Maksimov and Kolovsky, Equation (3)

Percentage Accurate: 73.6% → 99.8%

Time: 19.0s

Alternatives: 11

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.6% 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: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\right) \cdot t_0\right) \cdot \sqrt{1 + {\left(\frac{U}{t_0 \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;t_1 \leq 10^{+308}:\\ \;\;\;\;\left(J \cdot \left(-2 \cdot t_0\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot t_0\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* (* -2.0 J) t_0)
          (sqrt (+ 1.0 (pow (/ U (* t_0 (* J 2.0))) 2.0))))))
   (if (<= t_1 (- INFINITY))
     (- U)
     (if (<= t_1 1e+308)
       (* (* J (* -2.0 t_0)) (hypot 1.0 (/ U (* J (* 2.0 t_0)))))
       U))))

C

U = abs(U);
double code(double J, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * J) * t_0) * sqrt((1.0 + pow((U / (t_0 * (J * 2.0))), 2.0)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -U;
	} else if (t_1 <= 1e+308) {
		tmp = (J * (-2.0 * t_0)) * hypot(1.0, (U / (J * (2.0 * t_0))));
	} else {
		tmp = U;
	}
	return tmp;
}

Java

U = Math.abs(U);
public static double code(double J, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = ((-2.0 * J) * t_0) * Math.sqrt((1.0 + Math.pow((U / (t_0 * (J * 2.0))), 2.0)));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = -U;
	} else if (t_1 <= 1e+308) {
		tmp = (J * (-2.0 * t_0)) * Math.hypot(1.0, (U / (J * (2.0 * t_0))));
	} else {
		tmp = U;
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	t_0 = math.cos((K / 2.0))
	t_1 = ((-2.0 * J) * t_0) * math.sqrt((1.0 + math.pow((U / (t_0 * (J * 2.0))), 2.0)))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = -U
	elif t_1 <= 1e+308:
		tmp = (J * (-2.0 * t_0)) * math.hypot(1.0, (U / (J * (2.0 * t_0))))
	else:
		tmp = U
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * J) * t_0) * sqrt(Float64(1.0 + (Float64(U / Float64(t_0 * Float64(J * 2.0))) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-U);
	elseif (t_1 <= 1e+308)
		tmp = Float64(Float64(J * Float64(-2.0 * t_0)) * hypot(1.0, Float64(U / Float64(J * Float64(2.0 * t_0)))));
	else
		tmp = U;
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	t_0 = cos((K / 2.0));
	t_1 = ((-2.0 * J) * t_0) * sqrt((1.0 + ((U / (t_0 * (J * 2.0))) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = -U;
	elseif (t_1 <= 1e+308)
		tmp = (J * (-2.0 * t_0)) * hypot(1.0, (U / (J * (2.0 * t_0))));
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U / N[(t$95$0 * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], (-U), If[LessEqual[t$95$1, 1e+308], N[(N[(J * N[(-2.0 * t$95$0), $MachinePrecision]), $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[(U / N[(J * N[(2.0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], U]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 5.5%

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

      1. *-commutative 5.5%

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

      2. associate-*l* 5.5%

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

      3. associate-*r* 5.5%

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

      4. *-commutative 5.5%

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

      5. associate-*l* 5.5%

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

      6. *-commutative 5.5%

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

      7. unpow2 5.5%

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

      8. hypot-1-def 55.9%

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

      9. *-commutative 55.9%

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

      10. associate-*l* 55.9%

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

   3. Simplified 55.9%

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

   4. Taylor expanded in J around 0 53.3%

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

      1. neg-mul-1 53.3%

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

   6. Simplified 53.3%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 -2 J) (cos.f64 (/.f64 K 2))) (sqrt.f64 (+.f64 1 (pow.f64 (/.f64 U (*.f64 (*.f64 2 J) (cos.f64 (/.f64 K 2)))) 2)))) < 1e308

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-*l* 99.8%

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

      3. unpow2 99.8%

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

      4. hypot-1-def 99.9%

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

      5. *-commutative 99.9%

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

      6. associate-*l* 99.9%

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

   3. Simplified 99.9%

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

   if 1e308 < (*.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 4.9%

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

      1. *-commutative 4.9%

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

      2. associate-*l* 4.9%

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

      3. associate-*r* 4.9%

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

      4. *-commutative 4.9%

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

      5. associate-*l* 4.9%

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

      6. *-commutative 4.9%

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

      7. unpow2 4.9%

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

      8. hypot-1-def 60.9%

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

      9. *-commutative 60.9%

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

      10. associate-*l* 60.9%

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

   3. Simplified 60.9%

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

   4. Taylor expanded in U around -inf 53.3%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 10^{+308}:\\ \;\;\;\;\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternative 2: 87.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;U \leq 2.05 \cdot 10^{+246}:\\ \;\;\;\;J \cdot \left(t_0 \cdot \left(-2 \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot t_0\right)}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= U 2.05e+246)
     (* J (* t_0 (* -2.0 (hypot 1.0 (/ U (* J (* 2.0 t_0)))))))
     (- U))))

C

U = abs(U);
double code(double J, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (U <= 2.05e+246) {
		tmp = J * (t_0 * (-2.0 * hypot(1.0, (U / (J * (2.0 * t_0))))));
	} else {
		tmp = -U;
	}
	return tmp;
}

Java

U = Math.abs(U);
public static double code(double J, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double tmp;
	if (U <= 2.05e+246) {
		tmp = J * (t_0 * (-2.0 * Math.hypot(1.0, (U / (J * (2.0 * t_0))))));
	} else {
		tmp = -U;
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if U <= 2.05e+246:
		tmp = J * (t_0 * (-2.0 * math.hypot(1.0, (U / (J * (2.0 * t_0))))))
	else:
		tmp = -U
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (U <= 2.05e+246)
		tmp = Float64(J * Float64(t_0 * Float64(-2.0 * hypot(1.0, Float64(U / Float64(J * Float64(2.0 * t_0)))))));
	else
		tmp = Float64(-U);
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	t_0 = cos((K / 2.0));
	tmp = 0.0;
	if (U <= 2.05e+246)
		tmp = J * (t_0 * (-2.0 * hypot(1.0, (U / (J * (2.0 * t_0))))));
	else
		tmp = -U;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[U, 2.05e+246], N[(J * N[(t$95$0 * N[(-2.0 * N[Sqrt[1.0 ^ 2 + N[(U / N[(J * N[(2.0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-U)]]

