Maksimov and Kolovsky, Equation (3)

Percentage Accurate: 73.3% → 99.2%

Time: 19.6s

Alternatives: 11

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.3% 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: 99.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(t_0 \cdot \left(-2 \cdot J\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{t_0 \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+300}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

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 2e+300) 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 <= 2e+300) {
		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 <= 2e+300) {
		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 <= 2e+300:
		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 <= 2e+300)
		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 <= 2e+300)
		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, 2e+300], 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 5.1%

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

   3. Simplified 5.1%

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

   4. Taylor expanded in J around 0 63.6%

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

      1. neg-mul-1 63.6%

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

   6. Simplified 63.6%

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

   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)))) < 2.0000000000000001e300

   1. Initial program 99.9%

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

   if 2.0000000000000001e300 < (*.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 5.9%

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

   3. Simplified 5.9%

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

   4. Taylor expanded in U around -inf 38.3%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;\left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 2 \cdot 10^{+300}:\\ \;\;\;\;\left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternative 2: 88.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ -2 \cdot \left(t_0 \cdot \left(J \cdot \mathsf{hypot}\left(1, \frac{\frac{U}{J \cdot 2}}{t_0}\right)\right)\right) \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (* -2.0 (* t_0 (* J (hypot 1.0 (/ (/ U (* J 2.0)) t_0)))))))

C

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

Java

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

Python

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

Julia

U = abs(U)
function code(J, K, U)
	t_0 = cos(Float64(K / 2.0))
	return Float64(-2.0 * Float64(t_0 * Float64(J * hypot(1.0, Float64(Float64(U / Float64(J * 2.0)) / t_0)))))
end

MATLAB

U = abs(U)
function tmp = code(J, K, U)
	t_0 = cos((K / 2.0));
	tmp = -2.0 * (t_0 * (J * hypot(1.0, ((U / (J * 2.0)) / t_0))));
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]}, 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]]

Derivation

1. Initial program 75.2%

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

3. Simplified 91.0%

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

4. Final simplification 91.0%

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

Alternative 3: 88.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ -2 \cdot \left(\left(J \cdot t_0\right) \cdot \mathsf{hypot}\left(1, \frac{\frac{U}{J \cdot 2}}{t_0}\right)\right) \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (* -2.0 (* (* J t_0) (hypot 1.0 (/ (/ U (* J 2.0)) t_0))))))

C

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

Java

U = Math.abs(U);
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.hypot(1.0, ((U / (J * 2.0)) / t_0)));
}

Python

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

Julia

U = abs(U)
function code(J, K, U)
	t_0 = cos(Float64(K / 2.0))
	return Float64(-2.0 * Float64(Float64(J * t_0) * hypot(1.0, Float64(Float64(U / Float64(J * 2.0)) / t_0))))
end

MATLAB

U = abs(U)
function tmp = code(J, K, U)
	t_0 = cos((K / 2.0));
	tmp = -2.0 * ((J * t_0) * hypot(1.0, ((U / (J * 2.0)) / t_0)));
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]}, N[(-2.0 * N[(N[(J * t$95$0), $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[(N[(U / N[(J * 2.0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 75.2%

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

   1. associate-*l* 75.2%

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

   2. associate-*l* 75.2%

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

   3. unpow2 75.2%

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

   4. hypot-1-def 91.1%

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

   5. associate-/r* 91.1%

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

   6. cos-neg 91.1%

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

   7. distribute-frac-neg 91.1%

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

   8. associate-/r* 91.1%

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

   9. associate-/r* 91.1%

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

   10. distribute-frac-neg 91.1%

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

   11. cos-neg 91.1%

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

3. Simplified 91.1%

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

4. Final simplification 91.1%

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

Alternative 4: 73.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} \mathbf{if}\;U \leq 4.5 \cdot 10^{+230}:\\ \;\;\;\;-2 \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \mathsf{hypot}\left(1, \frac{U \cdot 0.5}{J}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (if (<= U 4.5e+230)
   (* -2.0 (* (cos (/ K 2.0)) (* J (hypot 1.0 (/ (* U 0.5) J)))))
   U))

C

U = abs(U);
double code(double J, double K, double U) {
	double tmp;
	if (U <= 4.5e+230) {
		tmp = -2.0 * (cos((K / 2.0)) * (J * hypot(1.0, ((U * 0.5) / J))));
	} else {
		tmp = U;
	}
	return tmp;
}

