Maksimov and Kolovsky, Equation (3)

Percentage Accurate: 73.5% → 99.0%

Time: 12.3s

Alternatives: 11

Speedup: 0.5×

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: 99.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_0 \cdot \left(J\_m \cdot 2\right)}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+301}:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+306}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(-U\_m\right) \cdot \mathsf{fma}\left(-2, \frac{J\_m \cdot J\_m}{U\_m \cdot U\_m}, -1\right)\\ \end{array} \end{array} \end{array} \]

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 -5e+301)
      (- U_m)
      (if (<= t_1 2e+306)
        t_1
        (* (- U_m) (fma -2.0 (/ (* J_m J_m) (* 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 <= -5e+301) {
		tmp = -U_m;
	} else if (t_1 <= 2e+306) {
		tmp = t_1;
	} else {
		tmp = -U_m * fma(-2.0, ((J_m * J_m) / (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 <= -5e+301)
		tmp = Float64(-U_m);
	elseif (t_1 <= 2e+306)
		tmp = t_1;
	else
		tmp = Float64(Float64(-U_m) * fma(-2.0, Float64(Float64(J_m * J_m) / Float64(U_m * U_m)), -1.0));
	end
	return Float64(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, -5e+301], (-U$95$m), If[LessEqual[t$95$1, 2e+306], t$95$1, N[((-U$95$m) * N[(-2.0 * N[(N[(J$95$m * J$95$m), $MachinePrecision] / N[(U$95$m * U$95$m), $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))))) < -5.0000000000000004e301

   1. Initial program 12.7%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 40.1

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

   5. Simplified 40.1%

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

   if -5.0000000000000004e301 < (*.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))))) < 2.00000000000000003e306

   1. Initial program 99.7%

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

   if 2.00000000000000003e306 < (*.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. Add Preprocessing

   3. Taylor expanded in K around 0

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

      1. lower-*.f64 4.6

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

   5. Simplified 4.6%

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

   6. Taylor expanded in K around 0

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

      1. lower-*.f64 4.6

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

   8. Simplified 4.6%

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

   9. Taylor expanded in U around -inf

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

       1. associate-*r* N/A

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

       2. lower-*.f64 N/A

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

       3. mul-1-neg N/A

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

       4. lower-neg.f64 N/A

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

       5. sub-neg N/A

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

       6. metadata-eval N/A

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

       7. lower-fma.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 N/A

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

       11. unpow2 N/A

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

       12. lower-*.f64 63.1

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

   11. Simplified 63.1%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -5 \cdot 10^{+301}:\\ \;\;\;\;-U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(-U\right) \cdot \mathsf{fma}\left(-2, \frac{J \cdot J}{U \cdot U}, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(-2 \cdot J\_m\right) \cdot t\_0\\ t_2 := t\_1 \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_0 \cdot \left(J\_m \cdot 2\right)}\right)}^{2}}\\ t_3 := \left(-2 \cdot J\_m\right) \cdot \cos \left(K \cdot 0.5\right)\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+301}:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{+164}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{-68}:\\ \;\;\;\;t\_1 \cdot \sqrt{\mathsf{fma}\left(0.25, \frac{U\_m \cdot U\_m}{J\_m \cdot J\_m}, 1\right)}\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{-291}:\\ \;\;\;\;\left(-2 \cdot J\_m\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U\_m}{J\_m}, \frac{U\_m}{J\_m} \cdot 0.25, 1\right)}\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+306}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\left(-U\_m\right) \cdot \mathsf{fma}\left(-2, \frac{J\_m \cdot J\_m}{U\_m \cdot U\_m}, -1\right)\\ \end{array} \end{array} \end{array} \]

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))
        (t_2 (* t_1 (sqrt (+ 1.0 (pow (/ U_m (* t_0 (* J_m 2.0))) 2.0)))))
        (t_3 (* (* -2.0 J_m) (cos (* K 0.5)))))
   (*
    J_s
    (if (<= t_2 -5e+301)
      (- U_m)
      (if (<= t_2 -5e+164)
        t_3
        (if (<= t_2 -2e-68)
          (* t_1 (sqrt (fma 0.25 (/ (* U_m U_m) (* J_m J_m)) 1.0)))
          (if (<= t_2 -2e-291)
            (* (* -2.0 J_m) (sqrt (fma (/ U_m J_m) (* (/ U_m J_m) 0.25) 1.0)))
            (if (<= t_2 2e+306)
              t_3
              (* (- U_m) (fma -2.0 (/ (* J_m J_m) (* 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;
	double t_2 = t_1 * sqrt((1.0 + pow((U_m / (t_0 * (J_m * 2.0))), 2.0)));
	double t_3 = (-2.0 * J_m) * cos((K * 0.5));
	double tmp;
	if (t_2 <= -5e+301) {
		tmp = -U_m;
	} else if (t_2 <= -5e+164) {
		tmp = t_3;
	} else if (t_2 <= -2e-68) {
		tmp = t_1 * sqrt(fma(0.25, ((U_m * U_m) / (J_m * J_m)), 1.0));
	} else if (t_2 <= -2e-291) {
		tmp = (-2.0 * J_m) * sqrt(fma((U_m / J_m), ((U_m / J_m) * 0.25), 1.0));
	} else if (t_2 <= 2e+306) {
		tmp = t_3;
	} else {
		tmp = -U_m * fma(-2.0, ((J_m * J_m) / (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(-2.0 * J_m) * t_0)
	t_2 = Float64(t_1 * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_0 * Float64(J_m * 2.0))) ^ 2.0))))
	t_3 = Float64(Float64(-2.0 * J_m) * cos(Float64(K * 0.5)))
	tmp = 0.0
	if (t_2 <= -5e+301)
		tmp = Float64(-U_m);
	elseif (t_2 <= -5e+164)
		tmp = t_3;
	elseif (t_2 <= -2e-68)
		tmp = Float64(t_1 * sqrt(fma(0.25, Float64(Float64(U_m * U_m) / Float64(J_m * J_m)), 1.0)));
	elseif (t_2 <= -2e-291)
		tmp = Float64(Float64(-2.0 * J_m) * sqrt(fma(Float64(U_m / J_m), Float64(Float64(U_m / J_m) * 0.25), 1.0)));
	elseif (t_2 <= 2e+306)
		tmp = t_3;
	else
		tmp = Float64(Float64(-U_m) * fma(-2.0, Float64(Float64(J_m * J_m) / Float64(U_m * U_m)), -1.0));
	end
	return Float64(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[(-2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * 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]}, Block[{t$95$3 = N[(N[(-2.0 * J$95$m), $MachinePrecision] * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$2, -5e+301], (-U$95$m), If[LessEqual[t$95$2, -5e+164], t$95$3, If[LessEqual[t$95$2, -2e-68], N[(t$95$1 * N[Sqrt[N[(0.25 * N[(N[(U$95$m * U$95$m), $MachinePrecision] / N[(J$95$m * J$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -2e-291], N[(N[(-2.0 * J$95$m), $MachinePrecision] * N[Sqrt[N[(N[(U$95$m / J$95$m), $MachinePrecision] * N[(N[(U$95$m / J$95$m), $MachinePrecision] * 0.25), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+306], t$95$3, N[((-U$95$m) * N[(-2.0 * N[(N[(J$95$m * J$95$m), $MachinePrecision] / N[(U$95$m * U$95$m), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]]]]]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 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))))) < -5.0000000000000004e301

