Maksimov and Kolovsky, Equation (3)

Percentage Accurate: 72.5% → 99.8%

Time: 14.7s

Alternatives: 8

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: 72.5% 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, 0.4× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J_m = \left|J\right| \\ J_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := J\_m \cdot t\_0\\ t_2 := \left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_0 \cdot \left(J\_m \cdot 2\right)}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_2 \leq 10^{+307}:\\ \;\;\;\;-2 \cdot \left(t\_1 \cdot \mathsf{hypot}\left(1, \frac{\frac{U\_m}{2}}{t\_1}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \end{array} \]

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
         (*
          (* (* -2.0 J_m) t_0)
          (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 1e+307)
        (* -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 = ((-2.0 * J_m) * t_0) * 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 <= 1e+307) {
		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 = ((-2.0 * J_m) * t_0) * 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 <= 1e+307) {
		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 = ((-2.0 * J_m) * t_0) * 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 <= 1e+307:
		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(Float64(-2.0 * J_m) * t_0) * 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 <= 1e+307)
		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 = ((-2.0 * J_m) * t_0) * 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 <= 1e+307)
		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[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$0), $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, 1e+307], 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 5.9%

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

   3. Simplified 5.9%

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

   5. Taylor expanded in J around 0 49.2%

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

      1. mul-1-neg 49.2%

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

   7. Simplified 49.2%

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

   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)))) < 9.99999999999999986e306

   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 9.99999999999999986e306 < (*.f64 (*.f64 (*.f64 -2 J) (cos.f64 (/.f64 K 2))) (sqrt.f64 (+.f64 1 (pow.f64 (/.f64 U (*.f64 (*.f64 2 J) (cos.f64 (/.f64 K 2)))) 2))))

   1. Initial program 7.3%

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

   3. Simplified 7.3%

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

   5. Taylor expanded in U around -inf 35.5%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 10^{+307}:\\ \;\;\;\;-2 \cdot \left(\left(J \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{\frac{U}{2}}{J \cdot \cos \left(\frac{K}{2}\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 78.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J_m = \left|J\right| \\ J_s = \mathsf{copysign}\left(1, J\right) \\ J\_s \cdot \begin{array}{l} \mathbf{if}\;U\_m \leq 1.15 \cdot 10^{+171}:\\ \;\;\;\;-2 \cdot \left(\left(J\_m \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{\frac{U\_m}{2}}{J\_m}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-U\_m\\ \end{array} \end{array} \]

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 1.15e+171)
    (* -2.0 (* (* J_m (cos (/ K 2.0))) (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 tmp;
	if (U_m <= 1.15e+171) {
		tmp = -2.0 * ((J_m * cos((K / 2.0))) * 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 tmp;
	if (U_m <= 1.15e+171) {
		tmp = -2.0 * ((J_m * Math.cos((K / 2.0))) * 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):
	tmp = 0
	if U_m <= 1.15e+171:
		tmp = -2.0 * ((J_m * math.cos((K / 2.0))) * 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)
	tmp = 0.0
	if (U_m <= 1.15e+171)
		tmp = Float64(-2.0 * Float64(Float64(J_m * cos(Float64(K / 2.0))) * 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)
	tmp = 0.0;
	if (U_m <= 1.15e+171)
		tmp = -2.0 * ((J_m * cos((K / 2.0))) * 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_] := N[(J$95$s * If[LessEqual[U$95$m, 1.15e+171], N[(-2.0 * N[(N[(J$95$m * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $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 2 regimes