Java

U = Math.abs(U);
public static double code(double J, double K, double U) {
	double tmp;
	if (U <= 4.5e+230) {
		tmp = -2.0 * (Math.cos((K / 2.0)) * (J * Math.hypot(1.0, ((U * 0.5) / J))));
	} else {
		tmp = U;
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	tmp = 0
	if U <= 4.5e+230:
		tmp = -2.0 * (math.cos((K / 2.0)) * (J * math.hypot(1.0, ((U * 0.5) / J))))
	else:
		tmp = U
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	tmp = 0.0
	if (U <= 4.5e+230)
		tmp = Float64(-2.0 * Float64(cos(Float64(K / 2.0)) * Float64(J * hypot(1.0, Float64(Float64(U * 0.5) / J)))));
	else
		tmp = U;
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	tmp = 0.0;
	if (U <= 4.5e+230)
		tmp = -2.0 * (cos((K / 2.0)) * (J * hypot(1.0, ((U * 0.5) / J))));
	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, 4.5e+230], N[(-2.0 * N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(J * N[Sqrt[1.0 ^ 2 + N[(N[(U * 0.5), $MachinePrecision] / J), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], U]

Derivation

1. Split input into 2 regimes

2. if U < 4.4999999999999999e230

   1. Initial program 77.3%

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

   3. Simplified 92.1%

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

   4. Taylor expanded in K around 0 76.9%

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

      1. associate-*r/ 76.9%

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

   6. Simplified 76.9%

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

   if 4.4999999999999999e230 < U

   1. Initial program 31.6%

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

   3. Simplified 31.4%

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

   4. Taylor expanded in U around -inf 66.7%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 4.5 \cdot 10^{+230}:\\ \;\;\;\;-2 \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \mathsf{hypot}\left(1, \frac{U \cdot 0.5}{J}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternative 5: 62.9% accurate, 3.4× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} t_0 := \cos \left(K \cdot 0.5\right)\\ t_1 := \left(-2 \cdot J\right) \cdot t_0\\ t_2 := \left(-2 \cdot J\right) \cdot \mathsf{hypot}\left(1, 0.5 \cdot \frac{U}{J}\right)\\ \mathbf{if}\;K \leq 800000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;K \leq 5.7 \cdot 10^{+20}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;K \leq 4.8 \cdot 10^{+34}:\\ \;\;\;\;U\\ \mathbf{elif}\;K \leq 2.4 \cdot 10^{+241}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;K \leq 1.4 \cdot 10^{+255}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(J \cdot t_0\right) + -0.25 \cdot \left(U \cdot \frac{U}{J}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (cos (* K 0.5)))
        (t_1 (* (* -2.0 J) t_0))
        (t_2 (* (* -2.0 J) (hypot 1.0 (* 0.5 (/ U J))))))
   (if (<= K 800000.0)
     t_2
     (if (<= K 5.7e+20)
       t_1
       (if (<= K 4.8e+34)
         U
         (if (<= K 2.4e+241)
           t_1
           (if (<= K 1.4e+255)
             t_2
             (+ (* -2.0 (* J t_0)) (* -0.25 (* U (/ U J)))))))))))

C

U = abs(U);
double code(double J, double K, double U) {
	double t_0 = cos((K * 0.5));
	double t_1 = (-2.0 * J) * t_0;
	double t_2 = (-2.0 * J) * hypot(1.0, (0.5 * (U / J)));
	double tmp;
	if (K <= 800000.0) {
		tmp = t_2;
	} else if (K <= 5.7e+20) {
		tmp = t_1;
	} else if (K <= 4.8e+34) {
		tmp = U;
	} else if (K <= 2.4e+241) {
		tmp = t_1;
	} else if (K <= 1.4e+255) {
		tmp = t_2;
	} else {
		tmp = (-2.0 * (J * t_0)) + (-0.25 * (U * (U / J)));
	}
	return tmp;
}

