Maksimov and Kolovsky, Equation (3)

Percentage Accurate: 73.0% → 99.7%

Time: 16.2s

Alternatives: 10

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.0% 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

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

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* (* -2.0 J) t_0)
          (sqrt (+ 1.0 (pow (/ U (* t_0 (* J 2.0))) 2.0))))))
   (if (<= t_1 (- INFINITY))
     (- U)
     (if (<= t_1 2e+307)
       (* (* J (* -2.0 t_0)) (hypot 1.0 (/ U (* J (* 2.0 t_0)))))
       U))))

C

U = abs(U);
double code(double J, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * J) * t_0) * sqrt((1.0 + pow((U / (t_0 * (J * 2.0))), 2.0)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -U;
	} else if (t_1 <= 2e+307) {
		tmp = (J * (-2.0 * t_0)) * hypot(1.0, (U / (J * (2.0 * t_0))));
	} else {
		tmp = U;
	}
	return tmp;
}

Java

U = Math.abs(U);
public static double code(double J, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = ((-2.0 * J) * t_0) * Math.sqrt((1.0 + Math.pow((U / (t_0 * (J * 2.0))), 2.0)));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = -U;
	} else if (t_1 <= 2e+307) {
		tmp = (J * (-2.0 * t_0)) * Math.hypot(1.0, (U / (J * (2.0 * t_0))));
	} else {
		tmp = U;
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	t_0 = math.cos((K / 2.0))
	t_1 = ((-2.0 * J) * t_0) * math.sqrt((1.0 + math.pow((U / (t_0 * (J * 2.0))), 2.0)))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = -U
	elif t_1 <= 2e+307:
		tmp = (J * (-2.0 * t_0)) * math.hypot(1.0, (U / (J * (2.0 * t_0))))
	else:
		tmp = U
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * J) * t_0) * sqrt(Float64(1.0 + (Float64(U / Float64(t_0 * Float64(J * 2.0))) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-U);
	elseif (t_1 <= 2e+307)
		tmp = Float64(Float64(J * Float64(-2.0 * t_0)) * hypot(1.0, Float64(U / Float64(J * Float64(2.0 * t_0)))));
	else
		tmp = U;
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	t_0 = cos((K / 2.0));
	t_1 = ((-2.0 * J) * t_0) * sqrt((1.0 + ((U / (t_0 * (J * 2.0))) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = -U;
	elseif (t_1 <= 2e+307)
		tmp = (J * (-2.0 * t_0)) * hypot(1.0, (U / (J * (2.0 * t_0))));
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U / N[(t$95$0 * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], (-U), If[LessEqual[t$95$1, 2e+307], N[(N[(J * N[(-2.0 * t$95$0), $MachinePrecision]), $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[(U / N[(J * N[(2.0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], U]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 5.5%

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

      1. *-commutative 5.5%

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

      2. associate-*l* 5.5%

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

      3. unpow2 5.5%

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

      4. hypot-1-def 49.8%

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

      5. *-commutative 49.8%

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

      6. associate-*l* 49.8%

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

   3. Simplified 49.8%

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

   4. Taylor expanded in J around 0 47.0%

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

      1. neg-mul-1 47.0%

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

   6. Simplified 47.0%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 -2 J) (cos.f64 (/.f64 K 2))) (sqrt.f64 (+.f64 1 (pow.f64 (/.f64 U (*.f64 (*.f64 2 J) (cos.f64 (/.f64 K 2)))) 2)))) < 1.99999999999999997e307

   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. *-commutative 99.8%

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

      2. associate-*l* 99.8%

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

      3. unpow2 99.8%

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

      4. hypot-1-def 99.8%

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

      5. *-commutative 99.8%

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

      6. associate-*l* 99.8%

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

   3. Simplified 99.8%

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

   if 1.99999999999999997e307 < (*.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 6.3%

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

      1. *-commutative 6.3%

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

      2. associate-*l* 6.3%

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

      3. unpow2 6.3%

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

      4. hypot-1-def 31.4%

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

      5. *-commutative 31.4%

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

      6. associate-*l* 31.4%

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

   3. Simplified 31.4%

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

   4. Taylor expanded in U around -inf 56.1%

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

4. Final simplification 87.8%

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

Alternative 2: 76.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} t_0 := \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot 2}\right)\\ \mathbf{if}\;J \leq -2.1 \cdot 10^{-113}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq -8.4 \cdot 10^{-255}:\\ \;\;\;\;\mathsf{fma}\left(2, \frac{\frac{1 + \cos K}{2}}{\frac{U}{J \cdot J}}, U\right)\\ \mathbf{elif}\;J \leq 1.5 \cdot 10^{-170}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (* (* J (* -2.0 (cos (/ K 2.0)))) (hypot 1.0 (/ U (* J 2.0))))))
   (if (<= J -2.1e-113)
     t_0
     (if (<= J -8.4e-255)
       (fma 2.0 (/ (/ (+ 1.0 (cos K)) 2.0) (/ U (* J J))) U)
       (if (<= J 1.5e-170) (- U) t_0)))))

C

U = abs(U);
double code(double J, double K, double U) {
	double t_0 = (J * (-2.0 * cos((K / 2.0)))) * hypot(1.0, (U / (J * 2.0)));
	double tmp;
	if (J <= -2.1e-113) {
		tmp = t_0;
	} else if (J <= -8.4e-255) {
		tmp = fma(2.0, (((1.0 + cos(K)) / 2.0) / (U / (J * J))), U);
	} else if (J <= 1.5e-170) {
		tmp = -U;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

U = abs(U)
function code(J, K, U)
	t_0 = Float64(Float64(J * Float64(-2.0 * cos(Float64(K / 2.0)))) * hypot(1.0, Float64(U / Float64(J * 2.0))))
	tmp = 0.0
	if (J <= -2.1e-113)
		tmp = t_0;
	elseif (J <= -8.4e-255)
		tmp = fma(2.0, Float64(Float64(Float64(1.0 + cos(K)) / 2.0) / Float64(U / Float64(J * J))), U);
	elseif (J <= 1.5e-170)
		tmp = Float64(-U);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := Block[{t$95$0 = N[(N[(J * N[(-2.0 * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[(U / N[(J * 2.0), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[J, -2.1e-113], t$95$0, If[LessEqual[J, -8.4e-255], N[(2.0 * N[(N[(N[(1.0 + N[Cos[K], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] / N[(U / N[(J * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], If[LessEqual[J, 1.5e-170], (-U), t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if J < -2.1e-113 or 1.50000000000000007e-170 < J

   1. Initial program 86.4%

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

      1. *-commutative 86.4%

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

      2. associate-*l* 86.4%

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

      3. unpow2 86.4%

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

      4. hypot-1-def 92.9%

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

      5. *-commutative 92.9%

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

      6. associate-*l* 92.9%

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

   3. Simplified 92.9%

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

   4. Taylor expanded in K around 0 82.1%

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

   if -2.1e-113 < J < -8.3999999999999999e-255

   1. Initial program 39.1%

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

      1. *-commutative 39.1%

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

      2. associate-*l* 39.1%

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

      3. unpow2 39.1%

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

      4. hypot-1-def 53.0%

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

      5. *-commutative 53.0%

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

      6. associate-*l* 53.0%

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

   3. Simplified 53.0%

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

   4. Taylor expanded in U around -inf 54.7%

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

      1. fma-def 54.7%

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

      2. associate-/l* 54.7%

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

      3. unpow2 54.7%

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

   6. Simplified 54.7%

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

      1. unpow2 54.7%

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

      2. *-commutative 54.7%

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

      3. metadata-eval 54.7%

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

      4. div-inv 54.7%

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

      5. *-commutative 54.7%

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

      6. metadata-eval 54.7%

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

      7. div-inv 54.7%

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

      8. cos-mult 54.7%

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

      9. div-inv 54.7%

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

      10. metadata-eval 54.7%

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

      11. div-inv 54.7%

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

      12. metadata-eval 54.7%

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

      13. div-inv 54.7%

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

      14. metadata-eval 54.7%

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

      15. div-inv 54.7%

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

      16. metadata-eval 54.7%

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

   8. Applied egg-rr 54.7%

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

      1. +-commutative 54.7%

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

      2. +-inverses 54.7%

         \[\leadsto \mathsf{fma}\left(2, \frac{\frac{\cos \color{blue}{0} + \cos \left(K \cdot 0.5 + K \cdot 0.5\right)}{2}}{\frac{U}{J \cdot J}}, U\right) \]

