Maksimov and Kolovsky, Equation (3)

Percentage Accurate: 73.1% → 98.6%

Time: 18.4s

Alternatives: 10

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.1% 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: 98.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(t_0 \cdot \left(-2 \cdot J\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{t_0 \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;t_1 \leq 10^{+291}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

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
         (*
          (* t_0 (* -2.0 J))
          (sqrt (+ 1.0 (pow (/ U (* t_0 (* J 2.0))) 2.0))))))
   (if (<= t_1 (- INFINITY)) (- U) (if (<= t_1 1e+291) t_1 U))))

C

U = abs(U);
double code(double J, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = (t_0 * (-2.0 * J)) * 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+291) {
		tmp = t_1;
	} 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 = (t_0 * (-2.0 * J)) * 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+291) {
		tmp = t_1;
	} else {
		tmp = U;
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	t_0 = math.cos((K / 2.0))
	t_1 = (t_0 * (-2.0 * J)) * 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+291:
		tmp = t_1
	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(t_0 * Float64(-2.0 * J)) * 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+291)
		tmp = t_1;
	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 = (t_0 * (-2.0 * J)) * sqrt((1.0 + ((U / (t_0 * (J * 2.0))) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = -U;
	elseif (t_1 <= 1e+291)
		tmp = t_1;
	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[(t$95$0 * N[(-2.0 * J), $MachinePrecision]), $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+291], t$95$1, 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 5.8%

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

   2. Taylor expanded in J around 0 60.7%

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

      1. neg-mul-1 60.7%

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

   4. Simplified 60.7%

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

   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)))) < 9.9999999999999996e290

   1. Initial program 99.8%

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

   if 9.9999999999999996e290 < (*.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 7.9%

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

   2. Taylor expanded in U around -inf 48.0%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;\left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 10^{+291}:\\ \;\;\;\;\left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternative 2: 88.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;J \leq -3.2 \cdot 10^{-180} \lor \neg \left(J \leq -2.7 \cdot 10^{-301}\right):\\ \;\;\;\;-2 \cdot \left(t_0 \cdot \left(J \cdot \mathsf{hypot}\left(1, \frac{\frac{U}{J \cdot 2}}{t_0}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (or (<= J -3.2e-180) (not (<= J -2.7e-301)))
     (* -2.0 (* t_0 (* J (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 ((J <= -3.2e-180) || !(J <= -2.7e-301)) {
		tmp = -2.0 * (t_0 * (J * 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 ((J <= -3.2e-180) || !(J <= -2.7e-301)) {
		tmp = -2.0 * (t_0 * (J * 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 (J <= -3.2e-180) or not (J <= -2.7e-301):
		tmp = -2.0 * (t_0 * (J * 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 ((J <= -3.2e-180) || !(J <= -2.7e-301))
		tmp = Float64(-2.0 * Float64(t_0 * Float64(J * hypot(1.0, Float64(Float64(U / Float64(J * 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));
	tmp = 0.0;
	if ((J <= -3.2e-180) || ~((J <= -2.7e-301)))
		tmp = -2.0 * (t_0 * (J * 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[Or[LessEqual[J, -3.2e-180], N[Not[LessEqual[J, -2.7e-301]], $MachinePrecision]], N[(-2.0 * N[(t$95$0 * N[(J * N[Sqrt[1.0 ^ 2 + N[(N[(U / N[(J * 2.0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], U]]

