Maksimov and Kolovsky, Equation (3)

Percentage Accurate: 72.9% → 99.4%

Time: 12.5s

Alternatives: 8

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 72.9% 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: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := J \cdot t_0\\ t_2 := \left(\left(-2 \cdot J\right) \cdot t_0\right) \cdot \sqrt{1 + {\left(\frac{U_m}{t_0 \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;-2 \cdot \left(U_m \cdot 0.5\right)\\ \mathbf{elif}\;t_2 \leq 4 \cdot 10^{+302}:\\ \;\;\;\;-2 \cdot \left(t_1 \cdot \mathsf{hypot}\left(1, \frac{\frac{U_m}{2}}{t_1}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(U_m \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1 (* J t_0))
        (t_2
         (*
          (* (* -2.0 J) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* t_0 (* J 2.0))) 2.0))))))
   (if (<= t_2 (- INFINITY))
     (* -2.0 (* U_m 0.5))
     (if (<= t_2 4e+302)
       (* -2.0 (* t_1 (hypot 1.0 (/ (/ U_m 2.0) t_1))))
       (* -2.0 (* U_m -0.5))))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = J * t_0;
	double t_2 = ((-2.0 * J) * t_0) * sqrt((1.0 + pow((U_m / (t_0 * (J * 2.0))), 2.0)));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = -2.0 * (U_m * 0.5);
	} else if (t_2 <= 4e+302) {
		tmp = -2.0 * (t_1 * hypot(1.0, ((U_m / 2.0) / t_1)));
	} else {
		tmp = -2.0 * (U_m * -0.5);
	}
	return tmp;
}

Java

U_m = Math.abs(U);
public static double code(double J, double K, double U_m) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = J * t_0;
	double t_2 = ((-2.0 * J) * t_0) * Math.sqrt((1.0 + Math.pow((U_m / (t_0 * (J * 2.0))), 2.0)));
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = -2.0 * (U_m * 0.5);
	} else if (t_2 <= 4e+302) {
		tmp = -2.0 * (t_1 * Math.hypot(1.0, ((U_m / 2.0) / t_1)));
	} else {
		tmp = -2.0 * (U_m * -0.5);
	}
	return tmp;
}

Python

U_m = math.fabs(U)
def code(J, K, U_m):
	t_0 = math.cos((K / 2.0))
	t_1 = J * t_0
	t_2 = ((-2.0 * J) * t_0) * math.sqrt((1.0 + math.pow((U_m / (t_0 * (J * 2.0))), 2.0)))
	tmp = 0
	if t_2 <= -math.inf:
		tmp = -2.0 * (U_m * 0.5)
	elif t_2 <= 4e+302:
		tmp = -2.0 * (t_1 * math.hypot(1.0, ((U_m / 2.0) / t_1)))
	else:
		tmp = -2.0 * (U_m * -0.5)
	return tmp

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(J * t_0)
	t_2 = Float64(Float64(Float64(-2.0 * J) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_0 * Float64(J * 2.0))) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(-2.0 * Float64(U_m * 0.5));
	elseif (t_2 <= 4e+302)
		tmp = Float64(-2.0 * Float64(t_1 * hypot(1.0, Float64(Float64(U_m / 2.0) / t_1))));
	else
		tmp = Float64(-2.0 * Float64(U_m * -0.5));
	end
	return tmp
end

MATLAB

U_m = abs(U);
function tmp_2 = code(J, K, U_m)
	t_0 = cos((K / 2.0));
	t_1 = J * t_0;
	t_2 = ((-2.0 * J) * t_0) * sqrt((1.0 + ((U_m / (t_0 * (J * 2.0))) ^ 2.0)));
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = -2.0 * (U_m * 0.5);
	elseif (t_2 <= 4e+302)
		tmp = -2.0 * (t_1 * hypot(1.0, ((U_m / 2.0) / t_1)));
	else
		tmp = -2.0 * (U_m * -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(J * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(t$95$0 * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(-2.0 * N[(U$95$m * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 4e+302], N[(-2.0 * N[(t$95$1 * N[Sqrt[1.0 ^ 2 + N[(N[(U$95$m / 2.0), $MachinePrecision] / t$95$1), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(U$95$m * -0.5), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 6.5%

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

      1. associate-*l* 6.5%

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

      2. associate-*l* 6.5%

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

      3. unpow2 6.5%

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

      4. sqr-neg 6.5%

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

      5. distribute-frac-neg 6.5%

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

      6. distribute-frac-neg 6.5%

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

      7. unpow2 6.5%

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

   3. Simplified 67.3%

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

   5. Taylor expanded in U around inf 35.6%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 -2 J) (cos.f64 (/.f64 K 2))) (sqrt.f64 (+.f64 1 (pow.f64 (/.f64 U (*.f64 (*.f64 2 J) (cos.f64 (/.f64 K 2)))) 2)))) < 4.0000000000000003e302

