Maksimov and Kolovsky, Equation (3)

Percentage Accurate: 73.1% → 99.4%

Time: 11.9s

Alternatives: 10

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: 73.1% 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: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_0 \cdot \left(J\_m \cdot 2\right)}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;0 - U\_m\\ \mathbf{elif}\;t\_1 \leq 10^{+300}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \end{array} \]

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+300) 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+300) {
		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+300) {
		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+300:
		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+300)
		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+300)
		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+300], 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 5.0%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   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 51.5%

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

   5. Simplified 51.5%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 51.5%

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

   7. Applied egg-rr 51.5%

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

   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))))) < 1.0000000000000001e300

   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 1.0000000000000001e300 < (*.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 14.1%

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

   3. Taylor expanded in U around -inf

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

   5. Simplified 44.7%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -\infty:\\ \;\;\;\;0 - U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 10^{+300}:\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;U\_m \leq 1.2 \cdot 10^{+194}:\\ \;\;\;\;\left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \mathsf{hypot}\left(1, \frac{\frac{U\_m}{J\_m \cdot 2}}{t\_0}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{J\_m}{\frac{U\_m}{J\_m}}}{-0.5} - U\_m\\ \end{array} \end{array} \end{array} \]

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))))
   (*
    J_s
    (if (<= U_m 1.2e+194)
      (* (* (* -2.0 J_m) t_0) (hypot 1.0 (/ (/ U_m (* J_m 2.0)) t_0)))
      (- (/ (/ J_m (/ U_m J_m)) -0.5) U_m)))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (U_m <= 1.2e+194) {
		tmp = ((-2.0 * J_m) * t_0) * hypot(1.0, ((U_m / (J_m * 2.0)) / t_0));
	} else {
		tmp = ((J_m / (U_m / J_m)) / -0.5) - U_m;
	}
	return J_s * tmp;
}

Java

U_m = Math.abs(U);
J\_m = Math.abs(J);
J\_s = Math.copySign(1.0, J);
public static double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = Math.cos((K / 2.0));
	double tmp;
	if (U_m <= 1.2e+194) {
		tmp = ((-2.0 * J_m) * t_0) * Math.hypot(1.0, ((U_m / (J_m * 2.0)) / t_0));
	} else {
		tmp = ((J_m / (U_m / J_m)) / -0.5) - U_m;
	}
	return J_s * tmp;
}

Python

U_m = math.fabs(U)
J\_m = math.fabs(J)
J\_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if U_m <= 1.2e+194:
		tmp = ((-2.0 * J_m) * t_0) * math.hypot(1.0, ((U_m / (J_m * 2.0)) / t_0))
	else:
		tmp = ((J_m / (U_m / J_m)) / -0.5) - U_m
	return J_s * tmp

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (U_m <= 1.2e+194)
		tmp = Float64(Float64(Float64(-2.0 * J_m) * t_0) * hypot(1.0, Float64(Float64(U_m / Float64(J_m * 2.0)) / t_0)));
	else
		tmp = Float64(Float64(Float64(J_m / Float64(U_m / J_m)) / -0.5) - U_m);
	end
	return Float64(J_s * tmp)
end

MATLAB

U_m = abs(U);
J\_m = abs(J);
J\_s = sign(J) * abs(1.0);
function tmp_2 = code(J_s, J_m, K, U_m)
	t_0 = cos((K / 2.0));
	tmp = 0.0;
	if (U_m <= 1.2e+194)
		tmp = ((-2.0 * J_m) * t_0) * hypot(1.0, ((U_m / (J_m * 2.0)) / t_0));
	else
		tmp = ((J_m / (U_m / J_m)) / -0.5) - U_m;
	end
	tmp_2 = J_s * tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, N[(J$95$s * If[LessEqual[U$95$m, 1.2e+194], N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[(N[(U$95$m / N[(J$95$m * 2.0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(N[(N[(J$95$m / N[(U$95$m / J$95$m), $MachinePrecision]), $MachinePrecision] / -0.5), $MachinePrecision] - U$95$m), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U < 1.2e194

   1. Initial program 73.2%

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

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

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

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

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

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

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

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

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

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

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

      6. unpow2 N/A

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

      7. hypot-1-def N/A

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

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

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

      9. associate-/r* N/A

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

      10. /-lowering-/.f64 N/A

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

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

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

      12. *-commutative N/A

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

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

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

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

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

      15. /-lowering-/.f64 91.1%

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

   3. Simplified 91.1%

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

   if 1.2e194 < U

   1. Initial program 33.7%

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

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

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

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

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

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

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

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

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

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

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

      6. unpow2 N/A

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

      7. hypot-1-def N/A

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

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

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

      9. associate-/r* N/A

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

      10. /-lowering-/.f64 N/A

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

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

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

      12. *-commutative N/A

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

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

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

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

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

      15. /-lowering-/.f64 60.5%

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

   3. Simplified 60.5%

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

   5. Taylor expanded in K around 0

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

      1. associate-*r/ N/A

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

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

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 43.6%

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

   7. Simplified 43.6%

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

   8. Taylor expanded in K around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(-2 \cdot J\right)}, \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \frac{1}{2}\right), J\right)\right)\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 56.5%

