Maksimov and Kolovsky, Equation (3)

Percentage Accurate: 73.2% → 99.2%

Time: 17.6s

Alternatives: 9

Speedup: 0.3×

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: 73.2% 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

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(t_0 \cdot \left(-2 \cdot J\right)\right) \cdot \sqrt{1 + {\left(\frac{U_m}{t_0 \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;-U_m\\ \mathbf{elif}\;t_1 \leq 10^{+299}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;U_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* t_0 (* -2.0 J))
          (sqrt (+ 1.0 (pow (/ U_m (* t_0 (* J 2.0))) 2.0))))))
   (if (<= t_1 (- INFINITY)) (- U_m) (if (<= t_1 1e+299) t_1 U_m))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = (t_0 * (-2.0 * J)) * sqrt((1.0 + pow((U_m / (t_0 * (J * 2.0))), 2.0)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_1 <= 1e+299) {
		tmp = t_1;
	} else {
		tmp = U_m;
	}
	return tmp;
}

Java

U_m = Math.abs(U);
public static double code(double J, double K, double U_m) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = (t_0 * (-2.0 * J)) * Math.sqrt((1.0 + Math.pow((U_m / (t_0 * (J * 2.0))), 2.0)));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = -U_m;
	} else if (t_1 <= 1e+299) {
		tmp = t_1;
	} else {
		tmp = U_m;
	}
	return tmp;
}

Python

U_m = math.fabs(U)
def code(J, K, U_m):
	t_0 = math.cos((K / 2.0))
	t_1 = (t_0 * (-2.0 * J)) * math.sqrt((1.0 + math.pow((U_m / (t_0 * (J * 2.0))), 2.0)))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = -U_m
	elif t_1 <= 1e+299:
		tmp = t_1
	else:
		tmp = U_m
	return tmp

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(t_0 * Float64(-2.0 * J)) * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_0 * Float64(J * 2.0))) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_1 <= 1e+299)
		tmp = t_1;
	else
		tmp = U_m;
	end
	return tmp
end

MATLAB

U_m = abs(U);
function tmp_2 = code(J, K, U_m)
	t_0 = cos((K / 2.0));
	t_1 = (t_0 * (-2.0 * J)) * sqrt((1.0 + ((U_m / (t_0 * (J * 2.0))) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = -U_m;
	elseif (t_1 <= 1e+299)
		tmp = t_1;
	else
		tmp = U_m;
	end
	tmp_2 = tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * N[(-2.0 * J), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(t$95$0 * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], (-U$95$m), If[LessEqual[t$95$1, 1e+299], t$95$1, U$95$m]]]]

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 5.4%

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

   3. Simplified 5.4%

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

   5. Taylor expanded in J around 0 69.4%

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

      1. mul-1-neg 69.4%

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

   7. Simplified 69.4%

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

   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)))) < 1.0000000000000001e299

   1. Initial program 99.8%

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

   if 1.0000000000000001e299 < (*.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%

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

   3. Simplified 7.9%

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

   5. Taylor expanded in U around -inf 52.8%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;\left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 10^{+299}:\\ \;\;\;\;\left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 50.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := J \cdot \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;J \leq 4.2 \cdot 10^{-199}:\\ \;\;\;\;-U_m\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(t_0 \cdot \mathsf{hypot}\left(1, \frac{\frac{U_m}{2}}{t_0}\right)\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (* J (cos (/ K 2.0)))))
   (if (<= J 4.2e-199)
     (- U_m)
     (* -2.0 (* t_0 (hypot 1.0 (/ (/ U_m 2.0) t_0)))))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = J * cos((K / 2.0));
	double tmp;
	if (J <= 4.2e-199) {
		tmp = -U_m;
	} else {
		tmp = -2.0 * (t_0 * hypot(1.0, ((U_m / 2.0) / t_0)));
	}
	return tmp;
}

