Maksimov and Kolovsky, Equation (3)

Percentage Accurate: 73.7% → 98.5%

Time: 17.9s

Alternatives: 8

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: 73.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \left(\left(-2 \cdot J\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot t\_0}\right)}^{2}} \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J_m = \left|J\right| \\ J_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := J\_m \cdot t\_0\\ t_2 := \left(t\_0 \cdot \left(-2 \cdot J\_m\right)\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_0 \cdot \left(J\_m \cdot 2\right)}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+273}:\\ \;\;\;\;-2 \cdot \left(t\_1 \cdot \mathsf{hypot}\left(1, \frac{\frac{U\_m}{2}}{t\_1}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J_m = (fabs.f64 J)
J_s = (copysign.f64 1 J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1 (* J_m t_0))
        (t_2
         (*
          (* t_0 (* -2.0 J_m))
          (sqrt (+ 1.0 (pow (/ U_m (* t_0 (* J_m 2.0))) 2.0))))))
   (*
    J_s
    (if (<= t_2 (- INFINITY))
      (- U_m)
      (if (<= t_2 5e+273)
        (* -2.0 (* t_1 (hypot 1.0 (/ (/ U_m 2.0) t_1))))
        U_m)))))

C

U_m = fabs(U);
J_m = fabs(J);
J_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = J_m * t_0;
	double t_2 = (t_0 * (-2.0 * J_m)) * sqrt((1.0 + pow((U_m / (t_0 * (J_m * 2.0))), 2.0)));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_2 <= 5e+273) {
		tmp = -2.0 * (t_1 * hypot(1.0, ((U_m / 2.0) / t_1)));
	} else {
		tmp = U_m;
	}
	return J_s * tmp;
}

Java

U_m = Math.abs(U);
J_m = Math.abs(J);
J_s = Math.copySign(1.0, J);
public static double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = J_m * t_0;
	double t_2 = (t_0 * (-2.0 * J_m)) * Math.sqrt((1.0 + Math.pow((U_m / (t_0 * (J_m * 2.0))), 2.0)));
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = -U_m;
	} else if (t_2 <= 5e+273) {
		tmp = -2.0 * (t_1 * Math.hypot(1.0, ((U_m / 2.0) / t_1)));
	} else {
		tmp = U_m;
	}
	return J_s * tmp;
}

Python

U_m = math.fabs(U)
J_m = math.fabs(J)
J_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	t_0 = math.cos((K / 2.0))
	t_1 = J_m * t_0
	t_2 = (t_0 * (-2.0 * J_m)) * math.sqrt((1.0 + math.pow((U_m / (t_0 * (J_m * 2.0))), 2.0)))
	tmp = 0
	if t_2 <= -math.inf:
		tmp = -U_m
	elif t_2 <= 5e+273:
		tmp = -2.0 * (t_1 * math.hypot(1.0, ((U_m / 2.0) / t_1)))
	else:
		tmp = U_m
	return J_s * tmp

Julia

U_m = abs(U)
J_m = abs(J)
J_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(J_m * t_0)
	t_2 = Float64(Float64(t_0 * Float64(-2.0 * J_m)) * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_0 * Float64(J_m * 2.0))) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_2 <= 5e+273)
		tmp = Float64(-2.0 * Float64(t_1 * hypot(1.0, Float64(Float64(U_m / 2.0) / t_1))));
	else
		tmp = U_m;
	end
	return Float64(J_s * tmp)
end

MATLAB

U_m = abs(U);
J_m = abs(J);
J_s = sign(J) * abs(1.0);
function tmp_2 = code(J_s, J_m, K, U_m)
	t_0 = cos((K / 2.0));
	t_1 = J_m * t_0;
	t_2 = (t_0 * (-2.0 * J_m)) * sqrt((1.0 + ((U_m / (t_0 * (J_m * 2.0))) ^ 2.0)));
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = -U_m;
	elseif (t_2 <= 5e+273)
		tmp = -2.0 * (t_1 * hypot(1.0, ((U_m / 2.0) / t_1)));
	else
		tmp = U_m;
	end
	tmp_2 = J_s * tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J_m = N[Abs[J], $MachinePrecision]
J_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(J$95$m * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$0 * N[(-2.0 * J$95$m), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(t$95$0 * N[(J$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$2, (-Infinity)], (-U$95$m), If[LessEqual[t$95$2, 5e+273], N[(-2.0 * N[(t$95$1 * N[Sqrt[1.0 ^ 2 + N[(N[(U$95$m / 2.0), $MachinePrecision] / t$95$1), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], U$95$m]]), $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 6.6%

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

   3. Simplified 6.6%

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

   5. Taylor expanded in J around 0 56.1%

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

      1. neg-mul-1 56.1%

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

   7. Simplified 56.1%

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

   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)))) < 4.99999999999999961e273

   1. Initial program 99.8%

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

      1. associate-*l* 99.8%

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

      2. associate-*l* 99.8%

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

      3. unpow2 99.8%

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

      4. sqr-neg 99.8%

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

      5. distribute-frac-neg 99.8%

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

      6. distribute-frac-neg 99.8%

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

      7. unpow2 99.8%

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

   3. Simplified 99.8%

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

   if 4.99999999999999961e273 < (*.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 17.2%