Derivation

1. Split input into 2 regimes

2. if J < -3.20000000000000015e-180 or -2.7e-301 < J

   1. Initial program 74.2%

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

   3. Simplified 92.1%

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

   if -3.20000000000000015e-180 < J < -2.7e-301

   1. Initial program 32.2%

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

   2. Taylor expanded in U around -inf 40.5%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -3.2 \cdot 10^{-180} \lor \neg \left(J \leq -2.7 \cdot 10^{-301}\right):\\ \;\;\;\;-2 \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \mathsf{hypot}\left(1, \frac{\frac{U}{J \cdot 2}}{\cos \left(\frac{K}{2}\right)}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternative 3: 88.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \mathsf{hypot}\left(1, \frac{\frac{U}{J \cdot 2}}{t_0}\right)\\ \mathbf{if}\;J \leq -2.3 \cdot 10^{-180}:\\ \;\;\;\;-2 \cdot \left(t_0 \cdot \left(J \cdot t_1\right)\right)\\ \mathbf{elif}\;J \leq -2.7 \cdot 10^{-301}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(t_1 \cdot \left(J \cdot t_0\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))) (t_1 (hypot 1.0 (/ (/ U (* J 2.0)) t_0))))
   (if (<= J -2.3e-180)
     (* -2.0 (* t_0 (* J t_1)))
     (if (<= J -2.7e-301) U (* -2.0 (* t_1 (* J t_0)))))))

C

U = abs(U);
double code(double J, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = hypot(1.0, ((U / (J * 2.0)) / t_0));
	double tmp;
	if (J <= -2.3e-180) {
		tmp = -2.0 * (t_0 * (J * t_1));
	} else if (J <= -2.7e-301) {
		tmp = U;
	} else {
		tmp = -2.0 * (t_1 * (J * t_0));
	}
	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 = Math.hypot(1.0, ((U / (J * 2.0)) / t_0));
	double tmp;
	if (J <= -2.3e-180) {
		tmp = -2.0 * (t_0 * (J * t_1));
	} else if (J <= -2.7e-301) {
		tmp = U;
	} else {
		tmp = -2.0 * (t_1 * (J * t_0));
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	t_0 = math.cos((K / 2.0))
	t_1 = math.hypot(1.0, ((U / (J * 2.0)) / t_0))
	tmp = 0
	if J <= -2.3e-180:
		tmp = -2.0 * (t_0 * (J * t_1))
	elif J <= -2.7e-301:
		tmp = U
	else:
		tmp = -2.0 * (t_1 * (J * t_0))
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	t_0 = cos(Float64(K / 2.0))
	t_1 = hypot(1.0, Float64(Float64(U / Float64(J * 2.0)) / t_0))
	tmp = 0.0
	if (J <= -2.3e-180)
		tmp = Float64(-2.0 * Float64(t_0 * Float64(J * t_1)));
	elseif (J <= -2.7e-301)
		tmp = U;
	else
		tmp = Float64(-2.0 * Float64(t_1 * Float64(J * t_0)));
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	t_0 = cos((K / 2.0));
	t_1 = hypot(1.0, ((U / (J * 2.0)) / t_0));
	tmp = 0.0;
	if (J <= -2.3e-180)
		tmp = -2.0 * (t_0 * (J * t_1));
	elseif (J <= -2.7e-301)
		tmp = U;
	else
		tmp = -2.0 * (t_1 * (J * 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[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[1.0 ^ 2 + N[(N[(U / N[(J * 2.0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] ^ 2], $MachinePrecision]}, If[LessEqual[J, -2.3e-180], N[(-2.0 * N[(t$95$0 * N[(J * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[J, -2.7e-301], U, N[(-2.0 * N[(t$95$1 * N[(J * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if J < -2.29999999999999996e-180

   1. Initial program 81.7%

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

   3. Simplified 91.8%

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

   if -2.29999999999999996e-180 < J < -2.7e-301

   1. Initial program 32.2%

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

   2. Taylor expanded in U around -inf 40.5%

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

   if -2.7e-301 < J

   1. Initial program 69.1%

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

      1. associate-*l* 69.1%

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

      2. associate-*l* 69.1%

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

      3. unpow2 69.1%

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

      4. hypot-1-def 92.3%

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

      5. associate-/r* 92.3%

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

      6. cos-neg 92.3%

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

      7. distribute-frac-neg 92.3%

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

      8. associate-/r* 92.3%

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

      9. associate-/r* 92.3%

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

      10. distribute-frac-neg 92.3%

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

      11. cos-neg 92.3%

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

   3. Simplified 92.3%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -2.3 \cdot 10^{-180}:\\ \;\;\;\;-2 \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \mathsf{hypot}\left(1, \frac{\frac{U}{J \cdot 2}}{\cos \left(\frac{K}{2}\right)}\right)\right)\right)\\ \mathbf{elif}\;J \leq -2.7 \cdot 10^{-301}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\mathsf{hypot}\left(1, \frac{\frac{U}{J \cdot 2}}{\cos \left(\frac{K}{2}\right)}\right) \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)\right)\\ \end{array} \]

