Maksimov and Kolovsky, Equation (3)

Percentage Accurate: 73.8% → 99.5%

Time: 8.9s

Alternatives: 10

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: 73.8% 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: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \sqrt{{\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2} + 1}\\ t_2 := t\_1 \cdot \left(\left(J\_m \cdot -2\right) \cdot t\_0\right)\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+302}:\\ \;\;\;\;\left(\left(\cos \left(-0.5 \cdot K\right) \cdot J\_m\right) \cdot -2\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(-U\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1 (sqrt (+ (pow (/ U_m (* (* 2.0 J_m) t_0)) 2.0) 1.0)))
        (t_2 (* t_1 (* (* J_m -2.0) t_0))))
   (*
    J_s
    (if (<= t_2 (- INFINITY))
      (- U_m)
      (if (<= t_2 2e+302)
        (* (* (* (cos (* -0.5 K)) J_m) -2.0) t_1)
        (* -1.0 (- U_m)))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = sqrt((pow((U_m / ((2.0 * J_m) * t_0)), 2.0) + 1.0));
	double t_2 = t_1 * ((J_m * -2.0) * t_0);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_2 <= 2e+302) {
		tmp = ((cos((-0.5 * K)) * J_m) * -2.0) * t_1;
	} else {
		tmp = -1.0 * -U_m;
	}
	return J_s * tmp;
}

Java

U_m = Math.abs(U);
J\_m = Math.abs(J);
J\_s = Math.copySign(1.0, J);
public static double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = Math.sqrt((Math.pow((U_m / ((2.0 * J_m) * t_0)), 2.0) + 1.0));
	double t_2 = t_1 * ((J_m * -2.0) * t_0);
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = -U_m;
	} else if (t_2 <= 2e+302) {
		tmp = ((Math.cos((-0.5 * K)) * J_m) * -2.0) * t_1;
	} else {
		tmp = -1.0 * -U_m;
	}
	return J_s * tmp;
}

Python

U_m = math.fabs(U)
J\_m = math.fabs(J)
J\_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	t_0 = math.cos((K / 2.0))
	t_1 = math.sqrt((math.pow((U_m / ((2.0 * J_m) * t_0)), 2.0) + 1.0))
	t_2 = t_1 * ((J_m * -2.0) * t_0)
	tmp = 0
	if t_2 <= -math.inf:
		tmp = -U_m
	elif t_2 <= 2e+302:
		tmp = ((math.cos((-0.5 * K)) * J_m) * -2.0) * t_1
	else:
		tmp = -1.0 * -U_m
	return J_s * tmp

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = sqrt(Float64((Float64(U_m / Float64(Float64(2.0 * J_m) * t_0)) ^ 2.0) + 1.0))
	t_2 = Float64(t_1 * Float64(Float64(J_m * -2.0) * t_0))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_2 <= 2e+302)
		tmp = Float64(Float64(Float64(cos(Float64(-0.5 * K)) * J_m) * -2.0) * t_1);
	else
		tmp = Float64(-1.0 * 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 = sqrt((((U_m / ((2.0 * J_m) * t_0)) ^ 2.0) + 1.0));
	t_2 = t_1 * ((J_m * -2.0) * t_0);
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = -U_m;
	elseif (t_2 <= 2e+302)
		tmp = ((cos((-0.5 * K)) * J_m) * -2.0) * t_1;
	else
		tmp = -1.0 * -U_m;
	end
	tmp_2 = J_s * tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(N[(J$95$m * -2.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$2, (-Infinity)], (-U$95$m), If[LessEqual[t$95$2, 2e+302], N[(N[(N[(N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision] * J$95$m), $MachinePrecision] * -2.0), $MachinePrecision] * t$95$1), $MachinePrecision], N[(-1.0 * (-U$95$m)), $MachinePrecision]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 5.4%

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

   3. Taylor expanded in U around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 48.4

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

   5. Applied rewrites 48.4%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 2.0000000000000002e302

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 99.8

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

      8. lift-cos.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. metadata-eval N/A

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

      11. distribute-neg-frac2 N/A

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

      12. cos-neg N/A

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

      13. lower-cos.f64 N/A

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

      14. div-inv N/A

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

      15. lower-*.f64 N/A

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

      16. metadata-eval 99.8

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

   4. Applied rewrites 99.8%

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

   if 2.0000000000000002e302 < (*.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 8.0%