      3. cos-0 54.7%

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

      4. distribute-lft-out 54.7%

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

      5. metadata-eval 54.7%

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

      6. *-rgt-identity 54.7%

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

   10. Simplified 54.7%

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

   if -8.3999999999999999e-255 < J < 1.50000000000000007e-170

   1. Initial program 42.1%

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

      1. *-commutative 42.1%

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

      2. associate-*l* 42.1%

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

      3. unpow2 42.1%

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

      4. hypot-1-def 59.5%

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

      5. *-commutative 59.5%

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

      6. associate-*l* 59.5%

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

   3. Simplified 59.5%

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

   4. Taylor expanded in J around 0 37.9%

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

      1. neg-mul-1 37.9%

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

   6. Simplified 37.9%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -2.1 \cdot 10^{-113}:\\ \;\;\;\;\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot 2}\right)\\ \mathbf{elif}\;J \leq -8.4 \cdot 10^{-255}:\\ \;\;\;\;\mathsf{fma}\left(2, \frac{\frac{1 + \cos K}{2}}{\frac{U}{J \cdot J}}, U\right)\\ \mathbf{elif}\;J \leq 1.5 \cdot 10^{-170}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot 2}\right)\\ \end{array} \]

Alternative 3: 60.0% accurate, 3.6× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} t_0 := \left(-2 \cdot J\right) \cdot \mathsf{hypot}\left(1, -0.5 \cdot \frac{U}{J}\right)\\ t_1 := \left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{if}\;U \leq 3 \cdot 10^{-193}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;U \leq 3.2 \cdot 10^{-89}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;U \leq 1.45 \cdot 10^{+57}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;U \leq 1.8 \cdot 10^{+250}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;U \leq 3.4 \cdot 10^{+281}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (* (* -2.0 J) (hypot 1.0 (* -0.5 (/ U J)))))
        (t_1 (* (* -2.0 J) (cos (* K 0.5)))))
   (if (<= U 3e-193)
     t_1
     (if (<= U 3.2e-89)
       t_0
       (if (<= U 1.45e+57)
         t_1
         (if (<= U 1.8e+250) t_0 (if (<= U 3.4e+281) (- U) U)))))))

C

U = abs(U);
double code(double J, double K, double U) {
	double t_0 = (-2.0 * J) * hypot(1.0, (-0.5 * (U / J)));
	double t_1 = (-2.0 * J) * cos((K * 0.5));
	double tmp;
	if (U <= 3e-193) {
		tmp = t_1;
	} else if (U <= 3.2e-89) {
		tmp = t_0;
	} else if (U <= 1.45e+57) {
		tmp = t_1;
	} else if (U <= 1.8e+250) {
		tmp = t_0;
	} else if (U <= 3.4e+281) {
		tmp = -U;
	} else {
		tmp = U;
	}
	return tmp;
}

Java

U = Math.abs(U);
public static double code(double J, double K, double U) {
	double t_0 = (-2.0 * J) * Math.hypot(1.0, (-0.5 * (U / J)));
	double t_1 = (-2.0 * J) * Math.cos((K * 0.5));
	double tmp;
	if (U <= 3e-193) {
		tmp = t_1;
	} else if (U <= 3.2e-89) {
		tmp = t_0;
	} else if (U <= 1.45e+57) {
		tmp = t_1;
	} else if (U <= 1.8e+250) {
		tmp = t_0;
	} else if (U <= 3.4e+281) {
		tmp = -U;
	} else {
		tmp = U;
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	t_0 = (-2.0 * J) * math.hypot(1.0, (-0.5 * (U / J)))
	t_1 = (-2.0 * J) * math.cos((K * 0.5))
	tmp = 0
	if U <= 3e-193:
		tmp = t_1
	elif U <= 3.2e-89:
		tmp = t_0
	elif U <= 1.45e+57:
		tmp = t_1
	elif U <= 1.8e+250:
		tmp = t_0
	elif U <= 3.4e+281:
		tmp = -U
	else:
		tmp = U
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	t_0 = Float64(Float64(-2.0 * J) * hypot(1.0, Float64(-0.5 * Float64(U / J))))
	t_1 = Float64(Float64(-2.0 * J) * cos(Float64(K * 0.5)))
	tmp = 0.0
	if (U <= 3e-193)
		tmp = t_1;
	elseif (U <= 3.2e-89)
		tmp = t_0;
	elseif (U <= 1.45e+57)
		tmp = t_1;
	elseif (U <= 1.8e+250)
		tmp = t_0;
	elseif (U <= 3.4e+281)
		tmp = Float64(-U);
	else
		tmp = U;
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	t_0 = (-2.0 * J) * hypot(1.0, (-0.5 * (U / J)));
	t_1 = (-2.0 * J) * cos((K * 0.5));
	tmp = 0.0;
	if (U <= 3e-193)
		tmp = t_1;
	elseif (U <= 3.2e-89)
		tmp = t_0;
	elseif (U <= 1.45e+57)
		tmp = t_1;
	elseif (U <= 1.8e+250)
		tmp = t_0;
	elseif (U <= 3.4e+281)
		tmp = -U;
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := Block[{t$95$0 = N[(N[(-2.0 * J), $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[(-0.5 * N[(U / J), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(-2.0 * J), $MachinePrecision] * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[U, 3e-193], t$95$1, If[LessEqual[U, 3.2e-89], t$95$0, If[LessEqual[U, 1.45e+57], t$95$1, If[LessEqual[U, 1.8e+250], t$95$0, If[LessEqual[U, 3.4e+281], (-U), U]]]]]]]

Derivation

1. Split input into 4 regimes

2. if U < 2.9999999999999999e-193 or 3.19999999999999998e-89 < U < 1.4500000000000001e57

   1. Initial program 79.2%

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

      1. *-commutative 79.2%

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

      2. associate-*l* 79.2%

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

      3. unpow2 79.2%

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

      4. hypot-1-def 88.5%

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

      5. *-commutative 88.5%

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

      6. associate-*l* 88.5%

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

   3. Simplified 88.5%

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

   4. Taylor expanded in J around inf 58.6%

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

      1. *-commutative 58.6%

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

      2. *-commutative 58.6%

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

      3. associate-*r* 58.6%

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

      4. *-commutative 58.6%

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

   6. Simplified 58.6%

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

   if 2.9999999999999999e-193 < U < 3.19999999999999998e-89 or 1.4500000000000001e57 < U < 1.7999999999999999e250

   1. Initial program 77.4%

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

      1. *-commutative 77.4%

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

      2. associate-*l* 77.4%

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

      3. unpow2 77.4%

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

      4. hypot-1-def 83.9%

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

      5. *-commutative 83.9%

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

      6. associate-*l* 83.9%

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

   3. Simplified 83.9%

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

      1. add-cube-cbrt 82.1%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right)} \cdot \sqrt[3]{\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right)}\right) \cdot \sqrt[3]{\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right)}} \]

      2. pow3 82.1%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right)}\right)}^{3}} \]

   5. Applied egg-rr 82.1%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{J \cdot \left(-2 \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \mathsf{hypot}\left(1, \frac{\frac{U}{J}}{-2 \cdot \cos \left(K \cdot 0.5\right)}\right)\right)\right)}\right)}^{3}} \]
   6. Step-by-step derivation