Derivation

1. Split input into 2 regimes

2. if U < 2.04999999999999988e246

   1. Initial program 76.7%

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

      1. *-commutative 76.7%

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

      2. associate-*l* 76.7%

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

      3. associate-*r* 76.7%

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

      4. *-commutative 76.7%

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

      5. associate-*l* 76.7%

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

      6. *-commutative 76.7%

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

      7. unpow2 76.7%

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

      8. hypot-1-def 90.3%

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

      9. *-commutative 90.3%

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

      10. associate-*l* 90.3%

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

   3. Simplified 90.3%

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

   if 2.04999999999999988e246 < U

   1. Initial program 18.6%

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

      1. *-commutative 18.6%

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

      2. associate-*l* 18.6%

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

      3. associate-*r* 18.6%

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

      4. *-commutative 18.6%

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

      5. associate-*l* 18.6%

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

      6. *-commutative 18.6%

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

      7. unpow2 18.6%

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

      8. hypot-1-def 49.0%

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

      9. *-commutative 49.0%

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

      10. associate-*l* 49.0%

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

   3. Simplified 49.0%

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

   4. Taylor expanded in J around 0 77.8%

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

      1. neg-mul-1 77.8%

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

   6. Simplified 77.8%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 2.05 \cdot 10^{+246}:\\ \;\;\;\;J \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]

Alternative 3: 77.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} t_0 := \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot 2}\right)\\ \mathbf{if}\;J \leq -2.8 \cdot 10^{-234}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq -5 \cdot 10^{-310}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 3.1 \cdot 10^{-196}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (* (* J (* -2.0 (cos (/ K 2.0)))) (hypot 1.0 (/ U (* J 2.0))))))
   (if (<= J -2.8e-234)
     t_0
     (if (<= J -5e-310) U (if (<= J 3.1e-196) (- U) t_0)))))

C

U = abs(U);
double code(double J, double K, double U) {
	double t_0 = (J * (-2.0 * cos((K / 2.0)))) * hypot(1.0, (U / (J * 2.0)));
	double tmp;
	if (J <= -2.8e-234) {
		tmp = t_0;
	} else if (J <= -5e-310) {
		tmp = U;
	} else if (J <= 3.1e-196) {
		tmp = -U;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

U = Math.abs(U);
public static double code(double J, double K, double U) {
	double t_0 = (J * (-2.0 * Math.cos((K / 2.0)))) * Math.hypot(1.0, (U / (J * 2.0)));
	double tmp;
	if (J <= -2.8e-234) {
		tmp = t_0;
	} else if (J <= -5e-310) {
		tmp = U;
	} else if (J <= 3.1e-196) {
		tmp = -U;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	t_0 = (J * (-2.0 * math.cos((K / 2.0)))) * math.hypot(1.0, (U / (J * 2.0)))
	tmp = 0
	if J <= -2.8e-234:
		tmp = t_0
	elif J <= -5e-310:
		tmp = U
	elif J <= 3.1e-196:
		tmp = -U
	else:
		tmp = t_0
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	t_0 = Float64(Float64(J * Float64(-2.0 * cos(Float64(K / 2.0)))) * hypot(1.0, Float64(U / Float64(J * 2.0))))
	tmp = 0.0
	if (J <= -2.8e-234)
		tmp = t_0;
	elseif (J <= -5e-310)
		tmp = U;
	elseif (J <= 3.1e-196)
		tmp = Float64(-U);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	t_0 = (J * (-2.0 * cos((K / 2.0)))) * hypot(1.0, (U / (J * 2.0)));
	tmp = 0.0;
	if (J <= -2.8e-234)
		tmp = t_0;
	elseif (J <= -5e-310)
		tmp = U;
	elseif (J <= 3.1e-196)
		tmp = -U;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := Block[{t$95$0 = N[(N[(J * N[(-2.0 * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[(U / N[(J * 2.0), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[J, -2.8e-234], t$95$0, If[LessEqual[J, -5e-310], U, If[LessEqual[J, 3.1e-196], (-U), t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if J < -2.7999999999999999e-234 or 3.09999999999999993e-196 < J

   1. Initial program 80.7%

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

      1. *-commutative 80.7%

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

      2. associate-*l* 80.7%

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

      3. unpow2 80.7%

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

      4. hypot-1-def 93.9%

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

      5. *-commutative 93.9%

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

      6. associate-*l* 93.9%

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

   3. Simplified 93.9%

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

   4. Taylor expanded in K around 0 80.4%

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

   if -2.7999999999999999e-234 < J < -4.999999999999985e-310

   1. Initial program 36.3%

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

      1. *-commutative 36.3%

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

      2. associate-*l* 36.3%

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

      3. associate-*r* 36.3%

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

      4. *-commutative 36.3%

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

      5. associate-*l* 36.3%

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

      6. *-commutative 36.3%

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

      7. unpow2 36.3%

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

      8. hypot-1-def 62.0%

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

      9. *-commutative 62.0%

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

      10. associate-*l* 62.0%

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

   3. Simplified 62.0%

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

   4. Taylor expanded in U around -inf 54.3%

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

   if -4.999999999999985e-310 < J < 3.09999999999999993e-196

   1. Initial program 44.2%

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

      1. *-commutative 44.2%

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

      2. associate-*l* 44.2%

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

      3. associate-*r* 44.2%

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

      4. *-commutative 44.2%

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

      5. associate-*l* 44.1%

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

      6. *-commutative 44.1%

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

      7. unpow2 44.1%

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

      8. hypot-1-def 60.2%

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

      9. *-commutative 60.2%

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

      10. associate-*l* 60.2%

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

   3. Simplified 60.2%

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

   4. Taylor expanded in J around 0 55.0%

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

      1. neg-mul-1 55.0%

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

   6. Simplified 55.0%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -2.8 \cdot 10^{-234}:\\ \;\;\;\;\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot 2}\right)\\ \mathbf{elif}\;J \leq -5 \cdot 10^{-310}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 3.1 \cdot 10^{-196}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot 2}\right)\\ \end{array} \]

Alternative 4: 63.8% accurate, 3.5× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} t_0 := J \cdot \frac{J}{U}\\ t_1 := J \cdot \left(-2 \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{if}\;J \leq -2.1 \cdot 10^{-52}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;J \leq -1.1 \cdot 10^{-129}:\\ \;\;\;\;U + 2 \cdot t_0\\ \mathbf{elif}\;J \leq -2.3 \cdot 10^{-165}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;J \leq -2.75 \cdot 10^{-238}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq -5 \cdot 10^{-310}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 1.25 \cdot 10^{-40}:\\ \;\;\;\;-2 \cdot t_0 - U\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (* J (/ J U))) (t_1 (* J (* -2.0 (cos (* K 0.5))))))
   (if (<= J -2.1e-52)
     t_1
     (if (<= J -1.1e-129)
       (+ U (* 2.0 t_0))
       (if (<= J -2.3e-165)
         t_1
         (if (<= J -2.75e-238)
           (- U)
           (if (<= J -5e-310)
             U
             (if (<= J 1.25e-40) (- (* -2.0 t_0) U) t_1))))))))