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

   10. Simplified 56.5%

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

   11. Taylor expanded in J around 0

       \[\leadsto \color{blue}{-2 \cdot \frac{{J}^{2}}{U} + -1 \cdot U} \]
   12. 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. *-lowering-*.f64 N/A

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

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

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

       6. unpow2 N/A

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

       7. *-lowering-*.f64 47.7%

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

   13. Simplified 47.7%

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

       1. associate-/l* N/A

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

       2. associate-*r* N/A

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

       3. metadata-eval N/A

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

       4. metadata-eval N/A

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

       5. associate-/r/ N/A

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

       6. clear-num N/A

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

       7. associate-*l/ N/A

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

       8. associate-/l* N/A

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

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(\frac{J \cdot J}{U}\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), U\right) \]

       10. associate-/l* N/A

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(J \cdot \frac{J}{U}\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), U\right) \]

       11. clear-num N/A

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(J \cdot \frac{1}{\frac{U}{J}}\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), U\right) \]

       12. un-div-inv N/A

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(\frac{J}{\frac{U}{J}}\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), U\right) \]

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

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(J, \left(\frac{U}{J}\right)\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), U\right) \]

       14. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(J, \mathsf{/.f64}\left(U, J\right)\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), U\right) \]

       15. metadata-eval 54.1%

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

   15. Applied egg-rr 54.1%

       \[\leadsto \color{blue}{\frac{\frac{J}{\frac{U}{J}}}{-0.5}} - U \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 77.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ J\_s \cdot \begin{array}{l} \mathbf{if}\;U\_m \leq 1.15 \cdot 10^{+166}:\\ \;\;\;\;\cos \left(\frac{K}{2}\right) \cdot \left(\left(-2 \cdot J\_m\right) \cdot \mathsf{hypot}\left(1, \frac{U\_m}{\frac{J\_m}{0.5}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{J\_m}{\frac{U\_m}{J\_m}}}{-0.5} - U\_m\\ \end{array} \end{array} \]

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 1.15e+166)
    (* (cos (/ K 2.0)) (* (* -2.0 J_m) (hypot 1.0 (/ U_m (/ J_m 0.5)))))
    (- (/ (/ J_m (/ U_m J_m)) -0.5) 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+166) {
		tmp = cos((K / 2.0)) * ((-2.0 * J_m) * hypot(1.0, (U_m / (J_m / 0.5))));
	} else {
		tmp = ((J_m / (U_m / J_m)) / -0.5) - 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+166) {
		tmp = Math.cos((K / 2.0)) * ((-2.0 * J_m) * Math.hypot(1.0, (U_m / (J_m / 0.5))));
	} else {
		tmp = ((J_m / (U_m / J_m)) / -0.5) - 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+166:
		tmp = math.cos((K / 2.0)) * ((-2.0 * J_m) * math.hypot(1.0, (U_m / (J_m / 0.5))))
	else:
		tmp = ((J_m / (U_m / J_m)) / -0.5) - 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+166)
		tmp = Float64(cos(Float64(K / 2.0)) * Float64(Float64(-2.0 * J_m) * hypot(1.0, Float64(U_m / Float64(J_m / 0.5)))));
	else
		tmp = Float64(Float64(Float64(J_m / Float64(U_m / J_m)) / -0.5) - 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+166)
		tmp = cos((K / 2.0)) * ((-2.0 * J_m) * hypot(1.0, (U_m / (J_m / 0.5))));
	else
		tmp = ((J_m / (U_m / J_m)) / -0.5) - 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+166], N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[(-2.0 * J$95$m), $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[(U$95$m / N[(J$95$m / 0.5), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(J$95$m / N[(U$95$m / J$95$m), $MachinePrecision]), $MachinePrecision] / -0.5), $MachinePrecision] - U$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if U < 1.15000000000000004e166

   1. Initial program 74.7%

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

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

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

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

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

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

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

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

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

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

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

      6. unpow2 N/A

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

      7. hypot-1-def N/A

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

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

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

      9. associate-/r* N/A

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

      10. /-lowering-/.f64 N/A

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

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

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

      12. *-commutative N/A

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

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

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

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

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

      15. /-lowering-/.f64 92.1%

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

   3. Simplified 92.1%

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

   5. Taylor expanded in K around 0

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

      1. associate-*r/ N/A

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

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

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 79.3%

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

   7. Simplified 79.3%

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

      1. associate-*l* N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-neg-in N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