   1. Initial program 12.7%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 40.1

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

   5. Simplified 40.1%

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

   if -5.0000000000000004e301 < (*.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))))) < -4.9999999999999995e164 or -1.99999999999999992e-291 < (*.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))))) < 2.00000000000000003e306

   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

   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

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 70.2

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

   5. Simplified 70.2%

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

   if -4.9999999999999995e164 < (*.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))))) < -2.00000000000000013e-68

   1. Initial program 99.7%

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

   3. Taylor expanded in K around 0

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

      1. lower-sqrt.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 89.0

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\mathsf{fma}\left(0.25, \frac{U \cdot U}{\color{blue}{J \cdot J}}, 1\right)} \]

   5. Simplified 89.0%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\sqrt{\mathsf{fma}\left(0.25, \frac{U \cdot U}{J \cdot J}, 1\right)}} \]

   if -2.00000000000000013e-68 < (*.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))))) < -1.99999999999999992e-291

   1. Initial program 99.7%

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

   3. Taylor expanded in K around 0

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

      1. lower-*.f64 52.9

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

   5. Simplified 52.9%

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

   6. Taylor expanded in K around 0

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

      1. lower-*.f64 56.3

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

   8. Simplified 56.3%

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

      1. associate-/l/ N/A

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

      2. div-inv N/A

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

      3. metadata-eval N/A

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

      4. unpow-prod-down N/A

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

      5. pow2 N/A

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

      6. times-frac N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lift-/.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. times-frac N/A

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

      18. associate-*l* N/A

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

      19. lower-fma.f64 N/A

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

      20. lower-/.f64 N/A

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

      21. lower-*.f64 N/A

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

      22. lower-/.f64 56.3

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

   10. Applied egg-rr 56.3%

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

   if 2.00000000000000003e306 < (*.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. Add Preprocessing

   3. Taylor expanded in K around 0

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

      1. lower-*.f64 4.6

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

   5. Simplified 4.6%

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

   6. Taylor expanded in K around 0

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

      1. lower-*.f64 4.6

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

   8. Simplified 4.6%

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

   9. Taylor expanded in U around -inf

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

       1. associate-*r* N/A

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

       2. lower-*.f64 N/A

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

       3. mul-1-neg N/A

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

       4. lower-neg.f64 N/A

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

       5. sub-neg N/A

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

       6. metadata-eval N/A

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

       7. lower-fma.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 N/A

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

       11. unpow2 N/A

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

       12. lower-*.f64 63.1

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

   11. Simplified 63.1%

       \[\leadsto \color{blue}{\left(-U\right) \cdot \mathsf{fma}\left(-2, \frac{J \cdot J}{U \cdot U}, -1\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -5 \cdot 10^{+301}:\\ \;\;\;\;-U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -5 \cdot 10^{+164}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -2 \cdot 10^{-68}:\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\mathsf{fma}\left(0.25, \frac{U \cdot U}{J \cdot J}, 1\right)}\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -2 \cdot 10^{-291}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U}{J}, \frac{U}{J} \cdot 0.25, 1\right)}\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-U\right) \cdot \mathsf{fma}\left(-2, \frac{J \cdot J}{U \cdot U}, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_0 \cdot \left(J\_m \cdot 2\right)}\right)}^{2}}\\ t_2 := \left(-2 \cdot J\_m\right) \cdot \cos \left(K \cdot 0.5\right)\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+301}:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{+213}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-291}:\\ \;\;\;\;\left(-2 \cdot J\_m\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U\_m}{J\_m}, \frac{U\_m}{J\_m} \cdot 0.25, 1\right)}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+306}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\left(-U\_m\right) \cdot \mathsf{fma}\left(-2, \frac{J\_m \cdot J\_m}{U\_m \cdot U\_m}, -1\right)\\ \end{array} \end{array} \end{array} \]

