Maksimov and Kolovsky, Equation (3)

Percentage Accurate: 73.3% → 98.9%

Time: 9.5s

Alternatives: 13

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: 73.3% 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: 98.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(-0.5 \cdot K\right)\\ t_1 := \cos \left(\frac{K}{2}\right)\\ t_2 := \sqrt{{\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_1}\right)}^{2} + 1} \cdot \left(\left(J\_m \cdot -2\right) \cdot t\_1\right)\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{J\_m}{U\_m} \cdot \frac{J\_m}{U\_m}\right) \cdot {\cos \left(0.5 \cdot K\right)}^{2}, -2, -1\right) \cdot U\_m\\ \mathbf{elif}\;t\_2 \leq 10^{+287}:\\ \;\;\;\;\left(\sqrt{{\left(\frac{\frac{\frac{U\_m}{J\_m}}{-2}}{t\_0}\right)}^{2} + 1} \cdot \left(t\_0 \cdot -2\right)\right) \cdot J\_m\\ \mathbf{else}:\\ \;\;\;\;\left(-U\_m\right) \cdot -1\\ \end{array} \end{array} \end{array} \]

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 (* -0.5 K)))
        (t_1 (cos (/ K 2.0)))
        (t_2
         (*
          (sqrt (+ (pow (/ U_m (* (* 2.0 J_m) t_1)) 2.0) 1.0))
          (* (* J_m -2.0) t_1))))
   (*
    J_s
    (if (<= t_2 (- INFINITY))
      (*
       (fma
        (* (* (/ J_m U_m) (/ J_m U_m)) (pow (cos (* 0.5 K)) 2.0))
        -2.0
        -1.0)
       U_m)
      (if (<= t_2 1e+287)
        (*
         (* (sqrt (+ (pow (/ (/ (/ U_m J_m) -2.0) t_0) 2.0) 1.0)) (* t_0 -2.0))
         J_m)
        (* (- U_m) -1.0))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((-0.5 * K));
	double t_1 = cos((K / 2.0));
	double t_2 = sqrt((pow((U_m / ((2.0 * J_m) * t_1)), 2.0) + 1.0)) * ((J_m * -2.0) * t_1);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = fma((((J_m / U_m) * (J_m / U_m)) * pow(cos((0.5 * K)), 2.0)), -2.0, -1.0) * U_m;
	} else if (t_2 <= 1e+287) {
		tmp = (sqrt((pow((((U_m / J_m) / -2.0) / t_0), 2.0) + 1.0)) * (t_0 * -2.0)) * J_m;
	} else {
		tmp = -U_m * -1.0;
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(-0.5 * K))
	t_1 = cos(Float64(K / 2.0))
	t_2 = Float64(sqrt(Float64((Float64(U_m / Float64(Float64(2.0 * J_m) * t_1)) ^ 2.0) + 1.0)) * Float64(Float64(J_m * -2.0) * t_1))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(fma(Float64(Float64(Float64(J_m / U_m) * Float64(J_m / U_m)) * (cos(Float64(0.5 * K)) ^ 2.0)), -2.0, -1.0) * U_m);
	elseif (t_2 <= 1e+287)
		tmp = Float64(Float64(sqrt(Float64((Float64(Float64(Float64(U_m / J_m) / -2.0) / t_0) ^ 2.0) + 1.0)) * Float64(t_0 * -2.0)) * J_m);
	else
		tmp = Float64(Float64(-U_m) * -1.0);
	end
	return Float64(J_s * tmp)
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $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$1), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(N[(J$95$m * -2.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$2, (-Infinity)], N[(N[(N[(N[(N[(J$95$m / U$95$m), $MachinePrecision] * N[(J$95$m / U$95$m), $MachinePrecision]), $MachinePrecision] * N[Power[N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * -2.0 + -1.0), $MachinePrecision] * U$95$m), $MachinePrecision], If[LessEqual[t$95$2, 1e+287], N[(N[(N[Sqrt[N[(N[Power[N[(N[(N[(U$95$m / J$95$m), $MachinePrecision] / -2.0), $MachinePrecision] / t$95$0), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * -2.0), $MachinePrecision]), $MachinePrecision] * J$95$m), $MachinePrecision], N[((-U$95$m) * -1.0), $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 5.2%

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

   3. Taylor expanded in U around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 39.7%

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

   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))))) < 1.0000000000000001e287

