Maksimov and Kolovsky, Equation (3)

Percentage Accurate: 72.4% → 99.3%

Time: 18.9s

Alternatives: 9

Speedup: 0.4×

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: 72.4% 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: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(t_0 \cdot \left(-2 \cdot J\right)\right) \cdot \sqrt{1 + {\left(\frac{U_m}{t_0 \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ t_2 := J \cdot t_0\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;-U_m\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+299}:\\ \;\;\;\;-2 \cdot \left(t_2 \cdot \mathsf{hypot}\left(1, \frac{\frac{U_m}{2}}{t_2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U_m\\ \end{array} \end{array} \]

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)))))
        (t_2 (* J t_0)))
   (if (<= t_1 (- INFINITY))
     (- U_m)
     (if (<= t_1 2e+299)
       (* -2.0 (* t_2 (hypot 1.0 (/ (/ U_m 2.0) t_2))))
       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 t_2 = J * t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_1 <= 2e+299) {
		tmp = -2.0 * (t_2 * hypot(1.0, ((U_m / 2.0) / t_2)));
	} 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 t_2 = J * t_0;
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = -U_m;
	} else if (t_1 <= 2e+299) {
		tmp = -2.0 * (t_2 * Math.hypot(1.0, ((U_m / 2.0) / t_2)));
	} 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)))
	t_2 = J * t_0
	tmp = 0
	if t_1 <= -math.inf:
		tmp = -U_m
	elif t_1 <= 2e+299:
		tmp = -2.0 * (t_2 * math.hypot(1.0, ((U_m / 2.0) / t_2)))
	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))))
	t_2 = Float64(J * t_0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_1 <= 2e+299)
		tmp = Float64(-2.0 * Float64(t_2 * hypot(1.0, Float64(Float64(U_m / 2.0) / t_2))));
	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)));
	t_2 = J * t_0;
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = -U_m;
	elseif (t_1 <= 2e+299)
		tmp = -2.0 * (t_2 * hypot(1.0, ((U_m / 2.0) / t_2)));
	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]}, Block[{t$95$2 = N[(J * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], (-U$95$m), If[LessEqual[t$95$1, 2e+299], N[(-2.0 * N[(t$95$2 * N[Sqrt[1.0 ^ 2 + N[(N[(U$95$m / 2.0), $MachinePrecision] / t$95$2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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%

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

   2. Taylor expanded in J around 0 57.8%

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

      1. neg-mul-1 57.8%

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

   4. Simplified 57.8%

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

   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)))) < 2.0000000000000001e299

   1. Initial program 99.7%

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

      1. associate-*l* 99.7%

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

      2. associate-*l* 99.7%

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

      3. unpow2 99.7%

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

      4. sqr-neg 99.7%

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

      5. distribute-frac-neg 99.7%

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

      6. distribute-frac-neg 99.7%

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

      7. unpow2 99.7%

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

   3. Simplified 99.7%

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

   if 2.0000000000000001e299 < (*.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 7.8%

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

   2. Taylor expanded in U around -inf 50.3%

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

4. Final simplification 85.8%

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

Alternative 2: 51.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} \mathbf{if}\;J \leq 4 \cdot 10^{-179}:\\ \;\;\;\;-U_m\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \mathsf{hypot}\left(1, \frac{U_m}{J} \cdot \frac{0.5}{\cos \left(K \cdot 0.5\right)}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (if (<= J 4e-179)
   (- U_m)
   (*
    -2.0
    (*
     (cos (/ K 2.0))
     (* J (hypot 1.0 (* (/ U_m J) (/ 0.5 (cos (* K 0.5))))))))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double tmp;
	if (J <= 4e-179) {
		tmp = -U_m;
	} else {
		tmp = -2.0 * (cos((K / 2.0)) * (J * hypot(1.0, ((U_m / J) * (0.5 / 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 <= 4e-179) {
		tmp = -U_m;
	} else {
		tmp = -2.0 * (Math.cos((K / 2.0)) * (J * Math.hypot(1.0, ((U_m / J) * (0.5 / Math.cos((K * 0.5)))))));
	}
	return tmp;
}