C

U = abs(U);
double code(double J, double K, double U) {
	double t_0 = J * (J / U);
	double t_1 = J * (-2.0 * cos((K * 0.5)));
	double tmp;
	if (J <= -2.1e-52) {
		tmp = t_1;
	} else if (J <= -1.1e-129) {
		tmp = U + (2.0 * t_0);
	} else if (J <= -2.3e-165) {
		tmp = t_1;
	} else if (J <= -2.75e-238) {
		tmp = -U;
	} else if (J <= -5e-310) {
		tmp = U;
	} else if (J <= 1.25e-40) {
		tmp = (-2.0 * t_0) - U;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

NOTE: U should be positive before calling this function
real(8) function code(j, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = j * (j / u)
    t_1 = j * ((-2.0d0) * cos((k * 0.5d0)))
    if (j <= (-2.1d-52)) then
        tmp = t_1
    else if (j <= (-1.1d-129)) then
        tmp = u + (2.0d0 * t_0)
    else if (j <= (-2.3d-165)) then
        tmp = t_1
    else if (j <= (-2.75d-238)) then
        tmp = -u
    else if (j <= (-5d-310)) then
        tmp = u
    else if (j <= 1.25d-40) then
        tmp = ((-2.0d0) * t_0) - u
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

U = Math.abs(U);
public static double code(double J, double K, double U) {
	double t_0 = J * (J / U);
	double t_1 = J * (-2.0 * Math.cos((K * 0.5)));
	double tmp;
	if (J <= -2.1e-52) {
		tmp = t_1;
	} else if (J <= -1.1e-129) {
		tmp = U + (2.0 * t_0);
	} else if (J <= -2.3e-165) {
		tmp = t_1;
	} else if (J <= -2.75e-238) {
		tmp = -U;
	} else if (J <= -5e-310) {
		tmp = U;
	} else if (J <= 1.25e-40) {
		tmp = (-2.0 * t_0) - U;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	t_0 = J * (J / U)
	t_1 = J * (-2.0 * math.cos((K * 0.5)))
	tmp = 0
	if J <= -2.1e-52:
		tmp = t_1
	elif J <= -1.1e-129:
		tmp = U + (2.0 * t_0)
	elif J <= -2.3e-165:
		tmp = t_1
	elif J <= -2.75e-238:
		tmp = -U
	elif J <= -5e-310:
		tmp = U
	elif J <= 1.25e-40:
		tmp = (-2.0 * t_0) - U
	else:
		tmp = t_1
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	t_0 = Float64(J * Float64(J / U))
	t_1 = Float64(J * Float64(-2.0 * cos(Float64(K * 0.5))))
	tmp = 0.0
	if (J <= -2.1e-52)
		tmp = t_1;
	elseif (J <= -1.1e-129)
		tmp = Float64(U + Float64(2.0 * t_0));
	elseif (J <= -2.3e-165)
		tmp = t_1;
	elseif (J <= -2.75e-238)
		tmp = Float64(-U);
	elseif (J <= -5e-310)
		tmp = U;
	elseif (J <= 1.25e-40)
		tmp = Float64(Float64(-2.0 * t_0) - U);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	t_0 = J * (J / U);
	t_1 = J * (-2.0 * cos((K * 0.5)));
	tmp = 0.0;
	if (J <= -2.1e-52)
		tmp = t_1;
	elseif (J <= -1.1e-129)
		tmp = U + (2.0 * t_0);
	elseif (J <= -2.3e-165)
		tmp = t_1;
	elseif (J <= -2.75e-238)
		tmp = -U;
	elseif (J <= -5e-310)
		tmp = U;
	elseif (J <= 1.25e-40)
		tmp = (-2.0 * t_0) - U;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := Block[{t$95$0 = N[(J * N[(J / U), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(J * N[(-2.0 * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[J, -2.1e-52], t$95$1, If[LessEqual[J, -1.1e-129], N[(U + N[(2.0 * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[J, -2.3e-165], t$95$1, If[LessEqual[J, -2.75e-238], (-U), If[LessEqual[J, -5e-310], U, If[LessEqual[J, 1.25e-40], N[(N[(-2.0 * t$95$0), $MachinePrecision] - U), $MachinePrecision], t$95$1]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if J < -2.0999999999999999e-52 or -1.10000000000000001e-129 < J < -2.3e-165 or 1.24999999999999991e-40 < J

   1. Initial program 92.0%

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

      1. *-commutative 92.0%

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

      2. associate-*l* 92.0%

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

      3. associate-*r* 92.0%

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

      4. *-commutative 92.0%

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

      5. associate-*l* 91.9%

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

      6. *-commutative 91.9%

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

      7. unpow2 91.9%

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

      8. hypot-1-def 98.7%

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

      9. *-commutative 98.7%

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

      10. associate-*l* 98.7%

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

   3. Simplified 98.7%

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

   4. Taylor expanded in U around 0 74.7%

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

   if -2.0999999999999999e-52 < J < -1.10000000000000001e-129

   1. Initial program 65.2%

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

      1. *-commutative 65.2%

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

      2. associate-*l* 65.2%

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

      3. associate-*r* 65.2%

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

      4. *-commutative 65.2%

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

      5. associate-*l* 65.2%

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

      6. *-commutative 65.2%

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

      7. unpow2 65.2%

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

      8. hypot-1-def 99.7%

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

      9. *-commutative 99.7%

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

      10. associate-*l* 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in U around -inf 46.9%

