Maksimov and Kolovsky, Equation (3)

Percentage Accurate: 73.5% → 99.2%

Time: 11.7s

Alternatives: 12

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: 73.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_0 \cdot \left(J\_m \cdot 2\right)}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;0 - U\_m\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+293}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;U\_m \cdot \left(\frac{-2 \cdot \left(J\_m \cdot J\_m\right)}{0 - U\_m \cdot U\_m} - -1\right)\\ \end{array} \end{array} \end{array} \]

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 5e+293)
        t_1
        (* U_m (- (/ (* -2.0 (* J_m J_m)) (- 0.0 (* U_m U_m))) -1.0)))))))

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 <= 5e+293) {
		tmp = t_1;
	} else {
		tmp = U_m * (((-2.0 * (J_m * J_m)) / (0.0 - (U_m * U_m))) - -1.0);
	}
	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 <= 5e+293) {
		tmp = t_1;
	} else {
		tmp = U_m * (((-2.0 * (J_m * J_m)) / (0.0 - (U_m * U_m))) - -1.0);
	}
	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 <= 5e+293:
		tmp = t_1
	else:
		tmp = U_m * (((-2.0 * (J_m * J_m)) / (0.0 - (U_m * U_m))) - -1.0)
	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 <= 5e+293)
		tmp = t_1;
	else
		tmp = Float64(U_m * Float64(Float64(Float64(-2.0 * Float64(J_m * J_m)) / Float64(0.0 - Float64(U_m * U_m))) - -1.0));
	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 <= 5e+293)
		tmp = t_1;
	else
		tmp = U_m * (((-2.0 * (J_m * J_m)) / (0.0 - (U_m * U_m))) - -1.0);
	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, 5e+293], t$95$1, N[(U$95$m * N[(N[(N[(-2.0 * N[(J$95$m * J$95$m), $MachinePrecision]), $MachinePrecision] / N[(0.0 - N[(U$95$m * U$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]]]), $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 6.2%

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

   3. Taylor expanded in J around 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 41.1%

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

   5. Simplified 41.1%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 41.1%

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

   7. Applied egg-rr 41.1%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 5.00000000000000033e293

   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 5.00000000000000033e293 < (*.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 5.6%

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

      1. *-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 44.2%

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

   3. Simplified 44.2%

      \[\leadsto \color{blue}{\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{\frac{U}{J \cdot 2}}{\cos \left(\frac{K}{2}\right)}\right)} \]
   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 29.4%

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

   7. Simplified 29.4%

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

   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 34.3%

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

   10. Simplified 34.3%

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

   11. Taylor expanded in U around -inf

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

       1. associate-*r* N/A

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

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

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

       3. mul-1-neg N/A

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

       4. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(0 - U\right), \left(\color{blue}{-2 \cdot \frac{{J}^{2}}{{U}^{2}}} - 1\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, U\right), \left(\color{blue}{-2 \cdot \frac{{J}^{2}}{{U}^{2}}} - 1\right)\right) \]

       6. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, U\right), \left(-2 \cdot \frac{{J}^{2}}{{U}^{2}} + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

       7. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, U\right), \left(-2 \cdot \frac{{J}^{2}}{{U}^{2}} + -1\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, U\right), \mathsf{+.f64}\left(\left(-2 \cdot \frac{{J}^{2}}{{U}^{2}}\right), \color{blue}{-1}\right)\right) \]

       9. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, U\right), \mathsf{+.f64}\left(\left(\frac{-2 \cdot {J}^{2}}{{U}^{2}}\right), -1\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, U\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(-2 \cdot {J}^{2}\right), \left({U}^{2}\right)\right), -1\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, U\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \left({J}^{2}\right)\right), \left({U}^{2}\right)\right), -1\right)\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, U\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \left(J \cdot J\right)\right), \left({U}^{2}\right)\right), -1\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, U\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(J, J\right)\right), \left({U}^{2}\right)\right), -1\right)\right) \]

       14. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, U\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(J, J\right)\right), \left(U \cdot U\right)\right), -1\right)\right) \]

       15. *-lowering-*.f64 37.8%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, U\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(J, J\right)\right), \mathsf{*.f64}\left(U, U\right)\right), -1\right)\right) \]

   13. Simplified 37.8%

       \[\leadsto \color{blue}{\left(0 - U\right) \cdot \left(\frac{-2 \cdot \left(J \cdot J\right)}{U \cdot U} + -1\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -\infty:\\ \;\;\;\;0 - U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 5 \cdot 10^{+293}:\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(\frac{-2 \cdot \left(J \cdot J\right)}{0 - U \cdot U} - -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;U\_m \leq 1.55 \cdot 10^{+206}:\\ \;\;\;\;\frac{\mathsf{hypot}\left(1, \frac{\frac{U\_m}{\frac{J\_m}{0.5}}}{t\_0}\right)}{\frac{\frac{-0.5}{J\_m}}{t\_0}}\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot J\_m\right) \cdot \frac{J\_m \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \frac{K}{2}\right)\right)}{U\_m} - U\_m\\ \end{array} \end{array} \end{array} \]

