Maksimov and Kolovsky, Equation (3)

Percentage Accurate: 73.6% → 99.1%

Time: 17.8s

Alternatives: 8

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 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: 99.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := J \cdot t_0\\ t_2 := \left(\left(-2 \cdot J\right) \cdot t_0\right) \cdot \sqrt{1 + {\left(\frac{U}{t_0 \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;-2 \cdot \left(U \cdot 0.5\right)\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{+297}:\\ \;\;\;\;-2 \cdot \left(t_1 \cdot \mathsf{hypot}\left(1, \frac{U}{2 \cdot t_1}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1 (* J t_0))
        (t_2
         (*
          (* (* -2.0 J) t_0)
          (sqrt (+ 1.0 (pow (/ U (* t_0 (* J 2.0))) 2.0))))))
   (if (<= t_2 (- INFINITY))
     (* -2.0 (* U 0.5))
     (if (<= t_2 2e+297)
       (* -2.0 (* t_1 (hypot 1.0 (/ U (* 2.0 t_1)))))
       (* -2.0 (* U -0.5))))))

C

U = abs(U);
double code(double J, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = J * t_0;
	double t_2 = ((-2.0 * J) * t_0) * sqrt((1.0 + pow((U / (t_0 * (J * 2.0))), 2.0)));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = -2.0 * (U * 0.5);
	} else if (t_2 <= 2e+297) {
		tmp = -2.0 * (t_1 * hypot(1.0, (U / (2.0 * t_1))));
	} else {
		tmp = -2.0 * (U * -0.5);
	}
	return tmp;
}

Java

U = Math.abs(U);
public static double code(double J, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = J * t_0;
	double t_2 = ((-2.0 * J) * t_0) * Math.sqrt((1.0 + Math.pow((U / (t_0 * (J * 2.0))), 2.0)));
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = -2.0 * (U * 0.5);
	} else if (t_2 <= 2e+297) {
		tmp = -2.0 * (t_1 * Math.hypot(1.0, (U / (2.0 * t_1))));
	} else {
		tmp = -2.0 * (U * -0.5);
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	t_0 = math.cos((K / 2.0))
	t_1 = J * t_0
	t_2 = ((-2.0 * J) * t_0) * math.sqrt((1.0 + math.pow((U / (t_0 * (J * 2.0))), 2.0)))
	tmp = 0
	if t_2 <= -math.inf:
		tmp = -2.0 * (U * 0.5)
	elif t_2 <= 2e+297:
		tmp = -2.0 * (t_1 * math.hypot(1.0, (U / (2.0 * t_1))))
	else:
		tmp = -2.0 * (U * -0.5)
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(J * t_0)
	t_2 = Float64(Float64(Float64(-2.0 * J) * t_0) * sqrt(Float64(1.0 + (Float64(U / Float64(t_0 * Float64(J * 2.0))) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(-2.0 * Float64(U * 0.5));
	elseif (t_2 <= 2e+297)
		tmp = Float64(-2.0 * Float64(t_1 * hypot(1.0, Float64(U / Float64(2.0 * t_1)))));
	else
		tmp = Float64(-2.0 * Float64(U * -0.5));
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	t_0 = cos((K / 2.0));
	t_1 = J * t_0;
	t_2 = ((-2.0 * J) * t_0) * sqrt((1.0 + ((U / (t_0 * (J * 2.0))) ^ 2.0)));
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = -2.0 * (U * 0.5);
	elseif (t_2 <= 2e+297)
		tmp = -2.0 * (t_1 * hypot(1.0, (U / (2.0 * t_1))));
	else
		tmp = -2.0 * (U * -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(J * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U / N[(t$95$0 * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(-2.0 * N[(U * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+297], N[(-2.0 * N[(t$95$1 * N[Sqrt[1.0 ^ 2 + N[(U / N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(U * -0.5), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 5.1%

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

   3. Simplified 60.1%

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

   4. Taylor expanded in J around 0 58.1%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 -2 J) (cos.f64 (/.f64 K 2))) (sqrt.f64 (+.f64 1 (pow.f64 (/.f64 U (*.f64 (*.f64 2 J) (cos.f64 (/.f64 K 2)))) 2)))) < 2e297