Java

U_m = Math.abs(U);
public static double code(double J, double K, double U_m) {
	double t_0 = J * Math.cos((K / 2.0));
	double tmp;
	if (J <= 4.2e-199) {
		tmp = -U_m;
	} else {
		tmp = -2.0 * (t_0 * Math.hypot(1.0, ((U_m / 2.0) / t_0)));
	}
	return tmp;
}

Python

U_m = math.fabs(U)
def code(J, K, U_m):
	t_0 = J * math.cos((K / 2.0))
	tmp = 0
	if J <= 4.2e-199:
		tmp = -U_m
	else:
		tmp = -2.0 * (t_0 * math.hypot(1.0, ((U_m / 2.0) / t_0)))
	return tmp

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = Float64(J * cos(Float64(K / 2.0)))
	tmp = 0.0
	if (J <= 4.2e-199)
		tmp = Float64(-U_m);
	else
		tmp = Float64(-2.0 * Float64(t_0 * hypot(1.0, Float64(Float64(U_m / 2.0) / t_0))));
	end
	return tmp
end

MATLAB

U_m = abs(U);
function tmp_2 = code(J, K, U_m)
	t_0 = J * cos((K / 2.0));
	tmp = 0.0;
	if (J <= 4.2e-199)
		tmp = -U_m;
	else
		tmp = -2.0 * (t_0 * hypot(1.0, ((U_m / 2.0) / t_0)));
	end
	tmp_2 = tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := Block[{t$95$0 = N[(J * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[J, 4.2e-199], (-U$95$m), N[(-2.0 * N[(t$95$0 * N[Sqrt[1.0 ^ 2 + N[(N[(U$95$m / 2.0), $MachinePrecision] / t$95$0), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if J < 4.20000000000000004e-199

   1. Initial program 63.2%

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

   3. Simplified 63.2%

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

   5. Taylor expanded in J around 0 32.4%

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

      1. mul-1-neg 32.4%

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

   7. Simplified 32.4%

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

   if 4.20000000000000004e-199 < J

   1. Initial program 81.7%

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

      1. associate-*l* 81.7%

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

      2. associate-*l* 81.7%

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

      3. unpow2 81.7%

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

      4. sqr-neg 81.7%

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

      5. distribute-frac-neg 81.7%

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

      6. distribute-frac-neg 81.7%

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

      7. unpow2 81.7%

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

   3. Simplified 94.7%

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

4. Final simplification 58.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq 4.2 \cdot 10^{-199}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\left(J \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{\frac{U}{2}}{J \cdot \cos \left(\frac{K}{2}\right)}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 45.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} \mathbf{if}\;J \leq 9 \cdot 10^{-82}:\\ \;\;\;\;-U_m\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\left(J \cdot \mathsf{hypot}\left(1, U_m \cdot \frac{0.5}{J}\right)\right) \cdot \cos \left(K \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (if (<= J 9e-82)
   (- U_m)
   (* -2.0 (* (* J (hypot 1.0 (* U_m (/ 0.5 J)))) (cos (* K 0.5))))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double tmp;
	if (J <= 9e-82) {
		tmp = -U_m;
	} else {
		tmp = -2.0 * ((J * hypot(1.0, (U_m * (0.5 / J)))) * cos((K * 0.5)));
	}
	return tmp;
}

Java

U_m = Math.abs(U);
public static double code(double J, double K, double U_m) {
	double tmp;
	if (J <= 9e-82) {
		tmp = -U_m;
	} else {
		tmp = -2.0 * ((J * Math.hypot(1.0, (U_m * (0.5 / J)))) * Math.cos((K * 0.5)));
	}
	return tmp;
}

Python

U_m = math.fabs(U)
def code(J, K, U_m):
	tmp = 0
	if J <= 9e-82:
		tmp = -U_m
	else:
		tmp = -2.0 * ((J * math.hypot(1.0, (U_m * (0.5 / J)))) * math.cos((K * 0.5)))
	return tmp