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 1.55e+206)
      (/ (hypot 1.0 (/ (/ U_m (/ J_m 0.5)) t_0)) (/ (/ -0.5 J_m) t_0))
      (-
       (* (* -2.0 J_m) (/ (* J_m (+ 0.5 (* 0.5 (cos (* 2.0 (/ K 2.0)))))) 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 t_0 = cos((K / 2.0));
	double tmp;
	if (U_m <= 1.55e+206) {
		tmp = hypot(1.0, ((U_m / (J_m / 0.5)) / t_0)) / ((-0.5 / J_m) / t_0);
	} else {
		tmp = ((-2.0 * J_m) * ((J_m * (0.5 + (0.5 * cos((2.0 * (K / 2.0)))))) / U_m)) - 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 <= 1.55e+206) {
		tmp = Math.hypot(1.0, ((U_m / (J_m / 0.5)) / t_0)) / ((-0.5 / J_m) / t_0);
	} else {
		tmp = ((-2.0 * J_m) * ((J_m * (0.5 + (0.5 * Math.cos((2.0 * (K / 2.0)))))) / U_m)) - U_m;
	}
	return J_s * tmp;
}

Python

U_m = math.fabs(U)
J\_m = math.fabs(J)
J\_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if U_m <= 1.55e+206:
		tmp = math.hypot(1.0, ((U_m / (J_m / 0.5)) / t_0)) / ((-0.5 / J_m) / t_0)
	else:
		tmp = ((-2.0 * J_m) * ((J_m * (0.5 + (0.5 * math.cos((2.0 * (K / 2.0)))))) / U_m)) - U_m
	return J_s * tmp

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (U_m <= 1.55e+206)
		tmp = Float64(hypot(1.0, Float64(Float64(U_m / Float64(J_m / 0.5)) / t_0)) / Float64(Float64(-0.5 / J_m) / t_0));
	else
		tmp = Float64(Float64(Float64(-2.0 * J_m) * Float64(Float64(J_m * Float64(0.5 + Float64(0.5 * cos(Float64(2.0 * Float64(K / 2.0)))))) / U_m)) - U_m);
	end
	return Float64(J_s * tmp)
end

MATLAB

U_m = abs(U);
J\_m = abs(J);
J\_s = sign(J) * abs(1.0);
function tmp_2 = code(J_s, J_m, K, U_m)
	t_0 = cos((K / 2.0));
	tmp = 0.0;
	if (U_m <= 1.55e+206)
		tmp = hypot(1.0, ((U_m / (J_m / 0.5)) / t_0)) / ((-0.5 / J_m) / t_0);
	else
		tmp = ((-2.0 * J_m) * ((J_m * (0.5 + (0.5 * cos((2.0 * (K / 2.0)))))) / U_m)) - U_m;
	end
	tmp_2 = J_s * tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, N[(J$95$s * If[LessEqual[U$95$m, 1.55e+206], N[(N[Sqrt[1.0 ^ 2 + N[(N[(U$95$m / N[(J$95$m / 0.5), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] ^ 2], $MachinePrecision] / N[(N[(-0.5 / J$95$m), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * N[(N[(J$95$m * N[(0.5 + N[(0.5 * N[Cos[N[(2.0 * N[(K / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / U$95$m), $MachinePrecision]), $MachinePrecision] - U$95$m), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U < 1.54999999999999995e206

   1. Initial program 78.0%

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

      1. *-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 89.0%

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

   3. Simplified 89.0%

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

      1. associate-/l/ N/A

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

      2. associate-/r* N/A

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

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

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

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

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

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

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

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

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

      7. *-lowering-*.f64 89.0%

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

   6. Applied egg-rr 89.0%

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

      1. associate-*l* N/A

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

      2. metadata-eval N/A

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

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

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

   8. Applied egg-rr 88.9%

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

      1. associate-/l* N/A

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

      2. hypot-undefine N/A

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

      3. associate-/l* N/A

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

      4. metadata-eval N/A

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

      5. associate-/r* N/A

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

      6. *-commutative N/A

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

      7. associate-/l/ N/A

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

      8. div-inv N/A

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

      9. associate-/l/ N/A

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

      10. hypot-undefine N/A

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

      11. clear-num N/A

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

   10. Applied egg-rr 89.2%

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

   if 1.54999999999999995e206 < U

   1. Initial program 50.2%

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

      1. *-commutative N/A

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

      2. sqr-pow N/A

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

      3. hypot-1-def N/A

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

      4. associate-/r* N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. unpow1 N/A

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

      8. hypot-undefine N/A

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

      9. associate-*l* N/A

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

      10. associate-*l* N/A

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

   4. Applied egg-rr 56.0%

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

   5. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

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

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

   7. Simplified 30.7%

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

      1. associate-*r* N/A

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

      2. metadata-eval N/A

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

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

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

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

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

      7. *-commutative N/A

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

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

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

      9. metadata-eval N/A

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

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

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

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

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

   9. Applied egg-rr 61.0%

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

Alternative 3: 89.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;U\_m \leq 8 \cdot 10^{+206}:\\ \;\;\;\;\left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \mathsf{hypot}\left(1, \frac{\frac{U\_m}{J\_m \cdot 2}}{t\_0}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot J\_m\right) \cdot \frac{J\_m \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \frac{K}{2}\right)\right)}{U\_m} - U\_m\\ \end{array} \end{array} \end{array} \]