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)))))
        (t_2 (* (* -2.0 J_m) (cos (* K 0.5)))))
   (*
    J_s
    (if (<= t_1 -5e+301)
      (- U_m)
      (if (<= t_1 -1e+213)
        t_2
        (if (<= t_1 -2e-291)
          (* (* -2.0 J_m) (sqrt (fma (/ U_m J_m) (* (/ U_m J_m) 0.25) 1.0)))
          (if (<= t_1 2e+306)
            t_2
            (* (- U_m) (fma -2.0 (/ (* J_m J_m) (* 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 t_2 = (-2.0 * J_m) * cos((K * 0.5));
	double tmp;
	if (t_1 <= -5e+301) {
		tmp = -U_m;
	} else if (t_1 <= -1e+213) {
		tmp = t_2;
	} else if (t_1 <= -2e-291) {
		tmp = (-2.0 * J_m) * sqrt(fma((U_m / J_m), ((U_m / J_m) * 0.25), 1.0));
	} else if (t_1 <= 2e+306) {
		tmp = t_2;
	} else {
		tmp = -U_m * fma(-2.0, ((J_m * J_m) / (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))))
	t_2 = Float64(Float64(-2.0 * J_m) * cos(Float64(K * 0.5)))
	tmp = 0.0
	if (t_1 <= -5e+301)
		tmp = Float64(-U_m);
	elseif (t_1 <= -1e+213)
		tmp = t_2;
	elseif (t_1 <= -2e-291)
		tmp = Float64(Float64(-2.0 * J_m) * sqrt(fma(Float64(U_m / J_m), Float64(Float64(U_m / J_m) * 0.25), 1.0)));
	elseif (t_1 <= 2e+306)
		tmp = t_2;
	else
		tmp = Float64(Float64(-U_m) * fma(-2.0, Float64(Float64(J_m * J_m) / Float64(U_m * U_m)), -1.0));
	end
	return Float64(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]}, Block[{t$95$2 = N[(N[(-2.0 * J$95$m), $MachinePrecision] * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$1, -5e+301], (-U$95$m), If[LessEqual[t$95$1, -1e+213], t$95$2, If[LessEqual[t$95$1, -2e-291], N[(N[(-2.0 * J$95$m), $MachinePrecision] * N[Sqrt[N[(N[(U$95$m / J$95$m), $MachinePrecision] * N[(N[(U$95$m / J$95$m), $MachinePrecision] * 0.25), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+306], t$95$2, N[((-U$95$m) * N[(-2.0 * N[(N[(J$95$m * J$95$m), $MachinePrecision] / N[(U$95$m * U$95$m), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]]]]

Derivation

1. Split input into 4 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))))) < -5.0000000000000004e301

   1. Initial program 12.7%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 40.1

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

   5. Simplified 40.1%

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

   if -5.0000000000000004e301 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -9.99999999999999984e212 or -1.99999999999999992e-291 < (*.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))))) < 2.00000000000000003e306

   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

   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

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 70.0

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

   5. Simplified 70.0%

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

   if -9.99999999999999984e212 < (*.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))))) < -1.99999999999999992e-291

   1. Initial program 99.7%

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

   3. Taylor expanded in K around 0

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

      1. lower-*.f64 56.0

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

   5. Simplified 56.0%

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

   6. Taylor expanded in K around 0

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

      1. lower-*.f64 59.2

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

   8. Simplified 59.2%

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

      1. associate-/l/ N/A

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

      2. div-inv N/A

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

      3. metadata-eval N/A

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

      4. unpow-prod-down N/A

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

      5. pow2 N/A

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

      6. times-frac N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lift-/.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. times-frac N/A

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

      18. associate-*l* N/A

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

      19. lower-fma.f64 N/A

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

      20. lower-/.f64 N/A

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

      21. lower-*.f64 N/A

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

      22. lower-/.f64 59.2

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

   10. Applied egg-rr 59.2%

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

   if 2.00000000000000003e306 < (*.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. Add Preprocessing

   3. Taylor expanded in K around 0

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

      1. lower-*.f64 4.6

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

   5. Simplified 4.6%

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

   6. Taylor expanded in K around 0

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

      1. lower-*.f64 4.6

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

   8. Simplified 4.6%

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

   9. Taylor expanded in U around -inf

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

       1. associate-*r* N/A

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

       2. lower-*.f64 N/A

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

       3. mul-1-neg N/A

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

       4. lower-neg.f64 N/A

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

       5. sub-neg N/A

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

       6. metadata-eval N/A

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

       7. lower-fma.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 N/A

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

       11. unpow2 N/A

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

       12. lower-*.f64 63.1

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

   11. Simplified 63.1%

       \[\leadsto \color{blue}{\left(-U\right) \cdot \mathsf{fma}\left(-2, \frac{J \cdot J}{U \cdot U}, -1\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -5 \cdot 10^{+301}:\\ \;\;\;\;-U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -1 \cdot 10^{+213}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -2 \cdot 10^{-291}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U}{J}, \frac{U}{J} \cdot 0.25, 1\right)}\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-U\right) \cdot \mathsf{fma}\left(-2, \frac{J \cdot J}{U \cdot U}, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 64.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := U\_m \cdot \mathsf{fma}\left(\frac{J\_m}{U\_m}, -2 \cdot \frac{J\_m}{U\_m}, -1\right)\\ t_1 := \cos \left(\frac{K}{2}\right)\\ t_2 := \left(\left(-2 \cdot J\_m\right) \cdot t\_1\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_1 \cdot \left(J\_m \cdot 2\right)}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+291}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{-103}:\\ \;\;\;\;\left(-2 \cdot J\_m\right) \cdot \sqrt{\mathsf{fma}\left(0.25, \frac{U\_m \cdot U\_m}{J\_m \cdot J\_m}, 1\right)}\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{-292}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot J\_m\right) \cdot \frac{U\_m \cdot -0.5}{J\_m}\\ \end{array} \end{array} \end{array} \]