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

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

   9. Applied egg-rr 79.3%

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

   if 1.15000000000000004e166 < U

   1. Initial program 34.4%

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

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

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

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

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

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

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

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

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

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

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

      6. unpow2 N/A

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

      7. hypot-1-def N/A

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

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

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

      9. associate-/r* N/A

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

      10. /-lowering-/.f64 N/A

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

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

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

      12. *-commutative N/A

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

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

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

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

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

      15. /-lowering-/.f64 62.4%

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

   3. Simplified 62.4%

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

   5. Taylor expanded in K around 0

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

      1. associate-*r/ N/A

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

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

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 43.1%

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

   7. Simplified 43.1%

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

   8. Taylor expanded in K around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(-2 \cdot J\right)}, \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \frac{1}{2}\right), J\right)\right)\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 51.3%

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

   10. Simplified 51.3%

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

   11. Taylor expanded in J around 0

       \[\leadsto \color{blue}{-2 \cdot \frac{{J}^{2}}{U} + -1 \cdot U} \]
   12. 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. *-lowering-*.f64 N/A

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

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

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

       6. unpow2 N/A

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

       7. *-lowering-*.f64 47.5%

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

   13. Simplified 47.5%

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

       1. associate-/l* N/A

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

       2. associate-*r* N/A

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

       3. metadata-eval N/A

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

       4. metadata-eval N/A

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

       5. associate-/r/ N/A

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

       6. clear-num N/A

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

       7. associate-*l/ N/A

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

       8. associate-/l* N/A

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

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(\frac{J \cdot J}{U}\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), U\right) \]

       10. associate-/l* N/A

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(J \cdot \frac{J}{U}\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), U\right) \]

       11. clear-num N/A

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(J \cdot \frac{1}{\frac{U}{J}}\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), U\right) \]

       12. un-div-inv N/A

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(\frac{J}{\frac{U}{J}}\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), U\right) \]

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

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(J, \left(\frac{U}{J}\right)\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), U\right) \]

       14. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(J, \mathsf{/.f64}\left(U, J\right)\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), U\right) \]

       15. metadata-eval 52.3%

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

   15. Applied egg-rr 52.3%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 1.15 \cdot 10^{+166}:\\ \;\;\;\;\cos \left(\frac{K}{2}\right) \cdot \left(\left(-2 \cdot J\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\frac{J}{0.5}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{J}{\frac{U}{J}}}{-0.5} - U\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ J\_s \cdot \begin{array}{l} \mathbf{if}\;U\_m \leq 5.6 \cdot 10^{+165}:\\ \;\;\;\;\left(\left(-2 \cdot J\_m\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U\_m \cdot 0.5}{J\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{J\_m}{\frac{U\_m}{J\_m}}}{-0.5} - U\_m\\ \end{array} \end{array} \]

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 5.6e+165)
    (* (* (* -2.0 J_m) (cos (/ K 2.0))) (hypot 1.0 (/ (* U_m 0.5) J_m)))
    (- (/ (/ J_m (/ U_m J_m)) -0.5) 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 <= 5.6e+165) {
		tmp = ((-2.0 * J_m) * cos((K / 2.0))) * hypot(1.0, ((U_m * 0.5) / J_m));
	} else {
		tmp = ((J_m / (U_m / J_m)) / -0.5) - 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 <= 5.6e+165) {
		tmp = ((-2.0 * J_m) * Math.cos((K / 2.0))) * Math.hypot(1.0, ((U_m * 0.5) / J_m));
	} else {
		tmp = ((J_m / (U_m / J_m)) / -0.5) - 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 <= 5.6e+165:
		tmp = ((-2.0 * J_m) * math.cos((K / 2.0))) * math.hypot(1.0, ((U_m * 0.5) / J_m))
	else:
		tmp = ((J_m / (U_m / J_m)) / -0.5) - 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 <= 5.6e+165)
		tmp = Float64(Float64(Float64(-2.0 * J_m) * cos(Float64(K / 2.0))) * hypot(1.0, Float64(Float64(U_m * 0.5) / J_m)));
	else
		tmp = Float64(Float64(Float64(J_m / Float64(U_m / J_m)) / -0.5) - 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 <= 5.6e+165)
		tmp = ((-2.0 * J_m) * cos((K / 2.0))) * hypot(1.0, ((U_m * 0.5) / J_m));
	else
		tmp = ((J_m / (U_m / J_m)) / -0.5) - 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, 5.6e+165], N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[(N[(U$95$m * 0.5), $MachinePrecision] / J$95$m), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(N[(N[(J$95$m / N[(U$95$m / J$95$m), $MachinePrecision]), $MachinePrecision] / -0.5), $MachinePrecision] - U$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if U < 5.5999999999999996e165