   1. Initial program 99.8%

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

      1. associate-*l* 99.8%

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

      2. associate-*l* 99.8%

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

      3. unpow2 99.8%

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

      4. sqr-neg 99.8%

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

      5. distribute-frac-neg 99.8%

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

      6. distribute-frac-neg 99.8%

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

      7. unpow2 99.8%

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

   3. Simplified 99.8%

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

   if 2e297 < (*.f64 (*.f64 (*.f64 -2 J) (cos.f64 (/.f64 K 2))) (sqrt.f64 (+.f64 1 (pow.f64 (/.f64 U (*.f64 (*.f64 2 J) (cos.f64 (/.f64 K 2)))) 2))))

   1. Initial program 5.6%

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

   3. Simplified 52.7%

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

   4. Taylor expanded in U around -inf 32.7%

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

      1. *-commutative 32.7%

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

   6. Simplified 32.7%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -\infty:\\ \;\;\;\;-2 \cdot \left(U \cdot 0.5\right)\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 2 \cdot 10^{+297}:\\ \;\;\;\;-2 \cdot \left(\left(J \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5\right)\\ \end{array} \]

Alternative 2: 88.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := -2 \cdot \left(J \cdot \left(t_0 \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot t_0\right)}\right)\right)\right)\\ \mathbf{if}\;J \leq -1.9 \cdot 10^{-217}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;J \leq 7 \cdot 10^{-285}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5\right)\\ \mathbf{elif}\;J \leq 1.36 \cdot 10^{-220}:\\ \;\;\;\;-2 \cdot \left(U \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1 (* -2.0 (* J (* t_0 (hypot 1.0 (/ U (* J (* 2.0 t_0)))))))))
   (if (<= J -1.9e-217)
     t_1
     (if (<= J 7e-285)
       (* -2.0 (* U -0.5))
       (if (<= J 1.36e-220) (* -2.0 (* U 0.5)) t_1)))))

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 * hypot(1.0, (U / (J * (2.0 * t_0))))));
	double tmp;
	if (J <= -1.9e-217) {
		tmp = t_1;
	} else if (J <= 7e-285) {
		tmp = -2.0 * (U * -0.5);
	} else if (J <= 1.36e-220) {
		tmp = -2.0 * (U * 0.5);
	} else {
		tmp = t_1;
	}
	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.hypot(1.0, (U / (J * (2.0 * t_0))))));
	double tmp;
	if (J <= -1.9e-217) {
		tmp = t_1;
	} else if (J <= 7e-285) {
		tmp = -2.0 * (U * -0.5);
	} else if (J <= 1.36e-220) {
		tmp = -2.0 * (U * 0.5);
	} else {
		tmp = t_1;
	}
	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.hypot(1.0, (U / (J * (2.0 * t_0))))))
	tmp = 0
	if J <= -1.9e-217:
		tmp = t_1
	elif J <= 7e-285:
		tmp = -2.0 * (U * -0.5)
	elif J <= 1.36e-220:
		tmp = -2.0 * (U * 0.5)
	else:
		tmp = t_1
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(-2.0 * Float64(J * Float64(t_0 * hypot(1.0, Float64(U / Float64(J * Float64(2.0 * t_0)))))))
	tmp = 0.0
	if (J <= -1.9e-217)
		tmp = t_1;
	elseif (J <= 7e-285)
		tmp = Float64(-2.0 * Float64(U * -0.5));
	elseif (J <= 1.36e-220)
		tmp = Float64(-2.0 * Float64(U * 0.5));
	else
		tmp = t_1;
	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 * hypot(1.0, (U / (J * (2.0 * t_0))))));
	tmp = 0.0;
	if (J <= -1.9e-217)
		tmp = t_1;
	elseif (J <= 7e-285)
		tmp = -2.0 * (U * -0.5);
	elseif (J <= 1.36e-220)
		tmp = -2.0 * (U * 0.5);
	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[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(-2.0 * N[(J * N[(t$95$0 * N[Sqrt[1.0 ^ 2 + N[(U / N[(J * N[(2.0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[J, -1.9e-217], t$95$1, If[LessEqual[J, 7e-285], N[(-2.0 * N[(U * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[J, 1.36e-220], N[(-2.0 * N[(U * 0.5), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if J < -1.89999999999999993e-217 or 1.3600000000000001e-220 < J