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 (* U_m (fma (/ J_m U_m) (* -2.0 (/ J_m U_m)) -1.0)))
        (t_1 (cos (/ K 2.0)))
        (t_2
         (*
          (* (* -2.0 J_m) t_1)
          (sqrt (+ 1.0 (pow (/ U_m (* t_1 (* J_m 2.0))) 2.0))))))
   (*
    J_s
    (if (<= t_2 -5e+291)
      t_0
      (if (<= t_2 -1e-103)
        (* (* -2.0 J_m) (sqrt (fma 0.25 (/ (* U_m U_m) (* J_m J_m)) 1.0)))
        (if (<= t_2 -1e-292) t_0 (* (* -2.0 J_m) (/ (* 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 = U_m * fma((J_m / U_m), (-2.0 * (J_m / U_m)), -1.0);
	double t_1 = cos((K / 2.0));
	double t_2 = ((-2.0 * J_m) * t_1) * sqrt((1.0 + pow((U_m / (t_1 * (J_m * 2.0))), 2.0)));
	double tmp;
	if (t_2 <= -5e+291) {
		tmp = t_0;
	} else if (t_2 <= -1e-103) {
		tmp = (-2.0 * J_m) * sqrt(fma(0.25, ((U_m * U_m) / (J_m * J_m)), 1.0));
	} else if (t_2 <= -1e-292) {
		tmp = t_0;
	} else {
		tmp = (-2.0 * J_m) * ((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 = Float64(U_m * fma(Float64(J_m / U_m), Float64(-2.0 * Float64(J_m / U_m)), -1.0))
	t_1 = cos(Float64(K / 2.0))
	t_2 = Float64(Float64(Float64(-2.0 * J_m) * t_1) * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_1 * Float64(J_m * 2.0))) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= -5e+291)
		tmp = t_0;
	elseif (t_2 <= -1e-103)
		tmp = Float64(Float64(-2.0 * J_m) * sqrt(fma(0.25, Float64(Float64(U_m * U_m) / Float64(J_m * J_m)), 1.0)));
	elseif (t_2 <= -1e-292)
		tmp = t_0;
	else
		tmp = Float64(Float64(-2.0 * J_m) * Float64(Float64(U_m * -0.5) / J_m));
	end
	return Float64(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[(U$95$m * N[(N[(J$95$m / U$95$m), $MachinePrecision] * N[(-2.0 * N[(J$95$m / U$95$m), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$1), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(t$95$1 * N[(J$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$2, -5e+291], t$95$0, If[LessEqual[t$95$2, -1e-103], N[(N[(-2.0 * J$95$m), $MachinePrecision] * N[Sqrt[N[(0.25 * N[(N[(U$95$m * U$95$m), $MachinePrecision] / N[(J$95$m * J$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -1e-292], t$95$0, N[(N[(-2.0 * J$95$m), $MachinePrecision] * N[(N[(U$95$m * -0.5), $MachinePrecision] / J$95$m), $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))))) < -5.0000000000000001e291 or -9.99999999999999958e-104 < (*.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))))) < -1.0000000000000001e-292

   1. Initial program 45.6%

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

   3. Taylor expanded in K around 0

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

      1. lower-*.f64 28.6

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

   5. Simplified 28.6%

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

   6. Taylor expanded in K around 0

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

      1. lower-*.f64 30.2

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

   8. Simplified 30.2%

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

   9. Taylor expanded in U around inf

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

       1. lower-*.f64 N/A

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

       2. sub-neg N/A

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

       3. metadata-eval N/A

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

       4. lower-fma.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 26.8

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

   11. Simplified 26.8%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. *-commutative N/A

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

       5. lift-/.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. times-frac N/A

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

       9. associate-*l* N/A

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

       10. lower-fma.f64 N/A

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

       11. lower-/.f64 N/A

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

       12. lower-*.f64 N/A

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

       13. lower-/.f64 37.8

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

   13. Applied egg-rr 37.8%

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

   if -5.0000000000000001e291 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -9.99999999999999958e-104

   1. Initial program 99.7%

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

   3. Taylor expanded in K around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 48.2

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

   5. Simplified 48.2%

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

   if -1.0000000000000001e-292 < (*.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 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 K around 0

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

      1. lower-*.f64 45.8

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

   5. Simplified 45.8%

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

   6. Taylor expanded in K around 0

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

      1. lower-*.f64 49.1

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

   8. Simplified 49.1%

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

   9. Taylor expanded in U around -inf

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

       1. associate-*r/ N/A

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

       2. lower-/.f64 N/A

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

       3. lower-*.f64 28.4

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

   11. Simplified 28.4%

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

4. Final simplification 36.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -5 \cdot 10^{+291}:\\ \;\;\;\;U \cdot \mathsf{fma}\left(\frac{J}{U}, -2 \cdot \frac{J}{U}, -1\right)\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -1 \cdot 10^{-103}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \sqrt{\mathsf{fma}\left(0.25, \frac{U \cdot U}{J \cdot J}, 1\right)}\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -1 \cdot 10^{-292}:\\ \;\;\;\;U \cdot \mathsf{fma}\left(\frac{J}{U}, -2 \cdot \frac{J}{U}, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \frac{U \cdot -0.5}{J}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 59.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_0 \cdot \left(J\_m \cdot 2\right)}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+301}:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-75}:\\ \;\;\;\;-2 \cdot J\_m\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-292}:\\ \;\;\;\;\frac{J\_m \cdot \left(-2 \cdot J\_m\right)}{U\_m} - U\_m\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot J\_m\right) \cdot \frac{U\_m \cdot -0.5}{J\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* (* -2.0 J_m) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* t_0 (* J_m 2.0))) 2.0))))))
   (*
    J_s
    (if (<= t_1 -5e+301)
      (- U_m)
      (if (<= t_1 -1e-75)
        (* -2.0 J_m)
        (if (<= t_1 -1e-292)
          (- (/ (* J_m (* -2.0 J_m)) U_m) U_m)
          (* (* -2.0 J_m) (/ (* 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 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 <= -5e+301) {
		tmp = -U_m;
	} else if (t_1 <= -1e-75) {
		tmp = -2.0 * J_m;
	} else if (t_1 <= -1e-292) {
		tmp = ((J_m * (-2.0 * J_m)) / U_m) - U_m;
	} else {
		tmp = (-2.0 * J_m) * ((U_m * -0.5) / J_m);
	}
	return J_s * tmp;
}

Fortran

U_m = abs(u)
J\_m = abs(j)
J\_s = copysign(1.0d0, j)
real(8) function code(j_s, j_m, k, u_m)
    real(8), intent (in) :: j_s
    real(8), intent (in) :: j_m
    real(8), intent (in) :: k
    real(8), intent (in) :: u_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    t_1 = (((-2.0d0) * j_m) * t_0) * sqrt((1.0d0 + ((u_m / (t_0 * (j_m * 2.0d0))) ** 2.0d0)))
    if (t_1 <= (-5d+301)) then
        tmp = -u_m
    else if (t_1 <= (-1d-75)) then
        tmp = (-2.0d0) * j_m
    else if (t_1 <= (-1d-292)) then
        tmp = ((j_m * ((-2.0d0) * j_m)) / u_m) - u_m
    else
        tmp = ((-2.0d0) * j_m) * ((u_m * (-0.5d0)) / j_m)
    end if
    code = j_s * tmp
end function