   1. Initial program 99.8%

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

      1. associate-*l* 99.8%

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

      2. associate-*l* 99.8%

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

      3. unpow2 99.8%

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

      4. sqr-neg 99.8%

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

      5. distribute-frac-neg 99.8%

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

      6. distribute-frac-neg 99.8%

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

      7. unpow2 99.8%

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

   3. Simplified 99.8%

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

   if 4.0000000000000003e302 < (*.f64 (*.f64 (*.f64 -2 J) (cos.f64 (/.f64 K 2))) (sqrt.f64 (+.f64 1 (pow.f64 (/.f64 U (*.f64 (*.f64 2 J) (cos.f64 (/.f64 K 2)))) 2))))

   1. Initial program 5.0%

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

      1. associate-*l* 5.0%

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

      2. associate-*l* 5.0%

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

      3. unpow2 5.0%

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

      4. sqr-neg 5.0%

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

      5. distribute-frac-neg 5.0%

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

      6. distribute-frac-neg 5.0%

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

      7. unpow2 5.0%

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

   3. Simplified 70.1%

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

   5. Taylor expanded in U around -inf 54.0%

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

      1. *-commutative 54.0%

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

   7. Simplified 54.0%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -\infty:\\ \;\;\;\;-2 \cdot \left(U \cdot 0.5\right)\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 4 \cdot 10^{+302}:\\ \;\;\;\;-2 \cdot \left(\left(J \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{\frac{U}{2}}{J \cdot \cos \left(\frac{K}{2}\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 74.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} \mathbf{if}\;U_m \leq 1.95 \cdot 10^{+223}:\\ \;\;\;\;-2 \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot \mathsf{hypot}\left(1, \frac{U_m}{J \cdot 2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(U_m \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (if (<= U_m 1.95e+223)
   (* -2.0 (* (cos (* K 0.5)) (* J (hypot 1.0 (/ U_m (* J 2.0))))))
   (* -2.0 (* U_m 0.5))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double tmp;
	if (U_m <= 1.95e+223) {
		tmp = -2.0 * (cos((K * 0.5)) * (J * hypot(1.0, (U_m / (J * 2.0)))));
	} else {
		tmp = -2.0 * (U_m * 0.5);
	}
	return tmp;
}

Java

U_m = Math.abs(U);
public static double code(double J, double K, double U_m) {
	double tmp;
	if (U_m <= 1.95e+223) {
		tmp = -2.0 * (Math.cos((K * 0.5)) * (J * Math.hypot(1.0, (U_m / (J * 2.0)))));
	} else {
		tmp = -2.0 * (U_m * 0.5);
	}
	return tmp;
}

Python

U_m = math.fabs(U)
def code(J, K, U_m):
	tmp = 0
	if U_m <= 1.95e+223:
		tmp = -2.0 * (math.cos((K * 0.5)) * (J * math.hypot(1.0, (U_m / (J * 2.0)))))
	else:
		tmp = -2.0 * (U_m * 0.5)
	return tmp

Julia

U_m = abs(U)
function code(J, K, U_m)
	tmp = 0.0
	if (U_m <= 1.95e+223)
		tmp = Float64(-2.0 * Float64(cos(Float64(K * 0.5)) * Float64(J * hypot(1.0, Float64(U_m / Float64(J * 2.0))))));
	else
		tmp = Float64(-2.0 * Float64(U_m * 0.5));
	end
	return tmp
end

MATLAB

U_m = abs(U);
function tmp_2 = code(J, K, U_m)
	tmp = 0.0;
	if (U_m <= 1.95e+223)
		tmp = -2.0 * (cos((K * 0.5)) * (J * hypot(1.0, (U_m / (J * 2.0)))));
	else
		tmp = -2.0 * (U_m * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := If[LessEqual[U$95$m, 1.95e+223], N[(-2.0 * N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(J * N[Sqrt[1.0 ^ 2 + N[(U$95$m / N[(J * 2.0), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(U$95$m * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U < 1.9499999999999999e223