      1. cbrt-prod 81.6%

         \[\leadsto {\color{blue}{\left(\sqrt[3]{J} \cdot \sqrt[3]{-2 \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \mathsf{hypot}\left(1, \frac{\frac{U}{J}}{-2 \cdot \cos \left(K \cdot 0.5\right)}\right)\right)}\right)}}^{3} \]

      2. unpow-prod-down 81.6%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{J}\right)}^{3} \cdot {\left(\sqrt[3]{-2 \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \mathsf{hypot}\left(1, \frac{\frac{U}{J}}{-2 \cdot \cos \left(K \cdot 0.5\right)}\right)\right)}\right)}^{3}} \]

      3. *-commutative 81.6%

         \[\leadsto {\left(\sqrt[3]{J}\right)}^{3} \cdot {\left(\sqrt[3]{-2 \cdot \color{blue}{\left(\mathsf{hypot}\left(1, \frac{\frac{U}{J}}{-2 \cdot \cos \left(K \cdot 0.5\right)}\right) \cdot \cos \left(K \cdot 0.5\right)\right)}}\right)}^{3} \]

      4. associate-/r* 81.6%

         \[\leadsto {\left(\sqrt[3]{J}\right)}^{3} \cdot {\left(\sqrt[3]{-2 \cdot \left(\mathsf{hypot}\left(1, \color{blue}{\frac{\frac{\frac{U}{J}}{-2}}{\cos \left(K \cdot 0.5\right)}}\right) \cdot \cos \left(K \cdot 0.5\right)\right)}\right)}^{3} \]

   7. Applied egg-rr 81.6%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{J}\right)}^{3} \cdot {\left(\sqrt[3]{-2 \cdot \left(\mathsf{hypot}\left(1, \frac{\frac{\frac{U}{J}}{-2}}{\cos \left(K \cdot 0.5\right)}\right) \cdot \cos \left(K \cdot 0.5\right)\right)}\right)}^{3}} \]

   8. Taylor expanded in K around 0 33.9%

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

      1. *-commutative 33.9%

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

      2. associate-*l* 33.9%

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

      3. *-commutative 33.9%

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

      4. unpow2 33.9%

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

      5. unpow2 33.9%

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

      6. times-frac 62.1%

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

      7. metadata-eval 62.1%

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

      8. swap-sqr 62.1%

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

      9. hypot-1-def 68.6%

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

      10. *-commutative 68.6%

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

      11. *-commutative 68.6%

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

   10. Simplified 68.6%

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

   if 1.7999999999999999e250 < U < 3.40000000000000011e281

   1. Initial program 47.1%

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

      1. *-commutative 47.1%

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

      2. associate-*l* 47.1%

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

      3. unpow2 47.1%

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

      4. hypot-1-def 53.7%

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

      5. *-commutative 53.7%

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

      6. associate-*l* 53.7%

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

   3. Simplified 53.7%

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

   4. Taylor expanded in J around 0 71.5%

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

      1. neg-mul-1 71.5%

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

   6. Simplified 71.5%

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

   if 3.40000000000000011e281 < U

   1. Initial program 59.8%

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

      1. *-commutative 59.8%

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

      2. associate-*l* 59.8%

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

      3. unpow2 59.8%

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

      4. hypot-1-def 70.0%

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

      5. *-commutative 70.0%

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

      6. associate-*l* 70.0%

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

   3. Simplified 70.0%

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

   4. Taylor expanded in U around -inf 83.6%

      \[\leadsto \color{blue}{U} \]
3. Recombined 4 regimes into one program.

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 3 \cdot 10^{-193}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{elif}\;U \leq 3.2 \cdot 10^{-89}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \mathsf{hypot}\left(1, -0.5 \cdot \frac{U}{J}\right)\\ \mathbf{elif}\;U \leq 1.45 \cdot 10^{+57}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{elif}\;U \leq 1.8 \cdot 10^{+250}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \mathsf{hypot}\left(1, -0.5 \cdot \frac{U}{J}\right)\\ \mathbf{elif}\;U \leq 3.4 \cdot 10^{+281}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternative 4: 58.9% accurate, 3.9× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} \mathbf{if}\;U \leq 1.66 \cdot 10^{+131}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{elif}\;U \leq 4.7 \cdot 10^{+231}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \frac{J}{U} - U\\ \mathbf{elif}\;U \leq 5.2 \cdot 10^{+255}:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq 4 \cdot 10^{+281}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (if (<= U 1.66e+131)
   (* (* -2.0 J) (cos (* K 0.5)))
   (if (<= U 4.7e+231)
     (- (* (* -2.0 J) (/ J U)) U)
     (if (<= U 5.2e+255) U (if (<= U 4e+281) (- U) U)))))

C

U = abs(U);
double code(double J, double K, double U) {
	double tmp;
	if (U <= 1.66e+131) {
		tmp = (-2.0 * J) * cos((K * 0.5));
	} else if (U <= 4.7e+231) {
		tmp = ((-2.0 * J) * (J / U)) - U;
	} else if (U <= 5.2e+255) {
		tmp = U;
	} else if (U <= 4e+281) {
		tmp = -U;
	} else {
		tmp = U;
	}
	return tmp;
}

Fortran

NOTE: U should be positive before calling this function
real(8) function code(j, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (u <= 1.66d+131) then
        tmp = ((-2.0d0) * j) * cos((k * 0.5d0))
    else if (u <= 4.7d+231) then
        tmp = (((-2.0d0) * j) * (j / u)) - u
    else if (u <= 5.2d+255) then
        tmp = u
    else if (u <= 4d+281) then
        tmp = -u
    else
        tmp = u
    end if
    code = tmp
end function

Java

U = Math.abs(U);
public static double code(double J, double K, double U) {
	double tmp;
	if (U <= 1.66e+131) {
		tmp = (-2.0 * J) * Math.cos((K * 0.5));
	} else if (U <= 4.7e+231) {
		tmp = ((-2.0 * J) * (J / U)) - U;
	} else if (U <= 5.2e+255) {
		tmp = U;
	} else if (U <= 4e+281) {
		tmp = -U;
	} else {
		tmp = U;
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	tmp = 0
	if U <= 1.66e+131:
		tmp = (-2.0 * J) * math.cos((K * 0.5))
	elif U <= 4.7e+231:
		tmp = ((-2.0 * J) * (J / U)) - U
	elif U <= 5.2e+255:
		tmp = U
	elif U <= 4e+281:
		tmp = -U
	else:
		tmp = U
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	tmp = 0.0
	if (U <= 1.66e+131)
		tmp = Float64(Float64(-2.0 * J) * cos(Float64(K * 0.5)));
	elseif (U <= 4.7e+231)
		tmp = Float64(Float64(Float64(-2.0 * J) * Float64(J / U)) - U);
	elseif (U <= 5.2e+255)
		tmp = U;
	elseif (U <= 4e+281)
		tmp = Float64(-U);
	else
		tmp = U;
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	tmp = 0.0;
	if (U <= 1.66e+131)
		tmp = (-2.0 * J) * cos((K * 0.5));
	elseif (U <= 4.7e+231)
		tmp = ((-2.0 * J) * (J / U)) - U;
	elseif (U <= 5.2e+255)
		tmp = U;
	elseif (U <= 4e+281)
		tmp = -U;
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := If[LessEqual[U, 1.66e+131], N[(N[(-2.0 * J), $MachinePrecision] * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[U, 4.7e+231], N[(N[(N[(-2.0 * J), $MachinePrecision] * N[(J / U), $MachinePrecision]), $MachinePrecision] - U), $MachinePrecision], If[LessEqual[U, 5.2e+255], U, If[LessEqual[U, 4e+281], (-U), U]]]]