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

   3. Simplified 17.2%

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

   5. Taylor expanded in U around -inf 42.7%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;\left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 5 \cdot 10^{+273}:\\ \;\;\;\;-2 \cdot \left(\left(J \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{\frac{U}{2}}{J \cdot \cos \left(\frac{K}{2}\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 70.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J_m = \left|J\right| \\ J_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -0.19:\\ \;\;\;\;U\_m\\ \mathbf{elif}\;t\_0 \leq 0.2 \lor \neg \left(t\_0 \leq 0.99995\right):\\ \;\;\;\;-2 \cdot \left(\left(J\_m \cdot t\_0\right) \cdot \mathsf{hypot}\left(1, \frac{\frac{U\_m}{2}}{J\_m}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-U\_m\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J_m = (fabs.f64 J)
J_s = (copysign.f64 1 J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (*
    J_s
    (if (<= t_0 -0.19)
      U_m
      (if (or (<= t_0 0.2) (not (<= t_0 0.99995)))
        (* -2.0 (* (* J_m t_0) (hypot 1.0 (/ (/ U_m 2.0) J_m))))
        (- U_m))))))

C

U_m = fabs(U);
J_m = fabs(J);
J_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= -0.19) {
		tmp = U_m;
	} else if ((t_0 <= 0.2) || !(t_0 <= 0.99995)) {
		tmp = -2.0 * ((J_m * t_0) * hypot(1.0, ((U_m / 2.0) / J_m)));
	} else {
		tmp = -U_m;
	}
	return J_s * tmp;
}

Java

U_m = Math.abs(U);
J_m = Math.abs(J);
J_s = Math.copySign(1.0, J);
public static double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = Math.cos((K / 2.0));
	double tmp;
	if (t_0 <= -0.19) {
		tmp = U_m;
	} else if ((t_0 <= 0.2) || !(t_0 <= 0.99995)) {
		tmp = -2.0 * ((J_m * t_0) * Math.hypot(1.0, ((U_m / 2.0) / J_m)));
	} else {
		tmp = -U_m;
	}
	return J_s * tmp;
}

Python

U_m = math.fabs(U)
J_m = math.fabs(J)
J_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if t_0 <= -0.19:
		tmp = U_m
	elif (t_0 <= 0.2) or not (t_0 <= 0.99995):
		tmp = -2.0 * ((J_m * t_0) * math.hypot(1.0, ((U_m / 2.0) / J_m)))
	else:
		tmp = -U_m
	return J_s * tmp

Julia

U_m = abs(U)
J_m = abs(J)
J_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= -0.19)
		tmp = U_m;
	elseif ((t_0 <= 0.2) || !(t_0 <= 0.99995))
		tmp = Float64(-2.0 * Float64(Float64(J_m * t_0) * hypot(1.0, Float64(Float64(U_m / 2.0) / J_m))));
	else
		tmp = Float64(-U_m);
	end
	return Float64(J_s * tmp)
end

MATLAB

U_m = abs(U);
J_m = abs(J);
J_s = sign(J) * abs(1.0);
function tmp_2 = code(J_s, J_m, K, U_m)
	t_0 = cos((K / 2.0));
	tmp = 0.0;
	if (t_0 <= -0.19)
		tmp = U_m;
	elseif ((t_0 <= 0.2) || ~((t_0 <= 0.99995)))
		tmp = -2.0 * ((J_m * t_0) * hypot(1.0, ((U_m / 2.0) / J_m)));
	else
		tmp = -U_m;
	end
	tmp_2 = J_s * tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J_m = N[Abs[J], $MachinePrecision]
J_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$0, -0.19], U$95$m, If[Or[LessEqual[t$95$0, 0.2], N[Not[LessEqual[t$95$0, 0.99995]], $MachinePrecision]], N[(-2.0 * N[(N[(J$95$m * t$95$0), $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[(N[(U$95$m / 2.0), $MachinePrecision] / J$95$m), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-U$95$m)]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 (/.f64 K 2)) < -0.19

   1. Initial program 65.1%

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

   3. Simplified 65.0%

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

   5. Taylor expanded in U around -inf 26.6%

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

   if -0.19 < (cos.f64 (/.f64 K 2)) < 0.20000000000000001 or 0.999950000000000006 < (cos.f64 (/.f64 K 2))