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

      1. fma-def 46.9%

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

      2. associate-/l* 46.9%

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

      3. unpow2 46.9%

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

   6. Simplified 46.9%

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

   7. Taylor expanded in K around 0 46.9%

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

      1. unpow2 46.9%

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

      2. associate-/l* 46.9%

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

   9. Simplified 46.9%

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

       1. fma-udef 46.9%

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

       2. div-inv 46.9%

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

       3. clear-num 46.9%

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

   11. Applied egg-rr 46.9%

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

   if -2.3e-165 < J < -2.74999999999999997e-238

   1. Initial program 44.5%

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

      1. *-commutative 44.5%

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

      2. associate-*l* 44.5%

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

      3. associate-*r* 44.5%

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

      4. *-commutative 44.5%

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

      5. associate-*l* 44.4%

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

      6. *-commutative 44.4%

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

      7. unpow2 44.4%

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

      8. hypot-1-def 69.8%

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

      9. *-commutative 69.8%

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

      10. associate-*l* 69.8%

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

   3. Simplified 69.8%

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

   4. Taylor expanded in J around 0 38.6%

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

      1. neg-mul-1 38.6%

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

   6. Simplified 38.6%

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

   if -2.74999999999999997e-238 < J < -4.999999999999985e-310

   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. *-commutative 44.4%

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

      2. associate-*l* 44.4%

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

      3. associate-*r* 44.4%

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

      4. *-commutative 44.4%

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

      5. associate-*l* 44.4%

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

      6. *-commutative 44.4%

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

      7. unpow2 44.4%

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

      8. hypot-1-def 68.4%

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

      9. *-commutative 68.4%

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

      10. associate-*l* 68.4%

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

   3. Simplified 68.4%

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

   4. Taylor expanded in U around -inf 51.1%

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

   if -4.999999999999985e-310 < J < 1.24999999999999991e-40

   1. Initial program 48.5%

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

      1. *-commutative 48.5%

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

      2. associate-*l* 48.5%

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

      3. unpow2 48.5%

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

      4. hypot-1-def 72.1%

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

      5. *-commutative 72.1%

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

      6. associate-*l* 72.1%

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

   3. Simplified 72.1%

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

   4. Taylor expanded in K around 0 56.6%

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

   5. Taylor expanded in K around 0 55.5%

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

   6. Taylor expanded in J around 0 43.8%

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

      1. fma-def 43.8%

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

      2. unpow2 43.8%

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

      3. associate-*r/ 43.7%

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

      4. neg-mul-1 43.7%

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

      5. fma-neg 43.7%

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

   8. Simplified 43.7%

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

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -2.1 \cdot 10^{-52}:\\ \;\;\;\;J \cdot \left(-2 \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{elif}\;J \leq -1.1 \cdot 10^{-129}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \frac{J}{U}\right)\\ \mathbf{elif}\;J \leq -2.3 \cdot 10^{-165}:\\ \;\;\;\;J \cdot \left(-2 \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{elif}\;J \leq -2.75 \cdot 10^{-238}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq -5 \cdot 10^{-310}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 1.25 \cdot 10^{-40}:\\ \;\;\;\;-2 \cdot \left(J \cdot \frac{J}{U}\right) - U\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(-2 \cdot \cos \left(K \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 5: 62.9% accurate, 3.7× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} \mathbf{if}\;K \leq 1.75 \cdot 10^{-6}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot 2}\right)\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(-2 \cdot \cos \left(K \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (if (<= K 1.75e-6)
   (* (* -2.0 J) (hypot 1.0 (/ U (* J 2.0))))
   (* J (* -2.0 (cos (* K 0.5))))))

C

U = abs(U);
double code(double J, double K, double U) {
	double tmp;
	if (K <= 1.75e-6) {
		tmp = (-2.0 * J) * hypot(1.0, (U / (J * 2.0)));
	} else {
		tmp = J * (-2.0 * cos((K * 0.5)));
	}
	return tmp;
}

Java

U = Math.abs(U);
public static double code(double J, double K, double U) {
	double tmp;
	if (K <= 1.75e-6) {
		tmp = (-2.0 * J) * Math.hypot(1.0, (U / (J * 2.0)));
	} else {
		tmp = J * (-2.0 * Math.cos((K * 0.5)));
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	tmp = 0
	if K <= 1.75e-6:
		tmp = (-2.0 * J) * math.hypot(1.0, (U / (J * 2.0)))
	else:
		tmp = J * (-2.0 * math.cos((K * 0.5)))
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	tmp = 0.0
	if (K <= 1.75e-6)
		tmp = Float64(Float64(-2.0 * J) * hypot(1.0, Float64(U / Float64(J * 2.0))));
	else
		tmp = Float64(J * Float64(-2.0 * cos(Float64(K * 0.5))));
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	tmp = 0.0;
	if (K <= 1.75e-6)
		tmp = (-2.0 * J) * hypot(1.0, (U / (J * 2.0)));
	else
		tmp = J * (-2.0 * cos((K * 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := If[LessEqual[K, 1.75e-6], N[(N[(-2.0 * J), $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[(U / N[(J * 2.0), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(J * N[(-2.0 * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if K < 1.74999999999999997e-6

   1. Initial program 74.9%

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

      1. *-commutative 74.9%

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

      2. associate-*l* 74.9%

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

      3. unpow2 74.9%

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

      4. hypot-1-def 89.5%

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

      5. *-commutative 89.5%

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

      6. associate-*l* 89.5%

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

   3. Simplified 89.5%

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

   4. Taylor expanded in K around 0 80.9%

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

   5. Taylor expanded in K around 0 72.3%

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

   if 1.74999999999999997e-6 < K

   1. Initial program 74.1%

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

      1. *-commutative 74.1%

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

      2. associate-*l* 74.1%

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

      3. associate-*r* 74.1%

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

      4. *-commutative 74.1%

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

      5. associate-*l* 74.1%

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

      6. *-commutative 74.1%

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

      7. unpow2 74.1%

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

      8. hypot-1-def 87.0%

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

      9. *-commutative 87.0%

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

      10. associate-*l* 87.0%

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

   3. Simplified 87.0%

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

   4. Taylor expanded in U around 0 50.2%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;K \leq 1.75 \cdot 10^{-6}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot 2}\right)\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(-2 \cdot \cos \left(K \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 6: 48.7% accurate, 27.7× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} \mathbf{if}\;J \leq -1.15 \cdot 10^{-38}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;J \leq 8 \cdot 10^{-309}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 7.8 \cdot 10^{+111}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(-2 + 0.25 \cdot \left(K \cdot K\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (if (<= J -1.15e-38)
   (* -2.0 J)
   (if (<= J 8e-309)
     U
     (if (<= J 7.8e+111) (- U) (* J (+ -2.0 (* 0.25 (* K K))))))))

C

U = abs(U);
double code(double J, double K, double U) {
	double tmp;
	if (J <= -1.15e-38) {
		tmp = -2.0 * J;
	} else if (J <= 8e-309) {
		tmp = U;
	} else if (J <= 7.8e+111) {
		tmp = -U;
	} else {
		tmp = J * (-2.0 + (0.25 * (K * K)));
	}
	return tmp;
}

Fortran

NOTE: U should be positive before calling this function
real(8) function code(j, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (j <= (-1.15d-38)) then
        tmp = (-2.0d0) * j
    else if (j <= 8d-309) then
        tmp = u
    else if (j <= 7.8d+111) then
        tmp = -u
    else
        tmp = j * ((-2.0d0) + (0.25d0 * (k * k)))
    end if
    code = tmp
end function

