Maksimov and Kolovsky, Equation (3)

Percentage Accurate: 73.9% → 99.8%

Time: 21.2s

Alternatives: 9

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.9% 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: 99.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(t_0 \cdot \left(-2 \cdot J\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{t_0 \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;t_1 \leq 10^{+308}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5 - \frac{J \cdot J}{U}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* t_0 (* -2.0 J))
          (sqrt (+ 1.0 (pow (/ U (* t_0 (* J 2.0))) 2.0))))))
   (if (<= t_1 (- INFINITY))
     (- U)
     (if (<= t_1 1e+308) t_1 (* -2.0 (- (* U -0.5) (/ (* J J) U)))))))

C

U = abs(U);
double code(double J, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = (t_0 * (-2.0 * J)) * sqrt((1.0 + pow((U / (t_0 * (J * 2.0))), 2.0)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -U;
	} else if (t_1 <= 1e+308) {
		tmp = t_1;
	} else {
		tmp = -2.0 * ((U * -0.5) - ((J * J) / U));
	}
	return tmp;
}

Java

U = Math.abs(U);
public static double code(double J, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = (t_0 * (-2.0 * J)) * Math.sqrt((1.0 + Math.pow((U / (t_0 * (J * 2.0))), 2.0)));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = -U;
	} else if (t_1 <= 1e+308) {
		tmp = t_1;
	} else {
		tmp = -2.0 * ((U * -0.5) - ((J * J) / U));
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	t_0 = math.cos((K / 2.0))
	t_1 = (t_0 * (-2.0 * J)) * math.sqrt((1.0 + math.pow((U / (t_0 * (J * 2.0))), 2.0)))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = -U
	elif t_1 <= 1e+308:
		tmp = t_1
	else:
		tmp = -2.0 * ((U * -0.5) - ((J * J) / U))
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(t_0 * Float64(-2.0 * J)) * sqrt(Float64(1.0 + (Float64(U / Float64(t_0 * Float64(J * 2.0))) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-U);
	elseif (t_1 <= 1e+308)
		tmp = t_1;
	else
		tmp = Float64(-2.0 * Float64(Float64(U * -0.5) - Float64(Float64(J * J) / U)));
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	t_0 = cos((K / 2.0));
	t_1 = (t_0 * (-2.0 * J)) * sqrt((1.0 + ((U / (t_0 * (J * 2.0))) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = -U;
	elseif (t_1 <= 1e+308)
		tmp = t_1;
	else
		tmp = -2.0 * ((U * -0.5) - ((J * J) / U));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * N[(-2.0 * J), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U / N[(t$95$0 * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], (-U), If[LessEqual[t$95$1, 1e+308], t$95$1, N[(-2.0 * N[(N[(U * -0.5), $MachinePrecision] - N[(N[(J * J), $MachinePrecision] / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 4.8%

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

   2. Taylor expanded in J around 0 40.6%

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

      1. neg-mul-1 40.6%

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

   4. Simplified 40.6%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 -2 J) (cos.f64 (/.f64 K 2))) (sqrt.f64 (+.f64 1 (pow.f64 (/.f64 U (*.f64 (*.f64 2 J) (cos.f64 (/.f64 K 2)))) 2)))) < 1e308

   1. Initial program 99.9%

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

   if 1e308 < (*.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 10.8%

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

   2. Taylor expanded in K around 0 7.2%

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

      1. *-commutative 7.2%

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

      2. *-commutative 7.2%

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

      3. unpow2 7.2%

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

      4. unpow2 7.2%

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

   4. Simplified 7.2%

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

   5. Taylor expanded in U around -inf 54.1%

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

      1. +-commutative 54.1%

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

      2. mul-1-neg 54.1%

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

      3. unsub-neg 54.1%

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

      4. *-commutative 54.1%

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

      5. unpow2 54.1%

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

   7. Simplified 54.1%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;\left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 10^{+308}:\\ \;\;\;\;\left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5 - \frac{J \cdot J}{U}\right)\\ \end{array} \]