Java

U = Math.abs(U);
public static double code(double J, double K, double U) {
	double t_0 = Math.cos((K * 0.5));
	double t_1 = (-2.0 * J) * t_0;
	double t_2 = (-2.0 * J) * Math.hypot(1.0, (0.5 * (U / J)));
	double tmp;
	if (K <= 800000.0) {
		tmp = t_2;
	} else if (K <= 5.7e+20) {
		tmp = t_1;
	} else if (K <= 4.8e+34) {
		tmp = U;
	} else if (K <= 2.4e+241) {
		tmp = t_1;
	} else if (K <= 1.4e+255) {
		tmp = t_2;
	} else {
		tmp = (-2.0 * (J * t_0)) + (-0.25 * (U * (U / J)));
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	t_0 = math.cos((K * 0.5))
	t_1 = (-2.0 * J) * t_0
	t_2 = (-2.0 * J) * math.hypot(1.0, (0.5 * (U / J)))
	tmp = 0
	if K <= 800000.0:
		tmp = t_2
	elif K <= 5.7e+20:
		tmp = t_1
	elif K <= 4.8e+34:
		tmp = U
	elif K <= 2.4e+241:
		tmp = t_1
	elif K <= 1.4e+255:
		tmp = t_2
	else:
		tmp = (-2.0 * (J * t_0)) + (-0.25 * (U * (U / J)))
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	t_0 = cos(Float64(K * 0.5))
	t_1 = Float64(Float64(-2.0 * J) * t_0)
	t_2 = Float64(Float64(-2.0 * J) * hypot(1.0, Float64(0.5 * Float64(U / J))))
	tmp = 0.0
	if (K <= 800000.0)
		tmp = t_2;
	elseif (K <= 5.7e+20)
		tmp = t_1;
	elseif (K <= 4.8e+34)
		tmp = U;
	elseif (K <= 2.4e+241)
		tmp = t_1;
	elseif (K <= 1.4e+255)
		tmp = t_2;
	else
		tmp = Float64(Float64(-2.0 * Float64(J * t_0)) + Float64(-0.25 * Float64(U * Float64(U / J))));
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	t_0 = cos((K * 0.5));
	t_1 = (-2.0 * J) * t_0;
	t_2 = (-2.0 * J) * hypot(1.0, (0.5 * (U / J)));
	tmp = 0.0;
	if (K <= 800000.0)
		tmp = t_2;
	elseif (K <= 5.7e+20)
		tmp = t_1;
	elseif (K <= 4.8e+34)
		tmp = U;
	elseif (K <= 2.4e+241)
		tmp = t_1;
	elseif (K <= 1.4e+255)
		tmp = t_2;
	else
		tmp = (-2.0 * (J * t_0)) + (-0.25 * (U * (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[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(-2.0 * J), $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[(0.5 * N[(U / J), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[K, 800000.0], t$95$2, If[LessEqual[K, 5.7e+20], t$95$1, If[LessEqual[K, 4.8e+34], U, If[LessEqual[K, 2.4e+241], t$95$1, If[LessEqual[K, 1.4e+255], t$95$2, N[(N[(-2.0 * N[(J * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(-0.25 * N[(U * N[(U / J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if K < 8e5 or 2.3999999999999999e241 < K < 1.40000000000000007e255

   1. Initial program 73.1%

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

   3. Simplified 73.0%

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

   4. Taylor expanded in K around 0 39.2%

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

      1. associate-*r* 39.2%

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

      2. *-commutative 39.2%

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

      3. metadata-eval 39.2%

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

      4. unpow2 39.2%

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

      5. unpow2 39.2%

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

      6. times-frac 52.4%

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

      7. swap-sqr 52.9%

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

      8. associate-*r/ 52.9%

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

      9. *-commutative 52.9%

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

      10. associate-*r/ 52.9%

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

      11. associate-*r/ 52.9%

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

      12. *-commutative 52.9%

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

      13. associate-*r/ 52.9%

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

      14. unpow2 52.9%

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

      15. associate-*r/ 52.9%

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

      16. *-commutative 52.9%

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

      17. associate-*r/ 52.9%

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

   6. Simplified 52.9%

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

      1. expm1-log1p-u 51.8%

         \[\leadsto \left(J \cdot -2\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 + {\left(0.5 \cdot \frac{U}{J}\right)}^{2}}\right)\right)} \]