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 8e+206)
      (* (* (* -2.0 J_m) t_0) (hypot 1.0 (/ (/ U_m (* J_m 2.0)) t_0)))
      (-
       (* (* -2.0 J_m) (/ (* J_m (+ 0.5 (* 0.5 (cos (* 2.0 (/ K 2.0)))))) 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 t_0 = cos((K / 2.0));
	double tmp;
	if (U_m <= 8e+206) {
		tmp = ((-2.0 * J_m) * t_0) * hypot(1.0, ((U_m / (J_m * 2.0)) / t_0));
	} else {
		tmp = ((-2.0 * J_m) * ((J_m * (0.5 + (0.5 * cos((2.0 * (K / 2.0)))))) / U_m)) - 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 <= 8e+206) {
		tmp = ((-2.0 * J_m) * t_0) * Math.hypot(1.0, ((U_m / (J_m * 2.0)) / t_0));
	} else {
		tmp = ((-2.0 * J_m) * ((J_m * (0.5 + (0.5 * Math.cos((2.0 * (K / 2.0)))))) / U_m)) - U_m;
	}
	return J_s * tmp;
}

Python

U_m = math.fabs(U)
J\_m = math.fabs(J)
J\_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if U_m <= 8e+206:
		tmp = ((-2.0 * J_m) * t_0) * math.hypot(1.0, ((U_m / (J_m * 2.0)) / t_0))
	else:
		tmp = ((-2.0 * J_m) * ((J_m * (0.5 + (0.5 * math.cos((2.0 * (K / 2.0)))))) / U_m)) - U_m
	return J_s * tmp

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (U_m <= 8e+206)
		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(-2.0 * J_m) * Float64(Float64(J_m * Float64(0.5 + Float64(0.5 * cos(Float64(2.0 * Float64(K / 2.0)))))) / U_m)) - U_m);
	end
	return Float64(J_s * tmp)
end

MATLAB

U_m = abs(U);
J\_m = abs(J);
J\_s = sign(J) * abs(1.0);
function tmp_2 = code(J_s, J_m, K, U_m)
	t_0 = cos((K / 2.0));
	tmp = 0.0;
	if (U_m <= 8e+206)
		tmp = ((-2.0 * J_m) * t_0) * hypot(1.0, ((U_m / (J_m * 2.0)) / t_0));
	else
		tmp = ((-2.0 * J_m) * ((J_m * (0.5 + (0.5 * cos((2.0 * (K / 2.0)))))) / U_m)) - U_m;
	end
	tmp_2 = J_s * tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, N[(J$95$s * If[LessEqual[U$95$m, 8e+206], 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[(-2.0 * J$95$m), $MachinePrecision] * N[(N[(J$95$m * N[(0.5 + N[(0.5 * N[Cos[N[(2.0 * N[(K / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / U$95$m), $MachinePrecision]), $MachinePrecision] - U$95$m), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U < 8.0000000000000003e206

   1. Initial program 78.0%

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

      1. *-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 89.0%

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

   3. Simplified 89.0%

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

   if 8.0000000000000003e206 < U

   1. Initial program 50.2%

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

      1. *-commutative N/A

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

      2. sqr-pow N/A

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

      3. hypot-1-def N/A

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

      4. associate-/r* N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. unpow1 N/A

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

      8. hypot-undefine N/A

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

      9. associate-*l* N/A

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

      10. associate-*l* N/A

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

   4. Applied egg-rr 56.0%

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

   5. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

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

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

   7. Simplified 30.7%

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

      1. associate-*r* N/A

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

      2. metadata-eval N/A

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

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

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

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

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

      7. *-commutative N/A

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

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

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

      9. metadata-eval N/A

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

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

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

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

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

   9. Applied egg-rr 61.0%

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

Alternative 4: 89.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;U\_m \leq 8 \cdot 10^{+205}:\\ \;\;\;\;\left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \mathsf{hypot}\left(1, \frac{\frac{U\_m}{t\_0}}{J\_m \cdot 2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot J\_m\right) \cdot \frac{J\_m \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \frac{K}{2}\right)\right)}{U\_m} - U\_m\\ \end{array} \end{array} \end{array} \]

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 8e+205)
      (* (* (* -2.0 J_m) t_0) (hypot 1.0 (/ (/ U_m t_0) (* J_m 2.0))))
      (-
       (* (* -2.0 J_m) (/ (* J_m (+ 0.5 (* 0.5 (cos (* 2.0 (/ K 2.0)))))) 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 t_0 = cos((K / 2.0));
	double tmp;
	if (U_m <= 8e+205) {
		tmp = ((-2.0 * J_m) * t_0) * hypot(1.0, ((U_m / t_0) / (J_m * 2.0)));
	} else {
		tmp = ((-2.0 * J_m) * ((J_m * (0.5 + (0.5 * cos((2.0 * (K / 2.0)))))) / U_m)) - 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 <= 8e+205) {
		tmp = ((-2.0 * J_m) * t_0) * Math.hypot(1.0, ((U_m / t_0) / (J_m * 2.0)));
	} else {
		tmp = ((-2.0 * J_m) * ((J_m * (0.5 + (0.5 * Math.cos((2.0 * (K / 2.0)))))) / U_m)) - U_m;
	}
	return J_s * tmp;
}

Python

U_m = math.fabs(U)
J\_m = math.fabs(J)
J\_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if U_m <= 8e+205:
		tmp = ((-2.0 * J_m) * t_0) * math.hypot(1.0, ((U_m / t_0) / (J_m * 2.0)))
	else:
		tmp = ((-2.0 * J_m) * ((J_m * (0.5 + (0.5 * math.cos((2.0 * (K / 2.0)))))) / U_m)) - U_m
	return J_s * tmp

