Maksimov and Kolovsky, Equation (3)

Percentage Accurate: 73.6% → 99.6%

Time: 11.2s

Alternatives: 9

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.6% 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: 99.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_0 \cdot \left(J\_m \cdot 2\right)}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;0 - U\_m\\ \mathbf{elif}\;t\_1 \leq 10^{+298}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* (* -2.0 J_m) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* t_0 (* J_m 2.0))) 2.0))))))
   (*
    J_s
    (if (<= t_1 (- INFINITY)) (- 0.0 U_m) (if (<= t_1 1e+298) 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 = ((-2.0 * J_m) * t_0) * sqrt((1.0 + pow((U_m / (t_0 * (J_m * 2.0))), 2.0)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = 0.0 - U_m;
	} else if (t_1 <= 1e+298) {
		tmp = 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 = ((-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_1 <= -Double.POSITIVE_INFINITY) {
		tmp = 0.0 - U_m;
	} else if (t_1 <= 1e+298) {
		tmp = 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 = ((-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_1 <= -math.inf:
		tmp = 0.0 - U_m
	elif t_1 <= 1e+298:
		tmp = 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(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_1 <= Float64(-Inf))
		tmp = Float64(0.0 - U_m);
	elseif (t_1 <= 1e+298)
		tmp = 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 = ((-2.0 * J_m) * t_0) * sqrt((1.0 + ((U_m / (t_0 * (J_m * 2.0))) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = 0.0 - U_m;
	elseif (t_1 <= 1e+298)
		tmp = 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[(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$1, (-Infinity)], N[(0.0 - U$95$m), $MachinePrecision], If[LessEqual[t$95$1, 1e+298], t$95$1, U$95$m]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -inf.0

   1. Initial program 5.1%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(U\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{U} \]

      3. --lowering--.f64 46.1%

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{U}\right) \]

   5. Simplified 46.1%

      \[\leadsto \color{blue}{0 - U} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(U\right) \]

      2. neg-lowering-neg.f64 46.1%

         \[\leadsto \mathsf{neg.f64}\left(U\right) \]

   7. Applied egg-rr 46.1%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 9.9999999999999996e297

   1. Initial program 99.8%

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

   if 9.9999999999999996e297 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 17.5%

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

   3. Taylor expanded in U around -inf

      \[\leadsto \color{blue}{U} \]
   4. Step-by-step derivation

   5. Simplified 53.0%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -\infty:\\ \;\;\;\;0 - U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 10^{+298}:\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 79.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ J\_s \cdot \begin{array}{l} \mathbf{if}\;J\_m \leq 5.4 \cdot 10^{-181}:\\ \;\;\;\;J\_m \cdot \frac{-2 \cdot J\_m}{U\_m} - U\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-2 \cdot J\_m\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 - \frac{U\_m}{\left(-2 \cdot J\_m\right) \cdot \left(\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \frac{K}{2}\right)\right) \cdot \frac{J\_m \cdot 2}{U\_m}\right)}}\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (*
  J_s
  (if (<= J_m 5.4e-181)
    (- (* J_m (/ (* -2.0 J_m) U_m)) U_m)
    (*
     (* (* -2.0 J_m) (cos (/ K 2.0)))
     (sqrt
      (-
       1.0
       (/
        U_m
        (*
         (* -2.0 J_m)
         (*
          (+ 0.5 (* 0.5 (cos (* 2.0 (/ K 2.0)))))
          (/ (* J_m 2.0) 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 (J_m <= 5.4e-181) {
		tmp = (J_m * ((-2.0 * J_m) / U_m)) - U_m;
	} else {
		tmp = ((-2.0 * J_m) * cos((K / 2.0))) * sqrt((1.0 - (U_m / ((-2.0 * J_m) * ((0.5 + (0.5 * cos((2.0 * (K / 2.0))))) * ((J_m * 2.0) / 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 (j_m <= 5.4d-181) then
        tmp = (j_m * (((-2.0d0) * j_m) / u_m)) - u_m
    else
        tmp = (((-2.0d0) * j_m) * cos((k / 2.0d0))) * sqrt((1.0d0 - (u_m / (((-2.0d0) * j_m) * ((0.5d0 + (0.5d0 * cos((2.0d0 * (k / 2.0d0))))) * ((j_m * 2.0d0) / 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 (J_m <= 5.4e-181) {
		tmp = (J_m * ((-2.0 * J_m) / U_m)) - U_m;
	} else {
		tmp = ((-2.0 * J_m) * Math.cos((K / 2.0))) * Math.sqrt((1.0 - (U_m / ((-2.0 * J_m) * ((0.5 + (0.5 * Math.cos((2.0 * (K / 2.0))))) * ((J_m * 2.0) / 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 J_m <= 5.4e-181:
		tmp = (J_m * ((-2.0 * J_m) / U_m)) - U_m
	else:
		tmp = ((-2.0 * J_m) * math.cos((K / 2.0))) * math.sqrt((1.0 - (U_m / ((-2.0 * J_m) * ((0.5 + (0.5 * math.cos((2.0 * (K / 2.0))))) * ((J_m * 2.0) / 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 (J_m <= 5.4e-181)
		tmp = Float64(Float64(J_m * Float64(Float64(-2.0 * J_m) / U_m)) - U_m);
	else
		tmp = Float64(Float64(Float64(-2.0 * J_m) * cos(Float64(K / 2.0))) * sqrt(Float64(1.0 - Float64(U_m / Float64(Float64(-2.0 * J_m) * Float64(Float64(0.5 + Float64(0.5 * cos(Float64(2.0 * Float64(K / 2.0))))) * Float64(Float64(J_m * 2.0) / 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 (J_m <= 5.4e-181)
		tmp = (J_m * ((-2.0 * J_m) / U_m)) - U_m;
	else
		tmp = ((-2.0 * J_m) * cos((K / 2.0))) * sqrt((1.0 - (U_m / ((-2.0 * J_m) * ((0.5 + (0.5 * cos((2.0 * (K / 2.0))))) * ((J_m * 2.0) / 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[J$95$m, 5.4e-181], N[(N[(J$95$m * N[(N[(-2.0 * J$95$m), $MachinePrecision] / U$95$m), $MachinePrecision]), $MachinePrecision] - U$95$m), $MachinePrecision], N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 - N[(U$95$m / N[(N[(-2.0 * J$95$m), $MachinePrecision] * N[(N[(0.5 + N[(0.5 * N[Cos[N[(2.0 * N[(K / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(J$95$m * 2.0), $MachinePrecision] / U$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if J < 5.3999999999999999e-181