   1. Initial program 74.0%

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

      1. associate-*l* 74.0%

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

      2. associate-*l* 74.0%

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

      3. unpow2 74.0%

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

      4. sqr-neg 74.0%

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

      5. distribute-frac-neg 74.0%

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

      6. distribute-frac-neg 74.0%

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

      7. unpow2 74.0%

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

   3. Simplified 92.4%

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

   5. Taylor expanded in K around 0 75.7%

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

   7. Applied egg-rr 31.7%

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

      1. expm1-def 45.8%

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

      2. expm1-log1p 75.7%

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

      3. *-commutative 75.7%

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

      4. *-commutative 75.7%

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

      5. *-commutative 75.7%

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

      6. associate-*l* 75.7%

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

      7. *-commutative 75.7%

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

   9. Simplified 75.7%

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

   if 1.9499999999999999e223 < U

   1. Initial program 44.4%

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

      1. associate-*l* 44.4%

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

      2. associate-*l* 44.4%

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

      3. unpow2 44.4%

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

      4. sqr-neg 44.4%

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

      5. distribute-frac-neg 44.4%

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

      6. distribute-frac-neg 44.4%

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

      7. unpow2 44.4%

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

   3. Simplified 63.0%

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

   5. Taylor expanded in U around inf 60.1%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 1.95 \cdot 10^{+223}:\\ \;\;\;\;-2 \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot 2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(U \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 41.3% accurate, 3.2× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := -2 \cdot \left(J \cdot \mathsf{hypot}\left(1, \frac{\frac{U_m}{2}}{J}\right)\right)\\ \mathbf{if}\;J \leq 1.7 \cdot 10^{-196}:\\ \;\;\;\;-2 \cdot \left(U_m \cdot 0.5\right)\\ \mathbf{elif}\;J \leq 9.6 \cdot 10^{-32}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq 6.2 \cdot 10^{-19}:\\ \;\;\;\;-2 \cdot \left(U_m \cdot -0.5\right)\\ \mathbf{elif}\;J \leq 4.6 \cdot 10^{+51}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (* -2.0 (* J (hypot 1.0 (/ (/ U_m 2.0) J))))))
   (if (<= J 1.7e-196)
     (* -2.0 (* U_m 0.5))
     (if (<= J 9.6e-32)
       t_0
       (if (<= J 6.2e-19)
         (* -2.0 (* U_m -0.5))
         (if (<= J 4.6e+51) t_0 (* -2.0 (* J (cos (* K 0.5))))))))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = -2.0 * (J * hypot(1.0, ((U_m / 2.0) / J)));
	double tmp;
	if (J <= 1.7e-196) {
		tmp = -2.0 * (U_m * 0.5);
	} else if (J <= 9.6e-32) {
		tmp = t_0;
	} else if (J <= 6.2e-19) {
		tmp = -2.0 * (U_m * -0.5);
	} else if (J <= 4.6e+51) {
		tmp = t_0;
	} else {
		tmp = -2.0 * (J * cos((K * 0.5)));
	}
	return tmp;
}

Java

U_m = Math.abs(U);
public static double code(double J, double K, double U_m) {
	double t_0 = -2.0 * (J * Math.hypot(1.0, ((U_m / 2.0) / J)));
	double tmp;
	if (J <= 1.7e-196) {
		tmp = -2.0 * (U_m * 0.5);
	} else if (J <= 9.6e-32) {
		tmp = t_0;
	} else if (J <= 6.2e-19) {
		tmp = -2.0 * (U_m * -0.5);
	} else if (J <= 4.6e+51) {
		tmp = t_0;
	} else {
		tmp = -2.0 * (J * Math.cos((K * 0.5)));
	}
	return tmp;
}

Python

U_m = math.fabs(U)
def code(J, K, U_m):
	t_0 = -2.0 * (J * math.hypot(1.0, ((U_m / 2.0) / J)))
	tmp = 0
	if J <= 1.7e-196:
		tmp = -2.0 * (U_m * 0.5)
	elif J <= 9.6e-32:
		tmp = t_0
	elif J <= 6.2e-19:
		tmp = -2.0 * (U_m * -0.5)
	elif J <= 4.6e+51:
		tmp = t_0
	else:
		tmp = -2.0 * (J * math.cos((K * 0.5)))
	return tmp

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = Float64(-2.0 * Float64(J * hypot(1.0, Float64(Float64(U_m / 2.0) / J))))
	tmp = 0.0
	if (J <= 1.7e-196)
		tmp = Float64(-2.0 * Float64(U_m * 0.5));
	elseif (J <= 9.6e-32)
		tmp = t_0;
	elseif (J <= 6.2e-19)
		tmp = Float64(-2.0 * Float64(U_m * -0.5));
	elseif (J <= 4.6e+51)
		tmp = t_0;
	else
		tmp = Float64(-2.0 * Float64(J * cos(Float64(K * 0.5))));
	end
	return tmp
end

MATLAB

U_m = abs(U);
function tmp_2 = code(J, K, U_m)
	t_0 = -2.0 * (J * hypot(1.0, ((U_m / 2.0) / J)));
	tmp = 0.0;
	if (J <= 1.7e-196)
		tmp = -2.0 * (U_m * 0.5);
	elseif (J <= 9.6e-32)
		tmp = t_0;
	elseif (J <= 6.2e-19)
		tmp = -2.0 * (U_m * -0.5);
	elseif (J <= 4.6e+51)
		tmp = t_0;
	else
		tmp = -2.0 * (J * cos((K * 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := Block[{t$95$0 = N[(-2.0 * N[(J * N[Sqrt[1.0 ^ 2 + N[(N[(U$95$m / 2.0), $MachinePrecision] / J), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[J, 1.7e-196], N[(-2.0 * N[(U$95$m * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[J, 9.6e-32], t$95$0, If[LessEqual[J, 6.2e-19], N[(-2.0 * N[(U$95$m * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[J, 4.6e+51], t$95$0, N[(-2.0 * N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if J < 1.7e-196