   1. Initial program 77.6%

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

      1. associate-*l* 77.6%

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

      2. associate-*l* 77.6%

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

      3. unpow2 77.6%

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

      4. sqr-neg 77.6%

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

      5. distribute-frac-neg 77.6%

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

      6. distribute-frac-neg 77.6%

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

      7. unpow2 77.6%

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

   3. Simplified 88.5%

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

   5. Taylor expanded in K around 0 88.5%

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

   if 0.20000000000000001 < (cos.f64 (/.f64 K 2)) < 0.999950000000000006

   1. Initial program 59.9%

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

   3. Simplified 59.8%

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

   5. Taylor expanded in J around 0 44.2%

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

      1. neg-mul-1 44.2%

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

   7. Simplified 44.2%

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

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.19:\\ \;\;\;\;U\\ \mathbf{elif}\;\cos \left(\frac{K}{2}\right) \leq 0.2 \lor \neg \left(\cos \left(\frac{K}{2}\right) \leq 0.99995\right):\\ \;\;\;\;-2 \cdot \left(\left(J \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{\frac{U}{2}}{J}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 65.5% accurate, 3.4× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J_m = \left|J\right| \\ J_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \left(-2 \cdot J\_m\right) \cdot \cos \left(K \cdot 0.5\right)\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;U\_m \leq 9.5 \cdot 10^{+43}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;U\_m \leq 7 \cdot 10^{+66}:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;U\_m \leq 2.2 \cdot 10^{+97}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;U\_m \leq 1.2 \cdot 10^{+119} \lor \neg \left(U\_m \leq 8.6 \cdot 10^{+138}\right):\\ \;\;\;\;-U\_m\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J_m = (fabs.f64 J)
J_s = (copysign.f64 1 J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (* (* -2.0 J_m) (cos (* K 0.5)))))
   (*
    J_s
    (if (<= U_m 9.5e+43)
      t_0
      (if (<= U_m 7e+66)
        (- U_m)
        (if (<= U_m 2.2e+97)
          t_0
          (if (or (<= U_m 1.2e+119) (not (<= U_m 8.6e+138))) (- U_m) U_m)))))))

C

U_m = fabs(U);
J_m = fabs(J);
J_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = (-2.0 * J_m) * cos((K * 0.5));
	double tmp;
	if (U_m <= 9.5e+43) {
		tmp = t_0;
	} else if (U_m <= 7e+66) {
		tmp = -U_m;
	} else if (U_m <= 2.2e+97) {
		tmp = t_0;
	} else if ((U_m <= 1.2e+119) || !(U_m <= 8.6e+138)) {
		tmp = -U_m;
	} else {
		tmp = U_m;
	}
	return J_s * tmp;
}

Fortran

U_m = abs(U)
J_m = abs(J)
J_s = copysign(1.0d0, J)
real(8) function code(j_s, j_m, k, u_m)
    real(8), intent (in) :: j_s
    real(8), intent (in) :: j_m
    real(8), intent (in) :: k
    real(8), intent (in) :: u_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((-2.0d0) * j_m) * cos((k * 0.5d0))
    if (u_m <= 9.5d+43) then
        tmp = t_0
    else if (u_m <= 7d+66) then
        tmp = -u_m
    else if (u_m <= 2.2d+97) then
        tmp = t_0
    else if ((u_m <= 1.2d+119) .or. (.not. (u_m <= 8.6d+138))) then
        tmp = -u_m
    else
        tmp = u_m
    end if
    code = j_s * tmp
end function

Java

U_m = Math.abs(U);
J_m = Math.abs(J);
J_s = Math.copySign(1.0, J);
public static double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = (-2.0 * J_m) * Math.cos((K * 0.5));
	double tmp;
	if (U_m <= 9.5e+43) {
		tmp = t_0;
	} else if (U_m <= 7e+66) {
		tmp = -U_m;
	} else if (U_m <= 2.2e+97) {
		tmp = t_0;
	} else if ((U_m <= 1.2e+119) || !(U_m <= 8.6e+138)) {
		tmp = -U_m;
	} else {
		tmp = U_m;
	}
	return J_s * tmp;
}

Python

U_m = math.fabs(U)
J_m = math.fabs(J)
J_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	t_0 = (-2.0 * J_m) * math.cos((K * 0.5))
	tmp = 0
	if U_m <= 9.5e+43:
		tmp = t_0
	elif U_m <= 7e+66:
		tmp = -U_m
	elif U_m <= 2.2e+97:
		tmp = t_0
	elif (U_m <= 1.2e+119) or not (U_m <= 8.6e+138):
		tmp = -U_m
	else:
		tmp = U_m
	return J_s * tmp

Julia

U_m = abs(U)
J_m = abs(J)
J_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = Float64(Float64(-2.0 * J_m) * cos(Float64(K * 0.5)))
	tmp = 0.0
	if (U_m <= 9.5e+43)
		tmp = t_0;
	elseif (U_m <= 7e+66)
		tmp = Float64(-U_m);
	elseif (U_m <= 2.2e+97)
		tmp = t_0;
	elseif ((U_m <= 1.2e+119) || !(U_m <= 8.6e+138))
		tmp = Float64(-U_m);
	else
		tmp = U_m;
	end
	return Float64(J_s * tmp)
end

MATLAB

U_m = abs(U);
J_m = abs(J);
J_s = sign(J) * abs(1.0);
function tmp_2 = code(J_s, J_m, K, U_m)
	t_0 = (-2.0 * J_m) * cos((K * 0.5));
	tmp = 0.0;
	if (U_m <= 9.5e+43)
		tmp = t_0;
	elseif (U_m <= 7e+66)
		tmp = -U_m;
	elseif (U_m <= 2.2e+97)
		tmp = t_0;
	elseif ((U_m <= 1.2e+119) || ~((U_m <= 8.6e+138)))
		tmp = -U_m;
	else
		tmp = U_m;
	end
	tmp_2 = J_s * tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J_m = N[Abs[J], $MachinePrecision]
J_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[(N[(-2.0 * J$95$m), $MachinePrecision] * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[U$95$m, 9.5e+43], t$95$0, If[LessEqual[U$95$m, 7e+66], (-U$95$m), If[LessEqual[U$95$m, 2.2e+97], t$95$0, If[Or[LessEqual[U$95$m, 1.2e+119], N[Not[LessEqual[U$95$m, 8.6e+138]], $MachinePrecision]], (-U$95$m), U$95$m]]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if U < 9.5000000000000004e43 or 6.9999999999999994e66 < U < 2.2000000000000001e97