   1. Initial program 74.7%

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

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

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

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

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

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

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

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

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

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

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

      6. unpow2 N/A

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

      7. hypot-1-def N/A

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

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

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

      9. associate-/r* N/A

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

      10. /-lowering-/.f64 N/A

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

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

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

      12. *-commutative N/A

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

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

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

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

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

      15. /-lowering-/.f64 92.1%

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

   3. Simplified 92.1%

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

   5. Taylor expanded in K around 0

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

      1. associate-*r/ N/A

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

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

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 79.3%

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

   7. Simplified 79.3%

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

   if 5.5999999999999996e165 < U

   1. Initial program 34.4%

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

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

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

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

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

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

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

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

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

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

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

      6. unpow2 N/A

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

      7. hypot-1-def N/A

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

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

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

      9. associate-/r* N/A

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

      10. /-lowering-/.f64 N/A

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

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

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

      12. *-commutative N/A

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

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

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

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

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

      15. /-lowering-/.f64 62.4%

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

   3. Simplified 62.4%

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

   5. Taylor expanded in K around 0

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

      1. associate-*r/ N/A

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

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

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 43.1%

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

   7. Simplified 43.1%

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

   8. Taylor expanded in K around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(-2 \cdot J\right)}, \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \frac{1}{2}\right), J\right)\right)\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 51.3%

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

   10. Simplified 51.3%

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

   11. Taylor expanded in J around 0

       \[\leadsto \color{blue}{-2 \cdot \frac{{J}^{2}}{U} + -1 \cdot U} \]
   12. 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. *-lowering-*.f64 N/A

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

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

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

       6. unpow2 N/A

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

       7. *-lowering-*.f64 47.5%

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

   13. Simplified 47.5%

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

       1. associate-/l* N/A

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

       2. associate-*r* N/A

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

       3. metadata-eval N/A

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

       4. metadata-eval N/A

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

       5. associate-/r/ N/A

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

       6. clear-num N/A

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

       7. associate-*l/ N/A

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

       8. associate-/l* N/A

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

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(\frac{J \cdot J}{U}\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), U\right) \]

       10. associate-/l* N/A

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(J \cdot \frac{J}{U}\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), U\right) \]

       11. clear-num N/A

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(J \cdot \frac{1}{\frac{U}{J}}\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), U\right) \]

       12. un-div-inv N/A

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(\frac{J}{\frac{U}{J}}\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), U\right) \]

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

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(J, \left(\frac{U}{J}\right)\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), U\right) \]

       14. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(J, \mathsf{/.f64}\left(U, J\right)\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), U\right) \]

       15. metadata-eval 52.3%

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

   15. Applied egg-rr 52.3%

       \[\leadsto \color{blue}{\frac{\frac{J}{\frac{U}{J}}}{-0.5}} - U \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 69.9% accurate, 3.5× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ J\_s \cdot \begin{array}{l} \mathbf{if}\;U\_m \leq 1.68 \cdot 10^{-9}:\\ \;\;\;\;\left(-2 \cdot J\_m\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{elif}\;U\_m \leq 2.35 \cdot 10^{+147}:\\ \;\;\;\;\left(-2 \cdot J\_m\right) \cdot \mathsf{hypot}\left(1, \frac{U\_m \cdot 0.5}{J\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{J\_m}{\frac{U\_m}{J\_m}}}{-0.5} - U\_m\\ \end{array} \end{array} \]