   1. Initial program 83.7%

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

   3. Simplified 94.9%

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

   if -1.89999999999999993e-217 < J < 7.0000000000000007e-285

   1. Initial program 34.7%

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

   3. Simplified 44.7%

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

   4. Taylor expanded in U around -inf 32.7%

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

      1. *-commutative 32.7%

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

   6. Simplified 32.7%

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

   if 7.0000000000000007e-285 < J < 1.3600000000000001e-220

   1. Initial program 23.3%

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

   3. Simplified 52.2%

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

   4. Taylor expanded in J around 0 60.4%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -1.9 \cdot 10^{-217}:\\ \;\;\;\;-2 \cdot \left(J \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right)\right)\right)\\ \mathbf{elif}\;J \leq 7 \cdot 10^{-285}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5\right)\\ \mathbf{elif}\;J \leq 1.36 \cdot 10^{-220}:\\ \;\;\;\;-2 \cdot \left(U \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(J \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right)\right)\right)\\ \end{array} \]

Alternative 3: 77.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} t_0 := -2 \cdot \left(\left(J \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot 2}\right)\right)\\ \mathbf{if}\;J \leq -6.4 \cdot 10^{-152}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq 7 \cdot 10^{-285}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5 - \frac{J}{\frac{U}{J}}\right)\\ \mathbf{elif}\;J \leq 5.8 \cdot 10^{-179}:\\ \;\;\;\;-2 \cdot \left(U \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (* -2.0 (* (* J (cos (/ K 2.0))) (hypot 1.0 (/ U (* J 2.0)))))))
   (if (<= J -6.4e-152)
     t_0
     (if (<= J 7e-285)
       (* -2.0 (- (* U -0.5) (/ J (/ U J))))
       (if (<= J 5.8e-179) (* -2.0 (* U 0.5)) t_0)))))

C

U = abs(U);
double code(double J, double K, double U) {
	double t_0 = -2.0 * ((J * cos((K / 2.0))) * hypot(1.0, (U / (J * 2.0))));
	double tmp;
	if (J <= -6.4e-152) {
		tmp = t_0;
	} else if (J <= 7e-285) {
		tmp = -2.0 * ((U * -0.5) - (J / (U / J)));
	} else if (J <= 5.8e-179) {
		tmp = -2.0 * (U * 0.5);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

U = Math.abs(U);
public static double code(double J, double K, double U) {
	double t_0 = -2.0 * ((J * Math.cos((K / 2.0))) * Math.hypot(1.0, (U / (J * 2.0))));
	double tmp;
	if (J <= -6.4e-152) {
		tmp = t_0;
	} else if (J <= 7e-285) {
		tmp = -2.0 * ((U * -0.5) - (J / (U / J)));
	} else if (J <= 5.8e-179) {
		tmp = -2.0 * (U * 0.5);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	t_0 = -2.0 * ((J * math.cos((K / 2.0))) * math.hypot(1.0, (U / (J * 2.0))))
	tmp = 0
	if J <= -6.4e-152:
		tmp = t_0
	elif J <= 7e-285:
		tmp = -2.0 * ((U * -0.5) - (J / (U / J)))
	elif J <= 5.8e-179:
		tmp = -2.0 * (U * 0.5)
	else:
		tmp = t_0
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	t_0 = Float64(-2.0 * Float64(Float64(J * cos(Float64(K / 2.0))) * hypot(1.0, Float64(U / Float64(J * 2.0)))))
	tmp = 0.0
	if (J <= -6.4e-152)
		tmp = t_0;
	elseif (J <= 7e-285)
		tmp = Float64(-2.0 * Float64(Float64(U * -0.5) - Float64(J / Float64(U / J))));
	elseif (J <= 5.8e-179)
		tmp = Float64(-2.0 * Float64(U * 0.5));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	t_0 = -2.0 * ((J * cos((K / 2.0))) * hypot(1.0, (U / (J * 2.0))));
	tmp = 0.0;
	if (J <= -6.4e-152)
		tmp = t_0;
	elseif (J <= 7e-285)
		tmp = -2.0 * ((U * -0.5) - (J / (U / J)));
	elseif (J <= 5.8e-179)
		tmp = -2.0 * (U * 0.5);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := Block[{t$95$0 = N[(-2.0 * N[(N[(J * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[(U / N[(J * 2.0), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[J, -6.4e-152], t$95$0, If[LessEqual[J, 7e-285], N[(-2.0 * N[(N[(U * -0.5), $MachinePrecision] - N[(J / N[(U / J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[J, 5.8e-179], N[(-2.0 * N[(U * 0.5), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if J < -6.40000000000000025e-152 or 5.7999999999999998e-179 < J

