Maksimov and Kolovsky, Equation (3)

Percentage Accurate: 72.7% → 99.7%

Time: 22.0s

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: 72.7% 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

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := J \cdot t_0\\ t_2 := \left(\left(-2 \cdot J\right) \cdot t_0\right) \cdot \sqrt{1 + {\left(\frac{U}{t_0 \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;-2 \cdot \left(U \cdot 0.5\right)\\ \mathbf{elif}\;t_2 \leq 10^{+307}:\\ \;\;\;\;-2 \cdot \left(t_1 \cdot \mathsf{hypot}\left(1, \frac{U}{2 \cdot t_1}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5\right)\\ \end{array} \end{array} \]

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 1e+307)
       (* -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 <= 1e+307) {
		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 <= 1e+307) {
		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 <= 1e+307:
		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 <= 1e+307)
		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 <= 1e+307)
		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, 1e+307], 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 5.9%

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

   3. Simplified 55.5%

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

   4. Taylor expanded in J around 0 47.8%

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

   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)))) < 9.99999999999999986e306

   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 9.99999999999999986e306 < (*.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 8.5%

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

   3. Simplified 76.4%

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

   4. Taylor expanded in U around -inf 50.4%

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

      1. *-commutative 50.4%

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

   6. Simplified 50.4%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -\infty:\\ \;\;\;\;-2 \cdot \left(U \cdot 0.5\right)\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 10^{+307}:\\ \;\;\;\;-2 \cdot \left(\left(J \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5\right)\\ \end{array} \]

Alternative 2: 87.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ -2 \cdot \left(J \cdot \left(t_0 \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot t_0\right)}\right)\right)\right) \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (* -2.0 (* J (* t_0 (hypot 1.0 (/ U (* J (* 2.0 t_0)))))))))

C

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

Java

U = Math.abs(U);
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.hypot(1.0, (U / (J * (2.0 * t_0))))));
}

Python

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

Julia

U = abs(U)
function code(J, K, U)
	t_0 = cos(Float64(K / 2.0))
	return Float64(-2.0 * Float64(J * Float64(t_0 * hypot(1.0, Float64(U / Float64(J * Float64(2.0 * t_0)))))))
end

MATLAB

U = abs(U)
function tmp = code(J, K, U)
	t_0 = cos((K / 2.0));
	tmp = -2.0 * (J * (t_0 * hypot(1.0, (U / (J * (2.0 * t_0))))));
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]}, 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]]

Derivation

1. Initial program 74.5%

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

3. Simplified 90.2%

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

4. Final simplification 90.2%

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

Alternative 3: 73.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} \mathbf{if}\;U \leq 3 \cdot 10^{+234}:\\ \;\;\;\;-2 \cdot \left(J \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot 2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(U \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (if (<= U 3e+234)
   (* -2.0 (* J (* (cos (/ K 2.0)) (hypot 1.0 (/ U (* J 2.0))))))
   (* -2.0 (* U 0.5))))

C

U = abs(U);
double code(double J, double K, double U) {
	double tmp;
	if (U <= 3e+234) {
		tmp = -2.0 * (J * (cos((K / 2.0)) * hypot(1.0, (U / (J * 2.0)))));
	} 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 tmp;
	if (U <= 3e+234) {
		tmp = -2.0 * (J * (Math.cos((K / 2.0)) * Math.hypot(1.0, (U / (J * 2.0)))));
	} else {
		tmp = -2.0 * (U * 0.5);
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	tmp = 0
	if U <= 3e+234:
		tmp = -2.0 * (J * (math.cos((K / 2.0)) * math.hypot(1.0, (U / (J * 2.0)))))
	else:
		tmp = -2.0 * (U * 0.5)
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	tmp = 0.0
	if (U <= 3e+234)
		tmp = Float64(-2.0 * Float64(J * Float64(cos(Float64(K / 2.0)) * hypot(1.0, Float64(U / Float64(J * 2.0))))));
	else
		tmp = Float64(-2.0 * Float64(U * 0.5));
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	tmp = 0.0;
	if (U <= 3e+234)
		tmp = -2.0 * (J * (cos((K / 2.0)) * hypot(1.0, (U / (J * 2.0)))));
	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_] := If[LessEqual[U, 3e+234], N[(-2.0 * N[(J * N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[(U / N[(J * 2.0), $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 < 2.9999999999999999e234