      2. expm1-udef 51.8%

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

      3. metadata-eval 51.8%

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

      4. unpow2 51.8%

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

      5. associate-*r/ 51.8%

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

      6. associate-*r/ 51.8%

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

      7. hypot-udef 65.8%

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

      8. associate-/l* 65.8%

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

   8. Applied egg-rr 65.8%

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

      1. expm1-def 65.8%

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

      2. expm1-log1p 68.6%

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

      3. associate-/l* 68.7%

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

      4. associate-*r/ 68.7%

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

   10. Simplified 68.7%

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

   if 8e5 < K < 5.7e20 or 4.79999999999999974e34 < K < 2.3999999999999999e241

   1. Initial program 81.2%

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

   3. Simplified 81.2%

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

   4. Taylor expanded in J around inf 57.0%

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

      1. associate-*r* 57.0%

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

      2. *-commutative 57.0%

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

   6. Simplified 57.0%

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

   if 5.7e20 < K < 4.79999999999999974e34

   1. Initial program 62.8%

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

   3. Simplified 62.5%

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

   4. Taylor expanded in U around -inf 20.8%

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

   if 1.40000000000000007e255 < K

   1. Initial program 85.0%

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

   3. Simplified 84.9%

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

   4. Taylor expanded in J around inf 50.9%

      \[\leadsto \color{blue}{-2 \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right) + -0.25 \cdot \frac{{U}^{2}}{J \cdot \cos \left(0.5 \cdot K\right)}} \]

   5. Taylor expanded in K around 0 50.3%

      \[\leadsto -2 \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right) + -0.25 \cdot \color{blue}{\frac{{U}^{2}}{J}} \]
   6. Step-by-step derivation

      1. div-inv 9.1%

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

      2. unpow2 9.1%

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

      3. associate-*l* 10.2%

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

      4. div-inv 10.2%

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

   7. Applied egg-rr 60.2%

      \[\leadsto -2 \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right) + -0.25 \cdot \color{blue}{\left(U \cdot \frac{U}{J}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;K \leq 800000:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \mathsf{hypot}\left(1, 0.5 \cdot \frac{U}{J}\right)\\ \mathbf{elif}\;K \leq 5.7 \cdot 10^{+20}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{elif}\;K \leq 4.8 \cdot 10^{+34}:\\ \;\;\;\;U\\ \mathbf{elif}\;K \leq 2.4 \cdot 10^{+241}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{elif}\;K \leq 1.4 \cdot 10^{+255}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \mathsf{hypot}\left(1, 0.5 \cdot \frac{U}{J}\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right) + -0.25 \cdot \left(U \cdot \frac{U}{J}\right)\\ \end{array} \]

Alternative 6: 63.0% accurate, 3.5× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} t_0 := \left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ t_1 := \left(-2 \cdot J\right) \cdot \mathsf{hypot}\left(1, 0.5 \cdot \frac{U}{J}\right)\\ \mathbf{if}\;K \leq 800000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;K \leq 3.5 \cdot 10^{+20}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;K \leq 4.8 \cdot 10^{+34}:\\ \;\;\;\;U\\ \mathbf{elif}\;K \leq 1.5 \cdot 10^{+241} \lor \neg \left(K \leq 1.85 \cdot 10^{+254}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (* (* -2.0 J) (cos (* K 0.5))))
        (t_1 (* (* -2.0 J) (hypot 1.0 (* 0.5 (/ U J))))))
   (if (<= K 800000.0)
     t_1
     (if (<= K 3.5e+20)
       t_0
       (if (<= K 4.8e+34)
         U
         (if (or (<= K 1.5e+241) (not (<= K 1.85e+254))) t_0 t_1))))))

C

U = abs(U);
double code(double J, double K, double U) {
	double t_0 = (-2.0 * J) * cos((K * 0.5));
	double t_1 = (-2.0 * J) * hypot(1.0, (0.5 * (U / J)));
	double tmp;
	if (K <= 800000.0) {
		tmp = t_1;
	} else if (K <= 3.5e+20) {
		tmp = t_0;
	} else if (K <= 4.8e+34) {
		tmp = U;
	} else if ((K <= 1.5e+241) || !(K <= 1.85e+254)) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	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));
	double t_1 = (-2.0 * J) * Math.hypot(1.0, (0.5 * (U / J)));
	double tmp;
	if (K <= 800000.0) {
		tmp = t_1;
	} else if (K <= 3.5e+20) {
		tmp = t_0;
	} else if (K <= 4.8e+34) {
		tmp = U;
	} else if ((K <= 1.5e+241) || !(K <= 1.85e+254)) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	t_0 = (-2.0 * J) * math.cos((K * 0.5))
	t_1 = (-2.0 * J) * math.hypot(1.0, (0.5 * (U / J)))
	tmp = 0
	if K <= 800000.0:
		tmp = t_1
	elif K <= 3.5e+20:
		tmp = t_0
	elif K <= 4.8e+34:
		tmp = U
	elif (K <= 1.5e+241) or not (K <= 1.85e+254):
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	t_0 = Float64(Float64(-2.0 * J) * cos(Float64(K * 0.5)))
	t_1 = Float64(Float64(-2.0 * J) * hypot(1.0, Float64(0.5 * Float64(U / J))))
	tmp = 0.0
	if (K <= 800000.0)
		tmp = t_1;
	elseif (K <= 3.5e+20)
		tmp = t_0;
	elseif (K <= 4.8e+34)
		tmp = U;
	elseif ((K <= 1.5e+241) || !(K <= 1.85e+254))
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	t_0 = (-2.0 * J) * cos((K * 0.5));
	t_1 = (-2.0 * J) * hypot(1.0, (0.5 * (U / J)));
	tmp = 0.0;
	if (K <= 800000.0)
		tmp = t_1;
	elseif (K <= 3.5e+20)
		tmp = t_0;
	elseif (K <= 4.8e+34)
		tmp = U;
	elseif ((K <= 1.5e+241) || ~((K <= 1.85e+254)))
		tmp = t_0;
	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[(N[(-2.0 * J), $MachinePrecision] * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(-2.0 * J), $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[(0.5 * N[(U / J), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[K, 800000.0], t$95$1, If[LessEqual[K, 3.5e+20], t$95$0, If[LessEqual[K, 4.8e+34], U, If[Or[LessEqual[K, 1.5e+241], N[Not[LessEqual[K, 1.85e+254]], $MachinePrecision]], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if K < 8e5 or 1.50000000000000008e241 < K < 1.8499999999999999e254