Python

U_m = math.fabs(U)
def code(J, K, U_m):
	tmp = 0
	if J <= 4e-179:
		tmp = -U_m
	else:
		tmp = -2.0 * (math.cos((K / 2.0)) * (J * math.hypot(1.0, ((U_m / J) * (0.5 / math.cos((K * 0.5)))))))
	return tmp

Julia

U_m = abs(U)
function code(J, K, U_m)
	tmp = 0.0
	if (J <= 4e-179)
		tmp = Float64(-U_m);
	else
		tmp = Float64(-2.0 * Float64(cos(Float64(K / 2.0)) * Float64(J * hypot(1.0, Float64(Float64(U_m / J) * Float64(0.5 / 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 <= 4e-179)
		tmp = -U_m;
	else
		tmp = -2.0 * (cos((K / 2.0)) * (J * hypot(1.0, ((U_m / J) * (0.5 / 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, 4e-179], (-U$95$m), N[(-2.0 * N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(J * N[Sqrt[1.0 ^ 2 + N[(N[(U$95$m / J), $MachinePrecision] * N[(0.5 / N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if J < 4.0000000000000001e-179

   1. Initial program 64.8%

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

   2. Taylor expanded in J around 0 28.7%

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

      1. neg-mul-1 28.7%

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

   4. Simplified 28.7%

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

   if 4.0000000000000001e-179 < J

   1. Initial program 83.2%

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

      1. associate-*l* 83.2%

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

      2. associate-*l* 83.2%

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

      3. *-commutative 83.2%

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

      4. unpow2 83.2%

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

      5. sqr-neg 83.2%

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

      6. distribute-frac-neg 83.2%

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

      7. distribute-frac-neg 83.2%

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

      8. unpow2 83.2%

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

   3. Simplified 94.6%

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

      1. div-inv 94.6%

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

      2. metadata-eval 94.6%

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

      3. times-frac 94.7%

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

      4. div-inv 94.7%

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

      5. metadata-eval 94.7%

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

   5. Applied egg-rr 94.7%

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

4. Final simplification 52.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq 4 \cdot 10^{-179}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \mathsf{hypot}\left(1, \frac{U}{J} \cdot \frac{0.5}{\cos \left(K \cdot 0.5\right)}\right)\right)\right)\\ \end{array} \]

Alternative 3: 46.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} \mathbf{if}\;J \leq 1.05 \cdot 10^{-162}:\\ \;\;\;\;-U_m\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \mathsf{hypot}\left(1, \frac{U_m}{J} \cdot 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (if (<= J 1.05e-162)
   (- U_m)
   (* -2.0 (* (cos (/ K 2.0)) (* J (hypot 1.0 (* (/ U_m J) 0.5)))))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double tmp;
	if (J <= 1.05e-162) {
		tmp = -U_m;
	} else {
		tmp = -2.0 * (cos((K / 2.0)) * (J * hypot(1.0, ((U_m / J) * 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 <= 1.05e-162) {
		tmp = -U_m;
	} else {
		tmp = -2.0 * (Math.cos((K / 2.0)) * (J * Math.hypot(1.0, ((U_m / J) * 0.5))));
	}
	return tmp;
}

Python

U_m = math.fabs(U)
def code(J, K, U_m):
	tmp = 0
	if J <= 1.05e-162:
		tmp = -U_m
	else:
		tmp = -2.0 * (math.cos((K / 2.0)) * (J * math.hypot(1.0, ((U_m / J) * 0.5))))
	return tmp

Julia

U_m = abs(U)
function code(J, K, U_m)
	tmp = 0.0
	if (J <= 1.05e-162)
		tmp = Float64(-U_m);
	else
		tmp = Float64(-2.0 * Float64(cos(Float64(K / 2.0)) * Float64(J * hypot(1.0, Float64(Float64(U_m / J) * 0.5)))));
	end
	return tmp
end