Alternative 2: 88.3% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.0%

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

3. Simplified 89.5%

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

4. Final simplification 89.5%

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

Alternative 3: 88.3% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.0%

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

   1. associate-*l* 77.0%

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

   2. associate-*l* 77.0%

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

   3. unpow2 77.0%

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

   4. sqr-neg 77.0%

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

   5. distribute-frac-neg 77.0%

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

   6. distribute-frac-neg 77.0%

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

   7. unpow2 77.0%

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

3. Simplified 89.6%

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

4. Final simplification 89.6%

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

Alternative 4: 78.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} t_0 := -2 \cdot \left(J \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot 2}\right)\right)\right)\\ \mathbf{if}\;J \leq -2.8 \cdot 10^{-166}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq -6.5 \cdot 10^{-306}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 1.75 \cdot 10^{-114}:\\ \;\;\;\;-2 \cdot \frac{{\left(J \cdot \cos \left(K \cdot 0.5\right)\right)}^{2}}{U} - U\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (* -2.0 (* J (* (cos (/ K 2.0)) (hypot 1.0 (/ U (* J 2.0))))))))
   (if (<= J -2.8e-166)
     t_0
     (if (<= J -6.5e-306)
       U
       (if (<= J 1.75e-114)
         (- (* -2.0 (/ (pow (* J (cos (* K 0.5))) 2.0) U)) U)
         t_0)))))

C

U = abs(U);
double code(double J, double K, double U) {
	double t_0 = -2.0 * (J * (cos((K / 2.0)) * hypot(1.0, (U / (J * 2.0)))));
	double tmp;
	if (J <= -2.8e-166) {
		tmp = t_0;
	} else if (J <= -6.5e-306) {
		tmp = U;
	} else if (J <= 1.75e-114) {
		tmp = (-2.0 * (pow((J * cos((K * 0.5))), 2.0) / U)) - U;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

U = Math.abs(U);
public static double code(double J, double K, double U) {
	double t_0 = -2.0 * (J * (Math.cos((K / 2.0)) * Math.hypot(1.0, (U / (J * 2.0)))));
	double tmp;
	if (J <= -2.8e-166) {
		tmp = t_0;
	} else if (J <= -6.5e-306) {
		tmp = U;
	} else if (J <= 1.75e-114) {
		tmp = (-2.0 * (Math.pow((J * Math.cos((K * 0.5))), 2.0) / U)) - U;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	t_0 = -2.0 * (J * (math.cos((K / 2.0)) * math.hypot(1.0, (U / (J * 2.0)))))
	tmp = 0
	if J <= -2.8e-166:
		tmp = t_0
	elif J <= -6.5e-306:
		tmp = U
	elif J <= 1.75e-114:
		tmp = (-2.0 * (math.pow((J * math.cos((K * 0.5))), 2.0) / U)) - U
	else:
		tmp = t_0
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	t_0 = Float64(-2.0 * Float64(J * Float64(cos(Float64(K / 2.0)) * hypot(1.0, Float64(U / Float64(J * 2.0))))))
	tmp = 0.0
	if (J <= -2.8e-166)
		tmp = t_0;
	elseif (J <= -6.5e-306)
		tmp = U;
	elseif (J <= 1.75e-114)
		tmp = Float64(Float64(-2.0 * Float64((Float64(J * cos(Float64(K * 0.5))) ^ 2.0) / U)) - U);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	t_0 = -2.0 * (J * (cos((K / 2.0)) * hypot(1.0, (U / (J * 2.0)))));
	tmp = 0.0;
	if (J <= -2.8e-166)
		tmp = t_0;
	elseif (J <= -6.5e-306)
		tmp = U;
	elseif (J <= 1.75e-114)
		tmp = (-2.0 * (((J * cos((K * 0.5))) ^ 2.0) / U)) - U;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := Block[{t$95$0 = N[(-2.0 * N[(J * N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[(U / N[(J * 2.0), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[J, -2.8e-166], t$95$0, If[LessEqual[J, -6.5e-306], U, If[LessEqual[J, 1.75e-114], N[(N[(-2.0 * N[(N[Power[N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / U), $MachinePrecision]), $MachinePrecision] - U), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if J < -2.7999999999999999e-166 or 1.75e-114 < J