Java

U_m = Math.abs(U);
J\_m = Math.abs(J);
J\_s = Math.copySign(1.0, J);
public static double code(double J_s, double J_m, double K, double U_m) {
	double 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 <= -5e+301) {
		tmp = -U_m;
	} else if (t_1 <= -1e-75) {
		tmp = -2.0 * J_m;
	} else if (t_1 <= -1e-292) {
		tmp = ((J_m * (-2.0 * J_m)) / U_m) - U_m;
	} else {
		tmp = (-2.0 * J_m) * ((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))
	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 <= -5e+301:
		tmp = -U_m
	elif t_1 <= -1e-75:
		tmp = -2.0 * J_m
	elif t_1 <= -1e-292:
		tmp = ((J_m * (-2.0 * J_m)) / U_m) - U_m
	else:
		tmp = (-2.0 * J_m) * ((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))
	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 <= -5e+301)
		tmp = Float64(-U_m);
	elseif (t_1 <= -1e-75)
		tmp = Float64(-2.0 * J_m);
	elseif (t_1 <= -1e-292)
		tmp = Float64(Float64(Float64(J_m * Float64(-2.0 * J_m)) / U_m) - U_m);
	else
		tmp = Float64(Float64(-2.0 * J_m) * 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));
	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 <= -5e+301)
		tmp = -U_m;
	elseif (t_1 <= -1e-75)
		tmp = -2.0 * J_m;
	elseif (t_1 <= -1e-292)
		tmp = ((J_m * (-2.0 * J_m)) / U_m) - U_m;
	else
		tmp = (-2.0 * J_m) * ((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]}, 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, -5e+301], (-U$95$m), If[LessEqual[t$95$1, -1e-75], N[(-2.0 * J$95$m), $MachinePrecision], If[LessEqual[t$95$1, -1e-292], N[(N[(N[(J$95$m * N[(-2.0 * J$95$m), $MachinePrecision]), $MachinePrecision] / U$95$m), $MachinePrecision] - U$95$m), $MachinePrecision], N[(N[(-2.0 * J$95$m), $MachinePrecision] * N[(N[(U$95$m * -0.5), $MachinePrecision] / J$95$m), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 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))))) < -5.0000000000000004e301

   1. Initial program 12.7%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 40.1

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

   5. Simplified 40.1%

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

   if -5.0000000000000004e301 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -9.9999999999999996e-76

   1. Initial program 99.7%

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

   3. Taylor expanded in K around 0

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

      1. lower-*.f64 54.1

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

   5. Simplified 54.1%

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

   6. Taylor expanded in J around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 43.4

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

   8. Simplified 43.4%

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

   if -9.9999999999999996e-76 < (*.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))))) < -1.0000000000000001e-292

   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

   3. Taylor expanded in K around 0

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

      1. lower-*.f64 58.1

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

   5. Simplified 58.1%

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

   6. Taylor expanded in K around 0

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

      1. lower-*.f64 61.8

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

   8. Simplified 61.8%

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

   9. Taylor expanded in J around 0

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

       1. mul-1-neg N/A

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

       2. unsub-neg N/A

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

       3. lower--.f64 N/A

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

       4. associate-*r/ N/A

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

       5. unpow2 N/A

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

       8. associate-*r* N/A

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

       9. lower-/.f64 N/A

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

       10. lower-*.f64 N/A

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

       11. *-commutative N/A

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

       12. lower-*.f64 30.7

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

   11. Simplified 30.7%

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

   if -1.0000000000000001e-292 < (*.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 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 K around 0

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

      1. lower-*.f64 45.8

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

   5. Simplified 45.8%

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

   6. Taylor expanded in K around 0

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

      1. lower-*.f64 49.1

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

   8. Simplified 49.1%

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

   9. Taylor expanded in U around -inf

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

       1. associate-*r/ N/A

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

       2. lower-/.f64 N/A

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

       3. lower-*.f64 28.4

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

   11. Simplified 28.4%

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

4. Final simplification 34.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -5 \cdot 10^{+301}:\\ \;\;\;\;-U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -1 \cdot 10^{-75}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -1 \cdot 10^{-292}:\\ \;\;\;\;\frac{J \cdot \left(-2 \cdot J\right)}{U} - U\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \frac{U \cdot -0.5}{J}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(-2 \cdot J\_m\right) \cdot t\_0\\ t_2 := t\_1 \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_0 \cdot \left(J\_m \cdot 2\right)}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+301}:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+306}:\\ \;\;\;\;t\_1 \cdot \sqrt{\mathsf{fma}\left(\frac{U\_m}{\left(J\_m \cdot 4\right) \cdot \mathsf{fma}\left(0.5, \cos K, 0.5\right)}, \frac{U\_m}{J\_m}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-U\_m\right) \cdot \mathsf{fma}\left(-2, \frac{J\_m \cdot J\_m}{U\_m \cdot U\_m}, -1\right)\\ \end{array} \end{array} \end{array} \]

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))
        (t_2 (* t_1 (sqrt (+ 1.0 (pow (/ U_m (* t_0 (* J_m 2.0))) 2.0))))))
   (*
    J_s
    (if (<= t_2 -5e+301)
      (- U_m)
      (if (<= t_2 2e+306)
        (*
         t_1
         (sqrt
          (fma (/ U_m (* (* J_m 4.0) (fma 0.5 (cos K) 0.5))) (/ U_m J_m) 1.0)))
        (* (- U_m) (fma -2.0 (/ (* J_m J_m) (* 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;
	double t_2 = t_1 * sqrt((1.0 + pow((U_m / (t_0 * (J_m * 2.0))), 2.0)));
	double tmp;
	if (t_2 <= -5e+301) {
		tmp = -U_m;
	} else if (t_2 <= 2e+306) {
		tmp = t_1 * sqrt(fma((U_m / ((J_m * 4.0) * fma(0.5, cos(K), 0.5))), (U_m / J_m), 1.0));
	} else {
		tmp = -U_m * fma(-2.0, ((J_m * J_m) / (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(-2.0 * J_m) * t_0)
	t_2 = Float64(t_1 * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_0 * Float64(J_m * 2.0))) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= -5e+301)
		tmp = Float64(-U_m);
	elseif (t_2 <= 2e+306)
		tmp = Float64(t_1 * sqrt(fma(Float64(U_m / Float64(Float64(J_m * 4.0) * fma(0.5, cos(K), 0.5))), Float64(U_m / J_m), 1.0)));
	else
		tmp = Float64(Float64(-U_m) * fma(-2.0, Float64(Float64(J_m * J_m) / Float64(U_m * U_m)), -1.0));
	end
	return Float64(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[(-2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(t$95$0 * N[(J$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$2, -5e+301], (-U$95$m), If[LessEqual[t$95$2, 2e+306], N[(t$95$1 * N[Sqrt[N[(N[(U$95$m / N[(N[(J$95$m * 4.0), $MachinePrecision] * N[(0.5 * N[Cos[K], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(U$95$m / J$95$m), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[((-U$95$m) * N[(-2.0 * N[(N[(J$95$m * J$95$m), $MachinePrecision] / N[(U$95$m * U$95$m), $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))))) < -5.0000000000000004e301