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

   3. Taylor expanded in U around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 45.9%

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

   6. Taylor expanded in U around inf

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

   8. Applied rewrites 45.9%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \leq 2 \cdot 10^{+302}:\\ \;\;\;\;\left(\left(\cos \left(-0.5 \cdot K\right) \cdot J\right) \cdot -2\right) \cdot \sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(-U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(J\_m \cdot -2\right) \cdot \cos \left(0.5 \cdot K\right)\\ t_2 := \sqrt{{\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2} + 1} \cdot \left(\left(J\_m \cdot -2\right) \cdot t\_0\right)\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{+204}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.125}{J\_m \cdot J\_m} \cdot U\_m, U\_m, 1\right) \cdot t\_1\\ \mathbf{elif}\;t\_2 \leq -4 \cdot 10^{-171}:\\ \;\;\;\;\left(J\_m \cdot -2\right) \cdot \sqrt{\mathsf{fma}\left(\frac{0.25 \cdot U\_m}{J\_m}, \frac{U\_m}{J\_m}, 1\right)}\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+302}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(-U\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1 (* (* J_m -2.0) (cos (* 0.5 K))))
        (t_2
         (*
          (sqrt (+ (pow (/ U_m (* (* 2.0 J_m) t_0)) 2.0) 1.0))
          (* (* J_m -2.0) t_0))))
   (*
    J_s
    (if (<= t_2 (- INFINITY))
      (- U_m)
      (if (<= t_2 -1e+204)
        (* (fma (* (/ 0.125 (* J_m J_m)) U_m) U_m 1.0) t_1)
        (if (<= t_2 -4e-171)
          (* (* J_m -2.0) (sqrt (fma (/ (* 0.25 U_m) J_m) (/ U_m J_m) 1.0)))
          (if (<= t_2 2e+302) t_1 (* -1.0 (- U_m)))))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = (J_m * -2.0) * cos((0.5 * K));
	double t_2 = sqrt((pow((U_m / ((2.0 * J_m) * t_0)), 2.0) + 1.0)) * ((J_m * -2.0) * t_0);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_2 <= -1e+204) {
		tmp = fma(((0.125 / (J_m * J_m)) * U_m), U_m, 1.0) * t_1;
	} else if (t_2 <= -4e-171) {
		tmp = (J_m * -2.0) * sqrt(fma(((0.25 * U_m) / J_m), (U_m / J_m), 1.0));
	} else if (t_2 <= 2e+302) {
		tmp = t_1;
	} else {
		tmp = -1.0 * -U_m;
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(J_m * -2.0) * cos(Float64(0.5 * K)))
	t_2 = Float64(sqrt(Float64((Float64(U_m / Float64(Float64(2.0 * J_m) * t_0)) ^ 2.0) + 1.0)) * Float64(Float64(J_m * -2.0) * t_0))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_2 <= -1e+204)
		tmp = Float64(fma(Float64(Float64(0.125 / Float64(J_m * J_m)) * U_m), U_m, 1.0) * t_1);
	elseif (t_2 <= -4e-171)
		tmp = Float64(Float64(J_m * -2.0) * sqrt(fma(Float64(Float64(0.25 * U_m) / J_m), Float64(U_m / J_m), 1.0)));
	elseif (t_2 <= 2e+302)
		tmp = t_1;
	else
		tmp = Float64(-1.0 * Float64(-U_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[(J$95$m * -2.0), $MachinePrecision] * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(N[(J$95$m * -2.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$2, (-Infinity)], (-U$95$m), If[LessEqual[t$95$2, -1e+204], N[(N[(N[(N[(0.125 / N[(J$95$m * J$95$m), $MachinePrecision]), $MachinePrecision] * U$95$m), $MachinePrecision] * U$95$m + 1.0), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$2, -4e-171], N[(N[(J$95$m * -2.0), $MachinePrecision] * N[Sqrt[N[(N[(N[(0.25 * U$95$m), $MachinePrecision] / J$95$m), $MachinePrecision] * N[(U$95$m / J$95$m), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+302], t$95$1, N[(-1.0 * (-U$95$m)), $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))))) < -inf.0

   1. Initial program 5.4%

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

   3. Taylor expanded in U around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 48.4

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

   5. Applied rewrites 48.4%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -9.99999999999999989e203

   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. associate-*r/ N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. times-frac N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 96.5

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

   5. Applied rewrites 96.5%

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

   6. Taylor expanded in U around 0

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

   8. Applied rewrites 84.7%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{fma}\left(\frac{0.125}{J \cdot J} \cdot U, \color{blue}{U}, 1\right) \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \color{blue}{\left(\frac{K}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{\frac{1}{8}}{J \cdot J} \cdot U, U, 1\right) \]

      2. div-inv N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \color{blue}{\left(K \cdot \frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{\frac{1}{8}}{J \cdot J} \cdot U, U, 1\right) \]