   1. Initial program 65.6%

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

      1. associate-*l* 65.6%

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

      2. associate-*l* 65.6%

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

      3. unpow2 65.6%

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

      4. sqr-neg 65.6%

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

      5. distribute-frac-neg 65.6%

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

      6. distribute-frac-neg 65.6%

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

      7. unpow2 65.6%

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

   3. Simplified 85.8%

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

   5. Taylor expanded in U around inf 22.5%

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

   if 1.7e-196 < J < 9.6000000000000005e-32 or 6.1999999999999998e-19 < J < 4.6000000000000001e51

   1. Initial program 61.8%

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

      1. associate-*l* 61.8%

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

      2. associate-*l* 61.8%

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

      3. unpow2 61.8%

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

      4. sqr-neg 61.8%

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

      5. distribute-frac-neg 61.8%

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

      6. distribute-frac-neg 61.8%

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

      7. unpow2 61.8%

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

   3. Simplified 93.1%

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

   5. Taylor expanded in K around 0 65.1%

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

   6. Taylor expanded in K around 0 67.8%

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

   if 9.6000000000000005e-32 < J < 6.1999999999999998e-19

   1. Initial program 80.4%

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

      1. associate-*l* 80.4%

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

      2. associate-*l* 80.4%

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

      3. unpow2 80.4%

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

      4. sqr-neg 80.4%

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

      5. distribute-frac-neg 80.4%

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

      6. distribute-frac-neg 80.4%

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

      7. unpow2 80.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in U around -inf 22.6%

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

      1. *-commutative 22.6%

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

   7. Simplified 22.6%

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

   if 4.6000000000000001e51 < J

   1. Initial program 94.0%

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

      1. associate-*l* 94.0%

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

      2. associate-*l* 94.0%

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

      3. unpow2 94.0%

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

      4. sqr-neg 94.0%

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

      5. distribute-frac-neg 94.0%

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

      6. distribute-frac-neg 94.0%

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

      7. unpow2 94.0%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in U around 0 77.7%

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

4. Final simplification 43.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq 1.7 \cdot 10^{-196}:\\ \;\;\;\;-2 \cdot \left(U \cdot 0.5\right)\\ \mathbf{elif}\;J \leq 9.6 \cdot 10^{-32}:\\ \;\;\;\;-2 \cdot \left(J \cdot \mathsf{hypot}\left(1, \frac{\frac{U}{2}}{J}\right)\right)\\ \mathbf{elif}\;J \leq 6.2 \cdot 10^{-19}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5\right)\\ \mathbf{elif}\;J \leq 4.6 \cdot 10^{+51}:\\ \;\;\;\;-2 \cdot \left(J \cdot \mathsf{hypot}\left(1, \frac{\frac{U}{2}}{J}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 41.1% accurate, 3.2× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := -2 \cdot \left(J \cdot \mathsf{hypot}\left(1, \frac{\frac{U_m}{2}}{J}\right)\right)\\ \mathbf{if}\;J \leq 6 \cdot 10^{-87}:\\ \;\;\;\;-2 \cdot \mathsf{fma}\left(0.5, U_m, J \cdot \frac{J}{U_m}\right)\\ \mathbf{elif}\;J \leq 9.5 \cdot 10^{-32}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq 6.2 \cdot 10^{-19}:\\ \;\;\;\;-2 \cdot \left(U_m \cdot -0.5\right)\\ \mathbf{elif}\;J \leq 3.2 \cdot 10^{+51}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (* -2.0 (* J (hypot 1.0 (/ (/ U_m 2.0) J))))))
   (if (<= J 6e-87)
     (* -2.0 (fma 0.5 U_m (* J (/ J U_m))))
     (if (<= J 9.5e-32)
       t_0
       (if (<= J 6.2e-19)
         (* -2.0 (* U_m -0.5))
         (if (<= J 3.2e+51) t_0 (* -2.0 (* J (cos (* K 0.5))))))))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = -2.0 * (J * hypot(1.0, ((U_m / 2.0) / J)));
	double tmp;
	if (J <= 6e-87) {
		tmp = -2.0 * fma(0.5, U_m, (J * (J / U_m)));
	} else if (J <= 9.5e-32) {
		tmp = t_0;
	} else if (J <= 6.2e-19) {
		tmp = -2.0 * (U_m * -0.5);
	} else if (J <= 3.2e+51) {
		tmp = t_0;
	} else {
		tmp = -2.0 * (J * cos((K * 0.5)));
	}
	return tmp;
}