   1. Initial program 87.1%

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

      1. associate-*l* 87.1%

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

      2. associate-*l* 87.1%

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

      3. unpow2 87.1%

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

      4. sqr-neg 87.1%

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

      5. distribute-frac-neg 87.1%

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

      6. distribute-frac-neg 87.1%

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

      7. unpow2 87.1%

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

   3. Simplified 96.7%

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

   4. Taylor expanded in K around 0 81.8%

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

   if -6.40000000000000025e-152 < J < 7.0000000000000007e-285

   1. Initial program 49.5%

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

      1. associate-*l* 49.5%

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

      2. associate-*l* 49.5%

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

      3. unpow2 49.5%

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

      4. sqr-neg 49.5%

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

      5. distribute-frac-neg 49.5%

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

      6. distribute-frac-neg 49.5%

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

      7. unpow2 49.5%

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

   3. Simplified 55.6%

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

   4. Taylor expanded in K around 0 35.1%

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

   5. Taylor expanded in K around 0 37.6%

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

   6. Taylor expanded in U around -inf 35.7%

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

      1. +-commutative 35.7%

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

      2. mul-1-neg 35.7%

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

      3. unsub-neg 35.7%

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

      4. *-commutative 35.7%

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

      5. unpow2 35.7%

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

      6. associate-/l* 35.7%

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

   8. Simplified 35.7%

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

   if 7.0000000000000007e-285 < J < 5.7999999999999998e-179

   1. Initial program 24.3%

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

   3. Simplified 69.9%

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

   4. Taylor expanded in J around 0 53.2%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -6.4 \cdot 10^{-152}:\\ \;\;\;\;-2 \cdot \left(\left(J \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot 2}\right)\right)\\ \mathbf{elif}\;J \leq 7 \cdot 10^{-285}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5 - \frac{J}{\frac{U}{J}}\right)\\ \mathbf{elif}\;J \leq 5.8 \cdot 10^{-179}:\\ \;\;\;\;-2 \cdot \left(U \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\left(J \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot 2}\right)\right)\\ \end{array} \]

Alternative 4: 62.6% accurate, 3.7× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 1000000000:\\ \;\;\;\;-2 \cdot \left(J \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot 2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (if (<= (/ K 2.0) 1000000000.0)
   (* -2.0 (* J (hypot 1.0 (/ U (* J 2.0)))))
   (* -2.0 (* J (cos (* K 0.5))))))

C

U = abs(U);
double code(double J, double K, double U) {
	double tmp;
	if ((K / 2.0) <= 1000000000.0) {
		tmp = -2.0 * (J * hypot(1.0, (U / (J * 2.0))));
	} else {
		tmp = -2.0 * (J * 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 / 2.0) <= 1000000000.0) {
		tmp = -2.0 * (J * Math.hypot(1.0, (U / (J * 2.0))));
	} else {
		tmp = -2.0 * (J * Math.cos((K * 0.5)));
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	tmp = 0
	if (K / 2.0) <= 1000000000.0:
		tmp = -2.0 * (J * math.hypot(1.0, (U / (J * 2.0))))
	else:
		tmp = -2.0 * (J * math.cos((K * 0.5)))
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	tmp = 0.0
	if (Float64(K / 2.0) <= 1000000000.0)
		tmp = Float64(-2.0 * Float64(J * hypot(1.0, Float64(U / Float64(J * 2.0)))));
	else
		tmp = Float64(-2.0 * Float64(J * 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 / 2.0) <= 1000000000.0)
		tmp = -2.0 * (J * hypot(1.0, (U / (J * 2.0))));
	else
		tmp = -2.0 * (J * 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[N[(K / 2.0), $MachinePrecision], 1000000000.0], N[(-2.0 * N[(J * N[Sqrt[1.0 ^ 2 + N[(U / N[(J * 2.0), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 K 2) < 1e9