      3. metadata-eval N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(K \cdot \color{blue}{\frac{1}{2}}\right)\right) \cdot \mathsf{fma}\left(\frac{\frac{1}{8}}{J \cdot J} \cdot U, U, 1\right) \]

      4. *-commutative N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \color{blue}{\left(\frac{1}{2} \cdot K\right)}\right) \cdot \mathsf{fma}\left(\frac{\frac{1}{8}}{J \cdot J} \cdot U, U, 1\right) \]

      5. lower-*.f64 84.7

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \color{blue}{\left(0.5 \cdot K\right)}\right) \cdot \mathsf{fma}\left(\frac{0.125}{J \cdot J} \cdot U, U, 1\right) \]

   10. Applied rewrites 84.7%

       \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\cos \left(0.5 \cdot K\right)}\right) \cdot \mathsf{fma}\left(\frac{0.125}{J \cdot J} \cdot U, U, 1\right) \]

   if -9.99999999999999989e203 < (*.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))))) < -3.9999999999999999e-171

   1. Initial program 99.9%

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

   3. Taylor expanded in 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. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-*r/ N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. times-frac N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 64.4

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

   5. Applied rewrites 64.4%

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

   if -3.9999999999999999e-171 < (*.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))))) < 2.0000000000000002e302

   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 U around 0

      \[\leadsto \color{blue}{-2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)} \]
   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. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 76.1

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

   5. Applied rewrites 76.1%

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

   if 2.0000000000000002e302 < (*.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 8.0%

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

   3. Taylor expanded in U around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 45.9%

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

   6. Taylor expanded in U around inf

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

   8. Applied rewrites 45.9%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \leq -1 \cdot 10^{+204}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.125}{J \cdot J} \cdot U, U, 1\right) \cdot \left(\left(J \cdot -2\right) \cdot \cos \left(0.5 \cdot K\right)\right)\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \leq -4 \cdot 10^{-171}:\\ \;\;\;\;\left(J \cdot -2\right) \cdot \sqrt{\mathsf{fma}\left(\frac{0.25 \cdot U}{J}, \frac{U}{J}, 1\right)}\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \leq 2 \cdot 10^{+302}:\\ \;\;\;\;\left(J \cdot -2\right) \cdot \cos \left(0.5 \cdot K\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(-U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \sqrt{{\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2} + 1} \cdot \left(\left(J\_m \cdot -2\right) \cdot t\_0\right)\\ t_2 := \left(J\_m \cdot -2\right) \cdot \cos \left(0.5 \cdot K\right)\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{+204}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -4 \cdot 10^{-171}:\\ \;\;\;\;\left(J\_m \cdot -2\right) \cdot \sqrt{\mathsf{fma}\left(\frac{0.25 \cdot U\_m}{J\_m}, \frac{U\_m}{J\_m}, 1\right)}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+302}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(-U\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (sqrt (+ (pow (/ U_m (* (* 2.0 J_m) t_0)) 2.0) 1.0))
          (* (* J_m -2.0) t_0)))
        (t_2 (* (* J_m -2.0) (cos (* 0.5 K)))))
   (*
    J_s
    (if (<= t_1 (- INFINITY))
      (- U_m)
      (if (<= t_1 -1e+204)
        t_2
        (if (<= t_1 -4e-171)
          (* (* J_m -2.0) (sqrt (fma (/ (* 0.25 U_m) J_m) (/ U_m J_m) 1.0)))
          (if (<= t_1 2e+302) t_2 (* -1.0 (- U_m)))))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = sqrt((pow((U_m / ((2.0 * J_m) * t_0)), 2.0) + 1.0)) * ((J_m * -2.0) * t_0);
	double t_2 = (J_m * -2.0) * cos((0.5 * K));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_1 <= -1e+204) {
		tmp = t_2;
	} else if (t_1 <= -4e-171) {
		tmp = (J_m * -2.0) * sqrt(fma(((0.25 * U_m) / J_m), (U_m / J_m), 1.0));
	} else if (t_1 <= 2e+302) {
		tmp = t_2;
	} else {
		tmp = -1.0 * -U_m;
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(sqrt(Float64((Float64(U_m / Float64(Float64(2.0 * J_m) * t_0)) ^ 2.0) + 1.0)) * Float64(Float64(J_m * -2.0) * t_0))
	t_2 = Float64(Float64(J_m * -2.0) * cos(Float64(0.5 * K)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_1 <= -1e+204)
		tmp = t_2;
	elseif (t_1 <= -4e-171)
		tmp = Float64(Float64(J_m * -2.0) * sqrt(fma(Float64(Float64(0.25 * U_m) / J_m), Float64(U_m / J_m), 1.0)));
	elseif (t_1 <= 2e+302)
		tmp = t_2;
	else
		tmp = Float64(-1.0 * Float64(-U_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[Sqrt[N[(N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(N[(J$95$m * -2.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(J$95$m * -2.0), $MachinePrecision] * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$1, (-Infinity)], (-U$95$m), If[LessEqual[t$95$1, -1e+204], t$95$2, If[LessEqual[t$95$1, -4e-171], N[(N[(J$95$m * -2.0), $MachinePrecision] * N[Sqrt[N[(N[(N[(0.25 * U$95$m), $MachinePrecision] / J$95$m), $MachinePrecision] * N[(U$95$m / J$95$m), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+302], t$95$2, N[(-1.0 * (-U$95$m)), $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))))) < -inf.0