   1. Initial program 78.3%

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

   3. Simplified 78.2%

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

   5. Taylor expanded in J around inf 57.1%

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

      1. associate-*r* 57.1%

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

      2. *-commutative 57.1%

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

   7. Simplified 57.1%

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

   if 9.5000000000000004e43 < U < 6.9999999999999994e66 or 2.2000000000000001e97 < U < 1.2e119 or 8.5999999999999996e138 < U

   1. Initial program 38.6%

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

   3. Simplified 38.5%

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

   5. Taylor expanded in J around 0 43.8%

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

      1. neg-mul-1 43.8%

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

   7. Simplified 43.8%

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

   if 1.2e119 < U < 8.5999999999999996e138

   1. Initial program 72.4%

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

   3. Simplified 72.4%

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

   5. Taylor expanded in U around -inf 44.0%

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

4. Final simplification 54.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 9.5 \cdot 10^{+43}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{elif}\;U \leq 7 \cdot 10^{+66}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \leq 2.2 \cdot 10^{+97}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{elif}\;U \leq 1.2 \cdot 10^{+119} \lor \neg \left(U \leq 8.6 \cdot 10^{+138}\right):\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.3% accurate, 3.5× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J_m = \left|J\right| \\ J_s = \mathsf{copysign}\left(1, J\right) \\ J\_s \cdot \begin{array}{l} \mathbf{if}\;J\_m \leq 8.5 \cdot 10^{-115}:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;J\_m \leq 9.2 \cdot 10^{+172}:\\ \;\;\;\;\left(-2 \cdot J\_m\right) \cdot \mathsf{hypot}\left(1, \frac{0.5}{\frac{J\_m}{U\_m}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot J\_m\right) \cdot \cos \left(K \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J_m = (fabs.f64 J)
J_s = (copysign.f64 1 J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (*
  J_s
  (if (<= J_m 8.5e-115)
    (- U_m)
    (if (<= J_m 9.2e+172)
      (* (* -2.0 J_m) (hypot 1.0 (/ 0.5 (/ J_m U_m))))
      (* (* -2.0 J_m) (cos (* K 0.5)))))))

C

U_m = fabs(U);
J_m = fabs(J);
J_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double tmp;
	if (J_m <= 8.5e-115) {
		tmp = -U_m;
	} else if (J_m <= 9.2e+172) {
		tmp = (-2.0 * J_m) * hypot(1.0, (0.5 / (J_m / U_m)));
	} else {
		tmp = (-2.0 * J_m) * cos((K * 0.5));
	}
	return J_s * tmp;
}

Java

U_m = Math.abs(U);
J_m = Math.abs(J);
J_s = Math.copySign(1.0, J);
public static double code(double J_s, double J_m, double K, double U_m) {
	double tmp;
	if (J_m <= 8.5e-115) {
		tmp = -U_m;
	} else if (J_m <= 9.2e+172) {
		tmp = (-2.0 * J_m) * Math.hypot(1.0, (0.5 / (J_m / U_m)));
	} else {
		tmp = (-2.0 * J_m) * Math.cos((K * 0.5));
	}
	return J_s * tmp;
}

Python

U_m = math.fabs(U)
J_m = math.fabs(J)
J_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	tmp = 0
	if J_m <= 8.5e-115:
		tmp = -U_m
	elif J_m <= 9.2e+172:
		tmp = (-2.0 * J_m) * math.hypot(1.0, (0.5 / (J_m / U_m)))
	else:
		tmp = (-2.0 * J_m) * math.cos((K * 0.5))
	return J_s * tmp

Julia

U_m = abs(U)
J_m = abs(J)
J_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	tmp = 0.0
	if (J_m <= 8.5e-115)
		tmp = Float64(-U_m);
	elseif (J_m <= 9.2e+172)
		tmp = Float64(Float64(-2.0 * J_m) * hypot(1.0, Float64(0.5 / Float64(J_m / U_m))));
	else
		tmp = Float64(Float64(-2.0 * J_m) * cos(Float64(K * 0.5)));
	end
	return Float64(J_s * tmp)
end

MATLAB

U_m = abs(U);
J_m = abs(J);
J_s = sign(J) * abs(1.0);
function tmp_2 = code(J_s, J_m, K, U_m)
	tmp = 0.0;
	if (J_m <= 8.5e-115)
		tmp = -U_m;
	elseif (J_m <= 9.2e+172)
		tmp = (-2.0 * J_m) * hypot(1.0, (0.5 / (J_m / U_m)));
	else
		tmp = (-2.0 * J_m) * cos((K * 0.5));
	end
	tmp_2 = J_s * tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J_m = N[Abs[J], $MachinePrecision]
J_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := N[(J$95$s * If[LessEqual[J$95$m, 8.5e-115], (-U$95$m), If[LessEqual[J$95$m, 9.2e+172], N[(N[(-2.0 * J$95$m), $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[(0.5 / N[(J$95$m / U$95$m), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 * J$95$m), $MachinePrecision] * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if J < 8.49999999999999953e-115