Alternative 4: 77.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} t_0 := -2 \cdot \left(J \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \mathsf{hypot}\left(1, U \cdot \frac{0.5}{J}\right)\right)\right)\\ \mathbf{if}\;J \leq -1.45 \cdot 10^{-69}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq -4 \cdot 10^{-310}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 2 \cdot 10^{-124}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (* -2.0 (* J (* (cos (* K 0.5)) (hypot 1.0 (* U (/ 0.5 J))))))))
   (if (<= J -1.45e-69)
     t_0
     (if (<= J -4e-310) U (if (<= J 2e-124) (- U) t_0)))))

C

U = abs(U);
double code(double J, double K, double U) {
	double t_0 = -2.0 * (J * (cos((K * 0.5)) * hypot(1.0, (U * (0.5 / J)))));
	double tmp;
	if (J <= -1.45e-69) {
		tmp = t_0;
	} else if (J <= -4e-310) {
		tmp = U;
	} else if (J <= 2e-124) {
		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 = -2.0 * (J * (Math.cos((K * 0.5)) * Math.hypot(1.0, (U * (0.5 / J)))));
	double tmp;
	if (J <= -1.45e-69) {
		tmp = t_0;
	} else if (J <= -4e-310) {
		tmp = U;
	} else if (J <= 2e-124) {
		tmp = -U;
	} 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)) * math.hypot(1.0, (U * (0.5 / J)))))
	tmp = 0
	if J <= -1.45e-69:
		tmp = t_0
	elif J <= -4e-310:
		tmp = U
	elif J <= 2e-124:
		tmp = -U
	else:
		tmp = t_0
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	t_0 = Float64(-2.0 * Float64(J * Float64(cos(Float64(K * 0.5)) * hypot(1.0, Float64(U * Float64(0.5 / J))))))
	tmp = 0.0
	if (J <= -1.45e-69)
		tmp = t_0;
	elseif (J <= -4e-310)
		tmp = U;
	elseif (J <= 2e-124)
		tmp = Float64(-U);
	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)) * hypot(1.0, (U * (0.5 / J)))));
	tmp = 0.0;
	if (J <= -1.45e-69)
		tmp = t_0;
	elseif (J <= -4e-310)
		tmp = U;
	elseif (J <= 2e-124)
		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[(-2.0 * N[(J * N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[(U * N[(0.5 / J), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[J, -1.45e-69], t$95$0, If[LessEqual[J, -4e-310], U, If[LessEqual[J, 2e-124], (-U), t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if J < -1.4499999999999999e-69 or 1.99999999999999987e-124 < J