   1. Initial program 5.4%

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

   3. Taylor expanded in U around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 48.4

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

   5. Applied rewrites 48.4%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -9.99999999999999989e203 or -3.9999999999999999e-171 < (*.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))))) < 2.0000000000000002e302

   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 U around 0

      \[\leadsto \color{blue}{-2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)} \]
   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. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 78.0

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

   5. Applied rewrites 78.0%

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

   if -9.99999999999999989e203 < (*.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))))) < -3.9999999999999999e-171

   1. Initial program 99.9%

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

   3. Taylor expanded in 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. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-*r/ N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. times-frac N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 64.4

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

   5. Applied rewrites 64.4%

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

   if 2.0000000000000002e302 < (*.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 8.0%

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

   3. Taylor expanded in U around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 45.9%

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

   6. Taylor expanded in U around inf

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

   8. Applied rewrites 45.9%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \leq -1 \cdot 10^{+204}:\\ \;\;\;\;\left(J \cdot -2\right) \cdot \cos \left(0.5 \cdot K\right)\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \leq -4 \cdot 10^{-171}:\\ \;\;\;\;\left(J \cdot -2\right) \cdot \sqrt{\mathsf{fma}\left(\frac{0.25 \cdot U}{J}, \frac{U}{J}, 1\right)}\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \leq 2 \cdot 10^{+302}:\\ \;\;\;\;\left(J \cdot -2\right) \cdot \cos \left(0.5 \cdot K\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(-U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 62.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \sqrt{{\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2} + 1} \cdot \left(\left(J\_m \cdot -2\right) \cdot t\_0\right)\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -4 \cdot 10^{-93}:\\ \;\;\;\;J\_m \cdot -2\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-277}:\\ \;\;\;\;-U\_m\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{J\_m}{U\_m} \cdot \frac{J\_m}{U\_m}, -2, -1\right) \cdot \left(-U\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (sqrt (+ (pow (/ U_m (* (* 2.0 J_m) t_0)) 2.0) 1.0))
          (* (* J_m -2.0) t_0))))
   (*
    J_s
    (if (<= t_1 (- INFINITY))
      (- U_m)
      (if (<= t_1 -4e-93)
        (* J_m -2.0)
        (if (<= t_1 -2e-277)
          (- U_m)
          (* (fma (* (/ J_m U_m) (/ J_m U_m)) -2.0 -1.0) (- U_m))))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = sqrt((pow((U_m / ((2.0 * J_m) * t_0)), 2.0) + 1.0)) * ((J_m * -2.0) * t_0);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_1 <= -4e-93) {
		tmp = J_m * -2.0;
	} else if (t_1 <= -2e-277) {
		tmp = -U_m;
	} else {
		tmp = fma(((J_m / U_m) * (J_m / U_m)), -2.0, -1.0) * -U_m;
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(sqrt(Float64((Float64(U_m / Float64(Float64(2.0 * J_m) * t_0)) ^ 2.0) + 1.0)) * Float64(Float64(J_m * -2.0) * t_0))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_1 <= -4e-93)
		tmp = Float64(J_m * -2.0);
	elseif (t_1 <= -2e-277)
		tmp = Float64(-U_m);
	else
		tmp = Float64(fma(Float64(Float64(J_m / U_m) * Float64(J_m / U_m)), -2.0, -1.0) * Float64(-U_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[Sqrt[N[(N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(N[(J$95$m * -2.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$1, (-Infinity)], (-U$95$m), If[LessEqual[t$95$1, -4e-93], N[(J$95$m * -2.0), $MachinePrecision], If[LessEqual[t$95$1, -2e-277], (-U$95$m), N[(N[(N[(N[(J$95$m / U$95$m), $MachinePrecision] * N[(J$95$m / U$95$m), $MachinePrecision]), $MachinePrecision] * -2.0 + -1.0), $MachinePrecision] * (-U$95$m)), $MachinePrecision]]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -inf.0 or -3.9999999999999996e-93 < (*.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))))) < -1.99999999999999994e-277