MATLAB

U_m = abs(U);
function tmp_2 = code(J, K, U_m)
	tmp = 0.0;
	if (J <= 1.05e-162)
		tmp = -U_m;
	else
		tmp = -2.0 * (cos((K / 2.0)) * (J * hypot(1.0, ((U_m / J) * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := If[LessEqual[J, 1.05e-162], (-U$95$m), N[(-2.0 * N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(J * N[Sqrt[1.0 ^ 2 + N[(N[(U$95$m / J), $MachinePrecision] * 0.5), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if J < 1.05e-162

   1. Initial program 64.3%

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

   2. Taylor expanded in J around 0 29.0%

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

      1. neg-mul-1 29.0%

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

   4. Simplified 29.0%

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

   if 1.05e-162 < J

   1. Initial program 85.5%

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

      1. associate-*l* 85.5%

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

      2. associate-*l* 85.5%

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

      3. *-commutative 85.5%

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

      4. unpow2 85.5%

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

      5. sqr-neg 85.5%

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

      6. distribute-frac-neg 85.5%

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

      7. distribute-frac-neg 85.5%

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

      8. unpow2 85.5%

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

   3. Simplified 95.4%

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

   4. Taylor expanded in K around 0 86.3%

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

4. Final simplification 48.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq 1.05 \cdot 10^{-162}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \mathsf{hypot}\left(1, \frac{U}{J} \cdot 0.5\right)\right)\right)\\ \end{array} \]

Alternative 4: 41.1% accurate, 3.7× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} \mathbf{if}\;J \leq 1.9 \cdot 10^{-58}:\\ \;\;\;\;-U_m\\ \mathbf{elif}\;J \leq 2.7 \cdot 10^{+17} \lor \neg \left(J \leq 2.1 \cdot 10^{+38}\right):\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{{J}^{2}}{U_m} - U_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (if (<= J 1.9e-58)
   (- U_m)
   (if (or (<= J 2.7e+17) (not (<= J 2.1e+38)))
     (* (* -2.0 J) (cos (* K 0.5)))
     (- (* -2.0 (/ (pow J 2.0) U_m)) U_m))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double tmp;
	if (J <= 1.9e-58) {
		tmp = -U_m;
	} else if ((J <= 2.7e+17) || !(J <= 2.1e+38)) {
		tmp = (-2.0 * J) * cos((K * 0.5));
	} else {
		tmp = (-2.0 * (pow(J, 2.0) / U_m)) - 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 (j <= 1.9d-58) then
        tmp = -u_m
    else if ((j <= 2.7d+17) .or. (.not. (j <= 2.1d+38))) then
        tmp = ((-2.0d0) * j) * cos((k * 0.5d0))
    else
        tmp = ((-2.0d0) * ((j ** 2.0d0) / u_m)) - 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 (J <= 1.9e-58) {
		tmp = -U_m;
	} else if ((J <= 2.7e+17) || !(J <= 2.1e+38)) {
		tmp = (-2.0 * J) * Math.cos((K * 0.5));
	} else {
		tmp = (-2.0 * (Math.pow(J, 2.0) / U_m)) - U_m;
	}
	return tmp;
}

Python

U_m = math.fabs(U)
def code(J, K, U_m):
	tmp = 0
	if J <= 1.9e-58:
		tmp = -U_m
	elif (J <= 2.7e+17) or not (J <= 2.1e+38):
		tmp = (-2.0 * J) * math.cos((K * 0.5))
	else:
		tmp = (-2.0 * (math.pow(J, 2.0) / U_m)) - U_m
	return tmp

Julia

U_m = abs(U)
function code(J, K, U_m)
	tmp = 0.0
	if (J <= 1.9e-58)
		tmp = Float64(-U_m);
	elseif ((J <= 2.7e+17) || !(J <= 2.1e+38))
		tmp = Float64(Float64(-2.0 * J) * cos(Float64(K * 0.5)));
	else
		tmp = Float64(Float64(-2.0 * Float64((J ^ 2.0) / U_m)) - U_m);
	end
	return tmp
end