   1. Initial program 73.1%

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

   3. Simplified 73.0%

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

   4. Taylor expanded in K around 0 39.2%

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

      1. associate-*r* 39.2%

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

      2. *-commutative 39.2%

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

      3. metadata-eval 39.2%

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

      4. unpow2 39.2%

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

      5. unpow2 39.2%

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

      6. times-frac 52.4%

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

      7. swap-sqr 52.9%

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

      8. associate-*r/ 52.9%

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

      9. *-commutative 52.9%

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

      10. associate-*r/ 52.9%

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

      11. associate-*r/ 52.9%

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

      12. *-commutative 52.9%

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

      13. associate-*r/ 52.9%

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

      14. unpow2 52.9%

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

      15. associate-*r/ 52.9%

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

      16. *-commutative 52.9%

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

      17. associate-*r/ 52.9%

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

   6. Simplified 52.9%

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

      1. expm1-log1p-u 51.8%

         \[\leadsto \left(J \cdot -2\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 + {\left(0.5 \cdot \frac{U}{J}\right)}^{2}}\right)\right)} \]

      2. expm1-udef 51.8%

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

      3. metadata-eval 51.8%

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

      4. unpow2 51.8%

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

      5. associate-*r/ 51.8%

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

      6. associate-*r/ 51.8%

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

      7. hypot-udef 65.8%

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

      8. associate-/l* 65.8%

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

   8. Applied egg-rr 65.8%

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

      1. expm1-def 65.8%

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

      2. expm1-log1p 68.6%

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

      3. associate-/l* 68.7%

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

      4. associate-*r/ 68.7%

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

   10. Simplified 68.7%

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

   if 8e5 < K < 3.5e20 or 4.79999999999999974e34 < K < 1.50000000000000008e241 or 1.8499999999999999e254 < K

   1. Initial program 82.3%

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

   3. Simplified 82.3%

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

   4. Taylor expanded in J around inf 57.9%

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

      1. associate-*r* 57.9%

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

      2. *-commutative 57.9%

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

   6. Simplified 57.9%

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

   if 3.5e20 < K < 4.79999999999999974e34

   1. Initial program 62.8%

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

   3. Simplified 62.5%

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

   4. Taylor expanded in U around -inf 20.8%

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

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;K \leq 800000:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \mathsf{hypot}\left(1, 0.5 \cdot \frac{U}{J}\right)\\ \mathbf{elif}\;K \leq 3.5 \cdot 10^{+20}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{elif}\;K \leq 4.8 \cdot 10^{+34}:\\ \;\;\;\;U\\ \mathbf{elif}\;K \leq 1.5 \cdot 10^{+241} \lor \neg \left(K \leq 1.85 \cdot 10^{+254}\right):\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \mathsf{hypot}\left(1, 0.5 \cdot \frac{U}{J}\right)\\ \end{array} \]

Alternative 7: 66.2% accurate, 3.7× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} t_0 := \left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{if}\;J \leq -1.36 \cdot 10^{-90}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq 1.55 \cdot 10^{-289}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 4.6 \cdot 10^{+14}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