   1. Initial program 77.4%

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

   3. Simplified 92.1%

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

   4. Taylor expanded in K around 0 74.5%

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

      1. *-commutative 74.5%

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

   6. Simplified 74.5%

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

   if 2.9999999999999999e234 < U

   1. Initial program 37.6%

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

   3. Simplified 66.4%

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

   4. Taylor expanded in J around 0 52.7%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 3 \cdot 10^{+234}:\\ \;\;\;\;-2 \cdot \left(J \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot 2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(U \cdot 0.5\right)\\ \end{array} \]

Alternative 4: 68.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t_0 \leq 0.232:\\ \;\;\;\;-2 \cdot \left(J \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(J \cdot \mathsf{hypot}\left(1, \frac{U \cdot 0.5}{J}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 0.232)
     (* -2.0 (* J t_0))
     (* -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 tmp;
	if (t_0 <= 0.232) {
		tmp = -2.0 * (J * t_0);
	} 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 tmp;
	if (t_0 <= 0.232) {
		tmp = -2.0 * (J * t_0);
	} 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))
	tmp = 0
	if t_0 <= 0.232:
		tmp = -2.0 * (J * t_0)
	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))
	tmp = 0.0
	if (t_0 <= 0.232)
		tmp = Float64(-2.0 * Float64(J * t_0));
	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));
	tmp = 0.0;
	if (t_0 <= 0.232)
		tmp = -2.0 * (J * t_0);
	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]}, If[LessEqual[t$95$0, 0.232], N[(-2.0 * N[(J * t$95$0), $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)) < 0.232000000000000012

   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 91.1%

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

   4. Taylor expanded in J around inf 53.7%

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

   if 0.232000000000000012 < (cos.f64 (/.f64 K 2))

   1. Initial program 74.1%

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

   3. Simplified 89.8%

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

   4. Taylor expanded in K around 0 46.6%

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

      1. unpow2 46.6%

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

      2. unpow2 46.6%

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

   6. Simplified 46.6%

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

      1. expm1-log1p-u 46.0%

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

      2. expm1-udef 46.0%

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

      3. add-sqr-sqrt 46.0%

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

      4. hypot-1-def 46.0%

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

      5. sqrt-prod 46.0%

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

      6. metadata-eval 46.0%

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

      7. times-frac 63.9%

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

      8. sqrt-prod 38.4%

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

      9. add-sqr-sqrt 77.9%

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

   8. Applied egg-rr 77.9%

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

      1. expm1-def 77.9%

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

      2. expm1-log1p 81.5%

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

      3. associate-*r/ 81.5%

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

   10. Simplified 81.5%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.232:\\ \;\;\;\;-2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(J \cdot \mathsf{hypot}\left(1, \frac{U \cdot 0.5}{J}\right)\right)\\ \end{array} \]

Alternative 5: 63.6% accurate, 3.6× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} t_0 := -2 \cdot \left(U \cdot -0.5\right)\\ t_1 := -2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)\\ t_2 := -2 \cdot \left(U \cdot 0.5\right)\\ \mathbf{if}\;J \leq -3.9 \cdot 10^{+57}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;J \leq -9 \cdot 10^{-82}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq -1.05 \cdot 10^{-168}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;J \leq -3.4 \cdot 10^{-301}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq 1.1 \cdot 10^{-49}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (* -2.0 (* U -0.5)))
        (t_1 (* -2.0 (* J (cos (/ K 2.0)))))
        (t_2 (* -2.0 (* U 0.5))))
   (if (<= J -3.9e+57)
     t_1
     (if (<= J -9e-82)
       t_0
       (if (<= J -1.05e-168)
         t_2
         (if (<= J -3.4e-301) t_0 (if (<= J 1.1e-49) t_2 t_1)))))))