Java

U = Math.abs(U);
public static double code(double J, double K, double U) {
	double tmp;
	if (J <= -1.15e-38) {
		tmp = -2.0 * J;
	} else if (J <= 8e-309) {
		tmp = U;
	} else if (J <= 7.8e+111) {
		tmp = -U;
	} else {
		tmp = J * (-2.0 + (0.25 * (K * K)));
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	tmp = 0
	if J <= -1.15e-38:
		tmp = -2.0 * J
	elif J <= 8e-309:
		tmp = U
	elif J <= 7.8e+111:
		tmp = -U
	else:
		tmp = J * (-2.0 + (0.25 * (K * K)))
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	tmp = 0.0
	if (J <= -1.15e-38)
		tmp = Float64(-2.0 * J);
	elseif (J <= 8e-309)
		tmp = U;
	elseif (J <= 7.8e+111)
		tmp = Float64(-U);
	else
		tmp = Float64(J * Float64(-2.0 + Float64(0.25 * Float64(K * K))));
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	tmp = 0.0;
	if (J <= -1.15e-38)
		tmp = -2.0 * J;
	elseif (J <= 8e-309)
		tmp = U;
	elseif (J <= 7.8e+111)
		tmp = -U;
	else
		tmp = J * (-2.0 + (0.25 * (K * K)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := If[LessEqual[J, -1.15e-38], N[(-2.0 * J), $MachinePrecision], If[LessEqual[J, 8e-309], U, If[LessEqual[J, 7.8e+111], (-U), N[(J * N[(-2.0 + N[(0.25 * N[(K * K), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if J < -1.15000000000000001e-38

   1. Initial program 93.0%

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

      1. *-commutative 93.0%

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

      2. associate-*l* 93.0%

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

      3. associate-*r* 93.0%

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

      4. *-commutative 93.0%

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

      5. associate-*l* 93.0%

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

      6. *-commutative 93.0%

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

      7. unpow2 93.0%

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

      8. hypot-1-def 99.8%

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

      9. *-commutative 99.8%

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

      10. associate-*l* 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in U around 0 80.6%

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

   5. Taylor expanded in K around 0 49.8%

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

   if -1.15000000000000001e-38 < J < 8.0000000000000003e-309

   1. Initial program 59.3%

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

      1. *-commutative 59.3%

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

      2. associate-*l* 59.3%

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

      3. associate-*r* 59.3%

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

      4. *-commutative 59.3%

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

      5. associate-*l* 59.3%

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

      6. *-commutative 59.3%

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

      7. unpow2 59.3%

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

      8. hypot-1-def 81.3%

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

      9. *-commutative 81.3%

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

      10. associate-*l* 81.3%

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

   3. Simplified 81.3%

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

   4. Taylor expanded in U around -inf 37.7%

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

   if 8.0000000000000003e-309 < J < 7.79999999999999958e111

   1. Initial program 63.7%

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

      1. *-commutative 63.7%

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

      2. associate-*l* 63.7%

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

      3. associate-*r* 63.7%

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

      4. *-commutative 63.7%

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

      5. associate-*l* 63.7%

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

      6. *-commutative 63.7%

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

      7. unpow2 63.7%

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

      8. hypot-1-def 82.4%

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

      9. *-commutative 82.4%

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

      10. associate-*l* 82.4%

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

   3. Simplified 82.4%

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

   4. Taylor expanded in J around 0 34.8%

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

      1. neg-mul-1 34.8%

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

   6. Simplified 34.8%

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

   if 7.79999999999999958e111 < J

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-*l* 99.9%

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

      3. associate-*r* 99.9%

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

      4. *-commutative 99.9%

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

      5. associate-*l* 99.9%

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

      6. *-commutative 99.9%

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

      7. unpow2 99.9%

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

      8. hypot-1-def 99.9%

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

      9. *-commutative 99.9%

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

      10. associate-*l* 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in U around 0 94.5%

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

   5. Taylor expanded in K around 0 63.1%

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

      1. associate-*r* 63.1%

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

      2. distribute-rgt-out 63.1%

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

      3. unpow2 63.1%

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

   7. Simplified 63.1%

      \[\leadsto \color{blue}{J \cdot \left(0.25 \cdot \left(K \cdot K\right) + -2\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 42.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -1.15 \cdot 10^{-38}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;J \leq 8 \cdot 10^{-309}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 7.8 \cdot 10^{+111}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(-2 + 0.25 \cdot \left(K \cdot K\right)\right)\\ \end{array} \]

Alternative 7: 48.8% accurate, 27.7× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} \mathbf{if}\;J \leq -2.15 \cdot 10^{-38}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;J \leq -5 \cdot 10^{-310}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \frac{J}{U}\right)\\ \mathbf{elif}\;J \leq 6.8 \cdot 10^{+111}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(-2 + 0.25 \cdot \left(K \cdot K\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (if (<= J -2.15e-38)
   (* -2.0 J)
   (if (<= J -5e-310)
     (+ U (* 2.0 (* J (/ J U))))
     (if (<= J 6.8e+111) (- U) (* J (+ -2.0 (* 0.25 (* K K))))))))

C

U = abs(U);
double code(double J, double K, double U) {
	double tmp;
	if (J <= -2.15e-38) {
		tmp = -2.0 * J;
	} else if (J <= -5e-310) {
		tmp = U + (2.0 * (J * (J / U)));
	} else if (J <= 6.8e+111) {
		tmp = -U;
	} else {
		tmp = J * (-2.0 + (0.25 * (K * K)));
	}
	return tmp;
}

Fortran

NOTE: U should be positive before calling this function
real(8) function code(j, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (j <= (-2.15d-38)) then
        tmp = (-2.0d0) * j
    else if (j <= (-5d-310)) then
        tmp = u + (2.0d0 * (j * (j / u)))
    else if (j <= 6.8d+111) then
        tmp = -u
    else
        tmp = j * ((-2.0d0) + (0.25d0 * (k * k)))
    end if
    code = tmp
end function