   1. Initial program 88.5%

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

   3. Simplified 96.9%

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

   4. Taylor expanded in K around 0 82.8%

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

   if -2.7999999999999999e-166 < J < -6.5000000000000004e-306

   1. Initial program 50.7%

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

   2. Taylor expanded in U around -inf 57.0%

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

   if -6.5000000000000004e-306 < J < 1.75e-114

   1. Initial program 40.3%

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

   2. Taylor expanded in J around 0 32.0%

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

      1. neg-mul-1 32.0%

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

      2. unsub-neg 32.0%

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

      3. *-commutative 32.0%

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

      4. *-commutative 32.0%

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

      5. unpow2 32.0%

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

      6. *-commutative 32.0%

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

      7. unpow2 32.0%

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

      8. swap-sqr 32.0%

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

      9. unpow2 32.0%

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

      10. *-commutative 32.0%

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

   4. Simplified 32.0%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -2.8 \cdot 10^{-166}:\\ \;\;\;\;-2 \cdot \left(J \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot 2}\right)\right)\right)\\ \mathbf{elif}\;J \leq -6.5 \cdot 10^{-306}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 1.75 \cdot 10^{-114}:\\ \;\;\;\;-2 \cdot \frac{{\left(J \cdot \cos \left(K \cdot 0.5\right)\right)}^{2}}{U} - U\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(J \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot 2}\right)\right)\right)\\ \end{array} \]

Alternative 5: 64.8% accurate, 3.7× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} \mathbf{if}\;U \leq 3.35 \cdot 10^{-134}:\\ \;\;\;\;\cos \left(K \cdot 0.5\right) \cdot \left(-2 \cdot J\right)\\ \mathbf{elif}\;U \leq 1.02 \cdot 10^{+185}:\\ \;\;\;\;-2 \cdot \left(J \cdot \mathsf{hypot}\left(1, 0.5 \cdot \frac{U}{J}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (if (<= U 3.35e-134)
   (* (cos (* K 0.5)) (* -2.0 J))
   (if (<= U 1.02e+185) (* -2.0 (* J (hypot 1.0 (* 0.5 (/ U J))))) U)))

C

U = abs(U);
double code(double J, double K, double U) {
	double tmp;
	if (U <= 3.35e-134) {
		tmp = cos((K * 0.5)) * (-2.0 * J);
	} else if (U <= 1.02e+185) {
		tmp = -2.0 * (J * hypot(1.0, (0.5 * (U / J))));
	} else {
		tmp = U;
	}
	return tmp;
}

Java

U = Math.abs(U);
public static double code(double J, double K, double U) {
	double tmp;
	if (U <= 3.35e-134) {
		tmp = Math.cos((K * 0.5)) * (-2.0 * J);
	} else if (U <= 1.02e+185) {
		tmp = -2.0 * (J * Math.hypot(1.0, (0.5 * (U / J))));
	} else {
		tmp = U;
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	tmp = 0
	if U <= 3.35e-134:
		tmp = math.cos((K * 0.5)) * (-2.0 * J)
	elif U <= 1.02e+185:
		tmp = -2.0 * (J * math.hypot(1.0, (0.5 * (U / J))))
	else:
		tmp = U
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	tmp = 0.0
	if (U <= 3.35e-134)
		tmp = Float64(cos(Float64(K * 0.5)) * Float64(-2.0 * J));
	elseif (U <= 1.02e+185)
		tmp = Float64(-2.0 * Float64(J * hypot(1.0, Float64(0.5 * Float64(U / J)))));
	else
		tmp = U;
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	tmp = 0.0;
	if (U <= 3.35e-134)
		tmp = cos((K * 0.5)) * (-2.0 * J);
	elseif (U <= 1.02e+185)
		tmp = -2.0 * (J * hypot(1.0, (0.5 * (U / J))));
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := If[LessEqual[U, 3.35e-134], N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(-2.0 * J), $MachinePrecision]), $MachinePrecision], If[LessEqual[U, 1.02e+185], N[(-2.0 * N[(J * N[Sqrt[1.0 ^ 2 + N[(0.5 * N[(U / J), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], U]]