C

U = abs(U);
double code(double J, double K, double U) {
	double t_0 = -2.0 * (U * -0.5);
	double t_1 = -2.0 * (J * cos((K / 2.0)));
	double t_2 = -2.0 * (U * 0.5);
	double tmp;
	if (J <= -3.9e+57) {
		tmp = t_1;
	} else if (J <= -9e-82) {
		tmp = t_0;
	} else if (J <= -1.05e-168) {
		tmp = t_2;
	} else if (J <= -3.4e-301) {
		tmp = t_0;
	} else if (J <= 1.1e-49) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	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) :: t_2
    real(8) :: tmp
    t_0 = (-2.0d0) * (u * (-0.5d0))
    t_1 = (-2.0d0) * (j * cos((k / 2.0d0)))
    t_2 = (-2.0d0) * (u * 0.5d0)
    if (j <= (-3.9d+57)) then
        tmp = t_1
    else if (j <= (-9d-82)) then
        tmp = t_0
    else if (j <= (-1.05d-168)) then
        tmp = t_2
    else if (j <= (-3.4d-301)) then
        tmp = t_0
    else if (j <= 1.1d-49) then
        tmp = t_2
    else
        tmp = t_1
    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 t_1 = -2.0 * (J * Math.cos((K / 2.0)));
	double t_2 = -2.0 * (U * 0.5);
	double tmp;
	if (J <= -3.9e+57) {
		tmp = t_1;
	} else if (J <= -9e-82) {
		tmp = t_0;
	} else if (J <= -1.05e-168) {
		tmp = t_2;
	} else if (J <= -3.4e-301) {
		tmp = t_0;
	} else if (J <= 1.1e-49) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	t_0 = -2.0 * (U * -0.5)
	t_1 = -2.0 * (J * math.cos((K / 2.0)))
	t_2 = -2.0 * (U * 0.5)
	tmp = 0
	if J <= -3.9e+57:
		tmp = t_1
	elif J <= -9e-82:
		tmp = t_0
	elif J <= -1.05e-168:
		tmp = t_2
	elif J <= -3.4e-301:
		tmp = t_0
	elif J <= 1.1e-49:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	t_0 = Float64(-2.0 * Float64(U * -0.5))
	t_1 = Float64(-2.0 * Float64(J * cos(Float64(K / 2.0))))
	t_2 = Float64(-2.0 * Float64(U * 0.5))
	tmp = 0.0
	if (J <= -3.9e+57)
		tmp = t_1;
	elseif (J <= -9e-82)
		tmp = t_0;
	elseif (J <= -1.05e-168)
		tmp = t_2;
	elseif (J <= -3.4e-301)
		tmp = t_0;
	elseif (J <= 1.1e-49)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	t_0 = -2.0 * (U * -0.5);
	t_1 = -2.0 * (J * cos((K / 2.0)));
	t_2 = -2.0 * (U * 0.5);
	tmp = 0.0;
	if (J <= -3.9e+57)
		tmp = t_1;
	elseif (J <= -9e-82)
		tmp = t_0;
	elseif (J <= -1.05e-168)
		tmp = t_2;
	elseif (J <= -3.4e-301)
		tmp = t_0;
	elseif (J <= 1.1e-49)
		tmp = t_2;
	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[(-2.0 * N[(U * -0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-2.0 * N[(J * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(-2.0 * N[(U * 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[J, -3.9e+57], t$95$1, If[LessEqual[J, -9e-82], t$95$0, If[LessEqual[J, -1.05e-168], t$95$2, If[LessEqual[J, -3.4e-301], t$95$0, If[LessEqual[J, 1.1e-49], t$95$2, t$95$1]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if J < -3.89999999999999968e57 or 1.09999999999999995e-49 < J

   1. Initial program 92.8%

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

      1. associate-*l* 92.8%

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

      2. associate-*l* 92.8%

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

      3. *-commutative 92.8%

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

      4. unpow2 92.8%

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

      5. sqr-neg 92.8%

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

      6. distribute-frac-neg 92.8%

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

      7. distribute-frac-neg 92.8%

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

      8. unpow2 92.8%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in J around inf 78.6%

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

   if -3.89999999999999968e57 < J < -8.9999999999999997e-82 or -1.04999999999999997e-168 < J < -3.4000000000000002e-301

   1. Initial program 54.9%

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

   3. Simplified 84.8%

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

   4. Taylor expanded in U around -inf 44.0%

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

      1. *-commutative 44.0%

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

   6. Simplified 44.0%

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

   if -8.9999999999999997e-82 < J < -1.04999999999999997e-168 or -3.4000000000000002e-301 < J < 1.09999999999999995e-49

   1. Initial program 55.5%

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

   3. Simplified 77.1%

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

   4. Taylor expanded in J around 0 46.4%

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

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -3.9 \cdot 10^{+57}:\\ \;\;\;\;-2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)\\ \mathbf{elif}\;J \leq -9 \cdot 10^{-82}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5\right)\\ \mathbf{elif}\;J \leq -1.05 \cdot 10^{-168}:\\ \;\;\;\;-2 \cdot \left(U \cdot 0.5\right)\\ \mathbf{elif}\;J \leq -3.4 \cdot 10^{-301}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5\right)\\ \mathbf{elif}\;J \leq 1.1 \cdot 10^{-49}:\\ \;\;\;\;-2 \cdot \left(U \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)\\ \end{array} \]