   1. Initial program 87.6%

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

      1. associate-*l* 87.6%

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

      2. associate-*l* 87.6%

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

      3. unpow2 87.6%

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

      4. hypot-1-def 98.2%

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

      5. associate-/r* 98.1%

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

      6. cos-neg 98.1%

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

      7. distribute-frac-neg 98.1%

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

      8. associate-/r* 98.2%

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

      9. associate-/r* 98.1%

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

      10. distribute-frac-neg 98.1%

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

      11. cos-neg 98.1%

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

   3. Simplified 98.1%

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

   4. Taylor expanded in K around 0 85.6%

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

      1. associate-*r/ 85.6%

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

   6. Simplified 85.6%

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

      1. expm1-log1p-u 52.4%

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

      2. expm1-udef 39.8%

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

   8. Applied egg-rr 39.8%

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

      1. expm1-def 52.4%

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

      2. expm1-log1p 85.6%

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

      3. *-commutative 85.6%

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

      4. *-commutative 85.6%

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

      5. associate-/r/ 85.5%

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

   10. Simplified 85.5%

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

   if -1.4499999999999999e-69 < J < -3.999999999999988e-310

   1. Initial program 37.9%

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

   2. Taylor expanded in U around -inf 39.3%

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

   if -3.999999999999988e-310 < J < 1.99999999999999987e-124

   1. Initial program 43.6%

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

   2. Taylor expanded in J around 0 52.5%

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

      1. neg-mul-1 52.5%

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

   4. Simplified 52.5%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -1.45 \cdot 10^{-69}:\\ \;\;\;\;-2 \cdot \left(J \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \mathsf{hypot}\left(1, U \cdot \frac{0.5}{J}\right)\right)\right)\\ \mathbf{elif}\;J \leq -4 \cdot 10^{-310}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 2 \cdot 10^{-124}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(J \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \mathsf{hypot}\left(1, U \cdot \frac{0.5}{J}\right)\right)\right)\\ \end{array} \]

Alternative 5: 77.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} t_0 := -2 \cdot \left(\left(J \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U \cdot 0.5}{J}\right)\right)\\ \mathbf{if}\;J \leq -1.45 \cdot 10^{-69}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq 6 \cdot 10^{-309}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 1.5 \cdot 10^{-124}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (* -2.0 (* (* J (cos (/ K 2.0))) (hypot 1.0 (/ (* U 0.5) J))))))
   (if (<= J -1.45e-69)
     t_0
     (if (<= J 6e-309) U (if (<= J 1.5e-124) (- U) 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 * 0.5) / J)));
	double tmp;
	if (J <= -1.45e-69) {
		tmp = t_0;
	} else if (J <= 6e-309) {
		tmp = U;
	} else if (J <= 1.5e-124) {
		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 = -2.0 * ((J * Math.cos((K / 2.0))) * Math.hypot(1.0, ((U * 0.5) / J)));
	double tmp;
	if (J <= -1.45e-69) {
		tmp = t_0;
	} else if (J <= 6e-309) {
		tmp = U;
	} else if (J <= 1.5e-124) {
		tmp = -U;
	} 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 * 0.5) / J)))
	tmp = 0
	if J <= -1.45e-69:
		tmp = t_0
	elif J <= 6e-309:
		tmp = U
	elif J <= 1.5e-124:
		tmp = -U
	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(Float64(U * 0.5) / J))))
	tmp = 0.0
	if (J <= -1.45e-69)
		tmp = t_0;
	elseif (J <= 6e-309)
		tmp = U;
	elseif (J <= 1.5e-124)
		tmp = Float64(-U);
	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 * 0.5) / J)));
	tmp = 0.0;
	if (J <= -1.45e-69)
		tmp = t_0;
	elseif (J <= 6e-309)
		tmp = U;
	elseif (J <= 1.5e-124)
		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[(-2.0 * N[(N[(J * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[(N[(U * 0.5), $MachinePrecision] / J), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[J, -1.45e-69], t$95$0, If[LessEqual[J, 6e-309], U, If[LessEqual[J, 1.5e-124], (-U), t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if J < -1.4499999999999999e-69 or 1.5e-124 < J

   1. Initial program 87.6%

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

      1. associate-*l* 87.6%

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

      2. associate-*l* 87.6%

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

      3. unpow2 87.6%

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

      4. hypot-1-def 98.2%

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

      5. associate-/r* 98.1%

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

      6. cos-neg 98.1%

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

      7. distribute-frac-neg 98.1%

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

      8. associate-/r* 98.2%

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

      9. associate-/r* 98.1%

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

      10. distribute-frac-neg 98.1%

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

      11. cos-neg 98.1%

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

   3. Simplified 98.1%

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

   4. Taylor expanded in K around 0 85.6%

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

      1. associate-*r/ 85.6%

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

   6. Simplified 85.6%

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

   if -1.4499999999999999e-69 < J < 6.000000000000001e-309

   1. Initial program 37.9%

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

   2. Taylor expanded in U around -inf 39.3%

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

   if 6.000000000000001e-309 < J < 1.5e-124

   1. Initial program 43.6%

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

   2. Taylor expanded in J around 0 52.5%

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

      1. neg-mul-1 52.5%

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

   4. Simplified 52.5%

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

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -1.45 \cdot 10^{-69}:\\ \;\;\;\;-2 \cdot \left(\left(J \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U \cdot 0.5}{J}\right)\right)\\ \mathbf{elif}\;J \leq 6 \cdot 10^{-309}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 1.5 \cdot 10^{-124}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\left(J \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U \cdot 0.5}{J}\right)\right)\\ \end{array} \]