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (U_m <= 8e+205)
		tmp = Float64(Float64(Float64(-2.0 * J_m) * t_0) * hypot(1.0, Float64(Float64(U_m / t_0) / Float64(J_m * 2.0))));
	else
		tmp = Float64(Float64(Float64(-2.0 * J_m) * Float64(Float64(J_m * Float64(0.5 + Float64(0.5 * cos(Float64(2.0 * Float64(K / 2.0)))))) / U_m)) - U_m);
	end
	return Float64(J_s * tmp)
end

MATLAB

U_m = abs(U);
J\_m = abs(J);
J\_s = sign(J) * abs(1.0);
function tmp_2 = code(J_s, J_m, K, U_m)
	t_0 = cos((K / 2.0));
	tmp = 0.0;
	if (U_m <= 8e+205)
		tmp = ((-2.0 * J_m) * t_0) * hypot(1.0, ((U_m / t_0) / (J_m * 2.0)));
	else
		tmp = ((-2.0 * J_m) * ((J_m * (0.5 + (0.5 * cos((2.0 * (K / 2.0)))))) / U_m)) - U_m;
	end
	tmp_2 = J_s * tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, N[(J$95$s * If[LessEqual[U$95$m, 8e+205], N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[(N[(U$95$m / t$95$0), $MachinePrecision] / N[(J$95$m * 2.0), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * N[(N[(J$95$m * N[(0.5 + N[(0.5 * N[Cos[N[(2.0 * N[(K / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / U$95$m), $MachinePrecision]), $MachinePrecision] - U$95$m), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U < 8.00000000000000013e205

   1. Initial program 78.0%

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

      1. *-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 89.0%

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

   3. Simplified 89.0%

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

      1. associate-/l/ N/A

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

      2. associate-/r* N/A

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

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

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

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

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

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

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

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

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

      7. *-lowering-*.f64 89.0%

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

   6. Applied egg-rr 89.0%

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

   if 8.00000000000000013e205 < U

   1. Initial program 50.2%

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

      1. *-commutative N/A

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

      2. sqr-pow N/A

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

      3. hypot-1-def N/A

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

      4. associate-/r* N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. unpow1 N/A

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

      8. hypot-undefine N/A

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

      9. associate-*l* N/A

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

      10. associate-*l* N/A

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

   4. Applied egg-rr 56.0%

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

   5. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

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

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

   7. Simplified 30.7%

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

      1. associate-*r* N/A

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

      2. metadata-eval N/A

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

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

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

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

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

      7. *-commutative N/A

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

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

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

      9. metadata-eval N/A

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

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

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

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

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

   9. Applied egg-rr 61.0%

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

Alternative 5: 77.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ J\_s \cdot \begin{array}{l} \mathbf{if}\;U\_m \leq 6.4 \cdot 10^{+143}:\\ \;\;\;\;\left(\left(-2 \cdot J\_m\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U\_m \cdot 0.5}{J\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot J\_m\right) \cdot \frac{J\_m \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \frac{K}{2}\right)\right)}{U\_m} - U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (*
  J_s
  (if (<= U_m 6.4e+143)
    (* (* (* -2.0 J_m) (cos (/ K 2.0))) (hypot 1.0 (/ (* U_m 0.5) J_m)))
    (-
     (* (* -2.0 J_m) (/ (* J_m (+ 0.5 (* 0.5 (cos (* 2.0 (/ K 2.0)))))) 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 (U_m <= 6.4e+143) {
		tmp = ((-2.0 * J_m) * cos((K / 2.0))) * hypot(1.0, ((U_m * 0.5) / J_m));
	} else {
		tmp = ((-2.0 * J_m) * ((J_m * (0.5 + (0.5 * cos((2.0 * (K / 2.0)))))) / U_m)) - 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 <= 6.4e+143) {
		tmp = ((-2.0 * J_m) * Math.cos((K / 2.0))) * Math.hypot(1.0, ((U_m * 0.5) / J_m));
	} else {
		tmp = ((-2.0 * J_m) * ((J_m * (0.5 + (0.5 * Math.cos((2.0 * (K / 2.0)))))) / U_m)) - U_m;
	}
	return J_s * tmp;
}

Python

U_m = math.fabs(U)
J\_m = math.fabs(J)
J\_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	tmp = 0
	if U_m <= 6.4e+143:
		tmp = ((-2.0 * J_m) * math.cos((K / 2.0))) * math.hypot(1.0, ((U_m * 0.5) / J_m))
	else:
		tmp = ((-2.0 * J_m) * ((J_m * (0.5 + (0.5 * math.cos((2.0 * (K / 2.0)))))) / U_m)) - U_m
	return J_s * tmp

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	tmp = 0.0
	if (U_m <= 6.4e+143)
		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(-2.0 * J_m) * Float64(Float64(J_m * Float64(0.5 + Float64(0.5 * cos(Float64(2.0 * Float64(K / 2.0)))))) / U_m)) - U_m);
	end
	return Float64(J_s * tmp)
end

MATLAB

U_m = abs(U);
J\_m = abs(J);
J\_s = sign(J) * abs(1.0);
function tmp_2 = code(J_s, J_m, K, U_m)
	tmp = 0.0;
	if (U_m <= 6.4e+143)
		tmp = ((-2.0 * J_m) * cos((K / 2.0))) * hypot(1.0, ((U_m * 0.5) / J_m));
	else
		tmp = ((-2.0 * J_m) * ((J_m * (0.5 + (0.5 * cos((2.0 * (K / 2.0)))))) / U_m)) - U_m;
	end
	tmp_2 = J_s * tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := N[(J$95$s * If[LessEqual[U$95$m, 6.4e+143], 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[(-2.0 * J$95$m), $MachinePrecision] * N[(N[(J$95$m * N[(0.5 + N[(0.5 * N[Cos[N[(2.0 * N[(K / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / U$95$m), $MachinePrecision]), $MachinePrecision] - U$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if U < 6.40000000000000033e143