   1. Initial program 36.1%

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

   3. Taylor expanded in U around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 41.5

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

   5. Applied rewrites 41.5%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -3.9999999999999996e-93

   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 U around 0

      \[\leadsto \color{blue}{-2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)} \]
   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. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 76.5

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

   5. Applied rewrites 76.5%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 43.2%

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

   if -1.99999999999999994e-277 < (*.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 67.4%

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

   3. Taylor expanded in U around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 25.3%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 25.3%

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

4. Final simplification 34.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \leq -4 \cdot 10^{-93}:\\ \;\;\;\;J \cdot -2\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \leq -2 \cdot 10^{-277}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{J}{U} \cdot \frac{J}{U}, -2, -1\right) \cdot \left(-U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 62.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \sqrt{{\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2} + 1} \cdot \left(\left(J\_m \cdot -2\right) \cdot t\_0\right)\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -4 \cdot 10^{-93}:\\ \;\;\;\;J\_m \cdot -2\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-277}:\\ \;\;\;\;-U\_m\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(-U\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (sqrt (+ (pow (/ U_m (* (* 2.0 J_m) t_0)) 2.0) 1.0))
          (* (* J_m -2.0) t_0))))
   (*
    J_s
    (if (<= t_1 (- INFINITY))
      (- U_m)
      (if (<= t_1 -4e-93)
        (* J_m -2.0)
        (if (<= t_1 -2e-277) (- U_m) (* -1.0 (- U_m))))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = sqrt((pow((U_m / ((2.0 * J_m) * t_0)), 2.0) + 1.0)) * ((J_m * -2.0) * t_0);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_1 <= -4e-93) {
		tmp = J_m * -2.0;
	} else if (t_1 <= -2e-277) {
		tmp = -U_m;
	} else {
		tmp = -1.0 * -U_m;
	}
	return J_s * tmp;
}

Java

U_m = Math.abs(U);
J\_m = Math.abs(J);
J\_s = Math.copySign(1.0, J);
public static double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = Math.sqrt((Math.pow((U_m / ((2.0 * J_m) * t_0)), 2.0) + 1.0)) * ((J_m * -2.0) * t_0);
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = -U_m;
	} else if (t_1 <= -4e-93) {
		tmp = J_m * -2.0;
	} else if (t_1 <= -2e-277) {
		tmp = -U_m;
	} else {
		tmp = -1.0 * -U_m;
	}
	return J_s * tmp;
}

Python

U_m = math.fabs(U)
J\_m = math.fabs(J)
J\_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	t_0 = math.cos((K / 2.0))
	t_1 = math.sqrt((math.pow((U_m / ((2.0 * J_m) * t_0)), 2.0) + 1.0)) * ((J_m * -2.0) * t_0)
	tmp = 0
	if t_1 <= -math.inf:
		tmp = -U_m
	elif t_1 <= -4e-93:
		tmp = J_m * -2.0
	elif t_1 <= -2e-277:
		tmp = -U_m
	else:
		tmp = -1.0 * -U_m
	return J_s * tmp

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(sqrt(Float64((Float64(U_m / Float64(Float64(2.0 * J_m) * t_0)) ^ 2.0) + 1.0)) * Float64(Float64(J_m * -2.0) * t_0))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_1 <= -4e-93)
		tmp = Float64(J_m * -2.0);
	elseif (t_1 <= -2e-277)
		tmp = Float64(-U_m);
	else
		tmp = Float64(-1.0 * 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 = sqrt((((U_m / ((2.0 * J_m) * t_0)) ^ 2.0) + 1.0)) * ((J_m * -2.0) * t_0);
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = -U_m;
	elseif (t_1 <= -4e-93)
		tmp = J_m * -2.0;
	elseif (t_1 <= -2e-277)
		tmp = -U_m;
	else
		tmp = -1.0 * -U_m;
	end
	tmp_2 = J_s * tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[(N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(N[(J$95$m * -2.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$1, (-Infinity)], (-U$95$m), If[LessEqual[t$95$1, -4e-93], N[(J$95$m * -2.0), $MachinePrecision], If[LessEqual[t$95$1, -2e-277], (-U$95$m), N[(-1.0 * (-U$95$m)), $MachinePrecision]]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -inf.0 or -3.9999999999999996e-93 < (*.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))))) < -1.99999999999999994e-277