   1. Initial program 78.3%

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

      1. associate-*l* 78.3%

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

      2. associate-*l* 78.3%

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

      3. unpow2 78.3%

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

      4. sqr-neg 78.3%

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

      5. distribute-frac-neg 78.3%

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

      6. distribute-frac-neg 78.3%

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

      7. unpow2 78.3%

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

   3. Simplified 90.4%

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

   4. Taylor expanded in K around 0 78.2%

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

   5. Taylor expanded in K around 0 65.4%

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

   if 1e9 < (/.f64 K 2)

   1. Initial program 75.7%

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

   3. Simplified 86.8%

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

   4. Taylor expanded in J around inf 49.6%

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

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 1000000000:\\ \;\;\;\;-2 \cdot \left(J \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot 2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 5: 64.6% accurate, 3.7× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} t_0 := -2 \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{if}\;J \leq -2 \cdot 10^{-117}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq 4.4 \cdot 10^{-284}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5\right)\\ \mathbf{elif}\;J \leq 9.5 \cdot 10^{-138}:\\ \;\;\;\;-2 \cdot \left(U \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (* -2.0 (* J (cos (* K 0.5))))))
   (if (<= J -2e-117)
     t_0
     (if (<= J 4.4e-284)
       (* -2.0 (* U -0.5))
       (if (<= J 9.5e-138) (* -2.0 (* U 0.5)) t_0)))))

C

U = abs(U);
double code(double J, double K, double U) {
	double t_0 = -2.0 * (J * cos((K * 0.5)));
	double tmp;
	if (J <= -2e-117) {
		tmp = t_0;
	} else if (J <= 4.4e-284) {
		tmp = -2.0 * (U * -0.5);
	} else if (J <= 9.5e-138) {
		tmp = -2.0 * (U * 0.5);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

NOTE: U should be positive before calling this function
real(8) function code(j, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (-2.0d0) * (j * cos((k * 0.5d0)))
    if (j <= (-2d-117)) then
        tmp = t_0
    else if (j <= 4.4d-284) then
        tmp = (-2.0d0) * (u * (-0.5d0))
    else if (j <= 9.5d-138) then
        tmp = (-2.0d0) * (u * 0.5d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

U = Math.abs(U);
public static double code(double J, double K, double U) {
	double t_0 = -2.0 * (J * Math.cos((K * 0.5)));
	double tmp;
	if (J <= -2e-117) {
		tmp = t_0;
	} else if (J <= 4.4e-284) {
		tmp = -2.0 * (U * -0.5);
	} else if (J <= 9.5e-138) {
		tmp = -2.0 * (U * 0.5);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	t_0 = -2.0 * (J * math.cos((K * 0.5)))
	tmp = 0
	if J <= -2e-117:
		tmp = t_0
	elif J <= 4.4e-284:
		tmp = -2.0 * (U * -0.5)
	elif J <= 9.5e-138:
		tmp = -2.0 * (U * 0.5)
	else:
		tmp = t_0
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	t_0 = Float64(-2.0 * Float64(J * cos(Float64(K * 0.5))))
	tmp = 0.0
	if (J <= -2e-117)
		tmp = t_0;
	elseif (J <= 4.4e-284)
		tmp = Float64(-2.0 * Float64(U * -0.5));
	elseif (J <= 9.5e-138)
		tmp = Float64(-2.0 * Float64(U * 0.5));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	t_0 = -2.0 * (J * cos((K * 0.5)));
	tmp = 0.0;
	if (J <= -2e-117)
		tmp = t_0;
	elseif (J <= 4.4e-284)
		tmp = -2.0 * (U * -0.5);
	elseif (J <= 9.5e-138)
		tmp = -2.0 * (U * 0.5);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := Block[{t$95$0 = N[(-2.0 * N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[J, -2e-117], t$95$0, If[LessEqual[J, 4.4e-284], N[(-2.0 * N[(U * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[J, 9.5e-138], N[(-2.0 * N[(U * 0.5), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if J < -2.00000000000000006e-117 or 9.49999999999999997e-138 < J