Java

U = Math.abs(U);
public static double code(double J, double K, double U) {
	double tmp;
	if (J <= -2.15e-38) {
		tmp = -2.0 * J;
	} else if (J <= -5e-310) {
		tmp = U + (2.0 * (J * (J / U)));
	} else if (J <= 6.8e+111) {
		tmp = -U;
	} else {
		tmp = J * (-2.0 + (0.25 * (K * K)));
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	tmp = 0
	if J <= -2.15e-38:
		tmp = -2.0 * J
	elif J <= -5e-310:
		tmp = U + (2.0 * (J * (J / U)))
	elif J <= 6.8e+111:
		tmp = -U
	else:
		tmp = J * (-2.0 + (0.25 * (K * K)))
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	tmp = 0.0
	if (J <= -2.15e-38)
		tmp = Float64(-2.0 * J);
	elseif (J <= -5e-310)
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(J / U))));
	elseif (J <= 6.8e+111)
		tmp = Float64(-U);
	else
		tmp = Float64(J * Float64(-2.0 + Float64(0.25 * Float64(K * K))));
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	tmp = 0.0;
	if (J <= -2.15e-38)
		tmp = -2.0 * J;
	elseif (J <= -5e-310)
		tmp = U + (2.0 * (J * (J / U)));
	elseif (J <= 6.8e+111)
		tmp = -U;
	else
		tmp = J * (-2.0 + (0.25 * (K * K)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := If[LessEqual[J, -2.15e-38], N[(-2.0 * J), $MachinePrecision], If[LessEqual[J, -5e-310], N[(U + N[(2.0 * N[(J * N[(J / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[J, 6.8e+111], (-U), N[(J * N[(-2.0 + N[(0.25 * N[(K * K), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if J < -2.1500000000000001e-38

   1. Initial program 93.0%

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

      1. *-commutative 93.0%

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

      2. associate-*l* 93.0%

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

      3. associate-*r* 93.0%

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

      4. *-commutative 93.0%

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

      5. associate-*l* 93.0%

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

      6. *-commutative 93.0%

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

      7. unpow2 93.0%

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

      8. hypot-1-def 99.8%

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

      9. *-commutative 99.8%

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

      10. associate-*l* 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in U around 0 80.6%

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

   5. Taylor expanded in K around 0 49.8%

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

   if -2.1500000000000001e-38 < J < -4.999999999999985e-310

   1. Initial program 59.3%

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

      1. *-commutative 59.3%

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

      2. associate-*l* 59.3%

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

      3. associate-*r* 59.3%

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

      4. *-commutative 59.3%

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

      5. associate-*l* 59.3%

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

      6. *-commutative 59.3%

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

      7. unpow2 59.3%

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

      8. hypot-1-def 81.3%

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

      9. *-commutative 81.3%

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

      10. associate-*l* 81.3%

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

   3. Simplified 81.3%

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

   4. Taylor expanded in U around -inf 38.0%

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

      1. fma-def 38.0%

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

      2. associate-/l* 38.0%

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

      3. unpow2 38.0%

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

   6. Simplified 38.0%

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

   7. Taylor expanded in K around 0 38.0%

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

      1. unpow2 38.0%

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

      2. associate-/l* 37.9%

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

   9. Simplified 37.9%

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

       1. fma-udef 37.9%

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

       2. div-inv 37.9%

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

       3. clear-num 37.9%

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

   11. Applied egg-rr 37.9%

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

   if -4.999999999999985e-310 < J < 6.8000000000000003e111

   1. Initial program 63.7%

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

      1. *-commutative 63.7%

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

      2. associate-*l* 63.7%

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

      3. associate-*r* 63.7%

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

      4. *-commutative 63.7%

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

      5. associate-*l* 63.7%

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

      6. *-commutative 63.7%

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

      7. unpow2 63.7%

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

      8. hypot-1-def 82.4%

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

      9. *-commutative 82.4%

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

      10. associate-*l* 82.4%

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

   3. Simplified 82.4%

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

   4. Taylor expanded in J around 0 34.8%

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

      1. neg-mul-1 34.8%

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

   6. Simplified 34.8%

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

   if 6.8000000000000003e111 < J

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-*l* 99.9%

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

      3. associate-*r* 99.9%

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

      4. *-commutative 99.9%

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

      5. associate-*l* 99.9%

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

      6. *-commutative 99.9%

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

      7. unpow2 99.9%

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

      8. hypot-1-def 99.9%

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

      9. *-commutative 99.9%

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

      10. associate-*l* 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in U around 0 94.5%

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

   5. Taylor expanded in K around 0 63.1%

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

      1. associate-*r* 63.1%

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

      2. distribute-rgt-out 63.1%

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

      3. unpow2 63.1%

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

   7. Simplified 63.1%

      \[\leadsto \color{blue}{J \cdot \left(0.25 \cdot \left(K \cdot K\right) + -2\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 42.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -2.15 \cdot 10^{-38}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;J \leq -5 \cdot 10^{-310}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \frac{J}{U}\right)\\ \mathbf{elif}\;J \leq 6.8 \cdot 10^{+111}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(-2 + 0.25 \cdot \left(K \cdot K\right)\right)\\ \end{array} \]

Alternative 8: 48.8% accurate, 27.7× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} t_0 := J \cdot \frac{J}{U}\\ \mathbf{if}\;J \leq -1.25 \cdot 10^{-38}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;J \leq -5 \cdot 10^{-310}:\\ \;\;\;\;U + 2 \cdot t_0\\ \mathbf{elif}\;J \leq 7.2 \cdot 10^{+111}:\\ \;\;\;\;-2 \cdot t_0 - U\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(-2 + 0.25 \cdot \left(K \cdot K\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (* J (/ J U))))
   (if (<= J -1.25e-38)
     (* -2.0 J)
     (if (<= J -5e-310)
       (+ U (* 2.0 t_0))
       (if (<= J 7.2e+111)
         (- (* -2.0 t_0) U)
         (* J (+ -2.0 (* 0.25 (* K K)))))))))

C

U = abs(U);
double code(double J, double K, double U) {
	double t_0 = J * (J / U);
	double tmp;
	if (J <= -1.25e-38) {
		tmp = -2.0 * J;
	} else if (J <= -5e-310) {
		tmp = U + (2.0 * t_0);
	} else if (J <= 7.2e+111) {
		tmp = (-2.0 * t_0) - U;
	} else {
		tmp = J * (-2.0 + (0.25 * (K * K)));
	}
	return tmp;
}

Fortran

NOTE: U should be positive before calling this function
real(8) function code(j, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = j * (j / u)
    if (j <= (-1.25d-38)) then
        tmp = (-2.0d0) * j
    else if (j <= (-5d-310)) then
        tmp = u + (2.0d0 * t_0)
    else if (j <= 7.2d+111) then
        tmp = ((-2.0d0) * t_0) - u
    else
        tmp = j * ((-2.0d0) + (0.25d0 * (k * k)))
    end if
    code = tmp
end function