Alternative 6: 66.2% accurate, 3.6× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} t_0 := \cos \left(K \cdot 0.5\right) \cdot \left(-2 \cdot J\right)\\ \mathbf{if}\;J \leq -1.7 \cdot 10^{-69}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq -4 \cdot 10^{-310}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 2.9 \cdot 10^{-95}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 2.15 \cdot 10^{-6} \lor \neg \left(J \leq 4.5 \cdot 10^{+23}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \left(\frac{J}{U} + 0.5 \cdot \frac{U}{J}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (* (cos (* K 0.5)) (* -2.0 J))))
   (if (<= J -1.7e-69)
     t_0
     (if (<= J -4e-310)
       U
       (if (<= J 2.9e-95)
         (- U)
         (if (or (<= J 2.15e-6) (not (<= J 4.5e+23)))
           t_0
           (* (* -2.0 J) (+ (/ J U) (* 0.5 (/ U J))))))))))

C

U = abs(U);
double code(double J, double K, double U) {
	double t_0 = cos((K * 0.5)) * (-2.0 * J);
	double tmp;
	if (J <= -1.7e-69) {
		tmp = t_0;
	} else if (J <= -4e-310) {
		tmp = U;
	} else if (J <= 2.9e-95) {
		tmp = -U;
	} else if ((J <= 2.15e-6) || !(J <= 4.5e+23)) {
		tmp = t_0;
	} else {
		tmp = (-2.0 * J) * ((J / U) + (0.5 * (U / J)));
	}
	return tmp;
}

Fortran

NOTE: U should be positive before calling this function
real(8) function code(j, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((k * 0.5d0)) * ((-2.0d0) * j)
    if (j <= (-1.7d-69)) then
        tmp = t_0
    else if (j <= (-4d-310)) then
        tmp = u
    else if (j <= 2.9d-95) then
        tmp = -u
    else if ((j <= 2.15d-6) .or. (.not. (j <= 4.5d+23))) then
        tmp = t_0
    else
        tmp = ((-2.0d0) * j) * ((j / u) + (0.5d0 * (u / j)))
    end if
    code = tmp
end function

Java

U = Math.abs(U);
public static double code(double J, double K, double U) {
	double t_0 = Math.cos((K * 0.5)) * (-2.0 * J);
	double tmp;
	if (J <= -1.7e-69) {
		tmp = t_0;
	} else if (J <= -4e-310) {
		tmp = U;
	} else if (J <= 2.9e-95) {
		tmp = -U;
	} else if ((J <= 2.15e-6) || !(J <= 4.5e+23)) {
		tmp = t_0;
	} else {
		tmp = (-2.0 * J) * ((J / U) + (0.5 * (U / J)));
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	t_0 = math.cos((K * 0.5)) * (-2.0 * J)
	tmp = 0
	if J <= -1.7e-69:
		tmp = t_0
	elif J <= -4e-310:
		tmp = U
	elif J <= 2.9e-95:
		tmp = -U
	elif (J <= 2.15e-6) or not (J <= 4.5e+23):
		tmp = t_0
	else:
		tmp = (-2.0 * J) * ((J / U) + (0.5 * (U / J)))
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	t_0 = Float64(cos(Float64(K * 0.5)) * Float64(-2.0 * J))
	tmp = 0.0
	if (J <= -1.7e-69)
		tmp = t_0;
	elseif (J <= -4e-310)
		tmp = U;
	elseif (J <= 2.9e-95)
		tmp = Float64(-U);
	elseif ((J <= 2.15e-6) || !(J <= 4.5e+23))
		tmp = t_0;
	else
		tmp = Float64(Float64(-2.0 * J) * Float64(Float64(J / U) + Float64(0.5 * Float64(U / J))));
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	t_0 = cos((K * 0.5)) * (-2.0 * J);
	tmp = 0.0;
	if (J <= -1.7e-69)
		tmp = t_0;
	elseif (J <= -4e-310)
		tmp = U;
	elseif (J <= 2.9e-95)
		tmp = -U;
	elseif ((J <= 2.15e-6) || ~((J <= 4.5e+23)))
		tmp = t_0;
	else
		tmp = (-2.0 * J) * ((J / U) + (0.5 * (U / J)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := Block[{t$95$0 = N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(-2.0 * J), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[J, -1.7e-69], t$95$0, If[LessEqual[J, -4e-310], U, If[LessEqual[J, 2.9e-95], (-U), If[Or[LessEqual[J, 2.15e-6], N[Not[LessEqual[J, 4.5e+23]], $MachinePrecision]], t$95$0, N[(N[(-2.0 * J), $MachinePrecision] * N[(N[(J / U), $MachinePrecision] + N[(0.5 * N[(U / J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if J < -1.70000000000000004e-69 or 2.90000000000000002e-95 < J < 2.15000000000000017e-6 or 4.49999999999999979e23 < J