Derivation

1. Split input into 3 regimes

2. if U < 3.34999999999999998e-134

   1. Initial program 80.9%

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

   2. Taylor expanded in J around inf 61.4%

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

      1. associate-*r* 61.4%

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

      2. *-commutative 61.4%

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

      3. *-commutative 61.4%

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

      4. associate-*r* 61.4%

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

      5. *-commutative 61.4%

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

   4. Simplified 61.4%

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

   if 3.34999999999999998e-134 < U < 1.0200000000000001e185

   1. Initial program 78.2%

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

   2. Taylor expanded in K around 0 42.1%

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

      1. *-commutative 42.1%

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

      2. *-commutative 42.1%

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

      3. unpow2 42.1%

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

      4. unpow2 42.1%

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

   4. Simplified 42.1%

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

      1. expm1-log1p-u 41.3%

         \[\leadsto \left(J \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 + 0.25 \cdot \frac{U \cdot U}{J \cdot J}}\right)\right)}\right) \cdot -2 \]

      2. expm1-udef 41.3%

         \[\leadsto \left(J \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{1 + 0.25 \cdot \frac{U \cdot U}{J \cdot J}}\right)} - 1\right)}\right) \cdot -2 \]

      3. add-sqr-sqrt 41.3%

         \[\leadsto \left(J \cdot \left(e^{\mathsf{log1p}\left(\sqrt{1 + \color{blue}{\sqrt{0.25 \cdot \frac{U \cdot U}{J \cdot J}} \cdot \sqrt{0.25 \cdot \frac{U \cdot U}{J \cdot J}}}}\right)} - 1\right)\right) \cdot -2 \]

      4. hypot-1-def 41.3%

         \[\leadsto \left(J \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\mathsf{hypot}\left(1, \sqrt{0.25 \cdot \frac{U \cdot U}{J \cdot J}}\right)}\right)} - 1\right)\right) \cdot -2 \]

      5. sqrt-prod 41.3%

         \[\leadsto \left(J \cdot \left(e^{\mathsf{log1p}\left(\mathsf{hypot}\left(1, \color{blue}{\sqrt{0.25} \cdot \sqrt{\frac{U \cdot U}{J \cdot J}}}\right)\right)} - 1\right)\right) \cdot -2 \]

      6. metadata-eval 41.3%

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

      7. times-frac 54.8%

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

      8. sqrt-prod 39.2%

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

      9. add-sqr-sqrt 65.8%

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

   6. Applied egg-rr 65.8%

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

      1. expm1-def 65.8%

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

      2. expm1-log1p 68.2%

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

   8. Simplified 68.2%

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

   if 1.0200000000000001e185 < U

   1. Initial program 43.2%

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

   2. Taylor expanded in U around -inf 63.1%

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

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 3.35 \cdot 10^{-134}:\\ \;\;\;\;\cos \left(K \cdot 0.5\right) \cdot \left(-2 \cdot J\right)\\ \mathbf{elif}\;U \leq 1.02 \cdot 10^{+185}:\\ \;\;\;\;-2 \cdot \left(J \cdot \mathsf{hypot}\left(1, 0.5 \cdot \frac{U}{J}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternative 6: 66.3% accurate, 3.7× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} t_0 := \cos \left(K \cdot 0.5\right) \cdot \left(-2 \cdot J\right)\\ \mathbf{if}\;J \leq -9.6 \cdot 10^{-98}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq -5.5 \cdot 10^{-306}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 1.05 \cdot 10^{-82}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (* (cos (* K 0.5)) (* -2.0 J))))
   (if (<= J -9.6e-98)
     t_0
     (if (<= J -5.5e-306) U (if (<= J 1.05e-82) (- U) t_0)))))

C

U = abs(U);
double code(double J, double K, double U) {
	double t_0 = cos((K * 0.5)) * (-2.0 * J);
	double tmp;
	if (J <= -9.6e-98) {
		tmp = t_0;
	} else if (J <= -5.5e-306) {
		tmp = U;
	} else if (J <= 1.05e-82) {
		tmp = -U;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

NOTE: U should be positive before calling this function
real(8) function code(j, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((k * 0.5d0)) * ((-2.0d0) * j)
    if (j <= (-9.6d-98)) then
        tmp = t_0
    else if (j <= (-5.5d-306)) then
        tmp = u
    else if (j <= 1.05d-82) then
        tmp = -u
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