2. if U < 1.15000000000000009e171

   1. Initial program 73.4%

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

      1. associate-*l* 73.4%

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

      2. associate-*l* 73.4%

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

      3. unpow2 73.4%

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

      4. sqr-neg 73.4%

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

      5. distribute-frac-neg 73.4%

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

      6. distribute-frac-neg 73.4%

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

      7. unpow2 73.4%

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

   3. Simplified 89.5%

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

   5. Taylor expanded in K around 0 76.3%

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

   if 1.15000000000000009e171 < U

   1. Initial program 36.4%

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

   3. Simplified 36.3%

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

   5. Taylor expanded in J around 0 45.0%

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

      1. mul-1-neg 45.0%

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

   7. Simplified 45.0%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 1.15 \cdot 10^{+171}:\\ \;\;\;\;-2 \cdot \left(\left(J \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{\frac{U}{2}}{J}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 70.1% accurate, 3.5× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J_m = \left|J\right| \\ J_s = \mathsf{copysign}\left(1, J\right) \\ J\_s \cdot \begin{array}{l} \mathbf{if}\;U\_m \leq 1.13 \cdot 10^{-71}:\\ \;\;\;\;-2 \cdot \left(J\_m \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{elif}\;U\_m \leq 2.55 \cdot 10^{+63}:\\ \;\;\;\;-2 \cdot \left(J\_m \cdot \mathsf{hypot}\left(1, \frac{\frac{U\_m}{2}}{J\_m}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-U\_m\\ \end{array} \end{array} \]

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 1.13e-71)
    (* -2.0 (* J_m (cos (* K 0.5))))
    (if (<= U_m 2.55e+63)
      (* -2.0 (* J_m (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 tmp;
	if (U_m <= 1.13e-71) {
		tmp = -2.0 * (J_m * cos((K * 0.5)));
	} else if (U_m <= 2.55e+63) {
		tmp = -2.0 * (J_m * 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 tmp;
	if (U_m <= 1.13e-71) {
		tmp = -2.0 * (J_m * Math.cos((K * 0.5)));
	} else if (U_m <= 2.55e+63) {
		tmp = -2.0 * (J_m * 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):
	tmp = 0
	if U_m <= 1.13e-71:
		tmp = -2.0 * (J_m * math.cos((K * 0.5)))
	elif U_m <= 2.55e+63:
		tmp = -2.0 * (J_m * 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)
	tmp = 0.0
	if (U_m <= 1.13e-71)
		tmp = Float64(-2.0 * Float64(J_m * cos(Float64(K * 0.5))));
	elseif (U_m <= 2.55e+63)
		tmp = Float64(-2.0 * Float64(J_m * 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)
	tmp = 0.0;
	if (U_m <= 1.13e-71)
		tmp = -2.0 * (J_m * cos((K * 0.5)));
	elseif (U_m <= 2.55e+63)
		tmp = -2.0 * (J_m * 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_] := N[(J$95$s * If[LessEqual[U$95$m, 1.13e-71], N[(-2.0 * N[(J$95$m * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[U$95$m, 2.55e+63], N[(-2.0 * N[(J$95$m * 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 U < 1.12999999999999999e-71

   1. Initial program 75.2%

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

      1. associate-*l* 75.2%

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

      2. associate-*l* 75.2%

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

      3. unpow2 75.2%

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

      4. sqr-neg 75.2%

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

      5. distribute-frac-neg 75.2%

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

      6. distribute-frac-neg 75.2%

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

      7. unpow2 75.2%

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

   3. Simplified 88.1%

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

   5. Taylor expanded in U around 0 57.1%

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

   if 1.12999999999999999e-71 < U < 2.5499999999999999e63

   1. Initial program 73.6%

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

      1. associate-*l* 73.6%

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

      2. associate-*l* 73.6%

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

      3. unpow2 73.6%

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

      4. sqr-neg 73.6%

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

      5. distribute-frac-neg 73.6%

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

      6. distribute-frac-neg 73.6%

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

      7. unpow2 73.6%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in K around 0 74.9%