   1. Initial program 66.2%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{-2 \cdot \left(J \cdot \sqrt{1 + \frac{1}{4} \cdot \frac{{U}^{2}}{{J}^{2}}}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{1 + \frac{1}{4} \cdot \frac{{U}^{2}}{{J}^{2}}}\right), \color{blue}{\left(-2 \cdot J\right)}\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(1 + \frac{1}{4} \cdot \frac{{U}^{2}}{{J}^{2}}\right)\right), \left(\color{blue}{-2} \cdot J\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{4} \cdot \frac{{U}^{2}}{{J}^{2}}\right)\right)\right), \left(-2 \cdot J\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{1}{4} \cdot {U}^{2}}{{J}^{2}}\right)\right)\right), \left(-2 \cdot J\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{1}{4} \cdot {U}^{2}\right), \left({J}^{2}\right)\right)\right)\right), \left(-2 \cdot J\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left({U}^{2} \cdot \frac{1}{4}\right), \left({J}^{2}\right)\right)\right)\right), \left(-2 \cdot J\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({U}^{2}\right), \frac{1}{4}\right), \left({J}^{2}\right)\right)\right)\right), \left(-2 \cdot J\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(U \cdot U\right), \frac{1}{4}\right), \left({J}^{2}\right)\right)\right)\right), \left(-2 \cdot J\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, U\right), \frac{1}{4}\right), \left({J}^{2}\right)\right)\right)\right), \left(-2 \cdot J\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, U\right), \frac{1}{4}\right), \left(J \cdot J\right)\right)\right)\right), \left(-2 \cdot J\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, U\right), \frac{1}{4}\right), \mathsf{*.f64}\left(J, J\right)\right)\right)\right), \left(-2 \cdot J\right)\right) \]

      14. *-lowering-*.f64 25.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, U\right), \frac{1}{4}\right), \mathsf{*.f64}\left(J, J\right)\right)\right)\right), \mathsf{*.f64}\left(-2, \color{blue}{J}\right)\right) \]

   5. Simplified 25.7%

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

   6. Taylor expanded in J around 0

      \[\leadsto \color{blue}{-2 \cdot \frac{{J}^{2}}{U} + -1 \cdot U} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto -2 \cdot \frac{{J}^{2}}{U} + \left(\mathsf{neg}\left(U\right)\right) \]

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(-2 \cdot \frac{{J}^{2}}{U}\right), \color{blue}{U}\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{-2 \cdot {J}^{2}}{U}\right), U\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(-2 \cdot {J}^{2}\right), U\right), U\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left({J}^{2} \cdot -2\right), U\right), U\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({J}^{2}\right), -2\right), U\right), U\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(J \cdot J\right), -2\right), U\right), U\right) \]

      9. *-lowering-*.f64 26.9%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, J\right), -2\right), U\right), U\right) \]