Derivation

1. Split input into 4 regimes

2. if U < 1.65999999999999992e131

   1. Initial program 80.7%

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

      1. *-commutative 80.7%

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

      2. associate-*l* 80.7%

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

      3. unpow2 80.7%

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

      4. hypot-1-def 89.2%

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

      5. *-commutative 89.2%

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

      6. associate-*l* 89.2%

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

   3. Simplified 89.2%

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

   4. Taylor expanded in J around inf 60.1%

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

      1. *-commutative 60.1%

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

      2. *-commutative 60.1%

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

      3. associate-*r* 60.1%

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

      4. *-commutative 60.1%

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

   6. Simplified 60.1%

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

   if 1.65999999999999992e131 < U < 4.70000000000000006e231

   1. Initial program 68.3%

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

      1. *-commutative 68.3%

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

      2. associate-*l* 68.3%

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

      3. unpow2 68.3%

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

      4. hypot-1-def 80.1%

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

      5. *-commutative 80.1%

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

      6. associate-*l* 80.1%

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

   3. Simplified 80.1%

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

   4. Taylor expanded in K around 0 10.9%

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

      1. associate-*r* 10.9%

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

      2. unpow2 10.9%

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

      3. unpow2 10.9%

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

   6. Simplified 10.9%

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

      1. clear-num 10.9%

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

      2. inv-pow 10.9%

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

   8. Applied egg-rr 10.9%

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

      1. unpow-1 10.9%

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

      2. times-frac 55.4%

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

   10. Simplified 55.4%

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

   11. Taylor expanded in J around 0 38.2%

       \[\leadsto \color{blue}{-2 \cdot \frac{{J}^{2}}{U} + -1 \cdot U} \]
   12. Step-by-step derivation

       1. mul-1-neg 38.2%

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

       2. unsub-neg 38.2%

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

       3. unpow2 38.2%

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

       4. associate-*r/ 46.1%

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

       5. associate-*r* 46.1%

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

       6. *-commutative 46.1%

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

   13. Simplified 46.1%

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

   if 4.70000000000000006e231 < U < 5.20000000000000019e255 or 4.0000000000000001e281 < U

   1. Initial program 51.3%

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

      1. *-commutative 51.3%

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

      2. associate-*l* 51.3%

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

      3. unpow2 51.3%

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

      4. hypot-1-def 56.7%

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

      5. *-commutative 56.7%

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

      6. associate-*l* 56.7%

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

   3. Simplified 56.7%

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

   4. Taylor expanded in U around -inf 68.3%

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

   if 5.20000000000000019e255 < U < 4.0000000000000001e281

   1. Initial program 53.8%

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

      1. *-commutative 53.8%

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

      2. associate-*l* 53.8%

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

      3. unpow2 53.8%

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

      4. hypot-1-def 61.5%

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

      5. *-commutative 61.5%

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

      6. associate-*l* 61.5%

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

   3. Simplified 61.5%

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

   4. Taylor expanded in J around 0 75.0%

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

      1. neg-mul-1 75.0%

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

   6. Simplified 75.0%

      \[\leadsto \color{blue}{-U} \]
3. Recombined 4 regimes into one program.

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 1.66 \cdot 10^{+131}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{elif}\;U \leq 4.7 \cdot 10^{+231}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \frac{J}{U} - U\\ \mathbf{elif}\;U \leq 5.2 \cdot 10^{+255}:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq 4 \cdot 10^{+281}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternative 5: 46.9% accurate, 21.7× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} \mathbf{if}\;J \leq -6 \cdot 10^{+142}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;J \leq -1.46 \cdot 10^{+41}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq -3.8 \cdot 10^{-32}:\\ \;\;\;\;J \cdot \left(-2 + 0.25 \cdot \left(K \cdot K\right)\right)\\ \mathbf{elif}\;J \leq -8.4 \cdot 10^{-255}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 7.2 \cdot 10^{+84}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \frac{J}{U} - U\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot J\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (if (<= J -6e+142)
   (* -2.0 J)
   (if (<= J -1.46e+41)
     U
     (if (<= J -3.8e-32)
       (* J (+ -2.0 (* 0.25 (* K K))))
       (if (<= J -8.4e-255)
         U
         (if (<= J 7.2e+84) (- (* (* -2.0 J) (/ J U)) U) (* -2.0 J)))))))

C

U = abs(U);
double code(double J, double K, double U) {
	double tmp;
	if (J <= -6e+142) {
		tmp = -2.0 * J;
	} else if (J <= -1.46e+41) {
		tmp = U;
	} else if (J <= -3.8e-32) {
		tmp = J * (-2.0 + (0.25 * (K * K)));
	} else if (J <= -8.4e-255) {
		tmp = U;
	} else if (J <= 7.2e+84) {
		tmp = ((-2.0 * J) * (J / U)) - U;
	} else {
		tmp = -2.0 * J;
	}
	return tmp;
}

Fortran

NOTE: U should be positive before calling this function
real(8) function code(j, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (j <= (-6d+142)) then
        tmp = (-2.0d0) * j
    else if (j <= (-1.46d+41)) then
        tmp = u
    else if (j <= (-3.8d-32)) then
        tmp = j * ((-2.0d0) + (0.25d0 * (k * k)))
    else if (j <= (-8.4d-255)) then
        tmp = u
    else if (j <= 7.2d+84) then
        tmp = (((-2.0d0) * j) * (j / u)) - u
    else
        tmp = (-2.0d0) * j
    end if
    code = tmp
end function

Java

U = Math.abs(U);
public static double code(double J, double K, double U) {
	double tmp;
	if (J <= -6e+142) {
		tmp = -2.0 * J;
	} else if (J <= -1.46e+41) {
		tmp = U;
	} else if (J <= -3.8e-32) {
		tmp = J * (-2.0 + (0.25 * (K * K)));
	} else if (J <= -8.4e-255) {
		tmp = U;
	} else if (J <= 7.2e+84) {
		tmp = ((-2.0 * J) * (J / U)) - U;
	} else {
		tmp = -2.0 * J;
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	tmp = 0
	if J <= -6e+142:
		tmp = -2.0 * J
	elif J <= -1.46e+41:
		tmp = U
	elif J <= -3.8e-32:
		tmp = J * (-2.0 + (0.25 * (K * K)))
	elif J <= -8.4e-255:
		tmp = U
	elif J <= 7.2e+84:
		tmp = ((-2.0 * J) * (J / U)) - U
	else:
		tmp = -2.0 * J
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	tmp = 0.0
	if (J <= -6e+142)
		tmp = Float64(-2.0 * J);
	elseif (J <= -1.46e+41)
		tmp = U;
	elseif (J <= -3.8e-32)
		tmp = Float64(J * Float64(-2.0 + Float64(0.25 * Float64(K * K))));
	elseif (J <= -8.4e-255)
		tmp = U;
	elseif (J <= 7.2e+84)
		tmp = Float64(Float64(Float64(-2.0 * J) * Float64(J / U)) - U);
	else
		tmp = Float64(-2.0 * J);
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	tmp = 0.0;
	if (J <= -6e+142)
		tmp = -2.0 * J;
	elseif (J <= -1.46e+41)
		tmp = U;
	elseif (J <= -3.8e-32)
		tmp = J * (-2.0 + (0.25 * (K * K)));
	elseif (J <= -8.4e-255)
		tmp = U;
	elseif (J <= 7.2e+84)
		tmp = ((-2.0 * J) * (J / U)) - U;
	else
		tmp = -2.0 * J;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := If[LessEqual[J, -6e+142], N[(-2.0 * J), $MachinePrecision], If[LessEqual[J, -1.46e+41], U, If[LessEqual[J, -3.8e-32], N[(J * N[(-2.0 + N[(0.25 * N[(K * K), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[J, -8.4e-255], U, If[LessEqual[J, 7.2e+84], N[(N[(N[(-2.0 * J), $MachinePrecision] * N[(J / U), $MachinePrecision]), $MachinePrecision] - U), $MachinePrecision], N[(-2.0 * J), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if J < -5.99999999999999949e142 or 7.1999999999999999e84 < J