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = Float64(-2.0 * Float64(J * hypot(1.0, Float64(Float64(U_m / 2.0) / J))))
	tmp = 0.0
	if (J <= 6e-87)
		tmp = Float64(-2.0 * fma(0.5, U_m, Float64(J * Float64(J / U_m))));
	elseif (J <= 9.5e-32)
		tmp = t_0;
	elseif (J <= 6.2e-19)
		tmp = Float64(-2.0 * Float64(U_m * -0.5));
	elseif (J <= 3.2e+51)
		tmp = t_0;
	else
		tmp = Float64(-2.0 * Float64(J * cos(Float64(K * 0.5))));
	end
	return tmp
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := Block[{t$95$0 = N[(-2.0 * N[(J * N[Sqrt[1.0 ^ 2 + N[(N[(U$95$m / 2.0), $MachinePrecision] / J), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[J, 6e-87], N[(-2.0 * N[(0.5 * U$95$m + N[(J * N[(J / U$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[J, 9.5e-32], t$95$0, If[LessEqual[J, 6.2e-19], N[(-2.0 * N[(U$95$m * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[J, 3.2e+51], t$95$0, N[(-2.0 * N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if J < 6.00000000000000033e-87

   1. Initial program 64.0%

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

      1. associate-*l* 64.0%

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

      2. associate-*l* 64.0%

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

      3. unpow2 64.0%

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

      4. sqr-neg 64.0%

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

      5. distribute-frac-neg 64.0%

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

      6. distribute-frac-neg 64.0%

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

      7. unpow2 64.0%

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

   3. Simplified 85.1%

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

   5. Taylor expanded in U around inf 24.2%

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

      1. fma-def 24.2%

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

      2. unpow2 24.2%

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

      3. *-commutative 24.2%

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

      4. unpow2 24.2%

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

      5. swap-sqr 24.2%

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

      6. unpow2 24.2%

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

      7. *-commutative 24.2%

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

   7. Simplified 24.2%

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

   8. Taylor expanded in K around 0 24.2%

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

      1. unpow2 24.2%

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

      2. *-un-lft-identity 24.2%

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

      3. times-frac 25.6%

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

   10. Applied egg-rr 25.6%

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

   if 6.00000000000000033e-87 < J < 9.4999999999999999e-32 or 6.1999999999999998e-19 < J < 3.2000000000000002e51

   1. Initial program 68.7%

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

      1. associate-*l* 68.7%

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

      2. associate-*l* 68.7%

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

      3. unpow2 68.7%

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

      4. sqr-neg 68.7%

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

      5. distribute-frac-neg 68.7%

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

      6. distribute-frac-neg 68.7%

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

      7. unpow2 68.7%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in K around 0 71.1%

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

   6. Taylor expanded in K around 0 71.3%

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

   if 9.4999999999999999e-32 < J < 6.1999999999999998e-19

   1. Initial program 80.4%

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

      1. associate-*l* 80.4%

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

      2. associate-*l* 80.4%

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

      3. unpow2 80.4%

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

      4. sqr-neg 80.4%

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

      5. distribute-frac-neg 80.4%

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

      6. distribute-frac-neg 80.4%

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

      7. unpow2 80.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in U around -inf 22.6%

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

      1. *-commutative 22.6%

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

   7. Simplified 22.6%

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

   if 3.2000000000000002e51 < J

   1. Initial program 94.0%

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

      1. associate-*l* 94.0%

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

      2. associate-*l* 94.0%

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

      3. unpow2 94.0%

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

      4. sqr-neg 94.0%

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

      5. distribute-frac-neg 94.0%

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

      6. distribute-frac-neg 94.0%

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

      7. unpow2 94.0%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in U around 0 77.7%

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

4. Final simplification 43.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq 6 \cdot 10^{-87}:\\ \;\;\;\;-2 \cdot \mathsf{fma}\left(0.5, U, J \cdot \frac{J}{U}\right)\\ \mathbf{elif}\;J \leq 9.5 \cdot 10^{-32}:\\ \;\;\;\;-2 \cdot \left(J \cdot \mathsf{hypot}\left(1, \frac{\frac{U}{2}}{J}\right)\right)\\ \mathbf{elif}\;J \leq 6.2 \cdot 10^{-19}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5\right)\\ \mathbf{elif}\;J \leq 3.2 \cdot 10^{+51}:\\ \;\;\;\;-2 \cdot \left(J \cdot \mathsf{hypot}\left(1, \frac{\frac{U}{2}}{J}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 39.5% accurate, 3.4× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := -2 \cdot \left(U_m \cdot 0.5\right)\\ \mathbf{if}\;J \leq 2.55 \cdot 10^{-33}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq 1.05 \cdot 10^{-18}:\\ \;\;\;\;-2 \cdot \left(U_m \cdot -0.5\right)\\ \mathbf{elif}\;J \leq 5.2 \cdot 10^{+14}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (* -2.0 (* U_m 0.5))))
   (if (<= J 2.55e-33)
     t_0
     (if (<= J 1.05e-18)
       (* -2.0 (* U_m -0.5))
       (if (<= J 5.2e+14) t_0 (* -2.0 (* J (cos (* K 0.5)))))))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = -2.0 * (U_m * 0.5);
	double tmp;
	if (J <= 2.55e-33) {
		tmp = t_0;
	} else if (J <= 1.05e-18) {
		tmp = -2.0 * (U_m * -0.5);
	} else if (J <= 5.2e+14) {
		tmp = t_0;
	} else {
		tmp = -2.0 * (J * cos((K * 0.5)));
	}
	return tmp;
}