Julia

U_m = abs(U)
function code(J, K, U_m)
	tmp = 0.0
	if (J <= 9e-82)
		tmp = Float64(-U_m);
	else
		tmp = Float64(-2.0 * Float64(Float64(J * hypot(1.0, Float64(U_m * Float64(0.5 / J)))) * cos(Float64(K * 0.5))));
	end
	return tmp
end

MATLAB

U_m = abs(U);
function tmp_2 = code(J, K, U_m)
	tmp = 0.0;
	if (J <= 9e-82)
		tmp = -U_m;
	else
		tmp = -2.0 * ((J * hypot(1.0, (U_m * (0.5 / J)))) * cos((K * 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := If[LessEqual[J, 9e-82], (-U$95$m), N[(-2.0 * N[(N[(J * N[Sqrt[1.0 ^ 2 + N[(U$95$m * N[(0.5 / J), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if J < 8.9999999999999997e-82

   1. Initial program 62.7%

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

   3. Simplified 62.7%

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

   5. Taylor expanded in J around 0 34.9%

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

      1. mul-1-neg 34.9%

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

   7. Simplified 34.9%

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

   if 8.9999999999999997e-82 < J

   1. Initial program 87.0%

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

      1. associate-*l* 87.0%

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

      2. associate-*l* 87.0%

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

      3. unpow2 87.0%

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

      4. sqr-neg 87.0%

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

      5. distribute-frac-neg 87.0%

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

      6. distribute-frac-neg 87.0%

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

      7. unpow2 87.0%

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

   3. Simplified 99.8%

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

      1. add-sqr-sqrt 68.9%

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

      2. pow2 68.9%

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

   6. Applied egg-rr 68.9%

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

   7. Taylor expanded in K around 0 62.6%

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

      1. unpow2 62.6%

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

      2. add-sqr-sqrt 85.5%

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

      3. *-commutative 85.5%

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

      4. *-commutative 85.5%

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

      5. associate-*r* 85.5%

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

      6. *-commutative 85.5%

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

      7. clear-num 85.5%

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

      8. div-inv 85.5%

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

      9. associate-/r/ 85.5%

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

   9. Applied egg-rr 85.5%

      \[\leadsto -2 \cdot \color{blue}{\left(\left(J \cdot \mathsf{hypot}\left(1, \frac{0.5}{J} \cdot U\right)\right) \cdot \cos \left(0.5 \cdot K\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq 9 \cdot 10^{-82}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\left(J \cdot \mathsf{hypot}\left(1, U \cdot \frac{0.5}{J}\right)\right) \cdot \cos \left(K \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 45.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} \mathbf{if}\;J \leq 9 \cdot 10^{-82}:\\ \;\;\;\;-U_m\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\left(J \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{\frac{U_m}{2}}{J}\right)\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (if (<= J 9e-82)
   (- U_m)
   (* -2.0 (* (* J (cos (/ K 2.0))) (hypot 1.0 (/ (/ U_m 2.0) J))))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double tmp;
	if (J <= 9e-82) {
		tmp = -U_m;
	} else {
		tmp = -2.0 * ((J * cos((K / 2.0))) * hypot(1.0, ((U_m / 2.0) / J)));
	}
	return tmp;
}

Java

U_m = Math.abs(U);
public static double code(double J, double K, double U_m) {
	double tmp;
	if (J <= 9e-82) {
		tmp = -U_m;
	} else {
		tmp = -2.0 * ((J * Math.cos((K / 2.0))) * Math.hypot(1.0, ((U_m / 2.0) / J)));
	}
	return tmp;
}

Python

U_m = math.fabs(U)
def code(J, K, U_m):
	tmp = 0
	if J <= 9e-82:
		tmp = -U_m
	else:
		tmp = -2.0 * ((J * math.cos((K / 2.0))) * math.hypot(1.0, ((U_m / 2.0) / J)))
	return tmp

Julia

U_m = abs(U)
function code(J, K, U_m)
	tmp = 0.0
	if (J <= 9e-82)
		tmp = Float64(-U_m);
	else
		tmp = Float64(-2.0 * Float64(Float64(J * cos(Float64(K / 2.0))) * hypot(1.0, Float64(Float64(U_m / 2.0) / J))));
	end
	return tmp
end