MATLAB

U_m = abs(U);
function tmp_2 = code(J, K, U_m)
	tmp = 0.0;
	if (J <= 1.9e-58)
		tmp = -U_m;
	elseif ((J <= 2.7e+17) || ~((J <= 2.1e+38)))
		tmp = (-2.0 * J) * cos((K * 0.5));
	else
		tmp = (-2.0 * ((J ^ 2.0) / U_m)) - U_m;
	end
	tmp_2 = tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := If[LessEqual[J, 1.9e-58], (-U$95$m), If[Or[LessEqual[J, 2.7e+17], N[Not[LessEqual[J, 2.1e+38]], $MachinePrecision]], N[(N[(-2.0 * J), $MachinePrecision] * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 * N[(N[Power[J, 2.0], $MachinePrecision] / U$95$m), $MachinePrecision]), $MachinePrecision] - U$95$m), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if J < 1.8999999999999999e-58

   1. Initial program 63.6%

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

   2. Taylor expanded in J around 0 30.8%

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

      1. neg-mul-1 30.8%

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

   4. Simplified 30.8%

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

   if 1.8999999999999999e-58 < J < 2.7e17 or 2.1e38 < J

   1. Initial program 93.9%

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

   2. Taylor expanded in J around inf 80.9%

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

      1. associate-*r* 80.9%

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

      2. *-commutative 80.9%

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

   4. Simplified 80.9%

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

   if 2.7e17 < J < 2.1e38

   1. Initial program 69.0%

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

   2. Taylor expanded in J around 0 67.0%

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

      1. neg-mul-1 67.0%

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

      2. unsub-neg 67.0%

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

      3. unpow2 67.0%

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

      4. *-commutative 67.0%

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

      5. unpow2 67.0%

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

      6. swap-sqr 67.0%

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

      7. unpow2 67.0%

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

      8. *-commutative 67.0%

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

   4. Simplified 67.0%

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

   5. Taylor expanded in K around 0 67.0%

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

4. Final simplification 43.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq 1.9 \cdot 10^{-58}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 2.7 \cdot 10^{+17} \lor \neg \left(J \leq 2.1 \cdot 10^{+38}\right):\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{{J}^{2}}{U} - U\\ \end{array} \]

Alternative 5: 63.5% accurate, 3.7× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} \mathbf{if}\;U_m \leq 1.45 \cdot 10^{-55}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{elif}\;U_m \leq 5.6 \cdot 10^{+101}:\\ \;\;\;\;-2 \cdot \left(J \cdot \mathsf{hypot}\left(1, \frac{0.5}{\frac{J}{U_m}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-U_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (if (<= U_m 1.45e-55)
   (* (* -2.0 J) (cos (* K 0.5)))
   (if (<= U_m 5.6e+101)
     (* -2.0 (* J (hypot 1.0 (/ 0.5 (/ J U_m)))))
     (- U_m))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double tmp;
	if (U_m <= 1.45e-55) {
		tmp = (-2.0 * J) * cos((K * 0.5));
	} else if (U_m <= 5.6e+101) {
		tmp = -2.0 * (J * hypot(1.0, (0.5 / (J / U_m))));
	} else {
		tmp = -U_m;
	}
	return tmp;
}

Java

U_m = Math.abs(U);
public static double code(double J, double K, double U_m) {
	double tmp;
	if (U_m <= 1.45e-55) {
		tmp = (-2.0 * J) * Math.cos((K * 0.5));
	} else if (U_m <= 5.6e+101) {
		tmp = -2.0 * (J * Math.hypot(1.0, (0.5 / (J / U_m))));
	} else {
		tmp = -U_m;
	}
	return tmp;
}