   1. Initial program 93.3%

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

   2. Taylor expanded in J around inf 76.3%

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

      1. associate-*r* 76.3%

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

      2. *-commutative 76.3%

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

   4. Simplified 76.3%

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

   if -1.70000000000000004e-69 < J < -3.999999999999988e-310

   1. Initial program 37.9%

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

   2. Taylor expanded in U around -inf 39.3%

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

   if -3.999999999999988e-310 < J < 2.90000000000000002e-95

   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. Taylor expanded in J around 0 53.5%

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

      1. neg-mul-1 53.5%

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

   4. Simplified 53.5%

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

   if 2.15000000000000017e-6 < J < 4.49999999999999979e23

   1. Initial program 28.2%

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

   2. Taylor expanded in K around 0 26.8%

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

      1. associate-*r* 26.8%

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

      2. *-commutative 26.8%

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

      3. metadata-eval 26.8%

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

      4. unpow2 26.8%

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

      5. unpow2 26.8%

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

      6. times-frac 26.8%

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

      7. swap-sqr 26.8%

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

      8. associate-*r/ 26.8%

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

      9. *-commutative 26.8%

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

      10. associate-*r/ 26.8%

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

      11. associate-*r/ 26.8%

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

      12. *-commutative 26.8%

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

      13. associate-*r/ 26.8%

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

      14. unpow2 26.8%

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

      15. associate-*r/ 26.8%

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

      16. *-commutative 26.8%

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

      17. associate-*r/ 26.8%

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

   4. Simplified 26.8%

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

   5. Taylor expanded in U around inf 45.0%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -1.7 \cdot 10^{-69}:\\ \;\;\;\;\cos \left(K \cdot 0.5\right) \cdot \left(-2 \cdot J\right)\\ \mathbf{elif}\;J \leq -4 \cdot 10^{-310}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 2.9 \cdot 10^{-95}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 2.15 \cdot 10^{-6} \lor \neg \left(J \leq 4.5 \cdot 10^{+23}\right):\\ \;\;\;\;\cos \left(K \cdot 0.5\right) \cdot \left(-2 \cdot J\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \left(\frac{J}{U} + 0.5 \cdot \frac{U}{J}\right)\\ \end{array} \]

Alternative 7: 64.9% accurate, 3.7× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} \mathbf{if}\;U \leq 1.66 \cdot 10^{-91}:\\ \;\;\;\;\cos \left(K \cdot 0.5\right) \cdot \left(-2 \cdot J\right)\\ \mathbf{elif}\;U \leq 1.46 \cdot 10^{+169}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \mathsf{hypot}\left(1, \frac{0.5}{\frac{J}{U}}\right)\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (if (<= U 1.66e-91)
   (* (cos (* K 0.5)) (* -2.0 J))
   (if (<= U 1.46e+169) (* (* -2.0 J) (hypot 1.0 (/ 0.5 (/ J U)))) (- U))))

C

U = abs(U);
double code(double J, double K, double U) {
	double tmp;
	if (U <= 1.66e-91) {
		tmp = cos((K * 0.5)) * (-2.0 * J);
	} else if (U <= 1.46e+169) {
		tmp = (-2.0 * J) * hypot(1.0, (0.5 / (J / U)));
	} else {
		tmp = -U;
	}
	return tmp;
}