Fortran

U_m = abs(U)
real(8) function code(j, k, u_m)
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (-2.0d0) * (u_m * 0.5d0)
    if (j <= 2.55d-33) then
        tmp = t_0
    else if (j <= 1.05d-18) then
        tmp = (-2.0d0) * (u_m * (-0.5d0))
    else if (j <= 5.2d+14) then
        tmp = t_0
    else
        tmp = (-2.0d0) * (j * cos((k * 0.5d0)))
    end if
    code = tmp
end function

Java

U_m = Math.abs(U);
public static double code(double J, double K, double U_m) {
	double t_0 = -2.0 * (U_m * 0.5);
	double tmp;
	if (J <= 2.55e-33) {
		tmp = t_0;
	} else if (J <= 1.05e-18) {
		tmp = -2.0 * (U_m * -0.5);
	} else if (J <= 5.2e+14) {
		tmp = t_0;
	} else {
		tmp = -2.0 * (J * Math.cos((K * 0.5)));
	}
	return tmp;
}

Python

U_m = math.fabs(U)
def code(J, K, U_m):
	t_0 = -2.0 * (U_m * 0.5)
	tmp = 0
	if J <= 2.55e-33:
		tmp = t_0
	elif J <= 1.05e-18:
		tmp = -2.0 * (U_m * -0.5)
	elif J <= 5.2e+14:
		tmp = t_0
	else:
		tmp = -2.0 * (J * math.cos((K * 0.5)))
	return tmp

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = Float64(-2.0 * Float64(U_m * 0.5))
	tmp = 0.0
	if (J <= 2.55e-33)
		tmp = t_0;
	elseif (J <= 1.05e-18)
		tmp = Float64(-2.0 * Float64(U_m * -0.5));
	elseif (J <= 5.2e+14)
		tmp = t_0;
	else
		tmp = Float64(-2.0 * Float64(J * cos(Float64(K * 0.5))));
	end
	return tmp
end

MATLAB

U_m = abs(U);
function tmp_2 = code(J, K, U_m)
	t_0 = -2.0 * (U_m * 0.5);
	tmp = 0.0;
	if (J <= 2.55e-33)
		tmp = t_0;
	elseif (J <= 1.05e-18)
		tmp = -2.0 * (U_m * -0.5);
	elseif (J <= 5.2e+14)
		tmp = t_0;
	else
		tmp = -2.0 * (J * cos((K * 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := Block[{t$95$0 = N[(-2.0 * N[(U$95$m * 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[J, 2.55e-33], t$95$0, If[LessEqual[J, 1.05e-18], N[(-2.0 * N[(U$95$m * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[J, 5.2e+14], t$95$0, N[(-2.0 * N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if J < 2.55000000000000004e-33 or 1.05e-18 < J < 5.2e14

   1. Initial program 63.4%

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

      1. associate-*l* 63.4%

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

      2. associate-*l* 63.4%

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

      3. unpow2 63.4%

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

      4. sqr-neg 63.4%

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

      5. distribute-frac-neg 63.4%

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

      6. distribute-frac-neg 63.4%

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

      7. unpow2 63.4%

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

   3. Simplified 87.0%

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

   5. Taylor expanded in U around inf 26.7%

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

   if 2.55000000000000004e-33 < J < 1.05e-18

   1. Initial program 80.4%

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

      1. associate-*l* 80.4%

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

      2. associate-*l* 80.4%

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

      3. unpow2 80.4%

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

      4. sqr-neg 80.4%

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

      5. distribute-frac-neg 80.4%

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

      6. distribute-frac-neg 80.4%

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

      7. unpow2 80.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in U around -inf 22.6%

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

      1. *-commutative 22.6%

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

   7. Simplified 22.6%

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

   if 5.2e14 < J

   1. Initial program 94.5%

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

      1. associate-*l* 94.5%

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

      2. associate-*l* 94.5%

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

      3. unpow2 94.5%

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

      4. sqr-neg 94.5%

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

      5. distribute-frac-neg 94.5%

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

      6. distribute-frac-neg 94.5%

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

      7. unpow2 94.5%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in U around 0 77.2%