   1. Initial program 98.9%

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

      1. *-commutative 98.9%

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

      2. associate-*l* 98.9%

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

      3. unpow2 98.9%

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

      4. hypot-1-def 99.9%

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

      5. *-commutative 99.9%

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

      6. associate-*l* 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in K around 0 40.5%

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

      1. associate-*r* 40.5%

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

      2. unpow2 40.5%

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

      3. unpow2 40.5%

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

   6. Simplified 40.5%

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

   7. Taylor expanded in U around 0 46.2%

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

   if -5.99999999999999949e142 < J < -1.4600000000000001e41 or -3.80000000000000008e-32 < J < -8.3999999999999999e-255

   1. Initial program 60.1%

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

      1. *-commutative 60.1%

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

      2. associate-*l* 60.1%

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

      3. unpow2 60.1%

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

      4. hypot-1-def 71.7%

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

      5. *-commutative 71.7%

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

      6. associate-*l* 71.7%

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

   3. Simplified 71.7%

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

   4. Taylor expanded in U around -inf 42.2%

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

   if -1.4600000000000001e41 < J < -3.80000000000000008e-32

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-*l* 99.9%

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

      3. unpow2 99.9%

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

      4. hypot-1-def 99.9%

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

      5. *-commutative 99.9%

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

      6. associate-*l* 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in J around inf 99.9%

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

      1. *-commutative 99.9%

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

      2. *-commutative 99.9%

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

      3. associate-*r* 99.9%

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

      4. *-commutative 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in K around 0 72.3%

      \[\leadsto \color{blue}{0.25 \cdot \left({K}^{2} \cdot J\right) + -2 \cdot J} \]
   8. Step-by-step derivation

      1. +-commutative 72.3%

         \[\leadsto \color{blue}{-2 \cdot J + 0.25 \cdot \left({K}^{2} \cdot J\right)} \]

      2. associate-*r* 72.3%

         \[\leadsto -2 \cdot J + \color{blue}{\left(0.25 \cdot {K}^{2}\right) \cdot J} \]

      3. distribute-rgt-out 72.3%

         \[\leadsto \color{blue}{J \cdot \left(-2 + 0.25 \cdot {K}^{2}\right)} \]

      4. unpow2 72.3%

         \[\leadsto J \cdot \left(-2 + 0.25 \cdot \color{blue}{\left(K \cdot K\right)}\right) \]

   9. Simplified 72.3%

      \[\leadsto \color{blue}{J \cdot \left(-2 + 0.25 \cdot \left(K \cdot K\right)\right)} \]

   if -8.3999999999999999e-255 < J < 7.1999999999999999e84

   1. Initial program 60.6%

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

      1. *-commutative 60.6%

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

      2. associate-*l* 60.6%

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

      3. unpow2 60.6%

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

      4. hypot-1-def 76.4%

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

      5. *-commutative 76.4%

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

      6. associate-*l* 76.4%

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

   3. Simplified 76.4%

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

   4. Taylor expanded in K around 0 26.2%

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

      1. associate-*r* 26.2%

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

      2. unpow2 26.2%

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

      3. unpow2 26.2%

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

   6. Simplified 26.2%

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

      1. clear-num 26.2%

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

      2. inv-pow 26.2%

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

   8. Applied egg-rr 26.2%

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

      1. unpow-1 26.2%

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

      2. times-frac 40.5%

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

   10. Simplified 40.5%

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

   11. Taylor expanded in J around 0 36.3%

       \[\leadsto \color{blue}{-2 \cdot \frac{{J}^{2}}{U} + -1 \cdot U} \]
   12. Step-by-step derivation

       1. mul-1-neg 36.3%

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

       2. unsub-neg 36.3%

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

       3. unpow2 36.3%

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

       4. associate-*r/ 36.3%

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

       5. associate-*r* 36.3%

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

       6. *-commutative 36.3%

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

   13. Simplified 36.3%

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

4. Final simplification 43.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -6 \cdot 10^{+142}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;J \leq -1.46 \cdot 10^{+41}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq -3.8 \cdot 10^{-32}:\\ \;\;\;\;J \cdot \left(-2 + 0.25 \cdot \left(K \cdot K\right)\right)\\ \mathbf{elif}\;J \leq -8.4 \cdot 10^{-255}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 7.2 \cdot 10^{+84}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \frac{J}{U} - U\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot J\\ \end{array} \]

Alternative 6: 47.0% accurate, 21.7× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} \mathbf{if}\;J \leq -9.8 \cdot 10^{+142}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;J \leq -1.62 \cdot 10^{+34}:\\ \;\;\;\;J \cdot \left(-2 \cdot \left(\frac{U}{-2 \cdot J} - \frac{J}{U}\right)\right)\\ \mathbf{elif}\;J \leq -3.1 \cdot 10^{-32}:\\ \;\;\;\;J \cdot \left(-2 + 0.25 \cdot \left(K \cdot K\right)\right)\\ \mathbf{elif}\;J \leq -8.4 \cdot 10^{-255}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 6.5 \cdot 10^{+85}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \frac{J}{U} - U\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot J\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (if (<= J -9.8e+142)
   (* -2.0 J)
   (if (<= J -1.62e+34)
     (* J (* -2.0 (- (/ U (* -2.0 J)) (/ J U))))
     (if (<= J -3.1e-32)
       (* J (+ -2.0 (* 0.25 (* K K))))
       (if (<= J -8.4e-255)
         U
         (if (<= J 6.5e+85) (- (* (* -2.0 J) (/ J U)) U) (* -2.0 J)))))))

C

U = abs(U);
double code(double J, double K, double U) {
	double tmp;
	if (J <= -9.8e+142) {
		tmp = -2.0 * J;
	} else if (J <= -1.62e+34) {
		tmp = J * (-2.0 * ((U / (-2.0 * J)) - (J / U)));
	} else if (J <= -3.1e-32) {
		tmp = J * (-2.0 + (0.25 * (K * K)));
	} else if (J <= -8.4e-255) {
		tmp = U;
	} else if (J <= 6.5e+85) {
		tmp = ((-2.0 * J) * (J / U)) - U;
	} else {
		tmp = -2.0 * J;
	}
	return tmp;
}

Fortran

NOTE: U should be positive before calling this function
real(8) function code(j, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (j <= (-9.8d+142)) then
        tmp = (-2.0d0) * j
    else if (j <= (-1.62d+34)) then
        tmp = j * ((-2.0d0) * ((u / ((-2.0d0) * j)) - (j / u)))
    else if (j <= (-3.1d-32)) then
        tmp = j * ((-2.0d0) + (0.25d0 * (k * k)))
    else if (j <= (-8.4d-255)) then
        tmp = u
    else if (j <= 6.5d+85) then
        tmp = (((-2.0d0) * j) * (j / u)) - u
    else
        tmp = (-2.0d0) * j
    end if
    code = tmp
end function