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 1.68e-9)
    (* (* -2.0 J_m) (cos (* K 0.5)))
    (if (<= U_m 2.35e+147)
      (* (* -2.0 J_m) (hypot 1.0 (/ (* U_m 0.5) J_m)))
      (- (/ (/ J_m (/ U_m J_m)) -0.5) 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 <= 1.68e-9) {
		tmp = (-2.0 * J_m) * cos((K * 0.5));
	} else if (U_m <= 2.35e+147) {
		tmp = (-2.0 * J_m) * hypot(1.0, ((U_m * 0.5) / J_m));
	} else {
		tmp = ((J_m / (U_m / J_m)) / -0.5) - 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 <= 1.68e-9) {
		tmp = (-2.0 * J_m) * Math.cos((K * 0.5));
	} else if (U_m <= 2.35e+147) {
		tmp = (-2.0 * J_m) * Math.hypot(1.0, ((U_m * 0.5) / J_m));
	} else {
		tmp = ((J_m / (U_m / J_m)) / -0.5) - 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 <= 1.68e-9:
		tmp = (-2.0 * J_m) * math.cos((K * 0.5))
	elif U_m <= 2.35e+147:
		tmp = (-2.0 * J_m) * math.hypot(1.0, ((U_m * 0.5) / J_m))
	else:
		tmp = ((J_m / (U_m / J_m)) / -0.5) - 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 <= 1.68e-9)
		tmp = Float64(Float64(-2.0 * J_m) * cos(Float64(K * 0.5)));
	elseif (U_m <= 2.35e+147)
		tmp = Float64(Float64(-2.0 * J_m) * hypot(1.0, Float64(Float64(U_m * 0.5) / J_m)));
	else
		tmp = Float64(Float64(Float64(J_m / Float64(U_m / J_m)) / -0.5) - 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 <= 1.68e-9)
		tmp = (-2.0 * J_m) * cos((K * 0.5));
	elseif (U_m <= 2.35e+147)
		tmp = (-2.0 * J_m) * hypot(1.0, ((U_m * 0.5) / J_m));
	else
		tmp = ((J_m / (U_m / J_m)) / -0.5) - 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, 1.68e-9], N[(N[(-2.0 * J$95$m), $MachinePrecision] * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[U$95$m, 2.35e+147], N[(N[(-2.0 * J$95$m), $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[(N[(U$95$m * 0.5), $MachinePrecision] / J$95$m), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(N[(N[(J$95$m / N[(U$95$m / J$95$m), $MachinePrecision]), $MachinePrecision] / -0.5), $MachinePrecision] - U$95$m), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if U < 1.68e-9

   1. Initial program 77.5%

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

   3. Taylor expanded in J around 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. *-commutative N/A

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

      7. *-lowering-*.f64 65.6%

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

   5. Simplified 65.6%

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

   if 1.68e-9 < U < 2.3500000000000001e147

   1. Initial program 63.8%

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

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

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

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

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

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

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

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

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

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

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

      6. unpow2 N/A

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

      7. hypot-1-def N/A

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

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

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

      9. associate-/r* N/A

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

      10. /-lowering-/.f64 N/A

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

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

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

      12. *-commutative N/A

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

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

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

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

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

      15. /-lowering-/.f64 93.9%

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

   3. Simplified 93.9%

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

   5. Taylor expanded in K around 0

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

      1. associate-*r/ N/A

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

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

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 79.3%

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

   7. Simplified 79.3%

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

   8. Taylor expanded in K around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(-2 \cdot J\right)}, \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \frac{1}{2}\right), J\right)\right)\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 78.1%

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

   10. Simplified 78.1%

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

   if 2.3500000000000001e147 < U

   1. Initial program 35.4%

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

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

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

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

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

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

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

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

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

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

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

      6. unpow2 N/A

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

      7. hypot-1-def N/A

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

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

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

      9. associate-/r* N/A

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

      10. /-lowering-/.f64 N/A

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

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

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

      12. *-commutative N/A

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

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

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

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

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

      15. /-lowering-/.f64 64.6%

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

   3. Simplified 64.6%

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

   5. Taylor expanded in K around 0

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

      1. associate-*r/ N/A

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

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

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 45.7%

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

   7. Simplified 45.7%

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

   8. Taylor expanded in K around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(-2 \cdot J\right)}, \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \frac{1}{2}\right), J\right)\right)\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 50.1%

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

   10. Simplified 50.1%

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

   11. Taylor expanded in J around 0

       \[\leadsto \color{blue}{-2 \cdot \frac{{J}^{2}}{U} + -1 \cdot U} \]
   12. 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. *-lowering-*.f64 N/A

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

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

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

       6. unpow2 N/A

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

       7. *-lowering-*.f64 46.8%

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

   13. Simplified 46.8%

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

       1. associate-/l* N/A

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

       2. associate-*r* N/A

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

       3. metadata-eval N/A

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

       4. metadata-eval N/A

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

       5. associate-/r/ N/A

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

       6. clear-num N/A

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

       7. associate-*l/ N/A

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

       8. associate-/l* N/A

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

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(\frac{J \cdot J}{U}\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), U\right) \]

       10. associate-/l* N/A

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(J \cdot \frac{J}{U}\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), U\right) \]

       11. clear-num N/A

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(J \cdot \frac{1}{\frac{U}{J}}\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), U\right) \]

       12. un-div-inv N/A

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(\frac{J}{\frac{U}{J}}\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), U\right) \]

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

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(J, \left(\frac{U}{J}\right)\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), U\right) \]

       14. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(J, \mathsf{/.f64}\left(U, J\right)\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), U\right) \]

       15. metadata-eval 51.1%

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

   15. Applied egg-rr 51.1%

       \[\leadsto \color{blue}{\frac{\frac{J}{\frac{U}{J}}}{-0.5}} - U \]
3. Recombined 3 regimes into one program.