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)))))
   (if (<= J -1.36e-90)
     t_0
     (if (<= J 1.55e-289) U (if (<= J 4.6e+14) (- 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));
	double tmp;
	if (J <= -1.36e-90) {
		tmp = t_0;
	} else if (J <= 1.55e-289) {
		tmp = U;
	} else if (J <= 4.6e+14) {
		tmp = -U;
	} 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 <= (-1.36d-90)) then
        tmp = t_0
    else if (j <= 1.55d-289) then
        tmp = u
    else if (j <= 4.6d+14) then
        tmp = -u
    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 <= -1.36e-90) {
		tmp = t_0;
	} else if (J <= 1.55e-289) {
		tmp = U;
	} else if (J <= 4.6e+14) {
		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))
	tmp = 0
	if J <= -1.36e-90:
		tmp = t_0
	elif J <= 1.55e-289:
		tmp = U
	elif J <= 4.6e+14:
		tmp = -U
	else:
		tmp = t_0
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	t_0 = Float64(Float64(-2.0 * J) * cos(Float64(K * 0.5)))
	tmp = 0.0
	if (J <= -1.36e-90)
		tmp = t_0;
	elseif (J <= 1.55e-289)
		tmp = U;
	elseif (J <= 4.6e+14)
		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));
	tmp = 0.0;
	if (J <= -1.36e-90)
		tmp = t_0;
	elseif (J <= 1.55e-289)
		tmp = U;
	elseif (J <= 4.6e+14)
		tmp = -U;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := Block[{t$95$0 = N[(N[(-2.0 * J), $MachinePrecision] * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[J, -1.36e-90], t$95$0, If[LessEqual[J, 1.55e-289], U, If[LessEqual[J, 4.6e+14], (-U), t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if J < -1.36000000000000011e-90 or 4.6e14 < J

   1. Initial program 91.8%

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

   3. Simplified 91.8%

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

   4. Taylor expanded in J around inf 75.4%

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

      1. associate-*r* 75.4%

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

      2. *-commutative 75.4%

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

   6. Simplified 75.4%

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

   if -1.36000000000000011e-90 < J < 1.55e-289

   1. Initial program 57.5%

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

   3. Simplified 57.4%

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

   4. Taylor expanded in U around -inf 34.1%

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

   if 1.55e-289 < J < 4.6e14

   1. Initial program 47.8%

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

   3. Simplified 47.7%

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

   4. Taylor expanded in J around 0 47.4%

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

      1. neg-mul-1 47.4%

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

   6. Simplified 47.4%

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

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -1.36 \cdot 10^{-90}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{elif}\;J \leq 1.55 \cdot 10^{-289}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 4.6 \cdot 10^{+14}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \end{array} \]

Alternative 8: 49.5% accurate, 24.4× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} t_0 := -2 \cdot J + -0.25 \cdot \left(U \cdot \frac{U}{J}\right)\\ \mathbf{if}\;J \leq -1.75 \cdot 10^{+57}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq 1.55 \cdot 10^{-289}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 1.5 \cdot 10^{+30}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (+ (* -2.0 J) (* -0.25 (* U (/ U J))))))
   (if (<= J -1.75e+57)
     t_0
     (if (<= J 1.55e-289) U (if (<= J 1.5e+30) (- U) t_0)))))

C

U = abs(U);
double code(double J, double K, double U) {
	double t_0 = (-2.0 * J) + (-0.25 * (U * (U / J)));
	double tmp;
	if (J <= -1.75e+57) {
		tmp = t_0;
	} else if (J <= 1.55e-289) {
		tmp = U;
	} else if (J <= 1.5e+30) {
		tmp = -U;
	} 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) + ((-0.25d0) * (u * (u / j)))
    if (j <= (-1.75d+57)) then
        tmp = t_0
    else if (j <= 1.55d-289) then
        tmp = u
    else if (j <= 1.5d+30) then
        tmp = -u
    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) + (-0.25 * (U * (U / J)));
	double tmp;
	if (J <= -1.75e+57) {
		tmp = t_0;
	} else if (J <= 1.55e-289) {
		tmp = U;
	} else if (J <= 1.5e+30) {
		tmp = -U;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	t_0 = (-2.0 * J) + (-0.25 * (U * (U / J)))
	tmp = 0
	if J <= -1.75e+57:
		tmp = t_0
	elif J <= 1.55e-289:
		tmp = U
	elif J <= 1.5e+30:
		tmp = -U
	else:
		tmp = t_0
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	t_0 = Float64(Float64(-2.0 * J) + Float64(-0.25 * Float64(U * Float64(U / J))))
	tmp = 0.0
	if (J <= -1.75e+57)
		tmp = t_0;
	elseif (J <= 1.55e-289)
		tmp = U;
	elseif (J <= 1.5e+30)
		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) + (-0.25 * (U * (U / J)));
	tmp = 0.0;
	if (J <= -1.75e+57)
		tmp = t_0;
	elseif (J <= 1.55e-289)
		tmp = U;
	elseif (J <= 1.5e+30)
		tmp = -U;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := Block[{t$95$0 = N[(N[(-2.0 * J), $MachinePrecision] + N[(-0.25 * N[(U * N[(U / J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[J, -1.75e+57], t$95$0, If[LessEqual[J, 1.55e-289], U, If[LessEqual[J, 1.5e+30], (-U), t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if J < -1.7499999999999999e57 or 1.49999999999999989e30 < J