   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

   4. Applied rewrites 99.8%

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

   if 1.0000000000000001e287 < (*.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 22.1%

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

   3. Taylor expanded in U around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(U \cdot \left(-2 \cdot \frac{{J}^{2} \cdot {\cos \left(\frac{1}{2} \cdot K\right)}^{2}}{{U}^{2}} - 1\right)\right)} \]
   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 50.1%

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

   6. Taylor expanded in U around inf

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

   8. Applied rewrites 49.6%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{J}{U} \cdot \frac{J}{U}\right) \cdot {\cos \left(0.5 \cdot K\right)}^{2}, -2, -1\right) \cdot U\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \leq 10^{+287}:\\ \;\;\;\;\left(\sqrt{{\left(\frac{\frac{\frac{U}{J}}{-2}}{\cos \left(-0.5 \cdot K\right)}\right)}^{2} + 1} \cdot \left(\cos \left(-0.5 \cdot K\right) \cdot -2\right)\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;\left(-U\right) \cdot -1\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \sqrt{{\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2} + 1} \cdot \left(\left(J\_m \cdot -2\right) \cdot t\_0\right)\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-266}:\\ \;\;\;\;\left(J\_m \cdot -2\right) \cdot \sqrt{\mathsf{fma}\left(\frac{0.25 \cdot U\_m}{J\_m}, \frac{U\_m}{J\_m}, 1\right)}\\ \mathbf{elif}\;t\_1 \leq 10^{+287}:\\ \;\;\;\;\left(J\_m \cdot -2\right) \cdot \cos \left(0.5 \cdot K\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-U\_m\right) \cdot -1\\ \end{array} \end{array} \end{array} \]

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-266)
        (* (* J_m -2.0) (sqrt (fma (/ (* 0.25 U_m) J_m) (/ U_m J_m) 1.0)))
        (if (<= t_1 1e+287)
          (* (* J_m -2.0) (cos (* 0.5 K)))
          (* (- U_m) -1.0)))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = 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-266) {
		tmp = (J_m * -2.0) * sqrt(fma(((0.25 * U_m) / J_m), (U_m / J_m), 1.0));
	} else if (t_1 <= 1e+287) {
		tmp = (J_m * -2.0) * cos((0.5 * K));
	} else {
		tmp = -U_m * -1.0;
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(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-266)
		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 <= 1e+287)
		tmp = Float64(Float64(J_m * -2.0) * cos(Float64(0.5 * K)));
	else
		tmp = Float64(Float64(-U_m) * -1.0);
	end
	return Float64(J_s * tmp)
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[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-266], 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, 1e+287], N[(N[(J$95$m * -2.0), $MachinePrecision] * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[((-U$95$m) * -1.0), $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 5.2%

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

   3. Taylor expanded in 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 39.7

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

   5. Applied rewrites 39.7%

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

   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))))) < -2e-266

   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

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

      9. associate-*r* N/A

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

      10. unpow2 N/A

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

      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 70.0

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

   5. Applied rewrites 70.0%

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

   if -2e-266 < (*.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))))) < 1.0000000000000001e287

   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 66.9

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

   5. Applied rewrites 66.9%

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

   if 1.0000000000000001e287 < (*.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 22.1%

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

   3. Taylor expanded in U around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(U \cdot \left(-2 \cdot \frac{{J}^{2} \cdot {\cos \left(\frac{1}{2} \cdot K\right)}^{2}}{{U}^{2}} - 1\right)\right)} \]
   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 50.1%

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

   6. Taylor expanded in U around inf

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

   8. Applied rewrites 49.6%

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

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \leq -2 \cdot 10^{-266}:\\ \;\;\;\;\left(J \cdot -2\right) \cdot \sqrt{\mathsf{fma}\left(\frac{0.25 \cdot U}{J}, \frac{U}{J}, 1\right)}\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \leq 10^{+287}:\\ \;\;\;\;\left(J \cdot -2\right) \cdot \cos \left(0.5 \cdot K\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-U\right) \cdot -1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 60.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \sqrt{{\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2} + 1} \cdot \left(\left(J\_m \cdot -2\right) \cdot t\_0\right)\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+304}:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{+54}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.125}{J\_m \cdot J\_m} \cdot U\_m, U\_m, 1\right) \cdot \left(J\_m \cdot -2\right)\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-288}:\\ \;\;\;\;-U\_m\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{J\_m}{U\_m} \cdot \frac{J\_m}{U\_m}, -2, -1\right) \cdot \left(-U\_m\right)\\ \end{array} \end{array} \end{array} \]