   1. Initial program 36.1%

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

   3. Taylor expanded in U around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 41.5

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

   5. Applied rewrites 41.5%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -3.9999999999999996e-93

   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 U around 0

      \[\leadsto \color{blue}{-2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)} \]
   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. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 76.5

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

   5. Applied rewrites 76.5%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 43.2%

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

   if -1.99999999999999994e-277 < (*.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 67.4%

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

   3. Taylor expanded in U around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 25.3%

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

   6. Taylor expanded in U around inf

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

   8. Applied rewrites 25.0%

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

4. Final simplification 34.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \leq -4 \cdot 10^{-93}:\\ \;\;\;\;J \cdot -2\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \leq -2 \cdot 10^{-277}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(-U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 90.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(J\_m \cdot -2\right) \cdot t\_0\\ t_2 := \sqrt{{\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2} + 1} \cdot t\_1\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+302}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{0.25 \cdot U\_m}{J\_m}, \frac{U\_m}{J\_m}, 1\right)} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(-U\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1 (* (* J_m -2.0) t_0))
        (t_2 (* (sqrt (+ (pow (/ U_m (* (* 2.0 J_m) t_0)) 2.0) 1.0)) t_1)))
   (*
    J_s
    (if (<= t_2 (- INFINITY))
      (- U_m)
      (if (<= t_2 2e+302)
        (* (sqrt (fma (/ (* 0.25 U_m) J_m) (/ U_m J_m) 1.0)) t_1)
        (* -1.0 (- U_m)))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = (J_m * -2.0) * t_0;
	double t_2 = sqrt((pow((U_m / ((2.0 * J_m) * t_0)), 2.0) + 1.0)) * t_1;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_2 <= 2e+302) {
		tmp = sqrt(fma(((0.25 * U_m) / J_m), (U_m / J_m), 1.0)) * t_1;
	} else {
		tmp = -1.0 * -U_m;
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(J_m * -2.0) * t_0)
	t_2 = Float64(sqrt(Float64((Float64(U_m / Float64(Float64(2.0 * J_m) * t_0)) ^ 2.0) + 1.0)) * t_1)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_2 <= 2e+302)
		tmp = Float64(sqrt(fma(Float64(Float64(0.25 * U_m) / J_m), Float64(U_m / J_m), 1.0)) * t_1);
	else
		tmp = Float64(-1.0 * Float64(-U_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[(J$95$m * -2.0), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$2, (-Infinity)], (-U$95$m), If[LessEqual[t$95$2, 2e+302], N[(N[Sqrt[N[(N[(N[(0.25 * U$95$m), $MachinePrecision] / J$95$m), $MachinePrecision] * N[(U$95$m / J$95$m), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision], N[(-1.0 * (-U$95$m)), $MachinePrecision]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 5.4%

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

   3. Taylor expanded in U around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 48.4

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

   5. Applied rewrites 48.4%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 2.0000000000000002e302

   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

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\sqrt{1 + \frac{1}{4} \cdot \frac{{U}^{2}}{{J}^{2}}}} \]
   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. associate-*r/ N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. times-frac N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 89.6

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

   5. Applied rewrites 89.6%

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

   if 2.0000000000000002e302 < (*.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 8.0%