MATLAB

U_m = abs(U);
function tmp_2 = code(J, K, U_m)
	tmp = 0.0;
	if (J <= 9e-82)
		tmp = -U_m;
	else
		tmp = -2.0 * ((J * cos((K / 2.0))) * hypot(1.0, ((U_m / 2.0) / J)));
	end
	tmp_2 = tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := If[LessEqual[J, 9e-82], (-U$95$m), N[(-2.0 * N[(N[(J * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[(N[(U$95$m / 2.0), $MachinePrecision] / J), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if J < 8.9999999999999997e-82

   1. Initial program 62.7%

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

   3. Simplified 62.7%

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

   5. Taylor expanded in J around 0 34.9%

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

      1. mul-1-neg 34.9%

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

   7. Simplified 34.9%

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

   if 8.9999999999999997e-82 < J

   1. Initial program 87.0%

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

      1. associate-*l* 87.0%

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

      2. associate-*l* 87.0%

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

      3. unpow2 87.0%

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

      4. sqr-neg 87.0%

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

      5. distribute-frac-neg 87.0%

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

      6. distribute-frac-neg 87.0%

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

      7. unpow2 87.0%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in K around 0 85.5%

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

4. Final simplification 52.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq 9 \cdot 10^{-82}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\left(J \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{\frac{U}{2}}{J}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 40.9% accurate, 3.5× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} \mathbf{if}\;J \leq 2.45 \cdot 10^{-80}:\\ \;\;\;\;-U_m\\ \mathbf{elif}\;J \leq 2.1 \cdot 10^{+81}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \mathsf{hypot}\left(1, 0.5 \cdot \frac{U_m}{J}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(K \cdot 0.5\right) \cdot \left(-2 \cdot J\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (if (<= J 2.45e-80)
   (- U_m)
   (if (<= J 2.1e+81)
     (* (* -2.0 J) (hypot 1.0 (* 0.5 (/ U_m J))))
     (* (cos (* K 0.5)) (* -2.0 J)))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double tmp;
	if (J <= 2.45e-80) {
		tmp = -U_m;
	} else if (J <= 2.1e+81) {
		tmp = (-2.0 * J) * hypot(1.0, (0.5 * (U_m / J)));
	} else {
		tmp = cos((K * 0.5)) * (-2.0 * J);
	}
	return tmp;
}

Java

U_m = Math.abs(U);
public static double code(double J, double K, double U_m) {
	double tmp;
	if (J <= 2.45e-80) {
		tmp = -U_m;
	} else if (J <= 2.1e+81) {
		tmp = (-2.0 * J) * Math.hypot(1.0, (0.5 * (U_m / J)));
	} else {
		tmp = Math.cos((K * 0.5)) * (-2.0 * J);
	}
	return tmp;
}

Python

U_m = math.fabs(U)
def code(J, K, U_m):
	tmp = 0
	if J <= 2.45e-80:
		tmp = -U_m
	elif J <= 2.1e+81:
		tmp = (-2.0 * J) * math.hypot(1.0, (0.5 * (U_m / J)))
	else:
		tmp = math.cos((K * 0.5)) * (-2.0 * J)
	return tmp

Julia

U_m = abs(U)
function code(J, K, U_m)
	tmp = 0.0
	if (J <= 2.45e-80)
		tmp = Float64(-U_m);
	elseif (J <= 2.1e+81)
		tmp = Float64(Float64(-2.0 * J) * hypot(1.0, Float64(0.5 * Float64(U_m / J))));
	else
		tmp = Float64(cos(Float64(K * 0.5)) * Float64(-2.0 * J));
	end
	return tmp
end