Java

U = Math.abs(U);
public static double code(double J, double K, double U) {
	double t_0 = J * (J / U);
	double tmp;
	if (J <= -1.25e-38) {
		tmp = -2.0 * J;
	} else if (J <= -5e-310) {
		tmp = U + (2.0 * t_0);
	} else if (J <= 7.2e+111) {
		tmp = (-2.0 * t_0) - U;
	} else {
		tmp = J * (-2.0 + (0.25 * (K * K)));
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	t_0 = J * (J / U)
	tmp = 0
	if J <= -1.25e-38:
		tmp = -2.0 * J
	elif J <= -5e-310:
		tmp = U + (2.0 * t_0)
	elif J <= 7.2e+111:
		tmp = (-2.0 * t_0) - U
	else:
		tmp = J * (-2.0 + (0.25 * (K * K)))
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	t_0 = Float64(J * Float64(J / U))
	tmp = 0.0
	if (J <= -1.25e-38)
		tmp = Float64(-2.0 * J);
	elseif (J <= -5e-310)
		tmp = Float64(U + Float64(2.0 * t_0));
	elseif (J <= 7.2e+111)
		tmp = Float64(Float64(-2.0 * t_0) - U);
	else
		tmp = Float64(J * Float64(-2.0 + Float64(0.25 * Float64(K * K))));
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	t_0 = J * (J / U);
	tmp = 0.0;
	if (J <= -1.25e-38)
		tmp = -2.0 * J;
	elseif (J <= -5e-310)
		tmp = U + (2.0 * t_0);
	elseif (J <= 7.2e+111)
		tmp = (-2.0 * t_0) - U;
	else
		tmp = J * (-2.0 + (0.25 * (K * K)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := Block[{t$95$0 = N[(J * N[(J / U), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[J, -1.25e-38], N[(-2.0 * J), $MachinePrecision], If[LessEqual[J, -5e-310], N[(U + N[(2.0 * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[J, 7.2e+111], N[(N[(-2.0 * t$95$0), $MachinePrecision] - U), $MachinePrecision], N[(J * N[(-2.0 + N[(0.25 * N[(K * K), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if J < -1.25000000000000008e-38

   1. Initial program 93.0%

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

      1. *-commutative 93.0%

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

      2. associate-*l* 93.0%

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

      3. associate-*r* 93.0%

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

      4. *-commutative 93.0%

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

      5. associate-*l* 93.0%

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

      6. *-commutative 93.0%

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

      7. unpow2 93.0%

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

      8. hypot-1-def 99.8%

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

      9. *-commutative 99.8%

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

      10. associate-*l* 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in U around 0 80.6%

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

   5. Taylor expanded in K around 0 49.8%

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

   if -1.25000000000000008e-38 < J < -4.999999999999985e-310

   1. Initial program 59.3%

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

      1. *-commutative 59.3%

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

      2. associate-*l* 59.3%

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

      3. associate-*r* 59.3%

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

      4. *-commutative 59.3%

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

      5. associate-*l* 59.3%

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

      6. *-commutative 59.3%

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

      7. unpow2 59.3%

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

      8. hypot-1-def 81.3%

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

      9. *-commutative 81.3%

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

      10. associate-*l* 81.3%

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

   3. Simplified 81.3%

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

   4. Taylor expanded in U around -inf 38.0%

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

      1. fma-def 38.0%

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

      2. associate-/l* 38.0%

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

      3. unpow2 38.0%

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

   6. Simplified 38.0%

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

   7. Taylor expanded in K around 0 38.0%

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

      1. unpow2 38.0%

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

      2. associate-/l* 37.9%

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

   9. Simplified 37.9%

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

       1. fma-udef 37.9%

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

       2. div-inv 37.9%

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

       3. clear-num 37.9%

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

   11. Applied egg-rr 37.9%

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

   if -4.999999999999985e-310 < J < 7.2000000000000004e111

   1. Initial program 63.7%

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

      1. *-commutative 63.7%

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

      2. associate-*l* 63.7%

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

      3. unpow2 63.7%

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

      4. hypot-1-def 82.4%

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

      5. *-commutative 82.4%

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

      6. associate-*l* 82.4%

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

   3. Simplified 82.4%

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

   4. Taylor expanded in K around 0 65.1%

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

   5. Taylor expanded in K around 0 58.4%

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

   6. Taylor expanded in J around 0 34.7%

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

      1. fma-def 34.7%

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

      2. unpow2 34.7%

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

      3. associate-*r/ 34.6%

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

      4. neg-mul-1 34.6%

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

      5. fma-neg 34.6%

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

   8. Simplified 34.6%

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

   if 7.2000000000000004e111 < J

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-*l* 99.9%

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

      3. associate-*r* 99.9%

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

      4. *-commutative 99.9%

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

      5. associate-*l* 99.9%

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

      6. *-commutative 99.9%

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

      7. unpow2 99.9%

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

      8. hypot-1-def 99.9%

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

      9. *-commutative 99.9%

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

      10. associate-*l* 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in U around 0 94.5%

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

   5. Taylor expanded in K around 0 63.1%

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

      1. associate-*r* 63.1%

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

      2. distribute-rgt-out 63.1%

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

      3. unpow2 63.1%

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

   7. Simplified 63.1%

      \[\leadsto \color{blue}{J \cdot \left(0.25 \cdot \left(K \cdot K\right) + -2\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 42.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -1.25 \cdot 10^{-38}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;J \leq -5 \cdot 10^{-310}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \frac{J}{U}\right)\\ \mathbf{elif}\;J \leq 7.2 \cdot 10^{+111}:\\ \;\;\;\;-2 \cdot \left(J \cdot \frac{J}{U}\right) - U\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(-2 + 0.25 \cdot \left(K \cdot K\right)\right)\\ \end{array} \]

Alternative 9: 49.1% accurate, 45.7× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} \mathbf{if}\;J \leq -4 \cdot 10^{-40}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;J \leq -1.3 \cdot 10^{-305}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 1.05 \cdot 10^{+110}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot J\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (if (<= J -4e-40)
   (* -2.0 J)
   (if (<= J -1.3e-305) U (if (<= J 1.05e+110) (- U) (* -2.0 J)))))

C

U = abs(U);
double code(double J, double K, double U) {
	double tmp;
	if (J <= -4e-40) {
		tmp = -2.0 * J;
	} else if (J <= -1.3e-305) {
		tmp = U;
	} else if (J <= 1.05e+110) {
		tmp = -U;
	} else {
		tmp = -2.0 * J;
	}
	return tmp;
}

Fortran

NOTE: U should be positive before calling this function
real(8) function code(j, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (j <= (-4d-40)) then
        tmp = (-2.0d0) * j
    else if (j <= (-1.3d-305)) then
        tmp = u
    else if (j <= 1.05d+110) then
        tmp = -u
    else
        tmp = (-2.0d0) * j
    end if
    code = tmp
end function