   8. Simplified 26.9%

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

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\left(J \cdot J\right) \cdot -2}{U}\right), \color{blue}{U}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{J \cdot \left(J \cdot -2\right)}{U}\right), U\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{J \cdot \left(-2 \cdot J\right)}{U}\right), U\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(J \cdot \frac{-2 \cdot J}{U}\right), U\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(J, \left(\frac{-2 \cdot J}{U}\right)\right), U\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(J, \mathsf{/.f64}\left(\left(-2 \cdot J\right), U\right)\right), U\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(J, \mathsf{/.f64}\left(\left(J \cdot -2\right), U\right)\right), U\right) \]

      8. *-lowering-*.f64 29.4%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(J, \mathsf{/.f64}\left(\mathsf{*.f64}\left(J, -2\right), U\right)\right), U\right) \]

   10. Applied egg-rr 29.4%

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

   if 5.3999999999999999e-181 < J

   1. Initial program 86.5%

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

      1. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)\right)\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{\frac{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}{U}} \cdot \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)\right)\right)\right) \]

      3. frac-2neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{\frac{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}{U}} \cdot \frac{\mathsf{neg}\left(U\right)}{\mathsf{neg}\left(\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right)}\right)\right)\right)\right) \]

      4. frac-times N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1 \cdot \left(\mathsf{neg}\left(U\right)\right)}{\frac{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}{U} \cdot \left(\mathsf{neg}\left(\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right)\right)}\right)\right)\right)\right) \]

      5. frac-2neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(1 \cdot \left(\mathsf{neg}\left(U\right)\right)\right)}{\mathsf{neg}\left(\frac{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}{U} \cdot \left(\mathsf{neg}\left(\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right)\right)\right)}\right)\right)\right)\right) \]

      6. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(U\right)\right)\right)}{\mathsf{neg}\left(\frac{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}{U} \cdot \left(\mathsf{neg}\left(\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right)\right)\right)}\right)\right)\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{U}{\mathsf{neg}\left(\frac{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}{U} \cdot \left(\mathsf{neg}\left(\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right)\right)\right)}\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(U, \left(\mathsf{neg}\left(\frac{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}{U} \cdot \left(\mathsf{neg}\left(\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      9. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(U, \mathsf{neg.f64}\left(\left(\frac{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}{U} \cdot \left(\mathsf{neg}\left(\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(U, \mathsf{neg.f64}\left(\left(\frac{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}{U} \cdot \left(\mathsf{neg}\left(\cos \left(\frac{K}{2}\right) \cdot \left(2 \cdot J\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      11. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(U, \mathsf{neg.f64}\left(\left(\frac{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}{U} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(\mathsf{neg}\left(2 \cdot J\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 86.3%

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

4. Final simplification 51.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq 5.4 \cdot 10^{-181}:\\ \;\;\;\;J \cdot \frac{-2 \cdot J}{U} - U\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 - \frac{U}{\left(-2 \cdot J\right) \cdot \left(\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \frac{K}{2}\right)\right) \cdot \frac{J \cdot 2}{U}\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ J\_s \cdot \begin{array}{l} \mathbf{if}\;J\_m \leq 3.9 \cdot 10^{-128}:\\ \;\;\;\;J\_m \cdot \left(\left(-2 \cdot J\_m\right) \cdot \frac{0.5 + 0.5 \cdot \cos K}{U\_m}\right) - U\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-2 \cdot J\_m\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 - \frac{U\_m}{\left(-2 \cdot J\_m\right) \cdot \frac{J\_m \cdot 2}{U\_m}}}\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (*
  J_s
  (if (<= J_m 3.9e-128)
    (- (* J_m (* (* -2.0 J_m) (/ (+ 0.5 (* 0.5 (cos K))) U_m))) U_m)
    (*
     (* (* -2.0 J_m) (cos (/ K 2.0)))
     (sqrt (- 1.0 (/ U_m (* (* -2.0 J_m) (/ (* J_m 2.0) 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 (J_m <= 3.9e-128) {
		tmp = (J_m * ((-2.0 * J_m) * ((0.5 + (0.5 * cos(K))) / U_m))) - U_m;
	} else {
		tmp = ((-2.0 * J_m) * cos((K / 2.0))) * sqrt((1.0 - (U_m / ((-2.0 * J_m) * ((J_m * 2.0) / 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 (j_m <= 3.9d-128) then
        tmp = (j_m * (((-2.0d0) * j_m) * ((0.5d0 + (0.5d0 * cos(k))) / u_m))) - u_m
    else
        tmp = (((-2.0d0) * j_m) * cos((k / 2.0d0))) * sqrt((1.0d0 - (u_m / (((-2.0d0) * j_m) * ((j_m * 2.0d0) / 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 (J_m <= 3.9e-128) {
		tmp = (J_m * ((-2.0 * J_m) * ((0.5 + (0.5 * Math.cos(K))) / U_m))) - U_m;
	} else {
		tmp = ((-2.0 * J_m) * Math.cos((K / 2.0))) * Math.sqrt((1.0 - (U_m / ((-2.0 * J_m) * ((J_m * 2.0) / 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 J_m <= 3.9e-128:
		tmp = (J_m * ((-2.0 * J_m) * ((0.5 + (0.5 * math.cos(K))) / U_m))) - U_m
	else:
		tmp = ((-2.0 * J_m) * math.cos((K / 2.0))) * math.sqrt((1.0 - (U_m / ((-2.0 * J_m) * ((J_m * 2.0) / 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 (J_m <= 3.9e-128)
		tmp = Float64(Float64(J_m * Float64(Float64(-2.0 * J_m) * Float64(Float64(0.5 + Float64(0.5 * cos(K))) / U_m))) - U_m);
	else
		tmp = Float64(Float64(Float64(-2.0 * J_m) * cos(Float64(K / 2.0))) * sqrt(Float64(1.0 - Float64(U_m / Float64(Float64(-2.0 * J_m) * Float64(Float64(J_m * 2.0) / 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 (J_m <= 3.9e-128)
		tmp = (J_m * ((-2.0 * J_m) * ((0.5 + (0.5 * cos(K))) / U_m))) - U_m;
	else
		tmp = ((-2.0 * J_m) * cos((K / 2.0))) * sqrt((1.0 - (U_m / ((-2.0 * J_m) * ((J_m * 2.0) / 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[J$95$m, 3.9e-128], N[(N[(J$95$m * N[(N[(-2.0 * J$95$m), $MachinePrecision] * N[(N[(0.5 + N[(0.5 * N[Cos[K], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / U$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - U$95$m), $MachinePrecision], N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 - N[(U$95$m / N[(N[(-2.0 * J$95$m), $MachinePrecision] * N[(N[(J$95$m * 2.0), $MachinePrecision] / U$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if J < 3.89999999999999997e-128