Alternative 6: 47.7% accurate, 19.7× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} t_0 := -2 \cdot \left(U \cdot -0.5\right)\\ t_1 := -2 \cdot \left(J + 0.125 \cdot \frac{U}{\frac{J}{U}}\right)\\ t_2 := -2 \cdot \left(U \cdot 0.5\right)\\ \mathbf{if}\;J \leq -9.5 \cdot 10^{+63}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;J \leq -5.2 \cdot 10^{-85}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq -1.45 \cdot 10^{-168}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;J \leq -3.3 \cdot 10^{-301}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq 2.4 \cdot 10^{+29}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (* -2.0 (* U -0.5)))
        (t_1 (* -2.0 (+ J (* 0.125 (/ U (/ J U))))))
        (t_2 (* -2.0 (* U 0.5))))
   (if (<= J -9.5e+63)
     t_1
     (if (<= J -5.2e-85)
       t_0
       (if (<= J -1.45e-168)
         t_2
         (if (<= J -3.3e-301) t_0 (if (<= J 2.4e+29) t_2 t_1)))))))

C

U = abs(U);
double code(double J, double K, double U) {
	double t_0 = -2.0 * (U * -0.5);
	double t_1 = -2.0 * (J + (0.125 * (U / (J / U))));
	double t_2 = -2.0 * (U * 0.5);
	double tmp;
	if (J <= -9.5e+63) {
		tmp = t_1;
	} else if (J <= -5.2e-85) {
		tmp = t_0;
	} else if (J <= -1.45e-168) {
		tmp = t_2;
	} else if (J <= -3.3e-301) {
		tmp = t_0;
	} else if (J <= 2.4e+29) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	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) :: t_2
    real(8) :: tmp
    t_0 = (-2.0d0) * (u * (-0.5d0))
    t_1 = (-2.0d0) * (j + (0.125d0 * (u / (j / u))))
    t_2 = (-2.0d0) * (u * 0.5d0)
    if (j <= (-9.5d+63)) then
        tmp = t_1
    else if (j <= (-5.2d-85)) then
        tmp = t_0
    else if (j <= (-1.45d-168)) then
        tmp = t_2
    else if (j <= (-3.3d-301)) then
        tmp = t_0
    else if (j <= 2.4d+29) then
        tmp = t_2
    else
        tmp = t_1
    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 t_1 = -2.0 * (J + (0.125 * (U / (J / U))));
	double t_2 = -2.0 * (U * 0.5);
	double tmp;
	if (J <= -9.5e+63) {
		tmp = t_1;
	} else if (J <= -5.2e-85) {
		tmp = t_0;
	} else if (J <= -1.45e-168) {
		tmp = t_2;
	} else if (J <= -3.3e-301) {
		tmp = t_0;
	} else if (J <= 2.4e+29) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	t_0 = -2.0 * (U * -0.5)
	t_1 = -2.0 * (J + (0.125 * (U / (J / U))))
	t_2 = -2.0 * (U * 0.5)
	tmp = 0
	if J <= -9.5e+63:
		tmp = t_1
	elif J <= -5.2e-85:
		tmp = t_0
	elif J <= -1.45e-168:
		tmp = t_2
	elif J <= -3.3e-301:
		tmp = t_0
	elif J <= 2.4e+29:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	t_0 = Float64(-2.0 * Float64(U * -0.5))
	t_1 = Float64(-2.0 * Float64(J + Float64(0.125 * Float64(U / Float64(J / U)))))
	t_2 = Float64(-2.0 * Float64(U * 0.5))
	tmp = 0.0
	if (J <= -9.5e+63)
		tmp = t_1;
	elseif (J <= -5.2e-85)
		tmp = t_0;
	elseif (J <= -1.45e-168)
		tmp = t_2;
	elseif (J <= -3.3e-301)
		tmp = t_0;
	elseif (J <= 2.4e+29)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	t_0 = -2.0 * (U * -0.5);
	t_1 = -2.0 * (J + (0.125 * (U / (J / U))));
	t_2 = -2.0 * (U * 0.5);
	tmp = 0.0;
	if (J <= -9.5e+63)
		tmp = t_1;
	elseif (J <= -5.2e-85)
		tmp = t_0;
	elseif (J <= -1.45e-168)
		tmp = t_2;
	elseif (J <= -3.3e-301)
		tmp = t_0;
	elseif (J <= 2.4e+29)
		tmp = t_2;
	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[(-2.0 * N[(U * -0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-2.0 * N[(J + N[(0.125 * N[(U / N[(J / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(-2.0 * N[(U * 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[J, -9.5e+63], t$95$1, If[LessEqual[J, -5.2e-85], t$95$0, If[LessEqual[J, -1.45e-168], t$95$2, If[LessEqual[J, -3.3e-301], t$95$0, If[LessEqual[J, 2.4e+29], t$95$2, t$95$1]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if J < -9.5000000000000003e63 or 2.4000000000000001e29 < J