   1. Initial program 79.1%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. 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 89.4%

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

   3. Simplified 89.4%

      \[\leadsto \color{blue}{\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{\frac{U}{J \cdot 2}}{\cos \left(\frac{K}{2}\right)}\right)} \]
   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 79.4%

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

   7. Simplified 79.4%

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

   if 6.40000000000000033e143 < U

   1. Initial program 49.1%

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

      1. *-commutative N/A

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

      2. sqr-pow N/A

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

      3. hypot-1-def N/A

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

      4. associate-/r* N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. unpow1 N/A

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

      8. hypot-undefine N/A

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

      9. associate-*l* N/A

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

      10. associate-*l* N/A

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

   4. Applied egg-rr 64.5%

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

   5. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

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

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

   7. Simplified 40.3%

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

      1. associate-*r* N/A

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

      2. metadata-eval N/A

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

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

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

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

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

      7. *-commutative N/A

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

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

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

      9. metadata-eval N/A

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

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

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

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

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

   9. Applied egg-rr 63.2%

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

Alternative 6: 69.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 0.59:\\ \;\;\;\;-2 \cdot \left(J\_m \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot J\_m\right) \cdot \mathsf{hypot}\left(1, \frac{U\_m \cdot 0.5}{J\_m}\right)\\ \end{array} \end{array} \end{array} \]

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 (<= t_0 0.59)
      (* -2.0 (* J_m t_0))
      (* (* -2.0 J_m) (hypot 1.0 (/ (* U_m 0.5) 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 t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= 0.59) {
		tmp = -2.0 * (J_m * t_0);
	} else {
		tmp = (-2.0 * J_m) * hypot(1.0, ((U_m * 0.5) / J_m));
	}
	return J_s * tmp;
}

Java

U_m = Math.abs(U);
J\_m = Math.abs(J);
J\_s = Math.copySign(1.0, J);
public static double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = Math.cos((K / 2.0));
	double tmp;
	if (t_0 <= 0.59) {
		tmp = -2.0 * (J_m * t_0);
	} else {
		tmp = (-2.0 * J_m) * Math.hypot(1.0, ((U_m * 0.5) / 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):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if t_0 <= 0.59:
		tmp = -2.0 * (J_m * t_0)
	else:
		tmp = (-2.0 * J_m) * math.hypot(1.0, ((U_m * 0.5) / 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)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= 0.59)
		tmp = Float64(-2.0 * Float64(J_m * t_0));
	else
		tmp = Float64(Float64(-2.0 * J_m) * hypot(1.0, Float64(Float64(U_m * 0.5) / 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)
	t_0 = cos((K / 2.0));
	tmp = 0.0;
	if (t_0 <= 0.59)
		tmp = -2.0 * (J_m * t_0);
	else
		tmp = (-2.0 * J_m) * hypot(1.0, ((U_m * 0.5) / 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_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$0, 0.59], N[(-2.0 * N[(J$95$m * t$95$0), $MachinePrecision]), $MachinePrecision], 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]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.589999999999999969

   1. Initial program 77.4%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   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. *-lowering-*.f64 63.2%

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

   5. Simplified 63.2%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

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

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

      4. *-commutative N/A

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

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

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

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

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. div-inv N/A

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

      10. /-lowering-/.f64 64.1%

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

   7. Applied egg-rr 64.1%

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

   if 0.589999999999999969 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 75.8%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   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 87.9%

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

   3. Simplified 87.9%

      \[\leadsto \color{blue}{\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{\frac{U}{J \cdot 2}}{\cos \left(\frac{K}{2}\right)}\right)} \]
   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 78.6%

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

   7. Simplified 78.6%

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

   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 81.2%

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

   10. Simplified 81.2%

       \[\leadsto \color{blue}{\left(-2 \cdot J\right)} \cdot \mathsf{hypot}\left(1, \frac{U \cdot 0.5}{J}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.59:\\ \;\;\;\;-2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \mathsf{hypot}\left(1, \frac{U \cdot 0.5}{J}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 66.6% accurate, 3.7× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ J\_s \cdot \begin{array}{l} \mathbf{if}\;U\_m \leq 1.04 \cdot 10^{+76}:\\ \;\;\;\;-2 \cdot \left(J\_m \cdot \cos \left(\frac{K}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0 - U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (* J_s (if (<= U_m 1.04e+76) (* -2.0 (* J_m (cos (/ K 2.0)))) (- 0.0 U_m))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double tmp;
	if (U_m <= 1.04e+76) {
		tmp = -2.0 * (J_m * cos((K / 2.0)));
	} else {
		tmp = 0.0 - U_m;
	}
	return J_s * tmp;
}

Fortran

U_m = abs(u)
J\_m = abs(j)
J\_s = copysign(1.0d0, j)
real(8) function code(j_s, j_m, k, u_m)
    real(8), intent (in) :: j_s
    real(8), intent (in) :: j_m
    real(8), intent (in) :: k
    real(8), intent (in) :: u_m
    real(8) :: tmp
    if (u_m <= 1.04d+76) then
        tmp = (-2.0d0) * (j_m * cos((k / 2.0d0)))
    else
        tmp = 0.0d0 - u_m
    end if
    code = j_s * tmp
end function