   1. Initial program 66.4%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({J}^{2} \cdot \left(-2 \cdot \frac{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}{U}\right)\right), \color{blue}{U}\right) \]

   5. Simplified 28.2%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}{U} \cdot \left(\left(-2 \cdot J\right) \cdot J\right)\right), U\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(\frac{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}{U} \cdot \left(-2 \cdot J\right)\right) \cdot J\right), U\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(\frac{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}{U} \cdot \left(-2 \cdot J\right)\right), J\right), U\right) \]

   7. Applied egg-rr 30.6%

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

   if 3.89999999999999997e-128 < J

   1. Initial program 89.0%

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

      1. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)\right)\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{\frac{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}{U}} \cdot \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)\right)\right)\right) \]

      3. frac-2neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{\frac{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}{U}} \cdot \frac{\mathsf{neg}\left(U\right)}{\mathsf{neg}\left(\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right)}\right)\right)\right)\right) \]

      4. frac-times N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1 \cdot \left(\mathsf{neg}\left(U\right)\right)}{\frac{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}{U} \cdot \left(\mathsf{neg}\left(\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right)\right)}\right)\right)\right)\right) \]

      5. frac-2neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(1 \cdot \left(\mathsf{neg}\left(U\right)\right)\right)}{\mathsf{neg}\left(\frac{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}{U} \cdot \left(\mathsf{neg}\left(\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right)\right)\right)}\right)\right)\right)\right) \]

      6. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(U\right)\right)\right)}{\mathsf{neg}\left(\frac{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}{U} \cdot \left(\mathsf{neg}\left(\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right)\right)\right)}\right)\right)\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{U}{\mathsf{neg}\left(\frac{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}{U} \cdot \left(\mathsf{neg}\left(\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right)\right)\right)}\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(U, \left(\mathsf{neg}\left(\frac{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}{U} \cdot \left(\mathsf{neg}\left(\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      9. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(U, \mathsf{neg.f64}\left(\left(\frac{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}{U} \cdot \left(\mathsf{neg}\left(\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(U, \mathsf{neg.f64}\left(\left(\frac{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}{U} \cdot \left(\mathsf{neg}\left(\cos \left(\frac{K}{2}\right) \cdot \left(2 \cdot J\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      11. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(U, \mathsf{neg.f64}\left(\left(\frac{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}{U} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(\mathsf{neg}\left(2 \cdot J\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 88.8%

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

   5. Taylor expanded in K around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(U, \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(2 \cdot \frac{J}{U}\right)}, \mathsf{*.f64}\left(-2, J\right)\right)\right)\right)\right)\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(U, \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(\frac{2 \cdot J}{U}\right), \mathsf{*.f64}\left(-2, J\right)\right)\right)\right)\right)\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(U, \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot J\right), U\right), \mathsf{*.f64}\left(-2, J\right)\right)\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f64 79.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(U, \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, J\right), U\right), \mathsf{*.f64}\left(-2, J\right)\right)\right)\right)\right)\right)\right) \]