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

   6. Taylor expanded in K around 0 75.0%

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

   if 2.5499999999999999e63 < U

   1. Initial program 44.7%

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

   3. Simplified 44.6%

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

   5. Taylor expanded in J around 0 43.5%

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

      1. mul-1-neg 43.5%

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

   7. Simplified 43.5%

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

4. Final simplification 56.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 1.13 \cdot 10^{-71}:\\ \;\;\;\;-2 \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{elif}\;U \leq 2.55 \cdot 10^{+63}:\\ \;\;\;\;-2 \cdot \left(J \cdot \mathsf{hypot}\left(1, \frac{\frac{U}{2}}{J}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.1% accurate, 3.7× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J_m = \left|J\right| \\ J_s = \mathsf{copysign}\left(1, J\right) \\ J\_s \cdot \begin{array}{l} \mathbf{if}\;J\_m \leq 9.5 \cdot 10^{-26}:\\ \;\;\;\;\frac{-2 \cdot {J\_m}^{2}}{U\_m} - U\_m\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(J\_m \cdot \cos \left(K \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

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 9.5e-26)
    (- (/ (* -2.0 (pow J_m 2.0)) U_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 <= 9.5e-26) {
		tmp = ((-2.0 * pow(J_m, 2.0)) / U_m) - U_m;
	} else {
		tmp = -2.0 * (J_m * cos((K * 0.5)));
	}
	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 (j_m <= 9.5d-26) then
        tmp = (((-2.0d0) * (j_m ** 2.0d0)) / u_m) - u_m
    else
        tmp = (-2.0d0) * (j_m * cos((k * 0.5d0)))
    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 (J_m <= 9.5e-26) {
		tmp = ((-2.0 * Math.pow(J_m, 2.0)) / U_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 <= 9.5e-26:
		tmp = ((-2.0 * math.pow(J_m, 2.0)) / U_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 <= 9.5e-26)
		tmp = Float64(Float64(Float64(-2.0 * (J_m ^ 2.0)) / U_m) - U_m);
	else
		tmp = Float64(-2.0 * Float64(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 <= 9.5e-26)
		tmp = ((-2.0 * (J_m ^ 2.0)) / U_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, 9.5e-26], N[(N[(N[(-2.0 * N[Power[J$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / U$95$m), $MachinePrecision] - U$95$m), $MachinePrecision], N[(-2.0 * N[(J$95$m * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if J < 9.4999999999999995e-26

   1. Initial program 60.1%

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

   3. Simplified 60.1%

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

   5. Taylor expanded in J around 0 34.5%

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

      1. mul-1-neg 34.5%

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

      2. unsub-neg 34.5%

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

      3. associate-*r/ 34.5%

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

      4. associate-/l* 34.5%

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

      5. unpow2 34.5%

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

      6. *-commutative 34.5%

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

      7. unpow2 34.5%

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

      8. swap-sqr 34.5%

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

      9. unpow2 34.5%

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

      10. *-commutative 34.5%

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

   7. Simplified 34.5%

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

   8. Taylor expanded in K around 0 34.5%

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

      1. associate-*r/ 34.5%

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

   10. Simplified 34.5%

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

   if 9.4999999999999995e-26 < J

   1. Initial program 94.4%

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

      1. associate-*l* 94.4%

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

      2. associate-*l* 94.4%

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

      3. unpow2 94.4%

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

      4. sqr-neg 94.4%

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

      5. distribute-frac-neg 94.4%

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

      6. distribute-frac-neg 94.4%

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

      7. unpow2 94.4%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in U around 0 74.7%

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

4. Final simplification 45.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq 9.5 \cdot 10^{-26}:\\ \;\;\;\;\frac{-2 \cdot {J}^{2}}{U} - U\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.1% accurate, 3.7× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J_m = \left|J\right| \\ J_s = \mathsf{copysign}\left(1, J\right) \\ J\_s \cdot \begin{array}{l} \mathbf{if}\;J\_m \leq 5.2 \cdot 10^{-23}:\\ \;\;\;\;-U\_m\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(J\_m \cdot \cos \left(K \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