Python

U_m = math.fabs(U)
def code(J, K, U_m):
	tmp = 0
	if U_m <= 1.45e-55:
		tmp = (-2.0 * J) * math.cos((K * 0.5))
	elif U_m <= 5.6e+101:
		tmp = -2.0 * (J * math.hypot(1.0, (0.5 / (J / U_m))))
	else:
		tmp = -U_m
	return tmp

Julia

U_m = abs(U)
function code(J, K, U_m)
	tmp = 0.0
	if (U_m <= 1.45e-55)
		tmp = Float64(Float64(-2.0 * J) * cos(Float64(K * 0.5)));
	elseif (U_m <= 5.6e+101)
		tmp = Float64(-2.0 * Float64(J * hypot(1.0, Float64(0.5 / Float64(J / U_m)))));
	else
		tmp = Float64(-U_m);
	end
	return tmp
end

MATLAB

U_m = abs(U);
function tmp_2 = code(J, K, U_m)
	tmp = 0.0;
	if (U_m <= 1.45e-55)
		tmp = (-2.0 * J) * cos((K * 0.5));
	elseif (U_m <= 5.6e+101)
		tmp = -2.0 * (J * hypot(1.0, (0.5 / (J / 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[LessEqual[U$95$m, 1.45e-55], N[(N[(-2.0 * J), $MachinePrecision] * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[U$95$m, 5.6e+101], N[(-2.0 * N[(J * N[Sqrt[1.0 ^ 2 + N[(0.5 / N[(J / U$95$m), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-U$95$m)]]

Derivation

1. Split input into 3 regimes

2. if U < 1.45e-55

   1. Initial program 80.2%

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

   2. Taylor expanded in J around inf 62.6%

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

      1. associate-*r* 62.6%

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

      2. *-commutative 62.6%

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

   4. Simplified 62.6%

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

   if 1.45e-55 < U < 5.59999999999999962e101

   1. Initial program 78.0%

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

   2. Taylor expanded in K around 0 50.2%

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

      1. add-sqr-sqrt 50.2%

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

      2. hypot-1-def 50.2%

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

      3. sqrt-prod 50.2%

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

      4. metadata-eval 50.2%

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

      5. sqrt-div 53.3%

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

      6. unpow2 53.3%

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

      7. sqrt-prod 53.2%

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

      8. add-sqr-sqrt 53.3%

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

      9. unpow2 53.3%

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

      10. sqrt-prod 31.9%

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

      11. add-sqr-sqrt 62.7%

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

      12. clear-num 62.6%

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

      13. un-div-inv 62.6%

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

   4. Applied egg-rr 62.6%

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

   if 5.59999999999999962e101 < U

   1. Initial program 32.3%

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

   2. Taylor expanded in J around 0 50.6%

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

      1. neg-mul-1 50.6%

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

   4. Simplified 50.6%

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

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 1.45 \cdot 10^{-55}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{elif}\;U \leq 5.6 \cdot 10^{+101}:\\ \;\;\;\;-2 \cdot \left(J \cdot \mathsf{hypot}\left(1, \frac{0.5}{\frac{J}{U}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]

Alternative 6: 41.0% accurate, 3.7× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} \mathbf{if}\;J \leq 2.4 \cdot 10^{-57} \lor \neg \left(J \leq 2.7 \cdot 10^{+17}\right) \land J \leq 2.75 \cdot 10^{+38}:\\ \;\;\;\;-U_m\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (if (or (<= J 2.4e-57) (and (not (<= J 2.7e+17)) (<= J 2.75e+38)))
   (- U_m)
   (* (* -2.0 J) (cos (* K 0.5)))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double tmp;
	if ((J <= 2.4e-57) || (!(J <= 2.7e+17) && (J <= 2.75e+38))) {
		tmp = -U_m;
	} else {
		tmp = (-2.0 * J) * cos((K * 0.5));
	}
	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 <= 2.4d-57) .or. (.not. (j <= 2.7d+17)) .and. (j <= 2.75d+38)) then
        tmp = -u_m
    else
        tmp = ((-2.0d0) * j) * cos((k * 0.5d0))
    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 <= 2.4e-57) || (!(J <= 2.7e+17) && (J <= 2.75e+38))) {
		tmp = -U_m;
	} else {
		tmp = (-2.0 * J) * Math.cos((K * 0.5));
	}
	return tmp;
}