Java

U = Math.abs(U);
public static double code(double J, double K, double U) {
	double tmp;
	if (U <= 1.66e-91) {
		tmp = Math.cos((K * 0.5)) * (-2.0 * J);
	} else if (U <= 1.46e+169) {
		tmp = (-2.0 * J) * Math.hypot(1.0, (0.5 / (J / U)));
	} else {
		tmp = -U;
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	tmp = 0
	if U <= 1.66e-91:
		tmp = math.cos((K * 0.5)) * (-2.0 * J)
	elif U <= 1.46e+169:
		tmp = (-2.0 * J) * math.hypot(1.0, (0.5 / (J / U)))
	else:
		tmp = -U
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	tmp = 0.0
	if (U <= 1.66e-91)
		tmp = Float64(cos(Float64(K * 0.5)) * Float64(-2.0 * J));
	elseif (U <= 1.46e+169)
		tmp = Float64(Float64(-2.0 * J) * hypot(1.0, Float64(0.5 / Float64(J / U))));
	else
		tmp = Float64(-U);
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	tmp = 0.0;
	if (U <= 1.66e-91)
		tmp = cos((K * 0.5)) * (-2.0 * J);
	elseif (U <= 1.46e+169)
		tmp = (-2.0 * J) * hypot(1.0, (0.5 / (J / U)));
	else
		tmp = -U;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := If[LessEqual[U, 1.66e-91], N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(-2.0 * J), $MachinePrecision]), $MachinePrecision], If[LessEqual[U, 1.46e+169], N[(N[(-2.0 * J), $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[(0.5 / N[(J / U), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], (-U)]]

Derivation

1. Split input into 3 regimes

2. if U < 1.66e-91

   1. Initial program 79.2%

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

   2. Taylor expanded in J around inf 61.5%

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

      1. associate-*r* 61.5%

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

      2. *-commutative 61.5%

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

   4. Simplified 61.5%

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

   if 1.66e-91 < U < 1.45999999999999992e169

   1. Initial program 60.5%

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

   2. Taylor expanded in K around 0 34.7%

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

      1. associate-*r* 34.7%

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

      2. *-commutative 34.7%

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

      3. metadata-eval 34.7%

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

      4. unpow2 34.7%

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

      5. unpow2 34.7%

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

      6. times-frac 36.7%

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

      7. swap-sqr 36.7%

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

      8. associate-*r/ 36.7%

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

      9. *-commutative 36.7%

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

      10. associate-*r/ 36.7%

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

      11. associate-*r/ 36.7%

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

      12. *-commutative 36.7%

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

      13. associate-*r/ 36.6%

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

      14. unpow2 36.6%

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

      15. associate-*r/ 36.7%

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

      16. *-commutative 36.7%

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

      17. associate-*r/ 36.7%

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

   4. Simplified 36.7%

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

      1. metadata-eval 36.7%

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

      2. unpow2 36.7%

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

      3. associate-*r/ 36.7%

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

      4. associate-*r/ 36.7%

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

      5. hypot-udef 62.4%

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

      6. associate-/l* 62.3%

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

   6. Applied egg-rr 62.3%

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

   if 1.45999999999999992e169 < U

   1. Initial program 33.0%

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

   2. Taylor expanded in J around 0 55.4%

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

      1. neg-mul-1 55.4%

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

   4. Simplified 55.4%

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

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 1.66 \cdot 10^{-91}:\\ \;\;\;\;\cos \left(K \cdot 0.5\right) \cdot \left(-2 \cdot J\right)\\ \mathbf{elif}\;U \leq 1.46 \cdot 10^{+169}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \mathsf{hypot}\left(1, \frac{0.5}{\frac{J}{U}}\right)\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]

Alternative 8: 49.6% accurate, 45.7× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} \mathbf{if}\;J \leq -2.6 \cdot 10^{-69}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;J \leq -4 \cdot 10^{-310}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 3.2 \cdot 10^{+22}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot J\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (if (<= J -2.6e-69)
   (* -2.0 J)
   (if (<= J -4e-310) U (if (<= J 3.2e+22) (- U) (* -2.0 J)))))