Java

U_m = Math.abs(U);
J\_m = Math.abs(J);
J\_s = Math.copySign(1.0, J);
public static double code(double J_s, double J_m, double K, double U_m) {
	double tmp;
	if (U_m <= 1.04e+76) {
		tmp = -2.0 * (J_m * Math.cos((K / 2.0)));
	} else {
		tmp = 0.0 - U_m;
	}
	return J_s * tmp;
}

Python

U_m = math.fabs(U)
J\_m = math.fabs(J)
J\_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	tmp = 0
	if U_m <= 1.04e+76:
		tmp = -2.0 * (J_m * math.cos((K / 2.0)))
	else:
		tmp = 0.0 - U_m
	return J_s * tmp

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	tmp = 0.0
	if (U_m <= 1.04e+76)
		tmp = Float64(-2.0 * Float64(J_m * cos(Float64(K / 2.0))));
	else
		tmp = Float64(0.0 - U_m);
	end
	return Float64(J_s * tmp)
end

MATLAB

U_m = abs(U);
J\_m = abs(J);
J\_s = sign(J) * abs(1.0);
function tmp_2 = code(J_s, J_m, K, U_m)
	tmp = 0.0;
	if (U_m <= 1.04e+76)
		tmp = -2.0 * (J_m * cos((K / 2.0)));
	else
		tmp = 0.0 - U_m;
	end
	tmp_2 = J_s * tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := N[(J$95$s * If[LessEqual[U$95$m, 1.04e+76], N[(-2.0 * N[(J$95$m * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.0 - U$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if U < 1.03999999999999994e76

   1. Initial program 79.8%

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

   3. Taylor expanded in J around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

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

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

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

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

      6. *-lowering-*.f64 61.7%

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

   5. Simplified 61.7%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

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

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

      4. *-commutative N/A

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

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

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

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

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. div-inv N/A

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

      10. /-lowering-/.f64 62.0%

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

   7. Applied egg-rr 62.0%

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

   if 1.03999999999999994e76 < U

   1. Initial program 57.4%

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

   3. Taylor expanded in J around 0

      \[\leadsto \color{blue}{-1 \cdot U} \]
   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 47.5%

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

   5. Simplified 47.5%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 47.5%

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

   7. Applied egg-rr 47.5%

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

4. Final simplification 59.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 1.04 \cdot 10^{+76}:\\ \;\;\;\;-2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0 - U\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 66.6% accurate, 3.7× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ J\_s \cdot \begin{array}{l} \mathbf{if}\;U\_m \leq 1.24 \cdot 10^{+76}:\\ \;\;\;\;\left(-2 \cdot J\_m\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0 - U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (* J_s (if (<= U_m 1.24e+76) (* (* -2.0 J_m) (cos (* K 0.5))) (- 0.0 U_m))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double tmp;
	if (U_m <= 1.24e+76) {
		tmp = (-2.0 * J_m) * cos((K * 0.5));
	} else {
		tmp = 0.0 - U_m;
	}
	return J_s * tmp;
}

Fortran

U_m = abs(u)
J\_m = abs(j)
J\_s = copysign(1.0d0, j)
real(8) function code(j_s, j_m, k, u_m)
    real(8), intent (in) :: j_s
    real(8), intent (in) :: j_m
    real(8), intent (in) :: k
    real(8), intent (in) :: u_m
    real(8) :: tmp
    if (u_m <= 1.24d+76) then
        tmp = ((-2.0d0) * j_m) * cos((k * 0.5d0))
    else
        tmp = 0.0d0 - u_m
    end if
    code = j_s * tmp
end function

Java

U_m = Math.abs(U);
J\_m = Math.abs(J);
J\_s = Math.copySign(1.0, J);
public static double code(double J_s, double J_m, double K, double U_m) {
	double tmp;
	if (U_m <= 1.24e+76) {
		tmp = (-2.0 * J_m) * Math.cos((K * 0.5));
	} else {
		tmp = 0.0 - U_m;
	}
	return J_s * tmp;
}

Python

U_m = math.fabs(U)
J\_m = math.fabs(J)
J\_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	tmp = 0
	if U_m <= 1.24e+76:
		tmp = (-2.0 * J_m) * math.cos((K * 0.5))
	else:
		tmp = 0.0 - U_m
	return J_s * tmp

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	tmp = 0.0
	if (U_m <= 1.24e+76)
		tmp = Float64(Float64(-2.0 * J_m) * cos(Float64(K * 0.5)));
	else
		tmp = Float64(0.0 - U_m);
	end
	return Float64(J_s * tmp)
end

MATLAB

U_m = abs(U);
J\_m = abs(J);
J\_s = sign(J) * abs(1.0);
function tmp_2 = code(J_s, J_m, K, U_m)
	tmp = 0.0;
	if (U_m <= 1.24e+76)
		tmp = (-2.0 * J_m) * cos((K * 0.5));
	else
		tmp = 0.0 - U_m;
	end
	tmp_2 = J_s * tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := N[(J$95$s * If[LessEqual[U$95$m, 1.24e+76], N[(N[(-2.0 * J$95$m), $MachinePrecision] * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.0 - U$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if U < 1.24e76

   1. Initial program 79.8%

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

   3. Taylor expanded in J around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

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

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

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

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

      6. *-lowering-*.f64 61.7%

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

   5. Simplified 61.7%

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

   if 1.24e76 < U

   1. Initial program 57.4%

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

   3. Taylor expanded in J around 0

      \[\leadsto \color{blue}{-1 \cdot U} \]
   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 47.5%