   1. Initial program 65.6%

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

   3. Simplified 65.6%

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

   5. Taylor expanded in J around 0 33.3%

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

      1. neg-mul-1 33.3%

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

   7. Simplified 33.3%

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

   if 8.49999999999999953e-115 < J < 9.2000000000000003e172

   1. Initial program 76.9%

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

   3. Simplified 76.8%

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

   5. Taylor expanded in K around 0 46.8%

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

      1. associate-*r* 46.8%

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

      2. *-commutative 46.8%

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

      3. associate-*r/ 46.8%

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

      4. *-commutative 46.8%

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

      5. unpow2 46.8%

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

      6. associate-/r* 47.0%

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

      7. unpow2 47.0%

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

      8. metadata-eval 47.0%

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

      9. swap-sqr 47.0%

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

      10. associate-*r/ 51.8%

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

      11. associate-*r/ 51.8%

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

      12. associate-*l/ 55.0%

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

      13. associate-*r/ 54.9%

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

      14. unpow2 54.9%

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

      15. associate-*r/ 55.0%

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

      16. *-commutative 55.0%

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

      17. associate-*r/ 55.0%

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

   7. Simplified 55.0%

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

      1. unpow2 55.0%

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

      2. hypot-1-def 64.1%

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

      3. clear-num 64.0%

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

      4. un-div-inv 64.0%

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

   9. Applied egg-rr 64.0%

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

   if 9.2000000000000003e172 < J

   1. Initial program 99.9%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in J around inf 88.3%

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

      1. associate-*r* 88.3%

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

      2. *-commutative 88.3%

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

   7. Simplified 88.3%

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

4. Final simplification 45.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq 8.5 \cdot 10^{-115}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 9.2 \cdot 10^{+172}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \mathsf{hypot}\left(1, \frac{0.5}{\frac{J}{U}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.4% accurate, 3.5× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J_m = \left|J\right| \\ J_s = \mathsf{copysign}\left(1, J\right) \\ J\_s \cdot \begin{array}{l} \mathbf{if}\;J\_m \leq 2.05 \cdot 10^{-125}:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;J\_m \leq 9 \cdot 10^{+172}:\\ \;\;\;\;\left(-2 \cdot J\_m\right) \cdot \mathsf{hypot}\left(1, \frac{U\_m \cdot 0.5}{J\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot J\_m\right) \cdot \cos \left(K \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J_m = (fabs.f64 J)
J_s = (copysign.f64 1 J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (*
  J_s
  (if (<= J_m 2.05e-125)
    (- U_m)
    (if (<= J_m 9e+172)
      (* (* -2.0 J_m) (hypot 1.0 (/ (* U_m 0.5) J_m)))
      (* (* -2.0 J_m) (cos (* K 0.5)))))))

C

U_m = fabs(U);
J_m = fabs(J);
J_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double tmp;
	if (J_m <= 2.05e-125) {
		tmp = -U_m;
	} else if (J_m <= 9e+172) {
		tmp = (-2.0 * J_m) * hypot(1.0, ((U_m * 0.5) / J_m));
	} else {
		tmp = (-2.0 * J_m) * cos((K * 0.5));
	}
	return J_s * tmp;
}

Java

U_m = Math.abs(U);
J_m = Math.abs(J);
J_s = Math.copySign(1.0, J);
public static double code(double J_s, double J_m, double K, double U_m) {
	double tmp;
	if (J_m <= 2.05e-125) {
		tmp = -U_m;
	} else if (J_m <= 9e+172) {
		tmp = (-2.0 * J_m) * Math.hypot(1.0, ((U_m * 0.5) / J_m));
	} else {
		tmp = (-2.0 * J_m) * Math.cos((K * 0.5));
	}
	return J_s * tmp;
}

Python

U_m = math.fabs(U)
J_m = math.fabs(J)
J_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	tmp = 0
	if J_m <= 2.05e-125:
		tmp = -U_m
	elif J_m <= 9e+172:
		tmp = (-2.0 * J_m) * math.hypot(1.0, ((U_m * 0.5) / J_m))
	else:
		tmp = (-2.0 * J_m) * math.cos((K * 0.5))
	return J_s * tmp

Julia

U_m = abs(U)
J_m = abs(J)
J_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	tmp = 0.0
	if (J_m <= 2.05e-125)
		tmp = Float64(-U_m);
	elseif (J_m <= 9e+172)
		tmp = Float64(Float64(-2.0 * J_m) * hypot(1.0, Float64(Float64(U_m * 0.5) / J_m)));
	else
		tmp = Float64(Float64(-2.0 * J_m) * cos(Float64(K * 0.5)));
	end
	return Float64(J_s * tmp)
end

MATLAB

U_m = abs(U);
J_m = abs(J);
J_s = sign(J) * abs(1.0);
function tmp_2 = code(J_s, J_m, K, U_m)
	tmp = 0.0;
	if (J_m <= 2.05e-125)
		tmp = -U_m;
	elseif (J_m <= 9e+172)
		tmp = (-2.0 * J_m) * hypot(1.0, ((U_m * 0.5) / J_m));
	else
		tmp = (-2.0 * J_m) * cos((K * 0.5));
	end
	tmp_2 = J_s * tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J_m = N[Abs[J], $MachinePrecision]
J_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := N[(J$95$s * If[LessEqual[J$95$m, 2.05e-125], (-U$95$m), If[LessEqual[J$95$m, 9e+172], N[(N[(-2.0 * J$95$m), $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[(N[(U$95$m * 0.5), $MachinePrecision] / J$95$m), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 * J$95$m), $MachinePrecision] * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if J < 2.0499999999999999e-125