C

U = abs(U);
double code(double J, double K, double U) {
	double tmp;
	if (J <= -2.6e-69) {
		tmp = -2.0 * J;
	} else if (J <= -4e-310) {
		tmp = U;
	} else if (J <= 3.2e+22) {
		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 <= (-2.6d-69)) then
        tmp = (-2.0d0) * j
    else if (j <= (-4d-310)) then
        tmp = u
    else if (j <= 3.2d+22) 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 <= -2.6e-69) {
		tmp = -2.0 * J;
	} else if (J <= -4e-310) {
		tmp = U;
	} else if (J <= 3.2e+22) {
		tmp = -U;
	} else {
		tmp = -2.0 * J;
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	tmp = 0
	if J <= -2.6e-69:
		tmp = -2.0 * J
	elif J <= -4e-310:
		tmp = U
	elif J <= 3.2e+22:
		tmp = -U
	else:
		tmp = -2.0 * J
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	tmp = 0.0
	if (J <= -2.6e-69)
		tmp = Float64(-2.0 * J);
	elseif (J <= -4e-310)
		tmp = U;
	elseif (J <= 3.2e+22)
		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 <= -2.6e-69)
		tmp = -2.0 * J;
	elseif (J <= -4e-310)
		tmp = U;
	elseif (J <= 3.2e+22)
		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, -2.6e-69], N[(-2.0 * J), $MachinePrecision], If[LessEqual[J, -4e-310], U, If[LessEqual[J, 3.2e+22], (-U), N[(-2.0 * J), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if J < -2.6000000000000002e-69 or 3.2e22 < J

   1. Initial program 96.0%

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

   2. Taylor expanded in J around inf 78.5%

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

      1. associate-*r* 78.5%

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

      2. *-commutative 78.5%

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

   4. Simplified 78.5%

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

   5. Taylor expanded in K around 0 49.4%

      \[\leadsto \color{blue}{-2 \cdot J} \]
   6. Step-by-step derivation

      1. *-commutative 49.4%

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

   7. Simplified 49.4%

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

   if -2.6000000000000002e-69 < J < -3.999999999999988e-310

   1. Initial program 37.9%

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

   2. Taylor expanded in U around -inf 39.3%

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

   if -3.999999999999988e-310 < J < 3.2e22

   1. Initial program 51.5%

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

   2. Taylor expanded in J around 0 42.4%

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

      1. neg-mul-1 42.4%

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

   4. Simplified 42.4%

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

4. Final simplification 45.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -2.6 \cdot 10^{-69}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;J \leq -4 \cdot 10^{-310}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 3.2 \cdot 10^{+22}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot J\\ \end{array} \]

Alternative 9: 40.0% accurate, 103.4× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} \mathbf{if}\;J \leq -4 \cdot 10^{-310}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U) :precision binary64 (if (<= J -4e-310) U (- U)))

C

U = abs(U);
double code(double J, double K, double U) {
	double tmp;
	if (J <= -4e-310) {
		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 <= (-4d-310)) 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 <= -4e-310) {
		tmp = U;
	} else {
		tmp = -U;
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	tmp = 0
	if J <= -4e-310:
		tmp = U
	else:
		tmp = -U
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	tmp = 0.0
	if (J <= -4e-310)
		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 <= -4e-310)
		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, -4e-310], U, (-U)]

Derivation

1. Split input into 2 regimes

2. if J < -3.999999999999988e-310

   1. Initial program 68.8%

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

   2. Taylor expanded in U around -inf 27.4%

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

   if -3.999999999999988e-310 < J

   1. Initial program 70.4%

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

   2. Taylor expanded in J around 0 30.9%

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

      1. neg-mul-1 30.9%

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

   4. Simplified 30.9%

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

4. Final simplification 29.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -4 \cdot 10^{-310}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]

Alternative 10: 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 69.6%

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

2. Taylor expanded in U around -inf 25.6%

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

3. Final simplification 25.6%

   \[\leadsto U \]

Reproduce

herbie shell --seed 2023313 
(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)))))