Java

U = Math.abs(U);
public static double code(double J, double K, double U) {
	double tmp;
	if (J <= -9.8e+142) {
		tmp = -2.0 * J;
	} else if (J <= -1.62e+34) {
		tmp = J * (-2.0 * ((U / (-2.0 * J)) - (J / U)));
	} else if (J <= -3.1e-32) {
		tmp = J * (-2.0 + (0.25 * (K * K)));
	} else if (J <= -8.4e-255) {
		tmp = U;
	} else if (J <= 6.5e+85) {
		tmp = ((-2.0 * J) * (J / U)) - U;
	} else {
		tmp = -2.0 * J;
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	tmp = 0
	if J <= -9.8e+142:
		tmp = -2.0 * J
	elif J <= -1.62e+34:
		tmp = J * (-2.0 * ((U / (-2.0 * J)) - (J / U)))
	elif J <= -3.1e-32:
		tmp = J * (-2.0 + (0.25 * (K * K)))
	elif J <= -8.4e-255:
		tmp = U
	elif J <= 6.5e+85:
		tmp = ((-2.0 * J) * (J / U)) - U
	else:
		tmp = -2.0 * J
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	tmp = 0.0
	if (J <= -9.8e+142)
		tmp = Float64(-2.0 * J);
	elseif (J <= -1.62e+34)
		tmp = Float64(J * Float64(-2.0 * Float64(Float64(U / Float64(-2.0 * J)) - Float64(J / U))));
	elseif (J <= -3.1e-32)
		tmp = Float64(J * Float64(-2.0 + Float64(0.25 * Float64(K * K))));
	elseif (J <= -8.4e-255)
		tmp = U;
	elseif (J <= 6.5e+85)
		tmp = Float64(Float64(Float64(-2.0 * J) * Float64(J / U)) - U);
	else
		tmp = Float64(-2.0 * J);
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	tmp = 0.0;
	if (J <= -9.8e+142)
		tmp = -2.0 * J;
	elseif (J <= -1.62e+34)
		tmp = J * (-2.0 * ((U / (-2.0 * J)) - (J / U)));
	elseif (J <= -3.1e-32)
		tmp = J * (-2.0 + (0.25 * (K * K)));
	elseif (J <= -8.4e-255)
		tmp = U;
	elseif (J <= 6.5e+85)
		tmp = ((-2.0 * J) * (J / U)) - U;
	else
		tmp = -2.0 * J;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := If[LessEqual[J, -9.8e+142], N[(-2.0 * J), $MachinePrecision], If[LessEqual[J, -1.62e+34], N[(J * N[(-2.0 * N[(N[(U / N[(-2.0 * J), $MachinePrecision]), $MachinePrecision] - N[(J / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[J, -3.1e-32], N[(J * N[(-2.0 + N[(0.25 * N[(K * K), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[J, -8.4e-255], U, If[LessEqual[J, 6.5e+85], N[(N[(N[(-2.0 * J), $MachinePrecision] * N[(J / U), $MachinePrecision]), $MachinePrecision] - U), $MachinePrecision], N[(-2.0 * J), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if J < -9.80000000000000101e142 or 6.4999999999999994e85 < J

   1. Initial program 98.9%

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

      1. *-commutative 98.9%

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

      2. associate-*l* 98.9%

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

      3. unpow2 98.9%

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

      4. hypot-1-def 99.9%

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

      5. *-commutative 99.9%

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

      6. associate-*l* 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in K around 0 40.5%

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

      1. associate-*r* 40.5%

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

      2. unpow2 40.5%

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

      3. unpow2 40.5%

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

   6. Simplified 40.5%

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

   7. Taylor expanded in U around 0 46.2%

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

   if -9.80000000000000101e142 < J < -1.62000000000000006e34

   1. Initial program 93.8%

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

      1. *-commutative 93.8%

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

      2. associate-*l* 93.8%

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

      3. unpow2 93.8%

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

      4. hypot-1-def 99.8%

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

      5. *-commutative 99.8%

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

      6. associate-*l* 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in K around 0 43.4%

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

      1. associate-*r* 43.4%

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

      2. unpow2 43.4%

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

      3. unpow2 43.4%

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

   6. Simplified 43.4%

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

   7. Taylor expanded in U around -inf 29.0%

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

      1. mul-1-neg 29.0%

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

      2. unsub-neg 29.0%

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

      3. metadata-eval 29.0%

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

      4. times-frac 29.0%

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

      5. *-lft-identity 29.0%

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

   9. Simplified 29.0%

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

   if -1.62000000000000006e34 < J < -3.10000000000000011e-32

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-*l* 99.9%

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

      3. unpow2 99.9%

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

      4. hypot-1-def 99.9%

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

      5. *-commutative 99.9%

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

      6. associate-*l* 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in J around inf 99.9%

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

      1. *-commutative 99.9%

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

      2. *-commutative 99.9%

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

      3. associate-*r* 99.9%

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

      4. *-commutative 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in K around 0 72.3%

      \[\leadsto \color{blue}{0.25 \cdot \left({K}^{2} \cdot J\right) + -2 \cdot J} \]
   8. Step-by-step derivation

      1. +-commutative 72.3%

         \[\leadsto \color{blue}{-2 \cdot J + 0.25 \cdot \left({K}^{2} \cdot J\right)} \]

      2. associate-*r* 72.3%

         \[\leadsto -2 \cdot J + \color{blue}{\left(0.25 \cdot {K}^{2}\right) \cdot J} \]

      3. distribute-rgt-out 72.3%

         \[\leadsto \color{blue}{J \cdot \left(-2 + 0.25 \cdot {K}^{2}\right)} \]

      4. unpow2 72.3%

         \[\leadsto J \cdot \left(-2 + 0.25 \cdot \color{blue}{\left(K \cdot K\right)}\right) \]

   9. Simplified 72.3%

      \[\leadsto \color{blue}{J \cdot \left(-2 + 0.25 \cdot \left(K \cdot K\right)\right)} \]

   if -3.10000000000000011e-32 < J < -8.3999999999999999e-255

   1. Initial program 50.2%

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

      1. *-commutative 50.2%

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

      2. associate-*l* 50.2%

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

      3. unpow2 50.2%

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

      4. hypot-1-def 63.4%

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

      5. *-commutative 63.4%

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

      6. associate-*l* 63.4%

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

   3. Simplified 63.4%

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

   4. Taylor expanded in U around -inf 46.1%

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

   if -8.3999999999999999e-255 < J < 6.4999999999999994e85

   1. Initial program 60.6%

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

      1. *-commutative 60.6%

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

      2. associate-*l* 60.6%

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

      3. unpow2 60.6%

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

      4. hypot-1-def 76.4%

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

      5. *-commutative 76.4%

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

      6. associate-*l* 76.4%

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

   3. Simplified 76.4%

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

   4. Taylor expanded in K around 0 26.2%

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

      1. associate-*r* 26.2%

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

      2. unpow2 26.2%

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

      3. unpow2 26.2%

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

   6. Simplified 26.2%

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

      1. clear-num 26.2%

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

      2. inv-pow 26.2%

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

   8. Applied egg-rr 26.2%

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

      1. unpow-1 26.2%

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

      2. times-frac 40.5%

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

   10. Simplified 40.5%

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

   11. Taylor expanded in J around 0 36.3%

       \[\leadsto \color{blue}{-2 \cdot \frac{{J}^{2}}{U} + -1 \cdot U} \]
   12. Step-by-step derivation

       1. mul-1-neg 36.3%

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

       2. unsub-neg 36.3%

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

       3. unpow2 36.3%

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

       4. associate-*r/ 36.3%

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

       5. associate-*r* 36.3%

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

       6. *-commutative 36.3%

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

   13. Simplified 36.3%

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

4. Final simplification 43.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -9.8 \cdot 10^{+142}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;J \leq -1.62 \cdot 10^{+34}:\\ \;\;\;\;J \cdot \left(-2 \cdot \left(\frac{U}{-2 \cdot J} - \frac{J}{U}\right)\right)\\ \mathbf{elif}\;J \leq -3.1 \cdot 10^{-32}:\\ \;\;\;\;J \cdot \left(-2 + 0.25 \cdot \left(K \cdot K\right)\right)\\ \mathbf{elif}\;J \leq -8.4 \cdot 10^{-255}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 6.5 \cdot 10^{+85}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \frac{J}{U} - U\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot J\\ \end{array} \]

Alternative 7: 46.9% accurate, 27.7× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} \mathbf{if}\;J \leq -1.4 \cdot 10^{+142}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;J \leq -9.8 \cdot 10^{+36}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq -2.25 \cdot 10^{-32}:\\ \;\;\;\;J \cdot \left(-2 + 0.25 \cdot \left(K \cdot K\right)\right)\\ \mathbf{elif}\;J \leq -8.4 \cdot 10^{-255}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 5.8 \cdot 10^{+83}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot J\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (if (<= J -1.4e+142)
   (* -2.0 J)
   (if (<= J -9.8e+36)
     U
     (if (<= J -2.25e-32)
       (* J (+ -2.0 (* 0.25 (* K K))))
       (if (<= J -8.4e-255) U (if (<= J 5.8e+83) (- U) (* -2.0 J)))))))