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

   5. Simplified 47.5%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 47.5%

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

   7. Applied egg-rr 47.5%

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

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 1.24 \cdot 10^{+76}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0 - U\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 47.5% accurate, 26.2× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ J\_s \cdot \begin{array}{l} \mathbf{if}\;U\_m \leq 3.3 \cdot 10^{+75}:\\ \;\;\;\;\frac{\left(U\_m \cdot U\_m\right) \cdot -0.25}{J\_m} - J\_m \cdot 2\\ \mathbf{else}:\\ \;\;\;\;0 - U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (*
  J_s
  (if (<= U_m 3.3e+75)
    (- (/ (* (* U_m U_m) -0.25) J_m) (* J_m 2.0))
    (- 0.0 U_m))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double tmp;
	if (U_m <= 3.3e+75) {
		tmp = (((U_m * U_m) * -0.25) / J_m) - (J_m * 2.0);
	} else {
		tmp = 0.0 - U_m;
	}
	return J_s * tmp;
}

Fortran

U_m = abs(u)
J\_m = abs(j)
J\_s = copysign(1.0d0, j)
real(8) function code(j_s, j_m, k, u_m)
    real(8), intent (in) :: j_s
    real(8), intent (in) :: j_m
    real(8), intent (in) :: k
    real(8), intent (in) :: u_m
    real(8) :: tmp
    if (u_m <= 3.3d+75) then
        tmp = (((u_m * u_m) * (-0.25d0)) / j_m) - (j_m * 2.0d0)
    else
        tmp = 0.0d0 - u_m
    end if
    code = j_s * tmp
end function

Java

U_m = Math.abs(U);
J\_m = Math.abs(J);
J\_s = Math.copySign(1.0, J);
public static double code(double J_s, double J_m, double K, double U_m) {
	double tmp;
	if (U_m <= 3.3e+75) {
		tmp = (((U_m * U_m) * -0.25) / J_m) - (J_m * 2.0);
	} else {
		tmp = 0.0 - U_m;
	}
	return J_s * tmp;
}

Python

U_m = math.fabs(U)
J\_m = math.fabs(J)
J\_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	tmp = 0
	if U_m <= 3.3e+75:
		tmp = (((U_m * U_m) * -0.25) / J_m) - (J_m * 2.0)
	else:
		tmp = 0.0 - U_m
	return J_s * tmp

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	tmp = 0.0
	if (U_m <= 3.3e+75)
		tmp = Float64(Float64(Float64(Float64(U_m * U_m) * -0.25) / J_m) - Float64(J_m * 2.0));
	else
		tmp = Float64(0.0 - U_m);
	end
	return Float64(J_s * tmp)
end

MATLAB

U_m = abs(U);
J\_m = abs(J);
J\_s = sign(J) * abs(1.0);
function tmp_2 = code(J_s, J_m, K, U_m)
	tmp = 0.0;
	if (U_m <= 3.3e+75)
		tmp = (((U_m * U_m) * -0.25) / J_m) - (J_m * 2.0);
	else
		tmp = 0.0 - U_m;
	end
	tmp_2 = J_s * tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := N[(J$95$s * If[LessEqual[U$95$m, 3.3e+75], N[(N[(N[(N[(U$95$m * U$95$m), $MachinePrecision] * -0.25), $MachinePrecision] / J$95$m), $MachinePrecision] - N[(J$95$m * 2.0), $MachinePrecision]), $MachinePrecision], N[(0.0 - U$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if U < 3.29999999999999998e75

   1. Initial program 79.8%

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

   3. Taylor expanded in J around inf

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

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

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

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

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. associate-*l/ N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. times-frac N/A

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

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

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

   5. Simplified 61.3%

      \[\leadsto \color{blue}{J \cdot \left(-2 \cdot \cos \left(0.5 \cdot K\right) + \frac{\frac{U \cdot U}{J}}{\cos \left(0.5 \cdot K\right)} \cdot \frac{-0.25}{J}\right)} \]

   6. Taylor expanded in K around 0

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

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

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

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

      5. associate-*r/ N/A

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

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

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

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

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

      8. unpow2 N/A

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 34.3%

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

   8. Simplified 34.3%

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

      1. distribute-lft-in N/A

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

      2. *-commutative N/A

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

      3. fma-define N/A

         \[\leadsto \mathsf{fma}\left(J, \color{blue}{\frac{\frac{-1}{4} \cdot \left(U \cdot U\right)}{J \cdot J}}, -2 \cdot J\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(J, \frac{\frac{-1}{4} \cdot \left(U \cdot U\right)}{J \cdot J}, \left(\mathsf{neg}\left(2\right)\right) \cdot J\right) \]

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

         \[\leadsto \mathsf{fma}\left(J, \frac{\frac{-1}{4} \cdot \left(U \cdot U\right)}{J \cdot J}, \mathsf{neg}\left(2 \cdot J\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(J, \frac{\frac{-1}{4} \cdot \left(U \cdot U\right)}{J \cdot J}, \mathsf{neg}\left(J \cdot 2\right)\right) \]