   1. Initial program 96.5%

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

   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 45.4%

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

      1. unpow2 45.4%

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

      2. unpow2 45.4%

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

   6. Simplified 45.4%

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

   7. Taylor expanded in J around inf 45.9%

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

      1. unpow2 45.9%

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

      2. associate-/l* 49.5%

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

   9. Simplified 49.5%

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

   if -9.5000000000000003e63 < J < -5.20000000000000023e-85 or -1.4499999999999999e-168 < J < -3.3e-301

   1. Initial program 57.0%

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

   3. Simplified 85.5%

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

   4. Taylor expanded in U around -inf 42.0%

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

      1. *-commutative 42.0%

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

   6. Simplified 42.0%

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

   if -5.20000000000000023e-85 < J < -1.4499999999999999e-168 or -3.3e-301 < J < 2.4000000000000001e29

   1. Initial program 57.6%

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

   3. Simplified 80.6%

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

   4. Taylor expanded in J around 0 44.9%

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

4. Final simplification 46.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -9.5 \cdot 10^{+63}:\\ \;\;\;\;-2 \cdot \left(J + 0.125 \cdot \frac{U}{\frac{J}{U}}\right)\\ \mathbf{elif}\;J \leq -5.2 \cdot 10^{-85}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5\right)\\ \mathbf{elif}\;J \leq -1.45 \cdot 10^{-168}:\\ \;\;\;\;-2 \cdot \left(U \cdot 0.5\right)\\ \mathbf{elif}\;J \leq -3.3 \cdot 10^{-301}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5\right)\\ \mathbf{elif}\;J \leq 2.4 \cdot 10^{+29}:\\ \;\;\;\;-2 \cdot \left(U \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(J + 0.125 \cdot \frac{U}{\frac{J}{U}}\right)\\ \end{array} \]

Alternative 7: 47.5% accurate, 19.7× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} t_0 := -2 \cdot \left(J + 0.125 \cdot \frac{U}{\frac{J}{U}}\right)\\ t_1 := -2 \cdot \left(U \cdot 0.5\right)\\ \mathbf{if}\;J \leq -3 \cdot 10^{+62}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq -1.8 \cdot 10^{-80}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5 - J \cdot \frac{J}{U}\right)\\ \mathbf{elif}\;J \leq -8.8 \cdot 10^{-169}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;J \leq -3.3 \cdot 10^{-301}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5\right)\\ \mathbf{elif}\;J \leq 1.1 \cdot 10^{+30}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (* -2.0 (+ J (* 0.125 (/ U (/ J U)))))) (t_1 (* -2.0 (* U 0.5))))
   (if (<= J -3e+62)
     t_0
     (if (<= J -1.8e-80)
       (* -2.0 (- (* U -0.5) (* J (/ J U))))
       (if (<= J -8.8e-169)
         t_1
         (if (<= J -3.3e-301)
           (* -2.0 (* U -0.5))
           (if (<= J 1.1e+30) t_1 t_0)))))))