C

U = abs(U);
double code(double J, double K, double U) {
	double tmp;
	if (J <= -1.4e+142) {
		tmp = -2.0 * J;
	} else if (J <= -9.8e+36) {
		tmp = U;
	} else if (J <= -2.25e-32) {
		tmp = J * (-2.0 + (0.25 * (K * K)));
	} else if (J <= -8.4e-255) {
		tmp = U;
	} else if (J <= 5.8e+83) {
		tmp = -U;
	} else {
		tmp = -2.0 * J;
	}
	return tmp;
}

Fortran

NOTE: U should be positive before calling this function
real(8) function code(j, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (j <= (-1.4d+142)) then
        tmp = (-2.0d0) * j
    else if (j <= (-9.8d+36)) then
        tmp = u
    else if (j <= (-2.25d-32)) then
        tmp = j * ((-2.0d0) + (0.25d0 * (k * k)))
    else if (j <= (-8.4d-255)) then
        tmp = u
    else if (j <= 5.8d+83) then
        tmp = -u
    else
        tmp = (-2.0d0) * j
    end if
    code = tmp
end function

Java

U = Math.abs(U);
public static double code(double J, double K, double U) {
	double tmp;
	if (J <= -1.4e+142) {
		tmp = -2.0 * J;
	} else if (J <= -9.8e+36) {
		tmp = U;
	} else if (J <= -2.25e-32) {
		tmp = J * (-2.0 + (0.25 * (K * K)));
	} else if (J <= -8.4e-255) {
		tmp = U;
	} else if (J <= 5.8e+83) {
		tmp = -U;
	} else {
		tmp = -2.0 * J;
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	tmp = 0
	if J <= -1.4e+142:
		tmp = -2.0 * J
	elif J <= -9.8e+36:
		tmp = U
	elif J <= -2.25e-32:
		tmp = J * (-2.0 + (0.25 * (K * K)))
	elif J <= -8.4e-255:
		tmp = U
	elif J <= 5.8e+83:
		tmp = -U
	else:
		tmp = -2.0 * J
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	tmp = 0.0
	if (J <= -1.4e+142)
		tmp = Float64(-2.0 * J);
	elseif (J <= -9.8e+36)
		tmp = U;
	elseif (J <= -2.25e-32)
		tmp = Float64(J * Float64(-2.0 + Float64(0.25 * Float64(K * K))));
	elseif (J <= -8.4e-255)
		tmp = U;
	elseif (J <= 5.8e+83)
		tmp = Float64(-U);
	else
		tmp = Float64(-2.0 * J);
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	tmp = 0.0;
	if (J <= -1.4e+142)
		tmp = -2.0 * J;
	elseif (J <= -9.8e+36)
		tmp = U;
	elseif (J <= -2.25e-32)
		tmp = J * (-2.0 + (0.25 * (K * K)));
	elseif (J <= -8.4e-255)
		tmp = U;
	elseif (J <= 5.8e+83)
		tmp = -U;
	else
		tmp = -2.0 * J;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := If[LessEqual[J, -1.4e+142], N[(-2.0 * J), $MachinePrecision], If[LessEqual[J, -9.8e+36], U, If[LessEqual[J, -2.25e-32], N[(J * N[(-2.0 + N[(0.25 * N[(K * K), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[J, -8.4e-255], U, If[LessEqual[J, 5.8e+83], (-U), N[(-2.0 * J), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if J < -1.4e142 or 5.79999999999999999e83 < J

   1. Initial program 98.9%

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

      1. *-commutative 98.9%

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

      2. associate-*l* 98.9%

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

      3. unpow2 98.9%

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

      4. hypot-1-def 99.9%

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

      5. *-commutative 99.9%

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

      6. associate-*l* 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in K around 0 40.5%

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

      1. associate-*r* 40.5%

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

      2. unpow2 40.5%

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

      3. unpow2 40.5%

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

   6. Simplified 40.5%

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

   7. Taylor expanded in U around 0 46.2%

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

   if -1.4e142 < J < -9.79999999999999962e36 or -2.25000000000000002e-32 < J < -8.3999999999999999e-255

   1. Initial program 60.1%

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

      1. *-commutative 60.1%

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

      2. associate-*l* 60.1%

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

      3. unpow2 60.1%

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

      4. hypot-1-def 71.7%

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

      5. *-commutative 71.7%

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

      6. associate-*l* 71.7%

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

   3. Simplified 71.7%

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

   4. Taylor expanded in U around -inf 42.2%

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

   if -9.79999999999999962e36 < J < -2.25000000000000002e-32

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-*l* 99.9%

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

      3. unpow2 99.9%

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

      4. hypot-1-def 99.9%

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

      5. *-commutative 99.9%

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

      6. associate-*l* 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in J around inf 99.9%

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

      1. *-commutative 99.9%

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

      2. *-commutative 99.9%

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

      3. associate-*r* 99.9%

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

      4. *-commutative 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in K around 0 72.3%

      \[\leadsto \color{blue}{0.25 \cdot \left({K}^{2} \cdot J\right) + -2 \cdot J} \]
   8. Step-by-step derivation

      1. +-commutative 72.3%

         \[\leadsto \color{blue}{-2 \cdot J + 0.25 \cdot \left({K}^{2} \cdot J\right)} \]

      2. associate-*r* 72.3%

         \[\leadsto -2 \cdot J + \color{blue}{\left(0.25 \cdot {K}^{2}\right) \cdot J} \]

      3. distribute-rgt-out 72.3%

         \[\leadsto \color{blue}{J \cdot \left(-2 + 0.25 \cdot {K}^{2}\right)} \]

      4. unpow2 72.3%

         \[\leadsto J \cdot \left(-2 + 0.25 \cdot \color{blue}{\left(K \cdot K\right)}\right) \]

   9. Simplified 72.3%

      \[\leadsto \color{blue}{J \cdot \left(-2 + 0.25 \cdot \left(K \cdot K\right)\right)} \]

   if -8.3999999999999999e-255 < J < 5.79999999999999999e83

   1. Initial program 60.6%

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

      1. *-commutative 60.6%

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

      2. associate-*l* 60.6%

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

      3. unpow2 60.6%

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

      4. hypot-1-def 76.4%

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

      5. *-commutative 76.4%

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

      6. associate-*l* 76.4%

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

   3. Simplified 76.4%

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

   4. Taylor expanded in J around 0 36.3%

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

      1. neg-mul-1 36.3%

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

   6. Simplified 36.3%

      \[\leadsto \color{blue}{-U} \]
3. Recombined 4 regimes into one program.