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 1.68 \cdot 10^{-9}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{elif}\;U \leq 2.35 \cdot 10^{+147}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \mathsf{hypot}\left(1, \frac{U \cdot 0.5}{J}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{J}{\frac{U}{J}}}{-0.5} - U\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 64.9% accurate, 3.7× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ J\_s \cdot \begin{array}{l} \mathbf{if}\;U\_m \leq 3.25 \cdot 10^{+146}:\\ \;\;\;\;\left(-2 \cdot J\_m\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{J\_m}{\frac{U\_m}{J\_m}}}{-0.5} - U\_m\\ \end{array} \end{array} \]

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 3.25e+146)
    (* (* -2.0 J_m) (cos (* K 0.5)))
    (- (/ (/ J_m (/ U_m J_m)) -0.5) 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 <= 3.25e+146) {
		tmp = (-2.0 * J_m) * cos((K * 0.5));
	} else {
		tmp = ((J_m / (U_m / J_m)) / -0.5) - 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 <= 3.25d+146) then
        tmp = ((-2.0d0) * j_m) * cos((k * 0.5d0))
    else
        tmp = ((j_m / (u_m / j_m)) / (-0.5d0)) - 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 <= 3.25e+146) {
		tmp = (-2.0 * J_m) * Math.cos((K * 0.5));
	} else {
		tmp = ((J_m / (U_m / J_m)) / -0.5) - 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 <= 3.25e+146:
		tmp = (-2.0 * J_m) * math.cos((K * 0.5))
	else:
		tmp = ((J_m / (U_m / J_m)) / -0.5) - 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 <= 3.25e+146)
		tmp = Float64(Float64(-2.0 * J_m) * cos(Float64(K * 0.5)));
	else
		tmp = Float64(Float64(Float64(J_m / Float64(U_m / J_m)) / -0.5) - 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 <= 3.25e+146)
		tmp = (-2.0 * J_m) * cos((K * 0.5));
	else
		tmp = ((J_m / (U_m / J_m)) / -0.5) - 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, 3.25e+146], N[(N[(-2.0 * J$95$m), $MachinePrecision] * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(J$95$m / N[(U$95$m / J$95$m), $MachinePrecision]), $MachinePrecision] / -0.5), $MachinePrecision] - U$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if U < 3.2499999999999998e146

   1. Initial program 75.4%

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

   3. Taylor expanded in J around 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. *-commutative N/A

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

      7. *-lowering-*.f64 62.8%

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

   5. Simplified 62.8%

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

   if 3.2499999999999998e146 < U

   1. Initial program 35.4%

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

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

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

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

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

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

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

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

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

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

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

      6. unpow2 N/A

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

      7. hypot-1-def N/A

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

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

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

      9. associate-/r* N/A

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

      10. /-lowering-/.f64 N/A

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

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

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

      12. *-commutative N/A

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

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

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

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

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

      15. /-lowering-/.f64 64.6%

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

   3. Simplified 64.6%

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

   5. Taylor expanded in K around 0

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

      1. associate-*r/ N/A

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

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

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 45.7%

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

   7. Simplified 45.7%

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

   8. Taylor expanded in K around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(-2 \cdot J\right)}, \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \frac{1}{2}\right), J\right)\right)\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 50.1%

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

   10. Simplified 50.1%

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

   11. Taylor expanded in J around 0

       \[\leadsto \color{blue}{-2 \cdot \frac{{J}^{2}}{U} + -1 \cdot U} \]
   12. 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. *-lowering-*.f64 N/A

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

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

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

       6. unpow2 N/A

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

       7. *-lowering-*.f64 46.8%

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

   13. Simplified 46.8%

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

       1. associate-/l* N/A

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

       2. associate-*r* N/A

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

       3. metadata-eval N/A

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

       4. metadata-eval N/A

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

       5. associate-/r/ N/A

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

       6. clear-num N/A

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

       7. associate-*l/ N/A

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

       8. associate-/l* N/A

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

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(\frac{J \cdot J}{U}\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), U\right) \]

       10. associate-/l* N/A

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(J \cdot \frac{J}{U}\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), U\right) \]

       11. clear-num N/A

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(J \cdot \frac{1}{\frac{U}{J}}\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), U\right) \]

       12. un-div-inv N/A

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(\frac{J}{\frac{U}{J}}\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), U\right) \]

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

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(J, \left(\frac{U}{J}\right)\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), U\right) \]