C

U = abs(U);
double code(double J, double K, double U) {
	double t_0 = -2.0 * (J + (0.125 * (U / (J / U))));
	double t_1 = -2.0 * (U * 0.5);
	double tmp;
	if (J <= -3e+62) {
		tmp = t_0;
	} else if (J <= -1.8e-80) {
		tmp = -2.0 * ((U * -0.5) - (J * (J / U)));
	} else if (J <= -8.8e-169) {
		tmp = t_1;
	} else if (J <= -3.3e-301) {
		tmp = -2.0 * (U * -0.5);
	} else if (J <= 1.1e+30) {
		tmp = t_1;
	} 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 * (u / (j / u))))
    t_1 = (-2.0d0) * (u * 0.5d0)
    if (j <= (-3d+62)) then
        tmp = t_0
    else if (j <= (-1.8d-80)) then
        tmp = (-2.0d0) * ((u * (-0.5d0)) - (j * (j / u)))
    else if (j <= (-8.8d-169)) then
        tmp = t_1
    else if (j <= (-3.3d-301)) then
        tmp = (-2.0d0) * (u * (-0.5d0))
    else if (j <= 1.1d+30) then
        tmp = t_1
    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 * (U / (J / U))));
	double t_1 = -2.0 * (U * 0.5);
	double tmp;
	if (J <= -3e+62) {
		tmp = t_0;
	} else if (J <= -1.8e-80) {
		tmp = -2.0 * ((U * -0.5) - (J * (J / U)));
	} else if (J <= -8.8e-169) {
		tmp = t_1;
	} else if (J <= -3.3e-301) {
		tmp = -2.0 * (U * -0.5);
	} else if (J <= 1.1e+30) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	t_0 = -2.0 * (J + (0.125 * (U / (J / U))))
	t_1 = -2.0 * (U * 0.5)
	tmp = 0
	if J <= -3e+62:
		tmp = t_0
	elif J <= -1.8e-80:
		tmp = -2.0 * ((U * -0.5) - (J * (J / U)))
	elif J <= -8.8e-169:
		tmp = t_1
	elif J <= -3.3e-301:
		tmp = -2.0 * (U * -0.5)
	elif J <= 1.1e+30:
		tmp = t_1
	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(U / Float64(J / U)))))
	t_1 = Float64(-2.0 * Float64(U * 0.5))
	tmp = 0.0
	if (J <= -3e+62)
		tmp = t_0;
	elseif (J <= -1.8e-80)
		tmp = Float64(-2.0 * Float64(Float64(U * -0.5) - Float64(J * Float64(J / U))));
	elseif (J <= -8.8e-169)
		tmp = t_1;
	elseif (J <= -3.3e-301)
		tmp = Float64(-2.0 * Float64(U * -0.5));
	elseif (J <= 1.1e+30)
		tmp = t_1;
	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 * (U / (J / U))));
	t_1 = -2.0 * (U * 0.5);
	tmp = 0.0;
	if (J <= -3e+62)
		tmp = t_0;
	elseif (J <= -1.8e-80)
		tmp = -2.0 * ((U * -0.5) - (J * (J / U)));
	elseif (J <= -8.8e-169)
		tmp = t_1;
	elseif (J <= -3.3e-301)
		tmp = -2.0 * (U * -0.5);
	elseif (J <= 1.1e+30)
		tmp = t_1;
	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[(U / N[(J / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-2.0 * N[(U * 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[J, -3e+62], t$95$0, If[LessEqual[J, -1.8e-80], N[(-2.0 * N[(N[(U * -0.5), $MachinePrecision] - N[(J * N[(J / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[J, -8.8e-169], t$95$1, If[LessEqual[J, -3.3e-301], N[(-2.0 * N[(U * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[J, 1.1e+30], t$95$1, t$95$0]]]]]]]

Derivation

1. Split input into 4 regimes

2. if J < -3e62 or 1.1e30 < J

   1. Initial program 96.5%

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

   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 45.4%

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

      1. unpow2 45.4%

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

      2. unpow2 45.4%

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

   6. Simplified 45.4%

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

   7. Taylor expanded in J around inf 45.9%

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

      1. unpow2 45.9%

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

      2. associate-/l* 49.5%

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

   9. Simplified 49.5%

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

   if -3e62 < J < -1.8e-80

   1. Initial program 68.0%

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

   3. Simplified 97.6%

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

   4. Taylor expanded in K around 0 42.9%

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

      1. unpow2 42.9%

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

      2. unpow2 42.9%

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

   6. Simplified 42.9%

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

   7. Taylor expanded in U around -inf 35.4%

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

      1. +-commutative 35.4%

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

      2. mul-1-neg 35.4%

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

      3. unsub-neg 35.4%

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

   9. Simplified 35.4%

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

   10. Taylor expanded in J around 0 37.6%

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

       1. +-commutative 37.6%

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

       2. mul-1-neg 37.6%

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

       3. unsub-neg 37.6%

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

       4. *-commutative 37.6%

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

       5. unpow2 37.6%

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

       6. associate-*r/ 37.6%

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

   12. Simplified 37.6%

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

   if -1.8e-80 < J < -8.80000000000000029e-169 or -3.3e-301 < J < 1.1e30

   1. Initial program 57.6%

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

   3. Simplified 80.6%

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

   4. Taylor expanded in J around 0 44.9%

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

   if -8.80000000000000029e-169 < J < -3.3e-301

   1. Initial program 38.9%

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

   3. Simplified 65.7%

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

   4. Taylor expanded in U around -inf 48.9%

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

      1. *-commutative 48.9%

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

   6. Simplified 48.9%

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

4. Final simplification 46.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -3 \cdot 10^{+62}:\\ \;\;\;\;-2 \cdot \left(J + 0.125 \cdot \frac{U}{\frac{J}{U}}\right)\\ \mathbf{elif}\;J \leq -1.8 \cdot 10^{-80}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5 - J \cdot \frac{J}{U}\right)\\ \mathbf{elif}\;J \leq -8.8 \cdot 10^{-169}:\\ \;\;\;\;-2 \cdot \left(U \cdot 0.5\right)\\ \mathbf{elif}\;J \leq -3.3 \cdot 10^{-301}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5\right)\\ \mathbf{elif}\;J \leq 1.1 \cdot 10^{+30}:\\ \;\;\;\;-2 \cdot \left(U \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(J + 0.125 \cdot \frac{U}{\frac{J}{U}}\right)\\ \end{array} \]