   1. Initial program 66.0%

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

   3. Simplified 65.9%

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

   5. Taylor expanded in J around 0 33.0%

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

      1. neg-mul-1 33.0%

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

   7. Simplified 33.0%

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

   if 2.0499999999999999e-125 < J < 9.0000000000000004e172

   1. Initial program 75.7%

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

   3. Simplified 75.6%

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

   5. Taylor expanded in K around 0 46.1%

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

      1. associate-*r* 46.1%

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

      2. *-commutative 46.1%

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

      3. associate-*r/ 46.1%

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

      4. *-commutative 46.1%

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

      5. unpow2 46.1%

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

      6. associate-/r* 46.3%

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

      7. unpow2 46.3%

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

      8. metadata-eval 46.3%

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

      9. swap-sqr 46.3%

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

      10. associate-*r/ 51.0%

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

      11. associate-*r/ 51.0%

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

      12. associate-*l/ 54.1%

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

      13. associate-*r/ 54.1%

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

      14. unpow2 54.1%

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

      15. associate-*r/ 54.2%

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

      16. *-commutative 54.2%

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

      17. associate-*r/ 54.2%

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

   7. Simplified 54.2%

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

      1. add-cube-cbrt 53.8%

         \[\leadsto \left(J \cdot -2\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\sqrt{1 + {\left(0.5 \cdot \frac{U}{J}\right)}^{2}}} \cdot \sqrt[3]{\sqrt{1 + {\left(0.5 \cdot \frac{U}{J}\right)}^{2}}}\right) \cdot \sqrt[3]{\sqrt{1 + {\left(0.5 \cdot \frac{U}{J}\right)}^{2}}}\right)} \]

      2. pow3 53.7%

         \[\leadsto \left(J \cdot -2\right) \cdot \color{blue}{{\left(\sqrt[3]{\sqrt{1 + {\left(0.5 \cdot \frac{U}{J}\right)}^{2}}}\right)}^{3}} \]

      3. unpow2 53.7%

         \[\leadsto \left(J \cdot -2\right) \cdot {\left(\sqrt[3]{\sqrt{1 + \color{blue}{\left(0.5 \cdot \frac{U}{J}\right) \cdot \left(0.5 \cdot \frac{U}{J}\right)}}}\right)}^{3} \]

      4. hypot-1-def 64.0%

         \[\leadsto \left(J \cdot -2\right) \cdot {\left(\sqrt[3]{\color{blue}{\mathsf{hypot}\left(1, 0.5 \cdot \frac{U}{J}\right)}}\right)}^{3} \]

      5. clear-num 64.0%

         \[\leadsto \left(J \cdot -2\right) \cdot {\left(\sqrt[3]{\mathsf{hypot}\left(1, 0.5 \cdot \color{blue}{\frac{1}{\frac{J}{U}}}\right)}\right)}^{3} \]

      6. un-div-inv 64.0%

         \[\leadsto \left(J \cdot -2\right) \cdot {\left(\sqrt[3]{\mathsf{hypot}\left(1, \color{blue}{\frac{0.5}{\frac{J}{U}}}\right)}\right)}^{3} \]

   9. Applied egg-rr 64.0%

      \[\leadsto \left(J \cdot -2\right) \cdot \color{blue}{{\left(\sqrt[3]{\mathsf{hypot}\left(1, \frac{0.5}{\frac{J}{U}}\right)}\right)}^{3}} \]
   10. Step-by-step derivation

       1. expm1-log1p-u 11.8%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(J \cdot -2\right) \cdot {\left(\sqrt[3]{\mathsf{hypot}\left(1, \frac{0.5}{\frac{J}{U}}\right)}\right)}^{3}\right)\right)} \]

       2. expm1-udef 2.1%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(J \cdot -2\right) \cdot {\left(\sqrt[3]{\mathsf{hypot}\left(1, \frac{0.5}{\frac{J}{U}}\right)}\right)}^{3}\right)} - 1} \]

       3. rem-cube-cbrt 2.1%

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

       4. associate-*l* 2.1%

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

       5. div-inv 2.1%

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

       6. clear-num 2.1%

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

   11. Applied egg-rr 2.1%

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

       1. expm1-def 11.8%

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

       2. expm1-log1p 64.7%

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

       3. associate-*r* 64.7%

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

       4. associate-*r/ 64.7%

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

   13. Simplified 64.7%

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

   if 9.0000000000000004e172 < J

   1. Initial program 99.9%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in J around inf 88.3%

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

      1. associate-*r* 88.3%

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

      2. *-commutative 88.3%

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

   7. Simplified 88.3%

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

4. Final simplification 45.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq 2.05 \cdot 10^{-125}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 9 \cdot 10^{+172}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \mathsf{hypot}\left(1, \frac{U \cdot 0.5}{J}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 39.3% accurate, 24.6× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J_m = \left|J\right| \\ J_s = \mathsf{copysign}\left(1, J\right) \\ J\_s \cdot \begin{array}{l} \mathbf{if}\;K \leq 7.9 \cdot 10^{+62} \lor \neg \left(K \leq 4.3 \cdot 10^{+213}\right) \land K \leq 7 \cdot 10^{+280}:\\ \;\;\;\;-U\_m\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J_m = (fabs.f64 J)
J_s = (copysign.f64 1 J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (*
  J_s
  (if (or (<= K 7.9e+62) (and (not (<= K 4.3e+213)) (<= K 7e+280)))
    (- U_m)
    U_m)))