U = Math.abs(U);
public static double code(double J, double K, double U) {
	double t_0 = Math.cos((K * 0.5)) * (-2.0 * J);
	double tmp;
	if (J <= -9.6e-98) {
		tmp = t_0;
	} else if (J <= -5.5e-306) {
		tmp = U;
	} else if (J <= 1.05e-82) {
		tmp = -U;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	t_0 = math.cos((K * 0.5)) * (-2.0 * J)
	tmp = 0
	if J <= -9.6e-98:
		tmp = t_0
	elif J <= -5.5e-306:
		tmp = U
	elif J <= 1.05e-82:
		tmp = -U
	else:
		tmp = t_0
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	t_0 = Float64(cos(Float64(K * 0.5)) * Float64(-2.0 * J))
	tmp = 0.0
	if (J <= -9.6e-98)
		tmp = t_0;
	elseif (J <= -5.5e-306)
		tmp = U;
	elseif (J <= 1.05e-82)
		tmp = Float64(-U);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	t_0 = cos((K * 0.5)) * (-2.0 * J);
	tmp = 0.0;
	if (J <= -9.6e-98)
		tmp = t_0;
	elseif (J <= -5.5e-306)
		tmp = U;
	elseif (J <= 1.05e-82)
		tmp = -U;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := Block[{t$95$0 = N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(-2.0 * J), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[J, -9.6e-98], t$95$0, If[LessEqual[J, -5.5e-306], U, If[LessEqual[J, 1.05e-82], (-U), t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if J < -9.60000000000000019e-98 or 1.05e-82 < J

   1. Initial program 90.2%

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

   2. Taylor expanded in J around inf 71.9%

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

      1. associate-*r* 71.9%

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

      2. *-commutative 71.9%

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

      3. *-commutative 71.9%

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

      4. associate-*r* 71.9%

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

      5. *-commutative 71.9%

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

   4. Simplified 71.9%

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

   if -9.60000000000000019e-98 < J < -5.49999999999999992e-306

   1. Initial program 52.5%

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

   2. Taylor expanded in U around -inf 43.9%

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

   if -5.49999999999999992e-306 < J < 1.05e-82

   1. Initial program 47.0%

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

   2. Taylor expanded in J around 0 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%

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

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -9.6 \cdot 10^{-98}:\\ \;\;\;\;\cos \left(K \cdot 0.5\right) \cdot \left(-2 \cdot J\right)\\ \mathbf{elif}\;J \leq -5.5 \cdot 10^{-306}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 1.05 \cdot 10^{-82}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;\cos \left(K \cdot 0.5\right) \cdot \left(-2 \cdot J\right)\\ \end{array} \]

Alternative 7: 48.2% accurate, 45.7× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} \mathbf{if}\;J \leq -2.46 \cdot 10^{+120}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;J \leq -5.5 \cdot 10^{-306}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 3.7 \cdot 10^{-53}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot J\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (if (<= J -2.46e+120)
   (* -2.0 J)
   (if (<= J -5.5e-306) U (if (<= J 3.7e-53) (- U) (* -2.0 J)))))

C

U = abs(U);
double code(double J, double K, double U) {
	double tmp;
	if (J <= -2.46e+120) {
		tmp = -2.0 * J;
	} else if (J <= -5.5e-306) {
		tmp = U;
	} else if (J <= 3.7e-53) {
		tmp = -U;
	} else {
		tmp = -2.0 * J;
	}
	return tmp;
}

Fortran

NOTE: U should be positive before calling this function
real(8) function code(j, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (j <= (-2.46d+120)) then
        tmp = (-2.0d0) * j
    else if (j <= (-5.5d-306)) then
        tmp = u
    else if (j <= 3.7d-53) then
        tmp = -u
    else
        tmp = (-2.0d0) * j
    end if
    code = tmp
end function