   1. Initial program 12.7%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 40.1

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

   5. Simplified 40.1%

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

   if -5.0000000000000004e301 < (*.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))))) < 2.00000000000000003e306

   1. Initial program 99.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. div-inv N/A

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

      6. associate-*r/ N/A

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

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

      9. unpow-prod-down N/A

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

      10. pow2 N/A

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

      11. pow2 N/A

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

      12. lower-*.f64 N/A

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

   4. Applied egg-rr 97.7%

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

   5. Taylor expanded in K around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-cos.f64 97.7

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

   7. Simplified 97.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. /-rgt-identity N/A

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

      8. lift-/.f64 N/A

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

      9. clear-num N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. +-commutative N/A

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

   9. Applied egg-rr 99.6%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-/.f64 N/A

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

       5. lift-cos.f64 N/A

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

       6. lift-fma.f64 N/A

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

       7. lift-/.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-fma.f64 99.6

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

   11. Applied egg-rr 99.6%

       \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(\frac{U}{\left(J \cdot 4\right) \cdot \mathsf{fma}\left(0.5, \cos K, 0.5\right)}, \frac{U}{J}, 1\right)}} \]

   if 2.00000000000000003e306 < (*.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. Add Preprocessing

   3. Taylor expanded in K around 0

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

      1. lower-*.f64 4.6

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

   5. Simplified 4.6%

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

   6. Taylor expanded in K around 0

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

      1. lower-*.f64 4.6

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

   8. Simplified 4.6%

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

   9. Taylor expanded in U around -inf

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

       1. associate-*r* N/A

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

       2. lower-*.f64 N/A

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

       3. mul-1-neg N/A

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

       4. lower-neg.f64 N/A

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

       5. sub-neg N/A

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

       6. metadata-eval N/A

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

       7. lower-fma.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 N/A

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

       11. unpow2 N/A

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

       12. lower-*.f64 63.1

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

   11. Simplified 63.1%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -5 \cdot 10^{+301}:\\ \;\;\;\;-U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U}{\left(J \cdot 4\right) \cdot \mathsf{fma}\left(0.5, \cos K, 0.5\right)}, \frac{U}{J}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-U\right) \cdot \mathsf{fma}\left(-2, \frac{J \cdot J}{U \cdot U}, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 88.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(-2 \cdot J\_m\right) \cdot t\_0\\ t_2 := t\_1 \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_0 \cdot \left(J\_m \cdot 2\right)}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+301}:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+306}:\\ \;\;\;\;t\_1 \cdot \sqrt{\mathsf{fma}\left(0.25, \frac{U\_m \cdot \frac{U\_m}{J\_m}}{J\_m}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-U\_m\right) \cdot \mathsf{fma}\left(-2, \frac{J\_m \cdot J\_m}{U\_m \cdot U\_m}, -1\right)\\ \end{array} \end{array} \end{array} \]

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))
        (t_2 (* t_1 (sqrt (+ 1.0 (pow (/ U_m (* t_0 (* J_m 2.0))) 2.0))))))
   (*
    J_s
    (if (<= t_2 -5e+301)
      (- U_m)
      (if (<= t_2 2e+306)
        (* t_1 (sqrt (fma 0.25 (/ (* U_m (/ U_m J_m)) J_m) 1.0)))
        (* (- U_m) (fma -2.0 (/ (* J_m J_m) (* 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;
	double t_2 = t_1 * sqrt((1.0 + pow((U_m / (t_0 * (J_m * 2.0))), 2.0)));
	double tmp;
	if (t_2 <= -5e+301) {
		tmp = -U_m;
	} else if (t_2 <= 2e+306) {
		tmp = t_1 * sqrt(fma(0.25, ((U_m * (U_m / J_m)) / J_m), 1.0));
	} else {
		tmp = -U_m * fma(-2.0, ((J_m * J_m) / (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(-2.0 * J_m) * t_0)
	t_2 = Float64(t_1 * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_0 * Float64(J_m * 2.0))) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= -5e+301)
		tmp = Float64(-U_m);
	elseif (t_2 <= 2e+306)
		tmp = Float64(t_1 * sqrt(fma(0.25, Float64(Float64(U_m * Float64(U_m / J_m)) / J_m), 1.0)));
	else
		tmp = Float64(Float64(-U_m) * fma(-2.0, Float64(Float64(J_m * J_m) / Float64(U_m * U_m)), -1.0));
	end
	return Float64(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[(-2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(t$95$0 * N[(J$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$2, -5e+301], (-U$95$m), If[LessEqual[t$95$2, 2e+306], N[(t$95$1 * N[Sqrt[N[(0.25 * N[(N[(U$95$m * N[(U$95$m / J$95$m), $MachinePrecision]), $MachinePrecision] / J$95$m), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[((-U$95$m) * N[(-2.0 * N[(N[(J$95$m * J$95$m), $MachinePrecision] / N[(U$95$m * U$95$m), $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))))) < -5.0000000000000004e301

   1. Initial program 12.7%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 40.1

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

   5. Simplified 40.1%

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

   if -5.0000000000000004e301 < (*.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))))) < 2.00000000000000003e306

   1. Initial program 99.7%

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

   3. Taylor expanded in K around 0

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

      1. lower-sqrt.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 71.3

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\mathsf{fma}\left(0.25, \frac{U \cdot U}{\color{blue}{J \cdot J}}, 1\right)} \]