C

U_m = fabs(U);
J_m = fabs(J);
J_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double tmp;
	if ((K <= 7.9e+62) || (!(K <= 4.3e+213) && (K <= 7e+280))) {
		tmp = -U_m;
	} else {
		tmp = U_m;
	}
	return J_s * tmp;
}

Fortran

U_m = abs(U)
J_m = abs(J)
J_s = copysign(1.0d0, J)
real(8) function code(j_s, j_m, k, u_m)
    real(8), intent (in) :: j_s
    real(8), intent (in) :: j_m
    real(8), intent (in) :: k
    real(8), intent (in) :: u_m
    real(8) :: tmp
    if ((k <= 7.9d+62) .or. (.not. (k <= 4.3d+213)) .and. (k <= 7d+280)) then
        tmp = -u_m
    else
        tmp = u_m
    end if
    code = j_s * tmp
end function

Java

U_m = Math.abs(U);
J_m = Math.abs(J);
J_s = Math.copySign(1.0, J);
public static double code(double J_s, double J_m, double K, double U_m) {
	double tmp;
	if ((K <= 7.9e+62) || (!(K <= 4.3e+213) && (K <= 7e+280))) {
		tmp = -U_m;
	} else {
		tmp = U_m;
	}
	return J_s * tmp;
}

Python

U_m = math.fabs(U)
J_m = math.fabs(J)
J_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	tmp = 0
	if (K <= 7.9e+62) or (not (K <= 4.3e+213) and (K <= 7e+280)):
		tmp = -U_m
	else:
		tmp = U_m
	return J_s * tmp

Julia

U_m = abs(U)
J_m = abs(J)
J_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	tmp = 0.0
	if ((K <= 7.9e+62) || (!(K <= 4.3e+213) && (K <= 7e+280)))
		tmp = Float64(-U_m);
	else
		tmp = U_m;
	end
	return Float64(J_s * tmp)
end

MATLAB

U_m = abs(U);
J_m = abs(J);
J_s = sign(J) * abs(1.0);
function tmp_2 = code(J_s, J_m, K, U_m)
	tmp = 0.0;
	if ((K <= 7.9e+62) || (~((K <= 4.3e+213)) && (K <= 7e+280)))
		tmp = -U_m;
	else
		tmp = U_m;
	end
	tmp_2 = J_s * tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J_m = N[Abs[J], $MachinePrecision]
J_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := N[(J$95$s * If[Or[LessEqual[K, 7.9e+62], And[N[Not[LessEqual[K, 4.3e+213]], $MachinePrecision], LessEqual[K, 7e+280]]], (-U$95$m), U$95$m]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if K < 7.8999999999999997e62 or 4.29999999999999995e213 < K < 7.0000000000000002e280

   1. Initial program 70.9%

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

   3. Simplified 70.9%

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

   5. Taylor expanded in J around 0 29.1%

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

      1. neg-mul-1 29.1%

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

   7. Simplified 29.1%

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

   if 7.8999999999999997e62 < K < 4.29999999999999995e213 or 7.0000000000000002e280 < K

   1. Initial program 74.6%

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

   3. Simplified 74.6%

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

   5. Taylor expanded in U around -inf 17.4%

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

4. Final simplification 27.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;K \leq 7.9 \cdot 10^{+62} \lor \neg \left(K \leq 4.3 \cdot 10^{+213}\right) \land K \leq 7 \cdot 10^{+280}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 48.2% accurate, 24.6× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J_m = \left|J\right| \\ J_s = \mathsf{copysign}\left(1, J\right) \\ J\_s \cdot \begin{array}{l} \mathbf{if}\;U\_m \leq 2100:\\ \;\;\;\;-2 \cdot J\_m\\ \mathbf{elif}\;U\_m \leq 1.2 \cdot 10^{+119} \lor \neg \left(U\_m \leq 8.6 \cdot 10^{+138}\right):\\ \;\;\;\;-U\_m\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J_m = (fabs.f64 J)
J_s = (copysign.f64 1 J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (*
  J_s
  (if (<= U_m 2100.0)
    (* -2.0 J_m)
    (if (or (<= U_m 1.2e+119) (not (<= U_m 8.6e+138))) (- U_m) U_m))))

C

U_m = fabs(U);
J_m = fabs(J);
J_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double tmp;
	if (U_m <= 2100.0) {
		tmp = -2.0 * J_m;
	} else if ((U_m <= 1.2e+119) || !(U_m <= 8.6e+138)) {
		tmp = -U_m;
	} else {
		tmp = U_m;
	}
	return J_s * tmp;
}

Fortran

U_m = abs(U)
J_m = abs(J)
J_s = copysign(1.0d0, J)
real(8) function code(j_s, j_m, k, u_m)
    real(8), intent (in) :: j_s
    real(8), intent (in) :: j_m
    real(8), intent (in) :: k
    real(8), intent (in) :: u_m
    real(8) :: tmp
    if (u_m <= 2100.0d0) then
        tmp = (-2.0d0) * j_m
    else if ((u_m <= 1.2d+119) .or. (.not. (u_m <= 8.6d+138))) then
        tmp = -u_m
    else
        tmp = u_m
    end if
    code = j_s * tmp
end function

Java

U_m = Math.abs(U);
J_m = Math.abs(J);
J_s = Math.copySign(1.0, J);
public static double code(double J_s, double J_m, double K, double U_m) {
	double tmp;
	if (U_m <= 2100.0) {
		tmp = -2.0 * J_m;
	} else if ((U_m <= 1.2e+119) || !(U_m <= 8.6e+138)) {
		tmp = -U_m;
	} else {
		tmp = U_m;
	}
	return J_s * tmp;
}