   7. Simplified 79.8%

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

4. Final simplification 47.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq 3.9 \cdot 10^{-128}:\\ \;\;\;\;J \cdot \left(\left(-2 \cdot J\right) \cdot \frac{0.5 + 0.5 \cdot \cos K}{U}\right) - U\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 - \frac{U}{\left(-2 \cdot J\right) \cdot \frac{J \cdot 2}{U}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.6% accurate, 3.5× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ J\_s \cdot \begin{array}{l} \mathbf{if}\;J\_m \leq 9 \cdot 10^{-42}:\\ \;\;\;\;J\_m \cdot \left(\left(-2 \cdot J\_m\right) \cdot \frac{0.5 + 0.5 \cdot \cos K}{U\_m}\right) - U\_m\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot J\_m\right) \cdot \cos \left(K \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (*
  J_s
  (if (<= J_m 9e-42)
    (- (* J_m (* (* -2.0 J_m) (/ (+ 0.5 (* 0.5 (cos K))) 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 <= 9e-42) {
		tmp = (J_m * ((-2.0 * J_m) * ((0.5 + (0.5 * cos(K))) / 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 <= 9d-42) then
        tmp = (j_m * (((-2.0d0) * j_m) * ((0.5d0 + (0.5d0 * cos(k))) / 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 <= 9e-42) {
		tmp = (J_m * ((-2.0 * J_m) * ((0.5 + (0.5 * Math.cos(K))) / 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 <= 9e-42:
		tmp = (J_m * ((-2.0 * J_m) * ((0.5 + (0.5 * math.cos(K))) / 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 <= 9e-42)
		tmp = Float64(Float64(J_m * Float64(Float64(-2.0 * J_m) * Float64(Float64(0.5 + Float64(0.5 * cos(K))) / U_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 <= 9e-42)
		tmp = (J_m * ((-2.0 * J_m) * ((0.5 + (0.5 * cos(K))) / 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, 9e-42], N[(N[(J$95$m * N[(N[(-2.0 * J$95$m), $MachinePrecision] * N[(N[(0.5 + N[(0.5 * N[Cos[K], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / U$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - U$95$m), $MachinePrecision], N[(N[(-2.0 * J$95$m), $MachinePrecision] * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if J < 9e-42

   1. Initial program 66.5%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({J}^{2} \cdot \left(-2 \cdot \frac{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}{U}\right)\right), \color{blue}{U}\right) \]

   5. Simplified 27.5%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}{U} \cdot \left(\left(-2 \cdot J\right) \cdot J\right)\right), U\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(\frac{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}{U} \cdot \left(-2 \cdot J\right)\right) \cdot J\right), U\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(\frac{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}{U} \cdot \left(-2 \cdot J\right)\right), J\right), U\right) \]

   7. Applied egg-rr 29.7%

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

   if 9e-42 < J

   1. Initial program 95.7%

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

   3. Taylor expanded in J around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\cos \left(\frac{1}{2} \cdot K\right), \color{blue}{\left(-2 \cdot J\right)}\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot K\right)\right), \left(\color{blue}{-2} \cdot J\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, K\right)\right), \left(-2 \cdot J\right)\right) \]

      6. *-lowering-*.f64 76.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, K\right)\right), \mathsf{*.f64}\left(-2, \color{blue}{J}\right)\right) \]

   5. Simplified 76.1%

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

4. Final simplification 41.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq 9 \cdot 10^{-42}:\\ \;\;\;\;J \cdot \left(\left(-2 \cdot J\right) \cdot \frac{0.5 + 0.5 \cdot \cos K}{U}\right) - U\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.6% accurate, 3.7× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ J\_s \cdot \begin{array}{l} \mathbf{if}\;J\_m \leq 4.4 \cdot 10^{-43}:\\ \;\;\;\;J\_m \cdot \frac{-2 \cdot J\_m}{U\_m} - U\_m\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot J\_m\right) \cdot \cos \left(K \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (*
  J_s
  (if (<= J_m 4.4e-43)
    (- (* J_m (/ (* -2.0 J_m) 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 <= 4.4e-43) {
		tmp = (J_m * ((-2.0 * J_m) / 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 <= 4.4d-43) then
        tmp = (j_m * (((-2.0d0) * j_m) / 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 <= 4.4e-43) {
		tmp = (J_m * ((-2.0 * J_m) / 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 <= 4.4e-43:
		tmp = (J_m * ((-2.0 * J_m) / 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 <= 4.4e-43)
		tmp = Float64(Float64(J_m * Float64(Float64(-2.0 * J_m) / U_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 <= 4.4e-43)
		tmp = (J_m * ((-2.0 * J_m) / 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, 4.4e-43], N[(N[(J$95$m * N[(N[(-2.0 * J$95$m), $MachinePrecision] / U$95$m), $MachinePrecision]), $MachinePrecision] - U$95$m), $MachinePrecision], N[(N[(-2.0 * J$95$m), $MachinePrecision] * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if J < 4.39999999999999994e-43