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 5.2e-23) (- 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 <= 5.2e-23) {
		tmp = -U_m;
	} else {
		tmp = -2.0 * (J_m * cos((K * 0.5)));
	}
	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 (j_m <= 5.2d-23) then
        tmp = -u_m
    else
        tmp = (-2.0d0) * (j_m * cos((k * 0.5d0)))
    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 (J_m <= 5.2e-23) {
		tmp = -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 <= 5.2e-23:
		tmp = -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 <= 5.2e-23)
		tmp = Float64(-U_m);
	else
		tmp = Float64(-2.0 * Float64(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 <= 5.2e-23)
		tmp = -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, 5.2e-23], (-U$95$m), N[(-2.0 * N[(J$95$m * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if J < 5.2e-23

   1. Initial program 60.1%

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

   3. Simplified 60.1%

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

   5. Taylor expanded in J around 0 34.4%

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

      1. mul-1-neg 34.4%

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

   7. Simplified 34.4%

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

   if 5.2e-23 < J

   1. Initial program 94.4%

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

      1. associate-*l* 94.4%

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

      2. associate-*l* 94.4%

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

      3. unpow2 94.4%

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

      4. sqr-neg 94.4%

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

      5. distribute-frac-neg 94.4%

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

      6. distribute-frac-neg 94.4%

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

      7. unpow2 94.4%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in U around 0 74.7%

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

4. Final simplification 45.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq 5.2 \cdot 10^{-23}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 51.4% accurate, 52.4× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J_m = \left|J\right| \\ J_s = \mathsf{copysign}\left(1, J\right) \\ J\_s \cdot \begin{array}{l} \mathbf{if}\;U\_m \leq 4.8 \cdot 10^{-48}:\\ \;\;\;\;-2 \cdot J\_m\\ \mathbf{else}:\\ \;\;\;\;-U\_m\\ \end{array} \end{array} \]

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 4.8e-48) (* -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 tmp;
	if (U_m <= 4.8e-48) {
		tmp = -2.0 * J_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 <= 4.8d-48) then
        tmp = (-2.0d0) * j_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 <= 4.8e-48) {
		tmp = -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):
	tmp = 0
	if U_m <= 4.8e-48:
		tmp = -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)
	tmp = 0.0
	if (U_m <= 4.8e-48)
		tmp = Float64(-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)
	tmp = 0.0;
	if (U_m <= 4.8e-48)
		tmp = -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_] := N[(J$95$s * If[LessEqual[U$95$m, 4.8e-48], N[(-2.0 * J$95$m), $MachinePrecision], (-U$95$m)]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if U < 4.8e-48

   1. Initial program 75.6%

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

      1. associate-*l* 75.6%

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

      2. associate-*l* 75.6%

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

      3. unpow2 75.6%

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

      4. sqr-neg 75.6%

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

      5. distribute-frac-neg 75.6%

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

      6. distribute-frac-neg 75.6%

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

      7. unpow2 75.6%

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

   3. Simplified 88.3%

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

   5. Taylor expanded in U around 0 57.3%

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

   6. Taylor expanded in K around 0 32.1%

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

   if 4.8e-48 < U

   1. Initial program 51.6%

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

   3. Simplified 51.5%

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

   5. Taylor expanded in J around 0 39.1%

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

      1. mul-1-neg 39.1%

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

   7. Simplified 39.1%

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

4. Final simplification 34.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 4.8 \cdot 10^{-48}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 39.9% accurate, 210.0× speedup

Math

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

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 * Float64(-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 69.2%

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

3. Simplified 69.2%

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

5. Taylor expanded in J around 0 28.2%

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

   1. mul-1-neg 28.2%

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

7. Simplified 28.2%

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

8. Final simplification 28.2%

   \[\leadsto -U \]
9. Add Preprocessing

Alternative 8: 13.8% 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 69.2%

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

3. Simplified 69.2%

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

5. Taylor expanded in U around -inf 27.9%

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

6. Final simplification 27.9%

   \[\leadsto U \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024096 
(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)))))