Python

U_m = math.fabs(U)
J_m = math.fabs(J)
J_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	tmp = 0
	if U_m <= 2100.0:
		tmp = -2.0 * J_m
	elif (U_m <= 1.2e+119) or not (U_m <= 8.6e+138):
		tmp = -U_m
	else:
		tmp = U_m
	return J_s * tmp

Julia

U_m = abs(U)
J_m = abs(J)
J_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	tmp = 0.0
	if (U_m <= 2100.0)
		tmp = Float64(-2.0 * J_m);
	elseif ((U_m <= 1.2e+119) || !(U_m <= 8.6e+138))
		tmp = Float64(-U_m);
	else
		tmp = U_m;
	end
	return Float64(J_s * tmp)
end

MATLAB

U_m = abs(U);
J_m = abs(J);
J_s = sign(J) * abs(1.0);
function tmp_2 = code(J_s, J_m, K, U_m)
	tmp = 0.0;
	if (U_m <= 2100.0)
		tmp = -2.0 * J_m;
	elseif ((U_m <= 1.2e+119) || ~((U_m <= 8.6e+138)))
		tmp = -U_m;
	else
		tmp = U_m;
	end
	tmp_2 = J_s * tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J_m = N[Abs[J], $MachinePrecision]
J_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := N[(J$95$s * If[LessEqual[U$95$m, 2100.0], N[(-2.0 * J$95$m), $MachinePrecision], If[Or[LessEqual[U$95$m, 1.2e+119], N[Not[LessEqual[U$95$m, 8.6e+138]], $MachinePrecision]], (-U$95$m), U$95$m]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if U < 2100

   1. Initial program 78.3%

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

   3. Simplified 78.2%

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

   5. Taylor expanded in K around 0 38.9%

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

      1. associate-*r* 38.9%

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

      2. *-commutative 38.9%

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

      3. associate-*r/ 38.9%

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

      4. *-commutative 38.9%

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

      5. unpow2 38.9%

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

      6. associate-/r* 45.7%

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

      7. unpow2 45.7%

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

      8. metadata-eval 45.7%

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

      9. swap-sqr 45.7%

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

      10. associate-*r/ 51.0%

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

      11. associate-*r/ 50.9%

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

      12. associate-*l/ 53.3%

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

      13. associate-*r/ 53.3%

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

      14. unpow2 53.3%

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

      15. associate-*r/ 53.3%

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

      16. *-commutative 53.3%

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

      17. associate-*r/ 53.3%

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

   7. Simplified 53.3%

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

   8. Taylor expanded in J around inf 39.3%

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

      1. *-commutative 39.3%

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

   10. Simplified 39.3%

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

   if 2100 < U < 1.2e119 or 8.5999999999999996e138 < U

   1. Initial program 45.3%

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

   3. Simplified 45.3%

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

   5. Taylor expanded in J around 0 40.7%

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

      1. neg-mul-1 40.7%

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

   7. Simplified 40.7%

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

   if 1.2e119 < U < 8.5999999999999996e138

   1. Initial program 72.4%

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

   3. Simplified 72.4%

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

   5. Taylor expanded in U around -inf 44.0%

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

4. Final simplification 39.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 2100:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;U \leq 1.2 \cdot 10^{+119} \lor \neg \left(U \leq 8.6 \cdot 10^{+138}\right):\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 14.1% accurate, 420.0× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J_m = \left|J\right| \\ J_s = \mathsf{copysign}\left(1, J\right) \\ J\_s \cdot U\_m \end{array} \]

FPCore

U_m = (fabs.f64 U)
J_m = (fabs.f64 J)
J_s = (copysign.f64 1 J)
(FPCore (J_s J_m K U_m) :precision binary64 (* J_s U_m))

C

U_m = fabs(U);
J_m = fabs(J);
J_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	return J_s * U_m;
}

Fortran

U_m = abs(U)
J_m = abs(J)
J_s = copysign(1.0d0, J)
real(8) function code(j_s, j_m, k, u_m)
    real(8), intent (in) :: j_s
    real(8), intent (in) :: j_m
    real(8), intent (in) :: k
    real(8), intent (in) :: u_m
    code = j_s * u_m
end function

Java

U_m = Math.abs(U);
J_m = Math.abs(J);
J_s = Math.copySign(1.0, J);
public static double code(double J_s, double J_m, double K, double U_m) {
	return J_s * U_m;
}

Python

U_m = math.fabs(U)
J_m = math.fabs(J)
J_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	return J_s * U_m

Julia

U_m = abs(U)
J_m = abs(J)
J_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	return Float64(J_s * U_m)
end

MATLAB

U_m = abs(U);
J_m = abs(J);
J_s = sign(J) * abs(1.0);
function tmp = code(J_s, J_m, K, U_m)
	tmp = J_s * U_m;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J_m = N[Abs[J], $MachinePrecision]
J_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := N[(J$95$s * U$95$m), $MachinePrecision]

Derivation

1. Initial program 71.3%

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

3. Simplified 71.3%

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

5. Taylor expanded in U around -inf 26.0%

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

6. Final simplification 26.0%

   \[\leadsto U \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024040 
(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)))))