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

   3. Taylor expanded in U around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 45.9%

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

   6. Taylor expanded in U around inf

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

   8. Applied rewrites 45.9%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \leq 2 \cdot 10^{+302}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{0.25 \cdot U}{J}, \frac{U}{J}, 1\right)} \cdot \left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(-U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \sqrt{{\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2} + 1} \cdot \left(\left(J\_m \cdot -2\right) \cdot t\_0\right)\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-277}:\\ \;\;\;\;\left(J\_m \cdot -2\right) \cdot \sqrt{\mathsf{fma}\left(\frac{0.25 \cdot U\_m}{J\_m}, \frac{U\_m}{J\_m}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{-1}{\frac{U\_m}{J\_m}}}{U\_m}, \frac{-J\_m}{-0.5}, -1\right) \cdot \left(-U\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (sqrt (+ (pow (/ U_m (* (* 2.0 J_m) t_0)) 2.0) 1.0))
          (* (* J_m -2.0) t_0))))
   (*
    J_s
    (if (<= t_1 (- INFINITY))
      (- U_m)
      (if (<= t_1 -2e-277)
        (* (* J_m -2.0) (sqrt (fma (/ (* 0.25 U_m) J_m) (/ U_m J_m) 1.0)))
        (*
         (fma (/ (/ -1.0 (/ U_m J_m)) U_m) (/ (- J_m) -0.5) -1.0)
         (- U_m)))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = sqrt((pow((U_m / ((2.0 * J_m) * t_0)), 2.0) + 1.0)) * ((J_m * -2.0) * t_0);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_1 <= -2e-277) {
		tmp = (J_m * -2.0) * sqrt(fma(((0.25 * U_m) / J_m), (U_m / J_m), 1.0));
	} else {
		tmp = fma(((-1.0 / (U_m / J_m)) / U_m), (-J_m / -0.5), -1.0) * -U_m;
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(sqrt(Float64((Float64(U_m / Float64(Float64(2.0 * J_m) * t_0)) ^ 2.0) + 1.0)) * Float64(Float64(J_m * -2.0) * t_0))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_1 <= -2e-277)
		tmp = Float64(Float64(J_m * -2.0) * sqrt(fma(Float64(Float64(0.25 * U_m) / J_m), Float64(U_m / J_m), 1.0)));
	else
		tmp = Float64(fma(Float64(Float64(-1.0 / Float64(U_m / J_m)) / U_m), Float64(Float64(-J_m) / -0.5), -1.0) * Float64(-U_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[Sqrt[N[(N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(N[(J$95$m * -2.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$1, (-Infinity)], (-U$95$m), If[LessEqual[t$95$1, -2e-277], N[(N[(J$95$m * -2.0), $MachinePrecision] * N[Sqrt[N[(N[(N[(0.25 * U$95$m), $MachinePrecision] / J$95$m), $MachinePrecision] * N[(U$95$m / J$95$m), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-1.0 / N[(U$95$m / J$95$m), $MachinePrecision]), $MachinePrecision] / U$95$m), $MachinePrecision] * N[((-J$95$m) / -0.5), $MachinePrecision] + -1.0), $MachinePrecision] * (-U$95$m)), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 5.4%

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

   3. Taylor expanded in U around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 48.4

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

   5. Applied rewrites 48.4%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -1.99999999999999994e-277

   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

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

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-*r/ N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. times-frac N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 62.3

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

   5. Applied rewrites 62.3%

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

   if -1.99999999999999994e-277 < (*.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 67.4%

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

   3. Taylor expanded in U around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 25.3%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 23.8%

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

   10. Applied rewrites 25.3%

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

   12. Applied rewrites 25.3%

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

4. Final simplification 42.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \leq -2 \cdot 10^{-277}:\\ \;\;\;\;\left(J \cdot -2\right) \cdot \sqrt{\mathsf{fma}\left(\frac{0.25 \cdot U}{J}, \frac{U}{J}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{-1}{\frac{U}{J}}}{U}, \frac{-J}{-0.5}, -1\right) \cdot \left(-U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 77.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \sqrt{{\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2} + 1} \cdot \left(\left(J\_m \cdot -2\right) \cdot t\_0\right)\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-277}:\\ \;\;\;\;\left(J\_m \cdot -2\right) \cdot \sqrt{\mathsf{fma}\left(\frac{0.25 \cdot U\_m}{J\_m}, \frac{U\_m}{J\_m}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{J\_m}{U\_m} \cdot \frac{J\_m}{U\_m}, -2, -1\right) \cdot \left(-U\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (sqrt (+ (pow (/ U_m (* (* 2.0 J_m) t_0)) 2.0) 1.0))
          (* (* J_m -2.0) t_0))))
   (*
    J_s
    (if (<= t_1 (- INFINITY))
      (- U_m)
      (if (<= t_1 -2e-277)
        (* (* J_m -2.0) (sqrt (fma (/ (* 0.25 U_m) J_m) (/ U_m J_m) 1.0)))
        (* (fma (* (/ J_m U_m) (/ J_m U_m)) -2.0 -1.0) (- U_m)))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = sqrt((pow((U_m / ((2.0 * J_m) * t_0)), 2.0) + 1.0)) * ((J_m * -2.0) * t_0);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_1 <= -2e-277) {
		tmp = (J_m * -2.0) * sqrt(fma(((0.25 * U_m) / J_m), (U_m / J_m), 1.0));
	} else {
		tmp = fma(((J_m / U_m) * (J_m / U_m)), -2.0, -1.0) * -U_m;
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(sqrt(Float64((Float64(U_m / Float64(Float64(2.0 * J_m) * t_0)) ^ 2.0) + 1.0)) * Float64(Float64(J_m * -2.0) * t_0))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_1 <= -2e-277)
		tmp = Float64(Float64(J_m * -2.0) * sqrt(fma(Float64(Float64(0.25 * U_m) / J_m), Float64(U_m / J_m), 1.0)));
	else
		tmp = Float64(fma(Float64(Float64(J_m / U_m) * Float64(J_m / U_m)), -2.0, -1.0) * Float64(-U_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[Sqrt[N[(N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(N[(J$95$m * -2.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$1, (-Infinity)], (-U$95$m), If[LessEqual[t$95$1, -2e-277], N[(N[(J$95$m * -2.0), $MachinePrecision] * N[Sqrt[N[(N[(N[(0.25 * U$95$m), $MachinePrecision] / J$95$m), $MachinePrecision] * N[(U$95$m / J$95$m), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(J$95$m / U$95$m), $MachinePrecision] * N[(J$95$m / U$95$m), $MachinePrecision]), $MachinePrecision] * -2.0 + -1.0), $MachinePrecision] * (-U$95$m)), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 5.4%