Java

U = Math.abs(U);
public static double code(double J, double K, double U) {
	double tmp;
	if (J <= -2.46e+120) {
		tmp = -2.0 * J;
	} else if (J <= -5.5e-306) {
		tmp = U;
	} else if (J <= 3.7e-53) {
		tmp = -U;
	} else {
		tmp = -2.0 * J;
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	tmp = 0
	if J <= -2.46e+120:
		tmp = -2.0 * J
	elif J <= -5.5e-306:
		tmp = U
	elif J <= 3.7e-53:
		tmp = -U
	else:
		tmp = -2.0 * J
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	tmp = 0.0
	if (J <= -2.46e+120)
		tmp = Float64(-2.0 * J);
	elseif (J <= -5.5e-306)
		tmp = U;
	elseif (J <= 3.7e-53)
		tmp = Float64(-U);
	else
		tmp = Float64(-2.0 * J);
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	tmp = 0.0;
	if (J <= -2.46e+120)
		tmp = -2.0 * J;
	elseif (J <= -5.5e-306)
		tmp = U;
	elseif (J <= 3.7e-53)
		tmp = -U;
	else
		tmp = -2.0 * J;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := If[LessEqual[J, -2.46e+120], N[(-2.0 * J), $MachinePrecision], If[LessEqual[J, -5.5e-306], U, If[LessEqual[J, 3.7e-53], (-U), N[(-2.0 * J), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if J < -2.4600000000000001e120 or 3.69999999999999982e-53 < J

   1. Initial program 98.2%

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

   2. Taylor expanded in K around 0 51.6%

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

      1. *-commutative 51.6%

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

      2. *-commutative 51.6%

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

      3. unpow2 51.6%

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

      4. unpow2 51.6%

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

   4. Simplified 51.6%

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

   5. Taylor expanded in J around inf 54.3%

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

   if -2.4600000000000001e120 < J < -5.49999999999999992e-306

   1. Initial program 67.8%

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

   2. Taylor expanded in U around -inf 33.7%

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

   if -5.49999999999999992e-306 < J < 3.69999999999999982e-53

   1. Initial program 47.9%

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

   2. Taylor expanded in J around 0 34.2%

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

      1. neg-mul-1 34.2%

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

   4. Simplified 34.2%

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

4. Final simplification 42.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -2.46 \cdot 10^{+120}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;J \leq -5.5 \cdot 10^{-306}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 3.7 \cdot 10^{-53}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot J\\ \end{array} \]

Alternative 8: 38.8% accurate, 103.4× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} \mathbf{if}\;J \leq -5.5 \cdot 10^{-306}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U) :precision binary64 (if (<= J -5.5e-306) U (- U)))

C

U = abs(U);
double code(double J, double K, double U) {
	double tmp;
	if (J <= -5.5e-306) {
		tmp = U;
	} else {
		tmp = -U;
	}
	return tmp;
}

Fortran

NOTE: U should be positive before calling this function
real(8) function code(j, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (j <= (-5.5d-306)) then
        tmp = u
    else
        tmp = -u
    end if
    code = tmp
end function

Java

U = Math.abs(U);
public static double code(double J, double K, double U) {
	double tmp;
	if (J <= -5.5e-306) {
		tmp = U;
	} else {
		tmp = -U;
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	tmp = 0
	if J <= -5.5e-306:
		tmp = U
	else:
		tmp = -U
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	tmp = 0.0
	if (J <= -5.5e-306)
		tmp = U;
	else
		tmp = Float64(-U);
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	tmp = 0.0;
	if (J <= -5.5e-306)
		tmp = U;
	else
		tmp = -U;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := If[LessEqual[J, -5.5e-306], U, (-U)]

Derivation

1. Split input into 2 regimes

2. if J < -5.49999999999999992e-306

   1. Initial program 77.3%

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

   2. Taylor expanded in U around -inf 26.2%

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

   if -5.49999999999999992e-306 < J

   1. Initial program 76.7%

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

   2. Taylor expanded in J around 0 20.9%

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

      1. neg-mul-1 20.9%

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

   4. Simplified 20.9%

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

4. Final simplification 23.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -5.5 \cdot 10^{-306}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]

Alternative 9: 26.8% accurate, 420.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := U

Derivation

1. Initial program 77.0%

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

2. Taylor expanded in U around -inf 27.8%

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

3. Final simplification 27.8%

   \[\leadsto U \]

Reproduce

herbie shell --seed 2023275 
(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)))))