Python

U_m = math.fabs(U)
def code(J, K, U_m):
	tmp = 0
	if (J <= 2.4e-57) or (not (J <= 2.7e+17) and (J <= 2.75e+38)):
		tmp = -U_m
	else:
		tmp = (-2.0 * J) * math.cos((K * 0.5))
	return tmp

Julia

U_m = abs(U)
function code(J, K, U_m)
	tmp = 0.0
	if ((J <= 2.4e-57) || (!(J <= 2.7e+17) && (J <= 2.75e+38)))
		tmp = Float64(-U_m);
	else
		tmp = Float64(Float64(-2.0 * 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 <= 2.4e-57) || (~((J <= 2.7e+17)) && (J <= 2.75e+38)))
		tmp = -U_m;
	else
		tmp = (-2.0 * J) * cos((K * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := If[Or[LessEqual[J, 2.4e-57], And[N[Not[LessEqual[J, 2.7e+17]], $MachinePrecision], LessEqual[J, 2.75e+38]]], (-U$95$m), N[(N[(-2.0 * J), $MachinePrecision] * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if J < 2.40000000000000006e-57 or 2.7e17 < J < 2.7500000000000002e38

   1. Initial program 63.7%

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

   2. Taylor expanded in J around 0 31.3%

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

      1. neg-mul-1 31.3%

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

   4. Simplified 31.3%

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

   if 2.40000000000000006e-57 < J < 2.7e17 or 2.7500000000000002e38 < J

   1. Initial program 93.9%

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

   2. Taylor expanded in J around inf 80.9%

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

      1. associate-*r* 80.9%

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

      2. *-commutative 80.9%

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

   4. Simplified 80.9%

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

4. Final simplification 43.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq 2.4 \cdot 10^{-57} \lor \neg \left(J \leq 2.7 \cdot 10^{+17}\right) \land J \leq 2.75 \cdot 10^{+38}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \end{array} \]

Alternative 7: 33.3% accurate, 83.0× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} \mathbf{if}\;J \leq 1.45 \cdot 10^{+42}:\\ \;\;\;\;-U_m\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot J\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m) :precision binary64 (if (<= J 1.45e+42) (- U_m) (* -2.0 J)))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double tmp;
	if (J <= 1.45e+42) {
		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 <= 1.45d+42) 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 <= 1.45e+42) {
		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 <= 1.45e+42:
		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 <= 1.45e+42)
		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 <= 1.45e+42)
		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, 1.45e+42], (-U$95$m), N[(-2.0 * J), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if J < 1.4499999999999999e42

   1. Initial program 64.7%

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

   2. Taylor expanded in J around 0 31.4%

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

      1. neg-mul-1 31.4%

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

   4. Simplified 31.4%

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

   if 1.4499999999999999e42 < J

   1. Initial program 96.2%

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

   2. Taylor expanded in J around inf 83.9%

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

      1. associate-*r* 83.9%

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

      2. *-commutative 83.9%

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

   4. Simplified 83.9%

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

   5. Taylor expanded in K around 0 49.1%

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

      1. *-commutative 49.1%

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

   7. Simplified 49.1%

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

4. Final simplification 35.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq 1.45 \cdot 10^{+42}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot J\\ \end{array} \]

Alternative 8: 27.5% accurate, 210.0× 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 Float64(-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 71.3%

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

2. Taylor expanded in J around 0 27.4%

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

   1. neg-mul-1 27.4%

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

4. Simplified 27.4%

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

5. Final simplification 27.4%

   \[\leadsto -U \]

Alternative 9: 26.5% 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 71.3%

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

2. Taylor expanded in U around -inf 23.9%

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

3. Final simplification 23.9%

   \[\leadsto U \]

Reproduce

herbie shell --seed 2023322 
(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)))))