   1. Initial program 90.6%

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

   3. Simplified 97.8%

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

   4. Taylor expanded in J around inf 68.6%

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

   if -2.00000000000000006e-117 < J < 4.4000000000000001e-284

   1. Initial program 46.3%

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

   3. Simplified 58.6%

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

   4. Taylor expanded in U around -inf 34.8%

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

      1. *-commutative 34.8%

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

   6. Simplified 34.8%

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

   if 4.4000000000000001e-284 < J < 9.49999999999999997e-138

   1. Initial program 30.0%

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

   3. Simplified 74.3%

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

   4. Taylor expanded in J around 0 62.3%

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

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -2 \cdot 10^{-117}:\\ \;\;\;\;-2 \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{elif}\;J \leq 4.4 \cdot 10^{-284}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5\right)\\ \mathbf{elif}\;J \leq 9.5 \cdot 10^{-138}:\\ \;\;\;\;-2 \cdot \left(U \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 6: 48.1% accurate, 37.5× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} \mathbf{if}\;J \leq -2.6 \cdot 10^{+102}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;J \leq 7 \cdot 10^{-285}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5\right)\\ \mathbf{elif}\;J \leq 1.65 \cdot 10^{-78}:\\ \;\;\;\;-2 \cdot \left(U \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot J\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (if (<= J -2.6e+102)
   (* -2.0 J)
   (if (<= J 7e-285)
     (* -2.0 (* U -0.5))
     (if (<= J 1.65e-78) (* -2.0 (* U 0.5)) (* -2.0 J)))))

C

U = abs(U);
double code(double J, double K, double U) {
	double tmp;
	if (J <= -2.6e+102) {
		tmp = -2.0 * J;
	} else if (J <= 7e-285) {
		tmp = -2.0 * (U * -0.5);
	} else if (J <= 1.65e-78) {
		tmp = -2.0 * (U * 0.5);
	} else {
		tmp = -2.0 * J;
	}
	return tmp;
}

Fortran

NOTE: U should be positive before calling this function
real(8) function code(j, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (j <= (-2.6d+102)) then
        tmp = (-2.0d0) * j
    else if (j <= 7d-285) then
        tmp = (-2.0d0) * (u * (-0.5d0))
    else if (j <= 1.65d-78) then
        tmp = (-2.0d0) * (u * 0.5d0)
    else
        tmp = (-2.0d0) * j
    end if
    code = tmp
end function

Java

U = Math.abs(U);
public static double code(double J, double K, double U) {
	double tmp;
	if (J <= -2.6e+102) {
		tmp = -2.0 * J;
	} else if (J <= 7e-285) {
		tmp = -2.0 * (U * -0.5);
	} else if (J <= 1.65e-78) {
		tmp = -2.0 * (U * 0.5);
	} else {
		tmp = -2.0 * J;
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	tmp = 0
	if J <= -2.6e+102:
		tmp = -2.0 * J
	elif J <= 7e-285:
		tmp = -2.0 * (U * -0.5)
	elif J <= 1.65e-78:
		tmp = -2.0 * (U * 0.5)
	else:
		tmp = -2.0 * J
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	tmp = 0.0
	if (J <= -2.6e+102)
		tmp = Float64(-2.0 * J);
	elseif (J <= 7e-285)
		tmp = Float64(-2.0 * Float64(U * -0.5));
	elseif (J <= 1.65e-78)
		tmp = Float64(-2.0 * Float64(U * 0.5));
	else
		tmp = Float64(-2.0 * J);
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	tmp = 0.0;
	if (J <= -2.6e+102)
		tmp = -2.0 * J;
	elseif (J <= 7e-285)
		tmp = -2.0 * (U * -0.5);
	elseif (J <= 1.65e-78)
		tmp = -2.0 * (U * 0.5);
	else
		tmp = -2.0 * J;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := If[LessEqual[J, -2.6e+102], N[(-2.0 * J), $MachinePrecision], If[LessEqual[J, 7e-285], N[(-2.0 * N[(U * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[J, 1.65e-78], N[(-2.0 * N[(U * 0.5), $MachinePrecision]), $MachinePrecision], N[(-2.0 * J), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if J < -2.60000000000000006e102 or 1.64999999999999991e-78 < J