       14. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(J, \mathsf{/.f64}\left(U, J\right)\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), U\right) \]

       15. metadata-eval 51.1%

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

   15. Applied egg-rr 51.1%

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

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 3.25 \cdot 10^{+146}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{J}{\frac{U}{J}}}{-0.5} - U\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 50.2% accurate, 30.0× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ J\_s \cdot \begin{array}{l} \mathbf{if}\;J\_m \leq 1.55 \cdot 10^{+42}:\\ \;\;\;\;\frac{\frac{J\_m}{\frac{U\_m}{J\_m}}}{-0.5} - U\_m\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot J\_m\\ \end{array} \end{array} \]

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 1.55e+42) (- (/ (/ J_m (/ U_m J_m)) -0.5) U_m) (* -2.0 J_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 <= 1.55e+42) {
		tmp = ((J_m / (U_m / J_m)) / -0.5) - U_m;
	} else {
		tmp = -2.0 * J_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 <= 1.55d+42) then
        tmp = ((j_m / (u_m / j_m)) / (-0.5d0)) - u_m
    else
        tmp = (-2.0d0) * j_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 <= 1.55e+42) {
		tmp = ((J_m / (U_m / J_m)) / -0.5) - U_m;
	} else {
		tmp = -2.0 * J_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 <= 1.55e+42:
		tmp = ((J_m / (U_m / J_m)) / -0.5) - U_m
	else:
		tmp = -2.0 * J_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 <= 1.55e+42)
		tmp = Float64(Float64(Float64(J_m / Float64(U_m / J_m)) / -0.5) - U_m);
	else
		tmp = Float64(-2.0 * J_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 <= 1.55e+42)
		tmp = ((J_m / (U_m / J_m)) / -0.5) - U_m;
	else
		tmp = -2.0 * J_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, 1.55e+42], N[(N[(N[(J$95$m / N[(U$95$m / J$95$m), $MachinePrecision]), $MachinePrecision] / -0.5), $MachinePrecision] - U$95$m), $MachinePrecision], N[(-2.0 * J$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if J < 1.5500000000000001e42

   1. Initial program 59.9%

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

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

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

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

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

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

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

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

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

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

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

      6. unpow2 N/A

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

      7. hypot-1-def N/A

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

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

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

      9. associate-/r* N/A

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

      10. /-lowering-/.f64 N/A

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

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

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

      12. *-commutative N/A

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

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

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

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

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

      15. /-lowering-/.f64 84.1%

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

   3. Simplified 84.1%

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

   5. Taylor expanded in K around 0

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

      1. associate-*r/ N/A

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

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

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 68.1%

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

   7. Simplified 68.1%

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

   8. Taylor expanded in K around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(-2 \cdot J\right)}, \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \frac{1}{2}\right), J\right)\right)\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 55.4%

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

   10. Simplified 55.4%

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

   11. Taylor expanded in J around 0

       \[\leadsto \color{blue}{-2 \cdot \frac{{J}^{2}}{U} + -1 \cdot U} \]
   12. 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. *-lowering-*.f64 N/A

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

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

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

       6. unpow2 N/A

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

       7. *-lowering-*.f64 27.7%

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

   13. Simplified 27.7%

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

       1. associate-/l* N/A

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

       2. associate-*r* N/A

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

       3. metadata-eval N/A

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

       4. metadata-eval N/A

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

       5. associate-/r/ N/A

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

       6. clear-num N/A

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

       7. associate-*l/ N/A

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

       8. associate-/l* N/A

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

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(\frac{J \cdot J}{U}\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), U\right) \]

       10. associate-/l* N/A

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(J \cdot \frac{J}{U}\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), U\right) \]

       11. clear-num N/A

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(J \cdot \frac{1}{\frac{U}{J}}\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), U\right) \]

       12. un-div-inv N/A

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(\frac{J}{\frac{U}{J}}\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), U\right) \]

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

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(J, \left(\frac{U}{J}\right)\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), U\right) \]

       14. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(J, \mathsf{/.f64}\left(U, J\right)\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), U\right) \]

       15. metadata-eval 29.1%

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

   15. Applied egg-rr 29.1%

       \[\leadsto \color{blue}{\frac{\frac{J}{\frac{U}{J}}}{-0.5}} - U \]

   if 1.5500000000000001e42 < J

   1. Initial program 98.3%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. 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. *-commutative N/A

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

      7. *-lowering-*.f64 85.9%

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

   5. Simplified 85.9%

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

   6. Taylor expanded in K around 0

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

      1. *-lowering-*.f64 59.9%

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

   8. Simplified 59.9%

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

Alternative 8: 50.2% accurate, 52.4× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ J\_s \cdot \begin{array}{l} \mathbf{if}\;J\_m \leq 2.7 \cdot 10^{+42}:\\ \;\;\;\;0 - U\_m\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot J\_m\\ \end{array} \end{array} \]