      7. fmm-undef N/A

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

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

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

      9. *-commutative N/A

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

      10. div-inv N/A

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

      11. associate-*l* N/A

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

      12. pow2 N/A

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

      13. pow-flip N/A

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

      14. metadata-eval N/A

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

      15. pow-plus N/A

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

      16. metadata-eval N/A

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

      17. inv-pow N/A

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

      18. div-inv N/A

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

      19. /-lowering-/.f64 N/A

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

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

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

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

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

      22. *-lowering-*.f64 37.1%

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

   10. Applied egg-rr 37.1%

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

   if 3.29999999999999998e75 < U

   1. Initial program 57.4%

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

   3. Taylor expanded in J around 0

      \[\leadsto \color{blue}{-1 \cdot U} \]
   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 47.5%

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

   5. Simplified 47.5%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 47.5%

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

   7. Applied egg-rr 47.5%

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

4. Final simplification 38.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 3.3 \cdot 10^{+75}:\\ \;\;\;\;\frac{\left(U \cdot U\right) \cdot -0.25}{J} - J \cdot 2\\ \mathbf{else}:\\ \;\;\;\;0 - U\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 47.5% accurate, 52.4× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ J\_s \cdot \begin{array}{l} \mathbf{if}\;U\_m \leq 3.3 \cdot 10^{+75}:\\ \;\;\;\;-2 \cdot J\_m\\ \mathbf{else}:\\ \;\;\;\;0 - U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (* J_s (if (<= U_m 3.3e+75) (* -2.0 J_m) (- 0.0 U_m))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double tmp;
	if (U_m <= 3.3e+75) {
		tmp = -2.0 * J_m;
	} else {
		tmp = 0.0 - U_m;
	}
	return J_s * tmp;
}

Fortran

U_m = abs(u)
J\_m = abs(j)
J\_s = copysign(1.0d0, j)
real(8) function code(j_s, j_m, k, u_m)
    real(8), intent (in) :: j_s
    real(8), intent (in) :: j_m
    real(8), intent (in) :: k
    real(8), intent (in) :: u_m
    real(8) :: tmp
    if (u_m <= 3.3d+75) then
        tmp = (-2.0d0) * j_m
    else
        tmp = 0.0d0 - u_m
    end if
    code = j_s * tmp
end function

Java

U_m = Math.abs(U);
J\_m = Math.abs(J);
J\_s = Math.copySign(1.0, J);
public static double code(double J_s, double J_m, double K, double U_m) {
	double tmp;
	if (U_m <= 3.3e+75) {
		tmp = -2.0 * J_m;
	} else {
		tmp = 0.0 - U_m;
	}
	return J_s * tmp;
}

Python

U_m = math.fabs(U)
J\_m = math.fabs(J)
J\_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	tmp = 0
	if U_m <= 3.3e+75:
		tmp = -2.0 * J_m
	else:
		tmp = 0.0 - U_m
	return J_s * tmp

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	tmp = 0.0
	if (U_m <= 3.3e+75)
		tmp = Float64(-2.0 * J_m);
	else
		tmp = Float64(0.0 - U_m);
	end
	return Float64(J_s * tmp)
end

MATLAB

U_m = abs(U);
J\_m = abs(J);
J\_s = sign(J) * abs(1.0);
function tmp_2 = code(J_s, J_m, K, U_m)
	tmp = 0.0;
	if (U_m <= 3.3e+75)
		tmp = -2.0 * J_m;
	else
		tmp = 0.0 - U_m;
	end
	tmp_2 = J_s * tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := N[(J$95$s * If[LessEqual[U$95$m, 3.3e+75], N[(-2.0 * J$95$m), $MachinePrecision], N[(0.0 - U$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if U < 3.29999999999999998e75

   1. Initial program 79.8%

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

   3. Taylor expanded in J around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

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

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

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

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

      6. *-lowering-*.f64 61.7%

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

   5. Simplified 61.7%

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

   6. Taylor expanded in K around 0

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

      1. *-lowering-*.f64 37.6%

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

   8. Simplified 37.6%

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

   if 3.29999999999999998e75 < U

   1. Initial program 57.4%

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

   3. Taylor expanded in J around 0

      \[\leadsto \color{blue}{-1 \cdot U} \]
   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 47.5%

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

   5. Simplified 47.5%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 47.5%

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

   7. Applied egg-rr 47.5%

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

4. Final simplification 39.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 3.3 \cdot 10^{+75}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{else}:\\ \;\;\;\;0 - U\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 39.5% accurate, 52.4× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ J\_s \cdot \begin{array}{l} \mathbf{if}\;K \leq 1.6 \cdot 10^{+122}:\\ \;\;\;\;0 - U\_m\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \]

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 1.6e+122) (- 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 <= 1.6e+122) {
		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 <= 1.6d+122) 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 <= 1.6e+122) {
		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 <= 1.6e+122:
		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 <= 1.6e+122)
		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 <= 1.6e+122)
		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, 1.6e+122], N[(0.0 - U$95$m), $MachinePrecision], U$95$m]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if K < 1.60000000000000006e122

   1. Initial program 75.0%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   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 26.0%

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

   5. Simplified 26.0%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 26.0%

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

   7. Applied egg-rr 26.0%

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

   if 1.60000000000000006e122 < K

   1. Initial program 84.4%

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

   3. Taylor expanded in U around -inf

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

   5. Simplified 24.4%

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

4. Final simplification 25.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;K \leq 1.6 \cdot 10^{+122}:\\ \;\;\;\;0 - U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 14.3% 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 76.3%

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

3. Taylor expanded in U around -inf

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

5. Simplified 25.4%

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

Reproduce

herbie shell --seed 2024150 
(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)))))