   1. Initial program 97.8%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in J around inf 77.9%

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

   5. Taylor expanded in K around 0 45.5%

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

   if -2.60000000000000006e102 < J < 7.0000000000000007e-285

   1. Initial program 58.6%

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

   3. Simplified 77.4%

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

   4. Taylor expanded in U around -inf 25.4%

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

      1. *-commutative 25.4%

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

   6. Simplified 25.4%

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

   if 7.0000000000000007e-285 < J < 1.64999999999999991e-78

   1. Initial program 49.9%

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

   3. Simplified 79.6%

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

   4. Taylor expanded in J around 0 46.5%

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

4. Final simplification 39.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -2.6 \cdot 10^{+102}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;J \leq 7 \cdot 10^{-285}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5\right)\\ \mathbf{elif}\;J \leq 1.65 \cdot 10^{-78}:\\ \;\;\;\;-2 \cdot \left(U \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot J\\ \end{array} \]

Alternative 7: 39.4% accurate, 59.5× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} \mathbf{if}\;U \leq 3.4 \cdot 10^{-34}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(U \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (if (<= U 3.4e-34) (* -2.0 J) (* -2.0 (* U 0.5))))

C

U = abs(U);
double code(double J, double K, double U) {
	double tmp;
	if (U <= 3.4e-34) {
		tmp = -2.0 * J;
	} else {
		tmp = -2.0 * (U * 0.5);
	}
	return tmp;
}

Fortran

NOTE: U should be positive before calling this function
real(8) function code(j, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (u <= 3.4d-34) then
        tmp = (-2.0d0) * j
    else
        tmp = (-2.0d0) * (u * 0.5d0)
    end if
    code = tmp
end function

Java

U = Math.abs(U);
public static double code(double J, double K, double U) {
	double tmp;
	if (U <= 3.4e-34) {
		tmp = -2.0 * J;
	} else {
		tmp = -2.0 * (U * 0.5);
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	tmp = 0
	if U <= 3.4e-34:
		tmp = -2.0 * J
	else:
		tmp = -2.0 * (U * 0.5)
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	tmp = 0.0
	if (U <= 3.4e-34)
		tmp = Float64(-2.0 * J);
	else
		tmp = Float64(-2.0 * Float64(U * 0.5));
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	tmp = 0.0;
	if (U <= 3.4e-34)
		tmp = -2.0 * J;
	else
		tmp = -2.0 * (U * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := If[LessEqual[U, 3.4e-34], N[(-2.0 * J), $MachinePrecision], N[(-2.0 * N[(U * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U < 3.4000000000000001e-34

   1. Initial program 83.0%

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

   3. Simplified 91.8%

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

   4. Taylor expanded in J around inf 60.1%

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

   5. Taylor expanded in K around 0 36.6%

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

   if 3.4000000000000001e-34 < U

   1. Initial program 57.8%

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

   3. Simplified 80.6%

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

   4. Taylor expanded in J around 0 42.1%

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

4. Final simplification 37.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 3.4 \cdot 10^{-34}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(U \cdot 0.5\right)\\ \end{array} \]

Alternative 8: 29.2% accurate, 140.0× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: U should be positive before calling this function
real(8) function code(j, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = (-2.0d0) * j
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.7%

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

3. Simplified 89.5%

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

4. Taylor expanded in J around inf 54.1%

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

5. Taylor expanded in K around 0 32.4%

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

6. Final simplification 32.4%

   \[\leadsto -2 \cdot J \]

Reproduce

herbie shell --seed 2023276 
(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)))))