   1. Initial program 66.5%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{-2 \cdot \left(J \cdot \sqrt{1 + \frac{1}{4} \cdot \frac{{U}^{2}}{{J}^{2}}}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{1 + \frac{1}{4} \cdot \frac{{U}^{2}}{{J}^{2}}}\right), \color{blue}{\left(-2 \cdot J\right)}\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(1 + \frac{1}{4} \cdot \frac{{U}^{2}}{{J}^{2}}\right)\right), \left(\color{blue}{-2} \cdot J\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{4} \cdot \frac{{U}^{2}}{{J}^{2}}\right)\right)\right), \left(-2 \cdot J\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{1}{4} \cdot {U}^{2}}{{J}^{2}}\right)\right)\right), \left(-2 \cdot J\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{1}{4} \cdot {U}^{2}\right), \left({J}^{2}\right)\right)\right)\right), \left(-2 \cdot J\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left({U}^{2} \cdot \frac{1}{4}\right), \left({J}^{2}\right)\right)\right)\right), \left(-2 \cdot J\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({U}^{2}\right), \frac{1}{4}\right), \left({J}^{2}\right)\right)\right)\right), \left(-2 \cdot J\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(U \cdot U\right), \frac{1}{4}\right), \left({J}^{2}\right)\right)\right)\right), \left(-2 \cdot J\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, U\right), \frac{1}{4}\right), \left({J}^{2}\right)\right)\right)\right), \left(-2 \cdot J\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, U\right), \frac{1}{4}\right), \left(J \cdot J\right)\right)\right)\right), \left(-2 \cdot J\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, U\right), \frac{1}{4}\right), \mathsf{*.f64}\left(J, J\right)\right)\right)\right), \left(-2 \cdot J\right)\right) \]

      14. *-lowering-*.f64 27.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, U\right), \frac{1}{4}\right), \mathsf{*.f64}\left(J, J\right)\right)\right)\right), \mathsf{*.f64}\left(-2, \color{blue}{J}\right)\right) \]

   5. Simplified 27.0%

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

   6. Taylor expanded in J around 0

      \[\leadsto \color{blue}{-2 \cdot \frac{{J}^{2}}{U} + -1 \cdot U} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto -2 \cdot \frac{{J}^{2}}{U} + \left(\mathsf{neg}\left(U\right)\right) \]

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(-2 \cdot \frac{{J}^{2}}{U}\right), \color{blue}{U}\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{-2 \cdot {J}^{2}}{U}\right), U\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(-2 \cdot {J}^{2}\right), U\right), U\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left({J}^{2} \cdot -2\right), U\right), U\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({J}^{2}\right), -2\right), U\right), U\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(J \cdot J\right), -2\right), U\right), U\right) \]

      9. *-lowering-*.f64 27.5%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, J\right), -2\right), U\right), U\right) \]

   8. Simplified 27.5%

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

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\left(J \cdot J\right) \cdot -2}{U}\right), \color{blue}{U}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{J \cdot \left(J \cdot -2\right)}{U}\right), U\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{J \cdot \left(-2 \cdot J\right)}{U}\right), U\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(J \cdot \frac{-2 \cdot J}{U}\right), U\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(J, \left(\frac{-2 \cdot J}{U}\right)\right), U\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(J, \mathsf{/.f64}\left(\left(-2 \cdot J\right), U\right)\right), U\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(J, \mathsf{/.f64}\left(\left(J \cdot -2\right), U\right)\right), U\right) \]

      8. *-lowering-*.f64 29.7%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(J, \mathsf{/.f64}\left(\mathsf{*.f64}\left(J, -2\right), U\right)\right), U\right) \]

   10. Applied egg-rr 29.7%

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

   if 4.39999999999999994e-43 < J

   1. Initial program 95.7%

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

   3. Taylor expanded in J around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\cos \left(\frac{1}{2} \cdot K\right), \color{blue}{\left(-2 \cdot J\right)}\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot K\right)\right), \left(\color{blue}{-2} \cdot J\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, K\right)\right), \left(-2 \cdot J\right)\right) \]

      6. *-lowering-*.f64 76.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, K\right)\right), \mathsf{*.f64}\left(-2, \color{blue}{J}\right)\right) \]