MATLAB

U_m = abs(U);
function tmp_2 = code(J, K, U_m)
	tmp = 0.0;
	if (J <= 2.45e-80)
		tmp = -U_m;
	elseif (J <= 2.1e+81)
		tmp = (-2.0 * J) * hypot(1.0, (0.5 * (U_m / J)));
	else
		tmp = cos((K * 0.5)) * (-2.0 * J);
	end
	tmp_2 = tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := If[LessEqual[J, 2.45e-80], (-U$95$m), If[LessEqual[J, 2.1e+81], N[(N[(-2.0 * J), $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[(0.5 * N[(U$95$m / J), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(-2.0 * J), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if J < 2.44999999999999995e-80

   1. Initial program 62.7%

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

   3. Simplified 62.7%

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

   5. Taylor expanded in J around 0 34.9%

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

      1. mul-1-neg 34.9%

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

   7. Simplified 34.9%

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

   if 2.44999999999999995e-80 < J < 2.0999999999999999e81

   1. Initial program 64.8%

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

   3. Simplified 64.8%

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

   5. Taylor expanded in K around 0 25.9%

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

      1. associate-*r* 25.9%

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

      2. metadata-eval 25.9%

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

      3. unpow2 25.9%

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

      4. associate-/r* 25.8%

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

      5. unpow2 25.8%

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

      6. associate-*r/ 28.9%

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

      7. associate-*l/ 34.8%

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

      8. swap-sqr 34.8%

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

      9. unpow2 34.8%

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

   7. Simplified 34.8%

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

      1. expm1-log1p-u 14.2%

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

      2. expm1-udef 7.9%

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

      3. associate-*l* 7.9%

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

      4. unpow2 7.9%

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

      5. hypot-1-def 7.9%

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

   9. Applied egg-rr 7.9%

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

       1. expm1-def 14.2%

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

       2. expm1-log1p 61.3%

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

       3. associate-*r* 61.3%

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

       4. *-commutative 61.3%

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

   11. Simplified 61.3%

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

   if 2.0999999999999999e81 < J

   1. Initial program 99.8%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in J around inf 88.6%

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

      1. associate-*r* 88.6%

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

   7. Simplified 88.6%

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

4. Final simplification 49.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq 2.45 \cdot 10^{-80}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 2.1 \cdot 10^{+81}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \mathsf{hypot}\left(1, 0.5 \cdot \frac{U}{J}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(K \cdot 0.5\right) \cdot \left(-2 \cdot J\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 39.2% accurate, 3.7× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} \mathbf{if}\;J \leq 2.8 \cdot 10^{+38}:\\ \;\;\;\;-U_m\\ \mathbf{else}:\\ \;\;\;\;\cos \left(K \cdot 0.5\right) \cdot \left(-2 \cdot J\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (if (<= J 2.8e+38) (- U_m) (* (cos (* K 0.5)) (* -2.0 J))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double tmp;
	if (J <= 2.8e+38) {
		tmp = -U_m;
	} else {
		tmp = cos((K * 0.5)) * (-2.0 * J);
	}
	return tmp;
}

Fortran

U_m = abs(U)
real(8) function code(j, k, u_m)
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u_m
    real(8) :: tmp
    if (j <= 2.8d+38) then
        tmp = -u_m
    else
        tmp = cos((k * 0.5d0)) * ((-2.0d0) * j)
    end if
    code = tmp
end function

Java

U_m = Math.abs(U);
public static double code(double J, double K, double U_m) {
	double tmp;
	if (J <= 2.8e+38) {
		tmp = -U_m;
	} else {
		tmp = Math.cos((K * 0.5)) * (-2.0 * J);
	}
	return tmp;
}

Python

U_m = math.fabs(U)
def code(J, K, U_m):
	tmp = 0
	if J <= 2.8e+38:
		tmp = -U_m
	else:
		tmp = math.cos((K * 0.5)) * (-2.0 * J)
	return tmp

Julia

U_m = abs(U)
function code(J, K, U_m)
	tmp = 0.0
	if (J <= 2.8e+38)
		tmp = Float64(-U_m);
	else
		tmp = Float64(cos(Float64(K * 0.5)) * Float64(-2.0 * J));
	end
	return tmp
end