   5. Simplified 71.3%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\sqrt{\mathsf{fma}\left(0.25, \frac{U \cdot U}{J \cdot J}, 1\right)}} \]
   6. Step-by-step derivation

      1. times-frac N/A

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

      2. associate-*r/ N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 87.5

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\mathsf{fma}\left(0.25, \frac{\color{blue}{\frac{U}{J}} \cdot U}{J}, 1\right)} \]

   7. Applied egg-rr 87.5%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\mathsf{fma}\left(0.25, \color{blue}{\frac{\frac{U}{J} \cdot U}{J}}, 1\right)} \]

   if 2.00000000000000003e306 < (*.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. Add Preprocessing

   3. Taylor expanded in K around 0

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

      1. lower-*.f64 4.6

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

   5. Simplified 4.6%

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

   6. Taylor expanded in K around 0

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

      1. lower-*.f64 4.6

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

   8. Simplified 4.6%

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

   9. Taylor expanded in U around -inf

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

       1. associate-*r* N/A

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

       2. lower-*.f64 N/A

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

       3. mul-1-neg N/A

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

       4. lower-neg.f64 N/A

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

       5. sub-neg N/A

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

       6. metadata-eval N/A

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

       7. lower-fma.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 N/A

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

       11. unpow2 N/A

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

       12. lower-*.f64 63.1

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

   11. Simplified 63.1%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -5 \cdot 10^{+301}:\\ \;\;\;\;-U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\mathsf{fma}\left(0.25, \frac{U \cdot \frac{U}{J}}{J}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-U\right) \cdot \mathsf{fma}\left(-2, \frac{J \cdot J}{U \cdot U}, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_0 \cdot \left(J\_m \cdot 2\right)}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+301}:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-292}:\\ \;\;\;\;\left(-2 \cdot J\_m\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U\_m}{J\_m}, \frac{U\_m}{J\_m} \cdot 0.25, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot J\_m\right) \cdot \frac{U\_m \cdot -0.5}{J\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* (* -2.0 J_m) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* t_0 (* J_m 2.0))) 2.0))))))
   (*
    J_s
    (if (<= t_1 -5e+301)
      (- U_m)
      (if (<= t_1 -1e-292)
        (* (* -2.0 J_m) (sqrt (fma (/ U_m J_m) (* (/ U_m J_m) 0.25) 1.0)))
        (* (* -2.0 J_m) (/ (* 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 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 <= -5e+301) {
		tmp = -U_m;
	} else if (t_1 <= -1e-292) {
		tmp = (-2.0 * J_m) * sqrt(fma((U_m / J_m), ((U_m / J_m) * 0.25), 1.0));
	} else {
		tmp = (-2.0 * J_m) * ((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))
	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 <= -5e+301)
		tmp = Float64(-U_m);
	elseif (t_1 <= -1e-292)
		tmp = Float64(Float64(-2.0 * J_m) * sqrt(fma(Float64(U_m / J_m), Float64(Float64(U_m / J_m) * 0.25), 1.0)));
	else
		tmp = Float64(Float64(-2.0 * J_m) * Float64(Float64(U_m * -0.5) / J_m));
	end
	return Float64(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, -5e+301], (-U$95$m), If[LessEqual[t$95$1, -1e-292], N[(N[(-2.0 * J$95$m), $MachinePrecision] * N[Sqrt[N[(N[(U$95$m / J$95$m), $MachinePrecision] * N[(N[(U$95$m / J$95$m), $MachinePrecision] * 0.25), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 * J$95$m), $MachinePrecision] * N[(N[(U$95$m * -0.5), $MachinePrecision] / J$95$m), $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))))) < -5.0000000000000004e301

   1. Initial program 12.7%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 40.1

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

   5. Simplified 40.1%

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

   if -5.0000000000000004e301 < (*.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))))) < -1.0000000000000001e-292

   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

   3. Taylor expanded in K around 0

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

      1. lower-*.f64 55.1

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

   5. Simplified 55.1%

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

   6. Taylor expanded in K around 0

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

      1. lower-*.f64 57.7

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

   8. Simplified 57.7%

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

      1. associate-/l/ N/A

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

      2. div-inv N/A

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

      3. metadata-eval N/A

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

      4. unpow-prod-down N/A

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

      5. pow2 N/A

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

      6. times-frac N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lift-/.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. times-frac N/A

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

      18. associate-*l* N/A

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

      19. lower-fma.f64 N/A

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

      20. lower-/.f64 N/A

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

      21. lower-*.f64 N/A

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

      22. lower-/.f64 57.7

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

   10. Applied egg-rr 57.7%

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

   if -1.0000000000000001e-292 < (*.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 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 K around 0

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

      1. lower-*.f64 45.8

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

   5. Simplified 45.8%

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

   6. Taylor expanded in K around 0

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

      1. lower-*.f64 49.1

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

   8. Simplified 49.1%

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

   9. Taylor expanded in U around -inf

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

       1. associate-*r/ N/A

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

       2. lower-/.f64 N/A

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

       3. lower-*.f64 28.4

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

   11. Simplified 28.4%

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

4. Final simplification 42.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -5 \cdot 10^{+301}:\\ \;\;\;\;-U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -1 \cdot 10^{-292}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U}{J}, \frac{U}{J} \cdot 0.25, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \frac{U \cdot -0.5}{J}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 49.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_0 \cdot \left(J\_m \cdot 2\right)}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+301}:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-75}:\\ \;\;\;\;-2 \cdot J\_m\\ \mathbf{else}:\\ \;\;\;\;U\_m \cdot \left(0.5 \cdot \mathsf{fma}\left(0.25, K \cdot K, -2\right)\right)\\ \end{array} \end{array} \end{array} \]

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 -5e+301)
      (- U_m)
      (if (<= t_1 -1e-75)
        (* -2.0 J_m)
        (* U_m (* 0.5 (fma 0.25 (* K K) -2.0))))))))