   1. Initial program 98.2%

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

   3. Simplified 98.2%

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

   4. Taylor expanded in K around 0 44.5%

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

      1. associate-*r* 44.5%

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

      2. *-commutative 44.5%

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

      3. metadata-eval 44.5%

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

      4. unpow2 44.5%

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

      5. unpow2 44.5%

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

      6. times-frac 55.5%

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

      7. swap-sqr 55.5%

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

      8. associate-*r/ 55.5%

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

      9. *-commutative 55.5%

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

      10. associate-*r/ 55.5%

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

      11. associate-*r/ 55.5%

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

      12. *-commutative 55.5%

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

      13. associate-*r/ 55.5%

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

      14. unpow2 55.5%

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

      15. associate-*r/ 55.5%

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

      16. *-commutative 55.5%

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

      17. associate-*r/ 55.5%

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

   6. Simplified 55.5%

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

   7. Taylor expanded in J around inf 42.6%

      \[\leadsto \color{blue}{-2 \cdot J + -0.25 \cdot \frac{{U}^{2}}{J}} \]
   8. Step-by-step derivation

      1. div-inv 42.6%

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

      2. unpow2 42.6%

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

      3. associate-*l* 46.6%

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

      4. div-inv 46.6%

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

   9. Applied egg-rr 46.6%

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

   if -1.7499999999999999e57 < J < 1.55e-289

   1. Initial program 65.2%

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

   3. Simplified 65.1%

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

   4. Taylor expanded in U around -inf 27.4%

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

   if 1.55e-289 < J < 1.49999999999999989e30

   1. Initial program 49.4%

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

   3. Simplified 49.4%

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

   4. Taylor expanded in J around 0 45.9%

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

      1. neg-mul-1 45.9%

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

   6. Simplified 45.9%

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

4. Final simplification 39.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -1.75 \cdot 10^{+57}:\\ \;\;\;\;-2 \cdot J + -0.25 \cdot \left(U \cdot \frac{U}{J}\right)\\ \mathbf{elif}\;J \leq 1.55 \cdot 10^{-289}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 1.5 \cdot 10^{+30}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot J + -0.25 \cdot \left(U \cdot \frac{U}{J}\right)\\ \end{array} \]

Alternative 9: 49.3% accurate, 45.7× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} \mathbf{if}\;J \leq -2.65 \cdot 10^{+57}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;J \leq 1.55 \cdot 10^{-289}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 10^{+29}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot J\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (if (<= J -2.65e+57)
   (* -2.0 J)
   (if (<= J 1.55e-289) U (if (<= J 1e+29) (- U) (* -2.0 J)))))

C

U = abs(U);
double code(double J, double K, double U) {
	double tmp;
	if (J <= -2.65e+57) {
		tmp = -2.0 * J;
	} else if (J <= 1.55e-289) {
		tmp = U;
	} else if (J <= 1e+29) {
		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 <= (-2.65d+57)) then
        tmp = (-2.0d0) * j
    else if (j <= 1.55d-289) then
        tmp = u
    else if (j <= 1d+29) 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 <= -2.65e+57) {
		tmp = -2.0 * J;
	} else if (J <= 1.55e-289) {
		tmp = U;
	} else if (J <= 1e+29) {
		tmp = -U;
	} else {
		tmp = -2.0 * J;
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	tmp = 0
	if J <= -2.65e+57:
		tmp = -2.0 * J
	elif J <= 1.55e-289:
		tmp = U
	elif J <= 1e+29:
		tmp = -U
	else:
		tmp = -2.0 * J
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	tmp = 0.0
	if (J <= -2.65e+57)
		tmp = Float64(-2.0 * J);
	elseif (J <= 1.55e-289)
		tmp = U;
	elseif (J <= 1e+29)
		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 <= -2.65e+57)
		tmp = -2.0 * J;
	elseif (J <= 1.55e-289)
		tmp = U;
	elseif (J <= 1e+29)
		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, -2.65e+57], N[(-2.0 * J), $MachinePrecision], If[LessEqual[J, 1.55e-289], U, If[LessEqual[J, 1e+29], (-U), N[(-2.0 * J), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if J < -2.64999999999999993e57 or 9.99999999999999914e28 < J