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 2.7e+42) (- 0.0 U_m) (* -2.0 J_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 <= 2.7e+42) {
		tmp = 0.0 - U_m;
	} else {
		tmp = -2.0 * J_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 <= 2.7d+42) then
        tmp = 0.0d0 - u_m
    else
        tmp = (-2.0d0) * j_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 <= 2.7e+42) {
		tmp = 0.0 - U_m;
	} else {
		tmp = -2.0 * J_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 <= 2.7e+42:
		tmp = 0.0 - U_m
	else:
		tmp = -2.0 * J_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 <= 2.7e+42)
		tmp = Float64(0.0 - U_m);
	else
		tmp = Float64(-2.0 * J_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 <= 2.7e+42)
		tmp = 0.0 - U_m;
	else
		tmp = -2.0 * J_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, 2.7e+42], N[(0.0 - U$95$m), $MachinePrecision], N[(-2.0 * J$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if J < 2.7000000000000001e42

   1. Initial program 59.9%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   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 28.9%

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

   5. Simplified 28.9%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 28.9%

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

   7. Applied egg-rr 28.9%

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

   if 2.7000000000000001e42 < J

   1. Initial program 98.3%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. 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. *-commutative N/A

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

      7. *-lowering-*.f64 85.9%

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

   5. Simplified 85.9%

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

   6. Taylor expanded in K around 0

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

      1. *-lowering-*.f64 59.9%

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

   8. Simplified 59.9%

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

4. Final simplification 36.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq 2.7 \cdot 10^{+42}:\\ \;\;\;\;0 - U\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot J\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 38.7% accurate, 52.4× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ J\_s \cdot \begin{array}{l} \mathbf{if}\;K \leq 4 \cdot 10^{+166}:\\ \;\;\;\;0 - U\_m\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (* J_s (if (<= K 4e+166) (- 0.0 U_m) U_m)))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double tmp;
	if (K <= 4e+166) {
		tmp = 0.0 - U_m;
	} else {
		tmp = U_m;
	}
	return J_s * tmp;
}

Fortran

U_m = abs(u)
J\_m = abs(j)
J\_s = copysign(1.0d0, j)
real(8) function code(j_s, j_m, k, u_m)
    real(8), intent (in) :: j_s
    real(8), intent (in) :: j_m
    real(8), intent (in) :: k
    real(8), intent (in) :: u_m
    real(8) :: tmp
    if (k <= 4d+166) then
        tmp = 0.0d0 - u_m
    else
        tmp = u_m
    end if
    code = j_s * tmp
end function

Java

U_m = Math.abs(U);
J\_m = Math.abs(J);
J\_s = Math.copySign(1.0, J);
public static double code(double J_s, double J_m, double K, double U_m) {
	double tmp;
	if (K <= 4e+166) {
		tmp = 0.0 - U_m;
	} else {
		tmp = U_m;
	}
	return J_s * tmp;
}

Python

U_m = math.fabs(U)
J\_m = math.fabs(J)
J\_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	tmp = 0
	if K <= 4e+166:
		tmp = 0.0 - U_m
	else:
		tmp = U_m
	return J_s * tmp

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	tmp = 0.0
	if (K <= 4e+166)
		tmp = Float64(0.0 - U_m);
	else
		tmp = U_m;
	end
	return Float64(J_s * tmp)
end

MATLAB

U_m = abs(U);
J\_m = abs(J);
J\_s = sign(J) * abs(1.0);
function tmp_2 = code(J_s, J_m, K, U_m)
	tmp = 0.0;
	if (K <= 4e+166)
		tmp = 0.0 - U_m;
	else
		tmp = U_m;
	end
	tmp_2 = J_s * tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := N[(J$95$s * If[LessEqual[K, 4e+166], N[(0.0 - U$95$m), $MachinePrecision], U$95$m]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if K < 3.99999999999999976e166

   1. Initial program 69.1%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   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 24.3%

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

   5. Simplified 24.3%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 24.3%

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

   7. Applied egg-rr 24.3%

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

   if 3.99999999999999976e166 < K

   1. Initial program 66.7%

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

   3. Taylor expanded in U around -inf

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

   5. Simplified 32.3%

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

4. Final simplification 25.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;K \leq 4 \cdot 10^{+166}:\\ \;\;\;\;0 - U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 14.1% accurate, 420.0× speedup

Math

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

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #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 68.9%

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

3. Taylor expanded in U around -inf

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

5. Simplified 25.8%

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

Reproduce

herbie shell --seed 2024164 
(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)))))