4. Final simplification 43.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -1.4 \cdot 10^{+142}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;J \leq -9.8 \cdot 10^{+36}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq -2.25 \cdot 10^{-32}:\\ \;\;\;\;J \cdot \left(-2 + 0.25 \cdot \left(K \cdot K\right)\right)\\ \mathbf{elif}\;J \leq -8.4 \cdot 10^{-255}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 5.8 \cdot 10^{+83}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot J\\ \end{array} \]

Alternative 8: 46.6% accurate, 31.5× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} \mathbf{if}\;J \leq -4.4 \cdot 10^{+142}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;J \leq -1 \cdot 10^{+43}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq -1.85 \cdot 10^{-68}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;J \leq -8.4 \cdot 10^{-255}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 2 \cdot 10^{+84}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot J\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (if (<= J -4.4e+142)
   (* -2.0 J)
   (if (<= J -1e+43)
     U
     (if (<= J -1.85e-68)
       (* -2.0 J)
       (if (<= J -8.4e-255) U (if (<= J 2e+84) (- U) (* -2.0 J)))))))

C

U = abs(U);
double code(double J, double K, double U) {
	double tmp;
	if (J <= -4.4e+142) {
		tmp = -2.0 * J;
	} else if (J <= -1e+43) {
		tmp = U;
	} else if (J <= -1.85e-68) {
		tmp = -2.0 * J;
	} else if (J <= -8.4e-255) {
		tmp = U;
	} else if (J <= 2e+84) {
		tmp = -U;
	} else {
		tmp = -2.0 * J;
	}
	return tmp;
}

Fortran

NOTE: U should be positive before calling this function
real(8) function code(j, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (j <= (-4.4d+142)) then
        tmp = (-2.0d0) * j
    else if (j <= (-1d+43)) then
        tmp = u
    else if (j <= (-1.85d-68)) then
        tmp = (-2.0d0) * j
    else if (j <= (-8.4d-255)) then
        tmp = u
    else if (j <= 2d+84) then
        tmp = -u
    else
        tmp = (-2.0d0) * j
    end if
    code = tmp
end function

Java

U = Math.abs(U);
public static double code(double J, double K, double U) {
	double tmp;
	if (J <= -4.4e+142) {
		tmp = -2.0 * J;
	} else if (J <= -1e+43) {
		tmp = U;
	} else if (J <= -1.85e-68) {
		tmp = -2.0 * J;
	} else if (J <= -8.4e-255) {
		tmp = U;
	} else if (J <= 2e+84) {
		tmp = -U;
	} else {
		tmp = -2.0 * J;
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	tmp = 0
	if J <= -4.4e+142:
		tmp = -2.0 * J
	elif J <= -1e+43:
		tmp = U
	elif J <= -1.85e-68:
		tmp = -2.0 * J
	elif J <= -8.4e-255:
		tmp = U
	elif J <= 2e+84:
		tmp = -U
	else:
		tmp = -2.0 * J
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	tmp = 0.0
	if (J <= -4.4e+142)
		tmp = Float64(-2.0 * J);
	elseif (J <= -1e+43)
		tmp = U;
	elseif (J <= -1.85e-68)
		tmp = Float64(-2.0 * J);
	elseif (J <= -8.4e-255)
		tmp = U;
	elseif (J <= 2e+84)
		tmp = Float64(-U);
	else
		tmp = Float64(-2.0 * J);
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	tmp = 0.0;
	if (J <= -4.4e+142)
		tmp = -2.0 * J;
	elseif (J <= -1e+43)
		tmp = U;
	elseif (J <= -1.85e-68)
		tmp = -2.0 * J;
	elseif (J <= -8.4e-255)
		tmp = U;
	elseif (J <= 2e+84)
		tmp = -U;
	else
		tmp = -2.0 * J;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := If[LessEqual[J, -4.4e+142], N[(-2.0 * J), $MachinePrecision], If[LessEqual[J, -1e+43], U, If[LessEqual[J, -1.85e-68], N[(-2.0 * J), $MachinePrecision], If[LessEqual[J, -8.4e-255], U, If[LessEqual[J, 2e+84], (-U), N[(-2.0 * J), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if J < -4.39999999999999974e142 or -1.00000000000000001e43 < J < -1.85000000000000001e-68 or 2.00000000000000012e84 < J

   1. Initial program 98.1%

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

      1. *-commutative 98.1%

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

      2. associate-*l* 98.1%

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

      3. unpow2 98.1%

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

      4. hypot-1-def 99.9%

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

      5. *-commutative 99.9%

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

      6. associate-*l* 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in K around 0 43.4%

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

      1. associate-*r* 43.4%

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

      2. unpow2 43.4%

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

      3. unpow2 43.4%

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

   6. Simplified 43.4%

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

   7. Taylor expanded in U around 0 48.3%

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

   if -4.39999999999999974e142 < J < -1.00000000000000001e43 or -1.85000000000000001e-68 < J < -8.3999999999999999e-255

   1. Initial program 59.7%

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

      1. *-commutative 59.7%

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

      2. associate-*l* 59.7%

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

      3. unpow2 59.7%

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

      4. hypot-1-def 70.3%

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

      5. *-commutative 70.3%

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

      6. associate-*l* 70.3%

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

   3. Simplified 70.3%

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

   4. Taylor expanded in U around -inf 44.0%

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

   if -8.3999999999999999e-255 < J < 2.00000000000000012e84

   1. Initial program 60.6%

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

      1. *-commutative 60.6%

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

      2. associate-*l* 60.6%

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

      3. unpow2 60.6%

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

      4. hypot-1-def 76.4%

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

      5. *-commutative 76.4%

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

      6. associate-*l* 76.4%

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

   3. Simplified 76.4%

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

   4. Taylor expanded in J around 0 36.3%

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

      1. neg-mul-1 36.3%

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

   6. Simplified 36.3%

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

4. Final simplification 43.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -4.4 \cdot 10^{+142}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;J \leq -1 \cdot 10^{+43}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq -1.85 \cdot 10^{-68}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;J \leq -8.4 \cdot 10^{-255}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 2 \cdot 10^{+84}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot J\\ \end{array} \]

Alternative 9: 37.7% accurate, 103.4× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} \mathbf{if}\;J \leq -8.4 \cdot 10^{-255}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U) :precision binary64 (if (<= J -8.4e-255) U (- U)))

C

U = abs(U);
double code(double J, double K, double U) {
	double tmp;
	if (J <= -8.4e-255) {
		tmp = U;
	} else {
		tmp = -U;
	}
	return tmp;
}

Fortran

NOTE: U should be positive before calling this function
real(8) function code(j, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (j <= (-8.4d-255)) then
        tmp = u
    else
        tmp = -u
    end if
    code = tmp
end function

Java

U = Math.abs(U);
public static double code(double J, double K, double U) {
	double tmp;
	if (J <= -8.4e-255) {
		tmp = U;
	} else {
		tmp = -U;
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	tmp = 0
	if J <= -8.4e-255:
		tmp = U
	else:
		tmp = -U
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	tmp = 0.0
	if (J <= -8.4e-255)
		tmp = U;
	else
		tmp = Float64(-U);
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	tmp = 0.0;
	if (J <= -8.4e-255)
		tmp = U;
	else
		tmp = -U;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := If[LessEqual[J, -8.4e-255], U, (-U)]

Derivation

1. Split input into 2 regimes

2. if J < -8.3999999999999999e-255

   1. Initial program 77.2%

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

      1. *-commutative 77.2%

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

      2. associate-*l* 77.2%

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

      3. unpow2 77.2%

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

      4. hypot-1-def 83.8%

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

      5. *-commutative 83.8%

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

      6. associate-*l* 83.8%

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

   3. Simplified 83.8%

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

   4. Taylor expanded in U around -inf 30.7%

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

   if -8.3999999999999999e-255 < J

   1. Initial program 75.6%

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

      1. *-commutative 75.6%

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

      2. associate-*l* 75.6%

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

      3. unpow2 75.6%

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

      4. hypot-1-def 85.8%

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

      5. *-commutative 85.8%

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

      6. associate-*l* 85.8%

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

   3. Simplified 85.8%

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

   4. Taylor expanded in J around 0 28.5%

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

      1. neg-mul-1 28.5%

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

   6. Simplified 28.5%

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

4. Final simplification 29.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -8.4 \cdot 10^{-255}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]

Alternative 10: 27.2% accurate, 420.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.4%

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

   1. *-commutative 76.4%

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

   2. associate-*l* 76.4%

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

   3. unpow2 76.4%

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

   4. hypot-1-def 84.9%

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

   5. *-commutative 84.9%

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

   6. associate-*l* 84.9%

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

3. Simplified 84.9%

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

4. Taylor expanded in U around -inf 28.8%

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

5. Final simplification 28.8%

   \[\leadsto U \]

Reproduce

herbie shell --seed 2023215 
(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)))))