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 <= -5e+301) {
		tmp = -U_m;
	} else if (t_1 <= -1e-75) {
		tmp = -2.0 * J_m;
	} else {
		tmp = U_m * (0.5 * fma(0.25, (K * K), -2.0));
	}
	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 <= -5e+301)
		tmp = Float64(-U_m);
	elseif (t_1 <= -1e-75)
		tmp = Float64(-2.0 * J_m);
	else
		tmp = Float64(U_m * Float64(0.5 * fma(0.25, Float64(K * K), -2.0)));
	end
	return Float64(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, -5e+301], (-U$95$m), If[LessEqual[t$95$1, -1e-75], N[(-2.0 * J$95$m), $MachinePrecision], N[(U$95$m * N[(0.5 * N[(0.25 * N[(K * K), $MachinePrecision] + -2.0), $MachinePrecision]), $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))))) < -5.0000000000000004e301

   1. Initial program 12.7%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 40.1

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

   5. Simplified 40.1%

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

   if -5.0000000000000004e301 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -9.9999999999999996e-76

   1. Initial program 99.7%

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

   3. Taylor expanded in K around 0

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

      1. lower-*.f64 54.1

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

   5. Simplified 54.1%

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

   6. Taylor expanded in J around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 43.4

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

   8. Simplified 43.4%

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

   if -9.9999999999999996e-76 < (*.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 81.6%

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

   3. Taylor expanded in K around 0

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

      1. lower-sqrt.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 51.6

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\mathsf{fma}\left(0.25, \frac{U \cdot U}{\color{blue}{J \cdot J}}, 1\right)} \]

   5. Simplified 51.6%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\sqrt{\mathsf{fma}\left(0.25, \frac{U \cdot U}{J \cdot J}, 1\right)}} \]

   6. Taylor expanded in K around 0

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

      1. associate-*r* N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 30.1

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

   8. Simplified 30.1%

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

   9. Taylor expanded in J around 0

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*l* N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. sub-neg N/A

          \[\leadsto U \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{1}{4} \cdot {K}^{2} + \left(\mathsf{neg}\left(2\right)\right)\right)}\right) \]

       7. metadata-eval N/A

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

       8. lower-fma.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 15.3

           \[\leadsto U \cdot \left(0.5 \cdot \mathsf{fma}\left(0.25, \color{blue}{K \cdot K}, -2\right)\right) \]

   11. Simplified 15.3%

       \[\leadsto \color{blue}{U \cdot \left(0.5 \cdot \mathsf{fma}\left(0.25, K \cdot K, -2\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 27.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -5 \cdot 10^{+301}:\\ \;\;\;\;-U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -1 \cdot 10^{-75}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(0.5 \cdot \mathsf{fma}\left(0.25, K \cdot K, -2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 49.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_0 \cdot \left(J\_m \cdot 2\right)}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+301}:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-75}:\\ \;\;\;\;-2 \cdot J\_m\\ \mathbf{else}:\\ \;\;\;\;-U\_m\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #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 -5e+301) (- U_m) (if (<= t_1 -1e-75) (* -2.0 J_m) (- U_m))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double 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 <= -5e+301) {
		tmp = -U_m;
	} else if (t_1 <= -1e-75) {
		tmp = -2.0 * J_m;
	} else {
		tmp = -U_m;
	}
	return J_s * tmp;
}

Fortran

U_m = abs(u)
J\_m = abs(j)
J\_s = copysign(1.0d0, j)
real(8) function code(j_s, j_m, k, u_m)
    real(8), intent (in) :: j_s
    real(8), intent (in) :: j_m
    real(8), intent (in) :: k
    real(8), intent (in) :: u_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    t_1 = (((-2.0d0) * j_m) * t_0) * sqrt((1.0d0 + ((u_m / (t_0 * (j_m * 2.0d0))) ** 2.0d0)))
    if (t_1 <= (-5d+301)) then
        tmp = -u_m
    else if (t_1 <= (-1d-75)) then
        tmp = (-2.0d0) * j_m
    else
        tmp = -u_m
    end if
    code = j_s * tmp
end function

Java

U_m = Math.abs(U);
J\_m = Math.abs(J);
J\_s = Math.copySign(1.0, J);
public static double code(double J_s, double J_m, double K, double U_m) {
	double 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 <= -5e+301) {
		tmp = -U_m;
	} else if (t_1 <= -1e-75) {
		tmp = -2.0 * J_m;
	} else {
		tmp = -U_m;
	}
	return J_s * tmp;
}

Python

U_m = math.fabs(U)
J\_m = math.fabs(J)
J\_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	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 <= -5e+301:
		tmp = -U_m
	elif t_1 <= -1e-75:
		tmp = -2.0 * J_m
	else:
		tmp = -U_m
	return J_s * tmp

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	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 <= -5e+301)
		tmp = Float64(-U_m);
	elseif (t_1 <= -1e-75)
		tmp = Float64(-2.0 * J_m);
	else
		tmp = Float64(-U_m);
	end
	return Float64(J_s * tmp)
end

MATLAB

U_m = abs(U);
J\_m = abs(J);
J\_s = sign(J) * abs(1.0);
function tmp_2 = code(J_s, J_m, K, U_m)
	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 <= -5e+301)
		tmp = -U_m;
	elseif (t_1 <= -1e-75)
		tmp = -2.0 * J_m;
	else
		tmp = -U_m;
	end
	tmp_2 = J_s * tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := 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, -5e+301], (-U$95$m), If[LessEqual[t$95$1, -1e-75], N[(-2.0 * J$95$m), $MachinePrecision], (-U$95$m)]]), $MachinePrecision]]]

Derivation

1. Split input into 2 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))))) < -5.0000000000000004e301 or -9.9999999999999996e-76 < (*.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 66.9%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 26.7

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

   5. Simplified 26.7%

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

   if -5.0000000000000004e301 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -9.9999999999999996e-76

   1. Initial program 99.7%

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

   3. Taylor expanded in K around 0

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

      1. lower-*.f64 54.1

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

   5. Simplified 54.1%

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

   6. Taylor expanded in J around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 43.4

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

   8. Simplified 43.4%

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

4. Final simplification 31.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -5 \cdot 10^{+301}:\\ \;\;\;\;-U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -1 \cdot 10^{-75}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 39.6% accurate, 124.3× speedup

Math

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

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #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 * Float64(-U_m))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.9%

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

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

   2. lower-neg.f64 21.9

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

5. Simplified 21.9%

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

Reproduce

herbie shell --seed 2024208 
(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)))))