MATLAB

U_m = abs(U);
function tmp_2 = code(J, K, U_m)
	tmp = 0.0;
	if (J <= 2.8e+38)
		tmp = -U_m;
	else
		tmp = cos((K * 0.5)) * (-2.0 * J);
	end
	tmp_2 = tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := If[LessEqual[J, 2.8e+38], (-U$95$m), N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(-2.0 * J), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if J < 2.8e38

   1. Initial program 62.3%

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

   3. Simplified 62.3%

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

   5. Taylor expanded in J around 0 36.6%

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

      1. mul-1-neg 36.6%

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

   7. Simplified 36.6%

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

   if 2.8e38 < J

   1. Initial program 96.9%

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

   3. Simplified 96.9%

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

   5. Taylor expanded in J around inf 82.9%

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

      1. associate-*r* 82.9%

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

   7. Simplified 82.9%

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

4. Final simplification 48.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq 2.8 \cdot 10^{+38}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;\cos \left(K \cdot 0.5\right) \cdot \left(-2 \cdot J\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 26.3% accurate, 19.0× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} \mathbf{if}\;K \leq 3.2 \lor \neg \left(K \leq 2.7 \cdot 10^{+98}\right) \land \left(K \leq 7.8 \cdot 10^{+193} \lor \neg \left(K \leq 1.05 \cdot 10^{+225}\right)\right):\\ \;\;\;\;-U_m\\ \mathbf{else}:\\ \;\;\;\;U_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (if (or (<= K 3.2)
         (and (not (<= K 2.7e+98))
              (or (<= K 7.8e+193) (not (<= K 1.05e+225)))))
   (- U_m)
   U_m))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double tmp;
	if ((K <= 3.2) || (!(K <= 2.7e+98) && ((K <= 7.8e+193) || !(K <= 1.05e+225)))) {
		tmp = -U_m;
	} else {
		tmp = U_m;
	}
	return tmp;
}

Fortran

U_m = abs(U)
real(8) function code(j, k, u_m)
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u_m
    real(8) :: tmp
    if ((k <= 3.2d0) .or. (.not. (k <= 2.7d+98)) .and. (k <= 7.8d+193) .or. (.not. (k <= 1.05d+225))) then
        tmp = -u_m
    else
        tmp = u_m
    end if
    code = tmp
end function

Java

U_m = Math.abs(U);
public static double code(double J, double K, double U_m) {
	double tmp;
	if ((K <= 3.2) || (!(K <= 2.7e+98) && ((K <= 7.8e+193) || !(K <= 1.05e+225)))) {
		tmp = -U_m;
	} else {
		tmp = U_m;
	}
	return tmp;
}

Python

U_m = math.fabs(U)
def code(J, K, U_m):
	tmp = 0
	if (K <= 3.2) or (not (K <= 2.7e+98) and ((K <= 7.8e+193) or not (K <= 1.05e+225))):
		tmp = -U_m
	else:
		tmp = U_m
	return tmp

Julia

U_m = abs(U)
function code(J, K, U_m)
	tmp = 0.0
	if ((K <= 3.2) || (!(K <= 2.7e+98) && ((K <= 7.8e+193) || !(K <= 1.05e+225))))
		tmp = Float64(-U_m);
	else
		tmp = U_m;
	end
	return tmp
end

MATLAB

U_m = abs(U);
function tmp_2 = code(J, K, U_m)
	tmp = 0.0;
	if ((K <= 3.2) || (~((K <= 2.7e+98)) && ((K <= 7.8e+193) || ~((K <= 1.05e+225)))))
		tmp = -U_m;
	else
		tmp = U_m;
	end
	tmp_2 = tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := If[Or[LessEqual[K, 3.2], And[N[Not[LessEqual[K, 2.7e+98]], $MachinePrecision], Or[LessEqual[K, 7.8e+193], N[Not[LessEqual[K, 1.05e+225]], $MachinePrecision]]]], (-U$95$m), U$95$m]