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

4. Final simplification 40.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq 2.55 \cdot 10^{-33}:\\ \;\;\;\;-2 \cdot \left(U \cdot 0.5\right)\\ \mathbf{elif}\;J \leq 1.05 \cdot 10^{-18}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5\right)\\ \mathbf{elif}\;J \leq 5.2 \cdot 10^{+14}:\\ \;\;\;\;-2 \cdot \left(U \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 31.7% accurate, 21.0× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := -2 \cdot \left(U_m \cdot 0.5\right)\\ \mathbf{if}\;J \leq 9.6 \cdot 10^{-32}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq 6.6 \cdot 10^{-19}:\\ \;\;\;\;-2 \cdot \left(U_m \cdot -0.5\right)\\ \mathbf{elif}\;J \leq 5.2 \cdot 10^{+14}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot J\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (* -2.0 (* U_m 0.5))))
   (if (<= J 9.6e-32)
     t_0
     (if (<= J 6.6e-19)
       (* -2.0 (* U_m -0.5))
       (if (<= J 5.2e+14) t_0 (* -2.0 J))))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = -2.0 * (U_m * 0.5);
	double tmp;
	if (J <= 9.6e-32) {
		tmp = t_0;
	} else if (J <= 6.6e-19) {
		tmp = -2.0 * (U_m * -0.5);
	} else if (J <= 5.2e+14) {
		tmp = t_0;
	} else {
		tmp = -2.0 * J;
	}
	return tmp;
}

Fortran

U_m = abs(U)
real(8) function code(j, k, u_m)
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (-2.0d0) * (u_m * 0.5d0)
    if (j <= 9.6d-32) then
        tmp = t_0
    else if (j <= 6.6d-19) then
        tmp = (-2.0d0) * (u_m * (-0.5d0))
    else if (j <= 5.2d+14) then
        tmp = t_0
    else
        tmp = (-2.0d0) * j
    end if
    code = tmp
end function

Java

U_m = Math.abs(U);
public static double code(double J, double K, double U_m) {
	double t_0 = -2.0 * (U_m * 0.5);
	double tmp;
	if (J <= 9.6e-32) {
		tmp = t_0;
	} else if (J <= 6.6e-19) {
		tmp = -2.0 * (U_m * -0.5);
	} else if (J <= 5.2e+14) {
		tmp = t_0;
	} else {
		tmp = -2.0 * J;
	}
	return tmp;
}

Python

U_m = math.fabs(U)
def code(J, K, U_m):
	t_0 = -2.0 * (U_m * 0.5)
	tmp = 0
	if J <= 9.6e-32:
		tmp = t_0
	elif J <= 6.6e-19:
		tmp = -2.0 * (U_m * -0.5)
	elif J <= 5.2e+14:
		tmp = t_0
	else:
		tmp = -2.0 * J
	return tmp

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = Float64(-2.0 * Float64(U_m * 0.5))
	tmp = 0.0
	if (J <= 9.6e-32)
		tmp = t_0;
	elseif (J <= 6.6e-19)
		tmp = Float64(-2.0 * Float64(U_m * -0.5));
	elseif (J <= 5.2e+14)
		tmp = t_0;
	else
		tmp = Float64(-2.0 * J);
	end
	return tmp
end

MATLAB

U_m = abs(U);
function tmp_2 = code(J, K, U_m)
	t_0 = -2.0 * (U_m * 0.5);
	tmp = 0.0;
	if (J <= 9.6e-32)
		tmp = t_0;
	elseif (J <= 6.6e-19)
		tmp = -2.0 * (U_m * -0.5);
	elseif (J <= 5.2e+14)
		tmp = t_0;
	else
		tmp = -2.0 * J;
	end
	tmp_2 = tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := Block[{t$95$0 = N[(-2.0 * N[(U$95$m * 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[J, 9.6e-32], t$95$0, If[LessEqual[J, 6.6e-19], N[(-2.0 * N[(U$95$m * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[J, 5.2e+14], t$95$0, N[(-2.0 * J), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if J < 9.6000000000000005e-32 or 6.5999999999999995e-19 < J < 5.2e14

   1. Initial program 63.4%

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

      1. associate-*l* 63.4%

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

      2. associate-*l* 63.4%

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

      3. unpow2 63.4%

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

      4. sqr-neg 63.4%

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

      5. distribute-frac-neg 63.4%

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

      6. distribute-frac-neg 63.4%

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

      7. unpow2 63.4%

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

   3. Simplified 87.0%

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

   5. Taylor expanded in U around inf 26.7%

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

   if 9.6000000000000005e-32 < J < 6.5999999999999995e-19

   1. Initial program 80.4%

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

      1. associate-*l* 80.4%

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

      2. associate-*l* 80.4%

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

      3. unpow2 80.4%

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

      4. sqr-neg 80.4%

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

      5. distribute-frac-neg 80.4%

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

      6. distribute-frac-neg 80.4%

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

      7. unpow2 80.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in U around -inf 22.6%