   5. Simplified 76.1%

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

4. Final simplification 41.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq 4.4 \cdot 10^{-43}:\\ \;\;\;\;J \cdot \frac{-2 \cdot J}{U} - U\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 50.9% accurate, 30.0× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ J\_s \cdot \begin{array}{l} \mathbf{if}\;U\_m \leq 1.5 \cdot 10^{-89}:\\ \;\;\;\;-2 \cdot J\_m\\ \mathbf{else}:\\ \;\;\;\;J\_m \cdot \frac{-2 \cdot J\_m}{U\_m} - U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (*
  J_s
  (if (<= U_m 1.5e-89) (* -2.0 J_m) (- (* J_m (/ (* -2.0 J_m) 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 <= 1.5e-89) {
		tmp = -2.0 * J_m;
	} else {
		tmp = (J_m * ((-2.0 * J_m) / U_m)) - 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 <= 1.5d-89) then
        tmp = (-2.0d0) * j_m
    else
        tmp = (j_m * (((-2.0d0) * j_m) / u_m)) - 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 <= 1.5e-89) {
		tmp = -2.0 * J_m;
	} else {
		tmp = (J_m * ((-2.0 * J_m) / U_m)) - 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 <= 1.5e-89:
		tmp = -2.0 * J_m
	else:
		tmp = (J_m * ((-2.0 * J_m) / U_m)) - 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 <= 1.5e-89)
		tmp = Float64(-2.0 * J_m);
	else
		tmp = Float64(Float64(J_m * Float64(Float64(-2.0 * J_m) / U_m)) - 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 <= 1.5e-89)
		tmp = -2.0 * J_m;
	else
		tmp = (J_m * ((-2.0 * J_m) / U_m)) - 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, 1.5e-89], N[(-2.0 * J$95$m), $MachinePrecision], N[(N[(J$95$m * N[(N[(-2.0 * J$95$m), $MachinePrecision] / U$95$m), $MachinePrecision]), $MachinePrecision] - U$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if U < 1.5e-89

   1. Initial program 82.2%

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

   3. Taylor expanded in J around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\cos \left(\frac{1}{2} \cdot K\right), \color{blue}{\left(-2 \cdot J\right)}\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot K\right)\right), \left(\color{blue}{-2} \cdot J\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, K\right)\right), \left(-2 \cdot J\right)\right) \]

      6. *-lowering-*.f64 62.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, K\right)\right), \mathsf{*.f64}\left(-2, \color{blue}{J}\right)\right) \]

   5. Simplified 62.4%

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

   6. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 31.6%

         \[\leadsto \mathsf{*.f64}\left(J, \color{blue}{-2}\right) \]

   8. Simplified 31.6%

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

   if 1.5e-89 < U

   1. Initial program 56.1%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{-2 \cdot \left(J \cdot \sqrt{1 + \frac{1}{4} \cdot \frac{{U}^{2}}{{J}^{2}}}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{1 + \frac{1}{4} \cdot \frac{{U}^{2}}{{J}^{2}}}\right), \color{blue}{\left(-2 \cdot J\right)}\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(1 + \frac{1}{4} \cdot \frac{{U}^{2}}{{J}^{2}}\right)\right), \left(\color{blue}{-2} \cdot J\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{4} \cdot \frac{{U}^{2}}{{J}^{2}}\right)\right)\right), \left(-2 \cdot J\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{1}{4} \cdot {U}^{2}}{{J}^{2}}\right)\right)\right), \left(-2 \cdot J\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{1}{4} \cdot {U}^{2}\right), \left({J}^{2}\right)\right)\right)\right), \left(-2 \cdot J\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left({U}^{2} \cdot \frac{1}{4}\right), \left({J}^{2}\right)\right)\right)\right), \left(-2 \cdot J\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({U}^{2}\right), \frac{1}{4}\right), \left({J}^{2}\right)\right)\right)\right), \left(-2 \cdot J\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(U \cdot U\right), \frac{1}{4}\right), \left({J}^{2}\right)\right)\right)\right), \left(-2 \cdot J\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, U\right), \frac{1}{4}\right), \left({J}^{2}\right)\right)\right)\right), \left(-2 \cdot J\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, U\right), \frac{1}{4}\right), \left(J \cdot J\right)\right)\right)\right), \left(-2 \cdot J\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, U\right), \frac{1}{4}\right), \mathsf{*.f64}\left(J, J\right)\right)\right)\right), \left(-2 \cdot J\right)\right) \]

      14. *-lowering-*.f64 26.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, U\right), \frac{1}{4}\right), \mathsf{*.f64}\left(J, J\right)\right)\right)\right), \mathsf{*.f64}\left(-2, \color{blue}{J}\right)\right) \]

   5. Simplified 26.2%

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

   6. Taylor expanded in J around 0

      \[\leadsto \color{blue}{-2 \cdot \frac{{J}^{2}}{U} + -1 \cdot U} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto -2 \cdot \frac{{J}^{2}}{U} + \left(\mathsf{neg}\left(U\right)\right) \]

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(-2 \cdot \frac{{J}^{2}}{U}\right), \color{blue}{U}\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{-2 \cdot {J}^{2}}{U}\right), U\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(-2 \cdot {J}^{2}\right), U\right), U\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left({J}^{2} \cdot -2\right), U\right), U\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({J}^{2}\right), -2\right), U\right), U\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(J \cdot J\right), -2\right), U\right), U\right) \]

      9. *-lowering-*.f64 32.9%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, J\right), -2\right), U\right), U\right) \]