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 -5e+304)
      (- U_m)
      (if (<= t_1 -5e+54)
        (* (fma (* (/ 0.125 (* J_m J_m)) U_m) U_m 1.0) (* J_m -2.0))
        (if (<= t_1 -5e-288)
          (- 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 <= -5e+304) {
		tmp = -U_m;
	} else if (t_1 <= -5e+54) {
		tmp = fma(((0.125 / (J_m * J_m)) * U_m), U_m, 1.0) * (J_m * -2.0);
	} else if (t_1 <= -5e-288) {
		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 <= -5e+304)
		tmp = Float64(-U_m);
	elseif (t_1 <= -5e+54)
		tmp = Float64(fma(Float64(Float64(0.125 / Float64(J_m * J_m)) * U_m), U_m, 1.0) * Float64(J_m * -2.0));
	elseif (t_1 <= -5e-288)
		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, -5e+304], (-U$95$m), If[LessEqual[t$95$1, -5e+54], 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] * N[(J$95$m * -2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -5e-288], (-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))))) < -4.9999999999999997e304 or -5.00000000000000005e54 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -5.00000000000000011e-288

   1. Initial program 64.8%

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

   3. Taylor expanded in U around 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 27.0

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

   5. Applied rewrites 27.0%

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

   if -4.9999999999999997e304 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -5.00000000000000005e54

   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

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

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      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 95.9

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

   5. Applied rewrites 95.9%

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

   6. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 73.2

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

   8. Applied rewrites 73.2%

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

   9. Taylor expanded in U around 0

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

   11. Applied rewrites 51.8%

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

   if -5.00000000000000011e-288 < (*.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 70.3%

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

   3. Taylor expanded in U around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(U \cdot \left(-2 \cdot \frac{{J}^{2} \cdot {\cos \left(\frac{1}{2} \cdot K\right)}^{2}}{{U}^{2}} - 1\right)\right)} \]
   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 28.4%

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

   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 28.3%

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

4. Final simplification 33.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \leq -5 \cdot 10^{+304}:\\ \;\;\;\;-U\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \leq -5 \cdot 10^{+54}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.125}{J \cdot J} \cdot U, U, 1\right) \cdot \left(J \cdot -2\right)\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \leq -5 \cdot 10^{-288}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{J}{U} \cdot \frac{J}{U}, -2, -1\right) \cdot \left(-U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 60.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \sqrt{{\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2} + 1} \cdot \left(\left(J\_m \cdot -2\right) \cdot t\_0\right)\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+304}:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{+54}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.125}{J\_m \cdot J\_m} \cdot U\_m, U\_m, 1\right) \cdot \left(J\_m \cdot -2\right)\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-288}:\\ \;\;\;\;-U\_m\\ \mathbf{else}:\\ \;\;\;\;\left(-U\_m\right) \cdot -1\\ \end{array} \end{array} \end{array} \]

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 -5e+304)
      (- U_m)
      (if (<= t_1 -5e+54)
        (* (fma (* (/ 0.125 (* J_m J_m)) U_m) U_m 1.0) (* J_m -2.0))
        (if (<= t_1 -5e-288) (- U_m) (* (- U_m) -1.0)))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = 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 <= -5e+304) {
		tmp = -U_m;
	} else if (t_1 <= -5e+54) {
		tmp = fma(((0.125 / (J_m * J_m)) * U_m), U_m, 1.0) * (J_m * -2.0);
	} else if (t_1 <= -5e-288) {
		tmp = -U_m;
	} else {
		tmp = -U_m * -1.0;
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(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 <= -5e+304)
		tmp = Float64(-U_m);
	elseif (t_1 <= -5e+54)
		tmp = Float64(fma(Float64(Float64(0.125 / Float64(J_m * J_m)) * U_m), U_m, 1.0) * Float64(J_m * -2.0));
	elseif (t_1 <= -5e-288)
		tmp = Float64(-U_m);
	else
		tmp = Float64(Float64(-U_m) * -1.0);
	end
	return Float64(J_s * tmp)
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[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, -5e+304], (-U$95$m), If[LessEqual[t$95$1, -5e+54], 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] * N[(J$95$m * -2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -5e-288], (-U$95$m), N[((-U$95$m) * -1.0), $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))))) < -4.9999999999999997e304 or -5.00000000000000005e54 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -5.00000000000000011e-288