   1. Initial program 98.2%

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

   3. Simplified 98.2%

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

   4. Taylor expanded in K around 0 44.5%

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

      1. associate-*r* 44.5%

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

      2. *-commutative 44.5%

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

      3. metadata-eval 44.5%

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

      4. unpow2 44.5%

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

      5. unpow2 44.5%

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

      6. times-frac 55.5%

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

      7. swap-sqr 55.5%

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

      8. associate-*r/ 55.5%

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

      9. *-commutative 55.5%

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

      10. associate-*r/ 55.5%

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

      11. associate-*r/ 55.5%

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

      12. *-commutative 55.5%

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

      13. associate-*r/ 55.5%

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

      14. unpow2 55.5%

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

      15. associate-*r/ 55.5%

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

      16. *-commutative 55.5%

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

      17. associate-*r/ 55.5%

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

   6. Simplified 55.5%

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

   7. Taylor expanded in J around inf 46.3%

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

      1. *-commutative 46.3%

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

   9. Simplified 46.3%

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

   if -2.64999999999999993e57 < J < 1.55e-289

   1. Initial program 65.2%

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

   3. Simplified 65.1%

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

   4. Taylor expanded in U around -inf 27.4%

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

   if 1.55e-289 < J < 9.99999999999999914e28

   1. Initial program 49.4%

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

   3. Simplified 49.4%

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

   4. Taylor expanded in J around 0 45.9%

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

      1. neg-mul-1 45.9%

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

   6. Simplified 45.9%

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

4. Final simplification 39.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -2.65 \cdot 10^{+57}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;J \leq 1.55 \cdot 10^{-289}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 10^{+29}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot J\\ \end{array} \]

Alternative 10: 39.2% accurate, 103.4× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} \mathbf{if}\;J \leq 1.55 \cdot 10^{-289}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U) :precision binary64 (if (<= J 1.55e-289) U (- U)))

C

U = abs(U);
double code(double J, double K, double U) {
	double tmp;
	if (J <= 1.55e-289) {
		tmp = U;
	} else {
		tmp = -U;
	}
	return tmp;
}

Fortran

NOTE: U should be positive before calling this function
real(8) function code(j, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (j <= 1.55d-289) then
        tmp = u
    else
        tmp = -u
    end if
    code = tmp
end function

Java

U = Math.abs(U);
public static double code(double J, double K, double U) {
	double tmp;
	if (J <= 1.55e-289) {
		tmp = U;
	} else {
		tmp = -U;
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	tmp = 0
	if J <= 1.55e-289:
		tmp = U
	else:
		tmp = -U
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	tmp = 0.0
	if (J <= 1.55e-289)
		tmp = U;
	else
		tmp = Float64(-U);
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	tmp = 0.0;
	if (J <= 1.55e-289)
		tmp = U;
	else
		tmp = -U;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := If[LessEqual[J, 1.55e-289], U, (-U)]

Derivation

1. Split input into 2 regimes

2. if J < 1.55e-289

   1. Initial program 77.1%

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

   3. Simplified 77.0%

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

   4. Taylor expanded in U around -inf 21.8%

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

   if 1.55e-289 < J

   1. Initial program 73.1%

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

   3. Simplified 73.0%

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

   4. Taylor expanded in J around 0 27.8%

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

      1. neg-mul-1 27.8%

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

   6. Simplified 27.8%

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

4. Final simplification 24.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq 1.55 \cdot 10^{-289}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]

Alternative 11: 26.8% 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 75.2%

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

3. Simplified 75.1%

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

4. Taylor expanded in U around -inf 21.5%

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

5. Final simplification 21.5%

   \[\leadsto U \]

Reproduce

herbie shell --seed 2023318 
(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)))))