   8. Simplified 32.9%

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

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\left(J \cdot J\right) \cdot -2}{U}\right), \color{blue}{U}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{J \cdot \left(J \cdot -2\right)}{U}\right), U\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{J \cdot \left(-2 \cdot J\right)}{U}\right), U\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(J \cdot \frac{-2 \cdot J}{U}\right), U\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(J, \left(\frac{-2 \cdot J}{U}\right)\right), U\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(J, \mathsf{/.f64}\left(\left(-2 \cdot J\right), U\right)\right), U\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(J, \mathsf{/.f64}\left(\left(J \cdot -2\right), U\right)\right), U\right) \]

      8. *-lowering-*.f64 36.3%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(J, \mathsf{/.f64}\left(\mathsf{*.f64}\left(J, -2\right), U\right)\right), U\right) \]

   10. Applied egg-rr 36.3%

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

4. Final simplification 33.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 1.5 \cdot 10^{-89}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{else}:\\ \;\;\;\;J \cdot \frac{-2 \cdot J}{U} - U\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 50.6% accurate, 52.4× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ J\_s \cdot \begin{array}{l} \mathbf{if}\;U\_m \leq 10^{-93}:\\ \;\;\;\;-2 \cdot J\_m\\ \mathbf{else}:\\ \;\;\;\;0 - U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (* J_s (if (<= U_m 1e-93) (* -2.0 J_m) (- 0.0 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 <= 1e-93) {
		tmp = -2.0 * J_m;
	} else {
		tmp = 0.0 - 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 <= 1d-93) then
        tmp = (-2.0d0) * j_m
    else
        tmp = 0.0d0 - 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 <= 1e-93) {
		tmp = -2.0 * J_m;
	} else {
		tmp = 0.0 - 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 <= 1e-93:
		tmp = -2.0 * J_m
	else:
		tmp = 0.0 - 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 <= 1e-93)
		tmp = Float64(-2.0 * J_m);
	else
		tmp = Float64(0.0 - 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 <= 1e-93)
		tmp = -2.0 * J_m;
	else
		tmp = 0.0 - 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, 1e-93], N[(-2.0 * J$95$m), $MachinePrecision], N[(0.0 - U$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if U < 9.999999999999999e-94

   1. Initial program 82.2%

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

   3. Taylor expanded in J around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\cos \left(\frac{1}{2} \cdot K\right), \color{blue}{\left(-2 \cdot J\right)}\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot K\right)\right), \left(\color{blue}{-2} \cdot J\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, K\right)\right), \left(-2 \cdot J\right)\right) \]

      6. *-lowering-*.f64 62.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, K\right)\right), \mathsf{*.f64}\left(-2, \color{blue}{J}\right)\right) \]

   5. Simplified 62.4%

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

   6. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 31.6%

         \[\leadsto \mathsf{*.f64}\left(J, \color{blue}{-2}\right) \]

   8. Simplified 31.6%

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

   if 9.999999999999999e-94 < U

   1. Initial program 56.1%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(U\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{U} \]

      3. --lowering--.f64 36.2%

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{U}\right) \]

   5. Simplified 36.2%

      \[\leadsto \color{blue}{0 - U} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(U\right) \]

      2. neg-lowering-neg.f64 36.2%

         \[\leadsto \mathsf{neg.f64}\left(U\right) \]

   7. Applied egg-rr 36.2%

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

4. Final simplification 33.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 10^{-93}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{else}:\\ \;\;\;\;0 - U\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 39.1% accurate, 140.0× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ J\_s \cdot \left(0 - U\_m\right) \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m) :precision binary64 (* J_s (- 0.0 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 * (0.0 - 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 * (0.0d0 - 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 * (0.0 - 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 * (0.0 - 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(0.0 - 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 * (0.0 - 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 * N[(0.0 - U$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 74.0%

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

3. Taylor expanded in J around 0

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

   1. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(U\right) \]

   2. neg-sub0 N/A

      \[\leadsto 0 - \color{blue}{U} \]

   3. --lowering--.f64 27.1%

      \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{U}\right) \]

5. Simplified 27.1%

   \[\leadsto \color{blue}{0 - U} \]
6. Step-by-step derivation

   1. sub0-neg N/A

      \[\leadsto \mathsf{neg}\left(U\right) \]

   2. neg-lowering-neg.f64 27.1%

      \[\leadsto \mathsf{neg.f64}\left(U\right) \]

7. Applied egg-rr 27.1%

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

8. Final simplification 27.1%

   \[\leadsto 0 - U \]
9. Add Preprocessing

Alternative 9: 13.9% 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 #s(literal 1 binary64) 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 74.0%

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

3. Taylor expanded in U around -inf

   \[\leadsto \color{blue}{U} \]
4. Step-by-step derivation

5. Simplified 24.4%

   \[\leadsto \color{blue}{U} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024191 
(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)))))