Derivation

1. Split input into 2 regimes

2. if K < 3.2000000000000002 or 2.7e98 < K < 7.79999999999999999e193 or 1.05e225 < K

   1. Initial program 72.1%

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

   3. Simplified 72.1%

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

   5. Taylor expanded in J around 0 29.4%

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

      1. mul-1-neg 29.4%

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

   7. Simplified 29.4%

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

   if 3.2000000000000002 < K < 2.7e98 or 7.79999999999999999e193 < K < 1.05e225

   1. Initial program 64.0%

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

   3. Simplified 63.9%

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

   5. Taylor expanded in U around -inf 22.0%

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

4. Final simplification 28.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;K \leq 3.2 \lor \neg \left(K \leq 2.7 \cdot 10^{+98}\right) \land \left(K \leq 7.8 \cdot 10^{+193} \lor \neg \left(K \leq 1.05 \cdot 10^{+225}\right)\right):\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 31.7% accurate, 52.4× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} \mathbf{if}\;J \leq 5.9 \cdot 10^{+84}:\\ \;\;\;\;-U_m\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot J\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m) :precision binary64 (if (<= J 5.9e+84) (- U_m) (* -2.0 J)))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double tmp;
	if (J <= 5.9e+84) {
		tmp = -U_m;
	} else {
		tmp = -2.0 * J;
	}
	return tmp;
}

Fortran

U_m = abs(U)
real(8) function code(j, k, u_m)
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u_m
    real(8) :: tmp
    if (j <= 5.9d+84) then
        tmp = -u_m
    else
        tmp = (-2.0d0) * j
    end if
    code = tmp
end function

Java

U_m = Math.abs(U);
public static double code(double J, double K, double U_m) {
	double tmp;
	if (J <= 5.9e+84) {
		tmp = -U_m;
	} else {
		tmp = -2.0 * J;
	}
	return tmp;
}

Python

U_m = math.fabs(U)
def code(J, K, U_m):
	tmp = 0
	if J <= 5.9e+84:
		tmp = -U_m
	else:
		tmp = -2.0 * J
	return tmp

Julia

U_m = abs(U)
function code(J, K, U_m)
	tmp = 0.0
	if (J <= 5.9e+84)
		tmp = Float64(-U_m);
	else
		tmp = Float64(-2.0 * J);
	end
	return tmp
end

MATLAB

U_m = abs(U);
function tmp_2 = code(J, K, U_m)
	tmp = 0.0;
	if (J <= 5.9e+84)
		tmp = -U_m;
	else
		tmp = -2.0 * J;
	end
	tmp_2 = tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := If[LessEqual[J, 5.9e+84], (-U$95$m), N[(-2.0 * J), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if J < 5.89999999999999984e84

   1. Initial program 63.0%

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

   3. Simplified 63.0%

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

   5. Taylor expanded in J around 0 36.5%

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

      1. mul-1-neg 36.5%

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

   7. Simplified 36.5%

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

   if 5.89999999999999984e84 < J

   1. Initial program 99.8%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in K around 0 42.1%

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

      1. associate-*r* 42.1%

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

      2. metadata-eval 42.1%

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

      3. unpow2 42.1%

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

      4. associate-/r* 42.9%

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

      5. unpow2 42.9%

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

      6. associate-*r/ 50.3%

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

      7. associate-*l/ 51.9%

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

      8. swap-sqr 51.9%

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

      9. unpow2 51.9%

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

   7. Simplified 51.9%

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

   8. Taylor expanded in J around inf 47.9%

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

      1. *-commutative 47.9%

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

   10. Simplified 47.9%

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

4. Final simplification 39.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq 5.9 \cdot 10^{+84}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot J\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 27.4% accurate, 420.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := U$95$m

Derivation

1. Initial program 71.1%

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

3. Simplified 71.1%

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

5. Taylor expanded in U around -inf 23.5%

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

6. Final simplification 23.5%

   \[\leadsto U \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024024 
(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)))))