   1. Initial program 64.8%

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

   3. Taylor expanded in U around 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 27.0

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

   5. Applied rewrites 27.0%

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

   if -4.9999999999999997e304 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -5.00000000000000005e54

   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

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

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      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 95.9

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

   5. Applied rewrites 95.9%

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

   6. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 73.2

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

   8. Applied rewrites 73.2%

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

   9. Taylor expanded in U around 0

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

   11. Applied rewrites 51.8%

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

   if -5.00000000000000011e-288 < (*.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 70.3%

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

   3. Taylor expanded in U around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(U \cdot \left(-2 \cdot \frac{{J}^{2} \cdot {\cos \left(\frac{1}{2} \cdot K\right)}^{2}}{{U}^{2}} - 1\right)\right)} \]
   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 28.4%

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

   6. Taylor expanded in U around inf

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

   8. Applied rewrites 28.1%

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

4. Final simplification 32.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \leq -5 \cdot 10^{+304}:\\ \;\;\;\;-U\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \leq -5 \cdot 10^{+54}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.125}{J \cdot J} \cdot U, U, 1\right) \cdot \left(J \cdot -2\right)\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \leq -5 \cdot 10^{-288}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;\left(-U\right) \cdot -1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 60.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \sqrt{{\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2} + 1} \cdot \left(\left(J\_m \cdot -2\right) \cdot t\_0\right)\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+304}:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{+54}:\\ \;\;\;\;J\_m \cdot -2\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-288}:\\ \;\;\;\;-U\_m\\ \mathbf{else}:\\ \;\;\;\;\left(-U\_m\right) \cdot -1\\ \end{array} \end{array} \end{array} \]

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 -5e+304)
      (- U_m)
      (if (<= t_1 -5e+54)
        (* J_m -2.0)
        (if (<= t_1 -5e-288) (- U_m) (* (- U_m) -1.0)))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = 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 <= -5e+304) {
		tmp = -U_m;
	} else if (t_1 <= -5e+54) {
		tmp = J_m * -2.0;
	} else if (t_1 <= -5e-288) {
		tmp = -U_m;
	} else {
		tmp = -U_m * -1.0;
	}
	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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    t_1 = sqrt((((u_m / ((2.0d0 * j_m) * t_0)) ** 2.0d0) + 1.0d0)) * ((j_m * (-2.0d0)) * t_0)
    if (t_1 <= (-5d+304)) then
        tmp = -u_m
    else if (t_1 <= (-5d+54)) then
        tmp = j_m * (-2.0d0)
    else if (t_1 <= (-5d-288)) then
        tmp = -u_m
    else
        tmp = -u_m * (-1.0d0)
    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 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 <= -5e+304) {
		tmp = -U_m;
	} else if (t_1 <= -5e+54) {
		tmp = J_m * -2.0;
	} else if (t_1 <= -5e-288) {
		tmp = -U_m;
	} else {
		tmp = -U_m * -1.0;
	}
	return J_s * tmp;
}

Python

U_m = math.fabs(U)
J\_m = math.fabs(J)
J\_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	t_0 = math.cos((K / 2.0))
	t_1 = 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 <= -5e+304:
		tmp = -U_m
	elif t_1 <= -5e+54:
		tmp = J_m * -2.0
	elif t_1 <= -5e-288:
		tmp = -U_m
	else:
		tmp = -U_m * -1.0
	return J_s * tmp

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(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 <= -5e+304)
		tmp = Float64(-U_m);
	elseif (t_1 <= -5e+54)
		tmp = Float64(J_m * -2.0);
	elseif (t_1 <= -5e-288)
		tmp = Float64(-U_m);
	else
		tmp = Float64(Float64(-U_m) * -1.0);
	end
	return Float64(J_s * tmp)
end