Alternative 8: 47.5% accurate, 27.4× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} t_0 := -2 \cdot \left(U \cdot 0.5\right)\\ t_1 := -2 \cdot \left(U \cdot -0.5\right)\\ \mathbf{if}\;J \leq -5.6 \cdot 10^{+62}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;J \leq -2.8 \cdot 10^{-85}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;J \leq -1.45 \cdot 10^{-168}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq -4.7 \cdot 10^{-299}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;J \leq 1200000000000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot J\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (* -2.0 (* U 0.5))) (t_1 (* -2.0 (* U -0.5))))
   (if (<= J -5.6e+62)
     (* -2.0 J)
     (if (<= J -2.8e-85)
       t_1
       (if (<= J -1.45e-168)
         t_0
         (if (<= J -4.7e-299)
           t_1
           (if (<= J 1200000000000.0) t_0 (* -2.0 J))))))))

C

U = abs(U);
double code(double J, double K, double U) {
	double t_0 = -2.0 * (U * 0.5);
	double t_1 = -2.0 * (U * -0.5);
	double tmp;
	if (J <= -5.6e+62) {
		tmp = -2.0 * J;
	} else if (J <= -2.8e-85) {
		tmp = t_1;
	} else if (J <= -1.45e-168) {
		tmp = t_0;
	} else if (J <= -4.7e-299) {
		tmp = t_1;
	} else if (J <= 1200000000000.0) {
		tmp = t_0;
	} 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) :: t_1
    real(8) :: tmp
    t_0 = (-2.0d0) * (u * 0.5d0)
    t_1 = (-2.0d0) * (u * (-0.5d0))
    if (j <= (-5.6d+62)) then
        tmp = (-2.0d0) * j
    else if (j <= (-2.8d-85)) then
        tmp = t_1
    else if (j <= (-1.45d-168)) then
        tmp = t_0
    else if (j <= (-4.7d-299)) then
        tmp = t_1
    else if (j <= 1200000000000.0d0) then
        tmp = t_0
    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 t_1 = -2.0 * (U * -0.5);
	double tmp;
	if (J <= -5.6e+62) {
		tmp = -2.0 * J;
	} else if (J <= -2.8e-85) {
		tmp = t_1;
	} else if (J <= -1.45e-168) {
		tmp = t_0;
	} else if (J <= -4.7e-299) {
		tmp = t_1;
	} else if (J <= 1200000000000.0) {
		tmp = t_0;
	} else {
		tmp = -2.0 * J;
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	t_0 = -2.0 * (U * 0.5)
	t_1 = -2.0 * (U * -0.5)
	tmp = 0
	if J <= -5.6e+62:
		tmp = -2.0 * J
	elif J <= -2.8e-85:
		tmp = t_1
	elif J <= -1.45e-168:
		tmp = t_0
	elif J <= -4.7e-299:
		tmp = t_1
	elif J <= 1200000000000.0:
		tmp = t_0
	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))
	t_1 = Float64(-2.0 * Float64(U * -0.5))
	tmp = 0.0
	if (J <= -5.6e+62)
		tmp = Float64(-2.0 * J);
	elseif (J <= -2.8e-85)
		tmp = t_1;
	elseif (J <= -1.45e-168)
		tmp = t_0;
	elseif (J <= -4.7e-299)
		tmp = t_1;
	elseif (J <= 1200000000000.0)
		tmp = t_0;
	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);
	t_1 = -2.0 * (U * -0.5);
	tmp = 0.0;
	if (J <= -5.6e+62)
		tmp = -2.0 * J;
	elseif (J <= -2.8e-85)
		tmp = t_1;
	elseif (J <= -1.45e-168)
		tmp = t_0;
	elseif (J <= -4.7e-299)
		tmp = t_1;
	elseif (J <= 1200000000000.0)
		tmp = t_0;
	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]}, Block[{t$95$1 = N[(-2.0 * N[(U * -0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[J, -5.6e+62], N[(-2.0 * J), $MachinePrecision], If[LessEqual[J, -2.8e-85], t$95$1, If[LessEqual[J, -1.45e-168], t$95$0, If[LessEqual[J, -4.7e-299], t$95$1, If[LessEqual[J, 1200000000000.0], t$95$0, N[(-2.0 * J), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if J < -5.60000000000000029e62 or 1.2e12 < J