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

      1. *-commutative 22.6%

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

   7. Simplified 22.6%

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

   if 5.2e14 < J

   1. Initial program 94.5%

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

      1. associate-*l* 94.5%

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

      2. associate-*l* 94.5%

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

      3. unpow2 94.5%

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

      4. sqr-neg 94.5%

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

      5. distribute-frac-neg 94.5%

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

      6. distribute-frac-neg 94.5%

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

      7. unpow2 94.5%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in U around 0 77.2%

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

   6. Taylor expanded in K around 0 46.8%

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

4. Final simplification 32.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq 9.6 \cdot 10^{-32}:\\ \;\;\;\;-2 \cdot \left(U \cdot 0.5\right)\\ \mathbf{elif}\;J \leq 6.6 \cdot 10^{-19}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5\right)\\ \mathbf{elif}\;J \leq 5.2 \cdot 10^{+14}:\\ \;\;\;\;-2 \cdot \left(U \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot J\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 31.9% accurate, 41.9× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} \mathbf{if}\;J \leq 5.2 \cdot 10^{+14}:\\ \;\;\;\;-2 \cdot \left(U_m \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot J\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (if (<= J 5.2e+14) (* -2.0 (* U_m 0.5)) (* -2.0 J)))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double tmp;
	if (J <= 5.2e+14) {
		tmp = -2.0 * (U_m * 0.5);
	} else {
		tmp = -2.0 * J;
	}
	return tmp;
}

Fortran

U_m = abs(U)
real(8) function code(j, k, u_m)
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u_m
    real(8) :: tmp
    if (j <= 5.2d+14) then
        tmp = (-2.0d0) * (u_m * 0.5d0)
    else
        tmp = (-2.0d0) * j
    end if
    code = tmp
end function

Java

U_m = Math.abs(U);
public static double code(double J, double K, double U_m) {
	double tmp;
	if (J <= 5.2e+14) {
		tmp = -2.0 * (U_m * 0.5);
	} else {
		tmp = -2.0 * J;
	}
	return tmp;
}

Python

U_m = math.fabs(U)
def code(J, K, U_m):
	tmp = 0
	if J <= 5.2e+14:
		tmp = -2.0 * (U_m * 0.5)
	else:
		tmp = -2.0 * J
	return tmp

Julia

U_m = abs(U)
function code(J, K, U_m)
	tmp = 0.0
	if (J <= 5.2e+14)
		tmp = Float64(-2.0 * Float64(U_m * 0.5));
	else
		tmp = Float64(-2.0 * J);
	end
	return tmp
end

MATLAB

U_m = abs(U);
function tmp_2 = code(J, K, U_m)
	tmp = 0.0;
	if (J <= 5.2e+14)
		tmp = -2.0 * (U_m * 0.5);
	else
		tmp = -2.0 * J;
	end
	tmp_2 = tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := If[LessEqual[J, 5.2e+14], N[(-2.0 * N[(U$95$m * 0.5), $MachinePrecision]), $MachinePrecision], N[(-2.0 * J), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if J < 5.2e14

   1. Initial program 63.8%

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

      1. associate-*l* 63.8%

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

      2. associate-*l* 63.8%

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

      3. unpow2 63.8%

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

      4. sqr-neg 63.8%

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

      5. distribute-frac-neg 63.8%

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

      6. distribute-frac-neg 63.8%

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

      7. unpow2 63.8%

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

   3. Simplified 87.3%

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

   5. Taylor expanded in U around inf 27.1%

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

   if 5.2e14 < J

   1. Initial program 94.5%

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

      1. associate-*l* 94.5%

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

      2. associate-*l* 94.5%

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

      3. unpow2 94.5%

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

      4. sqr-neg 94.5%

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

      5. distribute-frac-neg 94.5%

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

      6. distribute-frac-neg 94.5%

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

      7. unpow2 94.5%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in U around 0 77.2%

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

   6. Taylor expanded in K around 0 46.8%

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

4. Final simplification 32.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq 5.2 \cdot 10^{+14}:\\ \;\;\;\;-2 \cdot \left(U \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot J\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 30.1% accurate, 140.0× speedup

Math

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

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m) :precision binary64 (* -2.0 J))

C

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

Fortran

U_m = abs(U)
real(8) function code(j, k, u_m)
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u_m
    code = (-2.0d0) * j
end function

Java

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

Python

U_m = math.fabs(U)
def code(J, K, U_m):
	return -2.0 * J

Julia

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

MATLAB

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

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := N[(-2.0 * J), $MachinePrecision]

Derivation

1. Initial program 72.2%

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

   1. associate-*l* 72.2%

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

   2. associate-*l* 72.2%

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

   3. unpow2 72.2%

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

   4. sqr-neg 72.2%

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

   5. distribute-frac-neg 72.2%

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

   6. distribute-frac-neg 72.2%

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

   7. unpow2 72.2%

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

3. Simplified 90.7%

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

5. Taylor expanded in U around 0 53.8%

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

6. Taylor expanded in K around 0 34.5%

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

7. Final simplification 34.5%

   \[\leadsto -2 \cdot J \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024020 
(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)))))