Java

U = Math.abs(U);
public static double code(double J, double K, double U) {
	double tmp;
	if (J <= -4e-40) {
		tmp = -2.0 * J;
	} else if (J <= -1.3e-305) {
		tmp = U;
	} else if (J <= 1.05e+110) {
		tmp = -U;
	} else {
		tmp = -2.0 * J;
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	tmp = 0
	if J <= -4e-40:
		tmp = -2.0 * J
	elif J <= -1.3e-305:
		tmp = U
	elif J <= 1.05e+110:
		tmp = -U
	else:
		tmp = -2.0 * J
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	tmp = 0.0
	if (J <= -4e-40)
		tmp = Float64(-2.0 * J);
	elseif (J <= -1.3e-305)
		tmp = U;
	elseif (J <= 1.05e+110)
		tmp = Float64(-U);
	else
		tmp = Float64(-2.0 * J);
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	tmp = 0.0;
	if (J <= -4e-40)
		tmp = -2.0 * J;
	elseif (J <= -1.3e-305)
		tmp = U;
	elseif (J <= 1.05e+110)
		tmp = -U;
	else
		tmp = -2.0 * J;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := If[LessEqual[J, -4e-40], N[(-2.0 * J), $MachinePrecision], If[LessEqual[J, -1.3e-305], U, If[LessEqual[J, 1.05e+110], (-U), N[(-2.0 * J), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if J < -3.9999999999999997e-40 or 1.05000000000000007e110 < J

   1. Initial program 94.3%

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

      1. *-commutative 94.3%

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

      2. associate-*l* 94.3%

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

      3. associate-*r* 94.3%

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

      4. *-commutative 94.3%

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

      5. associate-*l* 94.3%

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

      6. *-commutative 94.3%

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

      7. unpow2 94.3%

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

      8. hypot-1-def 99.9%

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

      9. *-commutative 99.9%

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

      10. associate-*l* 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in U around 0 84.2%

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

   5. Taylor expanded in K around 0 53.7%

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

   if -3.9999999999999997e-40 < J < -1.3000000000000001e-305

   1. Initial program 59.3%

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

      1. *-commutative 59.3%

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

      2. associate-*l* 59.3%

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

      3. associate-*r* 59.3%

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

      4. *-commutative 59.3%

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

      5. associate-*l* 59.3%

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

      6. *-commutative 59.3%

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

      7. unpow2 59.3%

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

      8. hypot-1-def 81.3%

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

      9. *-commutative 81.3%

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

      10. associate-*l* 81.3%

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

   3. Simplified 81.3%

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

   4. Taylor expanded in U around -inf 37.7%

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

   if -1.3000000000000001e-305 < J < 1.05000000000000007e110

   1. Initial program 63.9%

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

      1. *-commutative 63.9%

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

      2. associate-*l* 63.9%

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

      3. associate-*r* 63.9%

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

      4. *-commutative 63.9%

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

      5. associate-*l* 63.9%

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

      6. *-commutative 63.9%

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

      7. unpow2 63.9%

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

      8. hypot-1-def 82.0%

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

      9. *-commutative 82.0%

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

      10. associate-*l* 82.0%

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

   3. Simplified 82.0%

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

   4. Taylor expanded in J around 0 34.4%

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

      1. neg-mul-1 34.4%

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

   6. Simplified 34.4%

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

4. Final simplification 42.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -4 \cdot 10^{-40}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;J \leq -1.3 \cdot 10^{-305}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 1.05 \cdot 10^{+110}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot J\\ \end{array} \]

Alternative 10: 39.3% accurate, 103.4× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} \mathbf{if}\;J \leq -1.3 \cdot 10^{-305}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U) :precision binary64 (if (<= J -1.3e-305) U (- U)))

C

U = abs(U);
double code(double J, double K, double U) {
	double tmp;
	if (J <= -1.3e-305) {
		tmp = U;
	} else {
		tmp = -U;
	}
	return tmp;
}

Fortran

NOTE: U should be positive before calling this function
real(8) function code(j, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (j <= (-1.3d-305)) then
        tmp = u
    else
        tmp = -u
    end if
    code = tmp
end function

Java

U = Math.abs(U);
public static double code(double J, double K, double U) {
	double tmp;
	if (J <= -1.3e-305) {
		tmp = U;
	} else {
		tmp = -U;
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	tmp = 0
	if J <= -1.3e-305:
		tmp = U
	else:
		tmp = -U
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	tmp = 0.0
	if (J <= -1.3e-305)
		tmp = U;
	else
		tmp = Float64(-U);
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	tmp = 0.0;
	if (J <= -1.3e-305)
		tmp = U;
	else
		tmp = -U;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := If[LessEqual[J, -1.3e-305], U, (-U)]

Derivation

1. Split input into 2 regimes

2. if J < -1.3000000000000001e-305

   1. Initial program 77.1%

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

      1. *-commutative 77.1%

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

      2. associate-*l* 77.1%

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

      3. associate-*r* 77.1%

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

      4. *-commutative 77.1%

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

      5. associate-*l* 77.0%

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

      6. *-commutative 77.0%

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

      7. unpow2 77.0%

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

      8. hypot-1-def 91.1%

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

      9. *-commutative 91.1%

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

      10. associate-*l* 91.1%

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

   3. Simplified 91.1%

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

   4. Taylor expanded in U around -inf 23.9%

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

   if -1.3000000000000001e-305 < J

   1. Initial program 72.3%

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

      1. *-commutative 72.3%

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

      2. associate-*l* 72.3%

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

      3. associate-*r* 72.3%

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

      4. *-commutative 72.3%

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

      5. associate-*l* 72.3%

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

      6. *-commutative 72.3%

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

      7. unpow2 72.3%

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

      8. hypot-1-def 86.5%

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

      9. *-commutative 86.5%

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

      10. associate-*l* 86.5%

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

   3. Simplified 86.5%

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

   4. Taylor expanded in J around 0 27.3%

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

      1. neg-mul-1 27.3%

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

   6. Simplified 27.3%

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

4. Final simplification 25.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -1.3 \cdot 10^{-305}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]

Alternative 11: 26.4% accurate, 420.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.7%

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

   1. *-commutative 74.7%

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

   2. associate-*l* 74.7%

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

   3. associate-*r* 74.7%

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

   4. *-commutative 74.7%

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

   5. associate-*l* 74.7%

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

   6. *-commutative 74.7%

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

   7. unpow2 74.7%

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

   8. hypot-1-def 88.8%

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

   9. *-commutative 88.8%

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

   10. associate-*l* 88.8%

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

3. Simplified 88.8%

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

4. Taylor expanded in U around -inf 27.5%

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

5. Final simplification 27.5%

   \[\leadsto U \]

Reproduce

herbie shell --seed 2023240 
(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)))))