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

   3. Taylor expanded in U around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 48.4

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

   5. Applied rewrites 48.4%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -1.99999999999999994e-277

   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

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

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-*r/ N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. times-frac N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 62.3

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

   5. Applied rewrites 62.3%

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

   if -1.99999999999999994e-277 < (*.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 67.4%

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

   3. Taylor expanded in U around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 25.3%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 25.3%

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

4. Final simplification 42.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \leq -2 \cdot 10^{-277}:\\ \;\;\;\;\left(J \cdot -2\right) \cdot \sqrt{\mathsf{fma}\left(\frac{0.25 \cdot U}{J}, \frac{U}{J}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{J}{U} \cdot \frac{J}{U}, -2, -1\right) \cdot \left(-U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 49.6% accurate, 31.0× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ J\_s \cdot \begin{array}{l} \mathbf{if}\;J\_m \leq 1.55 \cdot 10^{+50}:\\ \;\;\;\;-U\_m\\ \mathbf{else}:\\ \;\;\;\;J\_m \cdot -2\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (* J_s (if (<= J_m 1.55e+50) (- U_m) (* J_m -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 tmp;
	if (J_m <= 1.55e+50) {
		tmp = -U_m;
	} else {
		tmp = J_m * -2.0;
	}
	return J_s * tmp;
}

Fortran

U_m = abs(u)
J\_m = abs(j)
J\_s = copysign(1.0d0, j)
real(8) function code(j_s, j_m, k, u_m)
    real(8), intent (in) :: j_s
    real(8), intent (in) :: j_m
    real(8), intent (in) :: k
    real(8), intent (in) :: u_m
    real(8) :: tmp
    if (j_m <= 1.55d+50) then
        tmp = -u_m
    else
        tmp = j_m * (-2.0d0)
    end if
    code = j_s * tmp
end function

Java

U_m = Math.abs(U);
J\_m = Math.abs(J);
J\_s = Math.copySign(1.0, J);
public static double code(double J_s, double J_m, double K, double U_m) {
	double tmp;
	if (J_m <= 1.55e+50) {
		tmp = -U_m;
	} else {
		tmp = J_m * -2.0;
	}
	return J_s * tmp;
}

Python

U_m = math.fabs(U)
J\_m = math.fabs(J)
J\_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	tmp = 0
	if J_m <= 1.55e+50:
		tmp = -U_m
	else:
		tmp = J_m * -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)
	tmp = 0.0
	if (J_m <= 1.55e+50)
		tmp = Float64(-U_m);
	else
		tmp = Float64(J_m * -2.0);
	end
	return Float64(J_s * tmp)
end

MATLAB

U_m = abs(U);
J\_m = abs(J);
J\_s = sign(J) * abs(1.0);
function tmp_2 = code(J_s, J_m, K, U_m)
	tmp = 0.0;
	if (J_m <= 1.55e+50)
		tmp = -U_m;
	else
		tmp = J_m * -2.0;
	end
	tmp_2 = J_s * tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := N[(J$95$s * If[LessEqual[J$95$m, 1.55e+50], (-U$95$m), N[(J$95$m * -2.0), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if J < 1.55000000000000001e50

   1. Initial program 65.4%

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

   3. Taylor expanded in U around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 31.7

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

   5. Applied rewrites 31.7%

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

   if 1.55000000000000001e50 < J

   1. Initial program 99.9%

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

   3. Taylor expanded in U around 0

      \[\leadsto \color{blue}{-2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)} \]
   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. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 88.4

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

   5. Applied rewrites 88.4%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 52.0%

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

Alternative 10: 38.6% 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 74.0%

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

3. Taylor expanded in U around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 26.0

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

5. Applied rewrites 26.0%

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

Reproduce

herbie shell --seed 2024270 
(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)))))