   1. Initial program 95.8%

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

   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 44.6%

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

      1. unpow2 44.6%

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

      2. unpow2 44.6%

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

   6. Simplified 44.6%

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

   7. Taylor expanded in J around inf 48.2%

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

   if -5.60000000000000029e62 < J < -2.80000000000000017e-85 or -1.4499999999999999e-168 < J < -4.6999999999999997e-299

   1. Initial program 57.0%

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

   3. Simplified 85.5%

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

   4. Taylor expanded in U around -inf 42.0%

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

      1. *-commutative 42.0%

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

   6. Simplified 42.0%

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

   if -2.80000000000000017e-85 < J < -1.4499999999999999e-168 or -4.6999999999999997e-299 < J < 1.2e12

   1. Initial program 57.6%

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

   3. Simplified 80.1%

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

   4. Taylor expanded in J around 0 43.5%

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

4. Final simplification 45.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -5.6 \cdot 10^{+62}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;J \leq -2.8 \cdot 10^{-85}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5\right)\\ \mathbf{elif}\;J \leq -1.45 \cdot 10^{-168}:\\ \;\;\;\;-2 \cdot \left(U \cdot 0.5\right)\\ \mathbf{elif}\;J \leq -4.7 \cdot 10^{-299}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5\right)\\ \mathbf{elif}\;J \leq 1200000000000:\\ \;\;\;\;-2 \cdot \left(U \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot J\\ \end{array} \]

Alternative 9: 40.2% accurate, 46.0× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} \mathbf{if}\;J \leq -7500000000000:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;J \leq 63000000000000:\\ \;\;\;\;-2 \cdot \left(U \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot J\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (if (<= J -7500000000000.0)
   (* -2.0 J)
   (if (<= J 63000000000000.0) (* -2.0 (* U 0.5)) (* -2.0 J))))

C

U = abs(U);
double code(double J, double K, double U) {
	double tmp;
	if (J <= -7500000000000.0) {
		tmp = -2.0 * J;
	} else if (J <= 63000000000000.0) {
		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) :: tmp
    if (j <= (-7500000000000.0d0)) then
        tmp = (-2.0d0) * j
    else if (j <= 63000000000000.0d0) 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 tmp;
	if (J <= -7500000000000.0) {
		tmp = -2.0 * J;
	} else if (J <= 63000000000000.0) {
		tmp = -2.0 * (U * 0.5);
	} else {
		tmp = -2.0 * J;
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	tmp = 0
	if J <= -7500000000000.0:
		tmp = -2.0 * J
	elif J <= 63000000000000.0:
		tmp = -2.0 * (U * 0.5)
	else:
		tmp = -2.0 * J
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	tmp = 0.0
	if (J <= -7500000000000.0)
		tmp = Float64(-2.0 * J);
	elseif (J <= 63000000000000.0)
		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)
	tmp = 0.0;
	if (J <= -7500000000000.0)
		tmp = -2.0 * J;
	elseif (J <= 63000000000000.0)
		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_] := If[LessEqual[J, -7500000000000.0], N[(-2.0 * J), $MachinePrecision], If[LessEqual[J, 63000000000000.0], N[(-2.0 * N[(U * 0.5), $MachinePrecision]), $MachinePrecision], N[(-2.0 * J), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if J < -7.5e12 or 6.3e13 < J

   1. Initial program 94.7%

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

   3. Simplified 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 47.1%

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

      1. unpow2 47.1%

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

      2. unpow2 47.1%

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

   6. Simplified 47.1%

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

   7. Taylor expanded in J around inf 46.7%

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

   if -7.5e12 < J < 6.3e13

   1. Initial program 54.8%

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

   3. Simplified 81.0%

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

   4. Taylor expanded in J around 0 41.4%

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

4. Final simplification 44.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -7500000000000:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;J \leq 63000000000000:\\ \;\;\;\;-2 \cdot \left(U \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot J\\ \end{array} \]

Alternative 10: 29.1% 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 74.5%

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

3. Simplified 90.2%

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

4. Taylor expanded in K around 0 31.2%

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

   1. unpow2 31.2%

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

   2. unpow2 31.2%

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

6. Simplified 31.2%

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

7. Taylor expanded in J around inf 29.1%

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

8. Final simplification 29.1%

   \[\leadsto -2 \cdot J \]

Reproduce

herbie shell --seed 2023285 
(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)))))