6 \cdot 10^{+63}:\\ \;\;\;\;-2 \cdot \left(U \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot J\\ \end{array} \]

Alternative 10: 40.9% accurate, 59.5× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} \mathbf{if}\;U \leq 1.2 \cdot 10^{-36}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(U \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (if (<= U 1.2e-36) (* -2.0 J) (* -2.0 (* U 0.5))))

C

U = abs(U);
double code(double J, double K, double U) {
	double tmp;
	if (U <= 1.2e-36) {
		tmp = -2.0 * J;
	} else {
		tmp = -2.0 * (U * 0.5);
	}
	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 (u <= 1.2d-36) then
        tmp = (-2.0d0) * j
    else
        tmp = (-2.0d0) * (u * 0.5d0)
    end if
    code = tmp
end function

Java

U = Math.abs(U);
public static double code(double J, double K, double U) {
	double tmp;
	if (U <= 1.2e-36) {
		tmp = -2.0 * J;
	} else {
		tmp = -2.0 * (U * 0.5);
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	tmp = 0
	if U <= 1.2e-36:
		tmp = -2.0 * J
	else:
		tmp = -2.0 * (U * 0.5)
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	tmp = 0.0
	if (U <= 1.2e-36)
		tmp = Float64(-2.0 * J);
	else
		tmp = Float64(-2.0 * Float64(U * 0.5));
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	tmp = 0.0;
	if (U <= 1.2e-36)
		tmp = -2.0 * J;
	else
		tmp = -2.0 * (U * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := If[LessEqual[U, 1.2e-36], N[(-2.0 * J), $MachinePrecision], N[(-2.0 * N[(U * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U < 1.2e-36

   1. Initial program 79.0%

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

   3. Simplified 93.9%

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

   4. Taylor expanded in K around 0 35.2%

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

      1. unpow2 35.2%

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

      2. unpow2 35.2%

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

   6. Simplified 35.2%

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

   7. Taylor expanded in J around inf 33.0%

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

   if 1.2e-36 < U

   1. Initial program 53.4%

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

   3. Simplified 76.1%

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

   4. Taylor expanded in J around 0 42.6%

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

4. Final simplification 35.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 1.2 \cdot 10^{-36}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(U \cdot 0.5\right)\\ \end{array} \]

Alternative 11: 29.7% accurate, 140.0× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ -2 \cdot J \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U) :precision binary64 (* -2.0 J))

C

U = abs(U);
double code(double J, double K, double U) {
	return -2.0 * J;
}

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 = (-2.0d0) * j
end function

Java

U = Math.abs(U);
public static double code(double J, double K, double U) {
	return -2.0 * J;
}

Python

U = abs(U)
def code(J, K, U):
	return -2.0 * J

Julia

U = abs(U)
function code(J, K, U)
	return Float64(-2.0 * J)
end

MATLAB

U = abs(U)
function tmp = code(J, K, U)
	tmp = -2.0 * J;
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := N[(-2.0 * J), $MachinePrecision]

Derivation

1. Initial program 71.1%

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

3. Simplified 88.4%

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

4. Taylor expanded in K around 0 30.5%

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

   1. unpow2 30.5%

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

   2. unpow2 30.5%

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

6. Simplified 30.5%

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

7. Taylor expanded in J around inf 28.2%

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

8. Final simplification 28.2%

   \[\leadsto -2 \cdot J \]

Reproduce

herbie shell --seed 2023271 
(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)))))