MATLAB

U_m = abs(U);
J\_m = abs(J);
J\_s = sign(J) * abs(1.0);
function tmp_2 = code(J_s, J_m, K, U_m)
	t_0 = cos((K / 2.0));
	t_1 = 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 <= -5e+304)
		tmp = -U_m;
	elseif (t_1 <= -5e+54)
		tmp = J_m * -2.0;
	elseif (t_1 <= -5e-288)
		tmp = -U_m;
	else
		tmp = -U_m * -1.0;
	end
	tmp_2 = J_s * tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[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, -5e+304], (-U$95$m), If[LessEqual[t$95$1, -5e+54], N[(J$95$m * -2.0), $MachinePrecision], If[LessEqual[t$95$1, -5e-288], (-U$95$m), N[((-U$95$m) * -1.0), $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))))) < -4.9999999999999997e304 or -5.00000000000000005e54 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -5.00000000000000011e-288

   1. Initial program 64.8%

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

   3. Taylor expanded in U around 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 27.0

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

   5. Applied rewrites 27.0%

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

   if -4.9999999999999997e304 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -5.00000000000000005e54

   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 78.4

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

   5. Applied rewrites 78.4%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 51.8%

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

   if -5.00000000000000011e-288 < (*.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 70.3%

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

   3. Taylor expanded in U around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(U \cdot \left(-2 \cdot \frac{{J}^{2} \cdot {\cos \left(\frac{1}{2} \cdot K\right)}^{2}}{{U}^{2}} - 1\right)\right)} \]
   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 28.4%

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

   6. Taylor expanded in U around inf

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

   8. Applied rewrites 28.1%

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

4. Final simplification 32.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \leq -5 \cdot 10^{+304}:\\ \;\;\;\;-U\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \leq -5 \cdot 10^{+54}:\\ \;\;\;\;J \cdot -2\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \leq -5 \cdot 10^{-288}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;\left(-U\right) \cdot -1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \sqrt{{\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2} + 1}\\ t_2 := t\_1 \cdot \left(\left(J\_m \cdot -2\right) \cdot t\_0\right)\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{J\_m}{U\_m} \cdot \frac{J\_m}{U\_m}\right) \cdot {\cos \left(0.5 \cdot K\right)}^{2}, -2, -1\right) \cdot U\_m\\ \mathbf{elif}\;t\_2 \leq 10^{+287}:\\ \;\;\;\;\left(\left(\cos \left(-0.5 \cdot K\right) \cdot J\_m\right) \cdot -2\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(-U\_m\right) \cdot -1\\ \end{array} \end{array} \end{array} \]

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))
      (*
       (fma
        (* (* (/ J_m U_m) (/ J_m U_m)) (pow (cos (* 0.5 K)) 2.0))
        -2.0
        -1.0)
       U_m)
      (if (<= t_2 1e+287)
        (* (* (* (cos (* -0.5 K)) J_m) -2.0) t_1)
        (* (- U_m) -1.0))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = 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 = fma((((J_m / U_m) * (J_m / U_m)) * pow(cos((0.5 * K)), 2.0)), -2.0, -1.0) * U_m;
	} else if (t_2 <= 1e+287) {
		tmp = ((cos((-0.5 * K)) * J_m) * -2.0) * t_1;
	} else {
		tmp = -U_m * -1.0;
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = 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(fma(Float64(Float64(Float64(J_m / U_m) * Float64(J_m / U_m)) * (cos(Float64(0.5 * K)) ^ 2.0)), -2.0, -1.0) * U_m);
	elseif (t_2 <= 1e+287)
		tmp = Float64(Float64(Float64(cos(Float64(-0.5 * K)) * J_m) * -2.0) * t_1);
	else
		tmp = Float64(Float64(-U_m) * -1.0);
	end
	return Float64(J_s * tmp)
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[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)], N[(N[(N[(N[(N[(J$95$m / U$95$m), $MachinePrecision] * N[(J$95$m / U$95$m), $MachinePrecision]), $MachinePrecision] * N[Power[N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * -2.0 + -1.0), $MachinePrecision] * U$95$m), $MachinePrecision], If[LessEqual[t$95$2, 1e+287], N[(N[(N[(N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision] * J$95$m), $MachinePrecision] * -2.0), $MachinePrecision] * t$95$1), $MachinePrecision], N[((-U$95$m) * -1.0), $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 5.2%

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

   3. Taylor expanded in U around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 39.7%

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

   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))))) < 1.0000000000000001e287

   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 1.0000000000000001e287 < (*.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 22.1%

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

   3. Taylor expanded in U around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(U \cdot \left(-2 \cdot \frac{{J}^{2} \cdot {\cos \left(\frac{1}{2} \cdot K\right)}^{2}}{{U}^{2}} - 1\right)\right)} \]
   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 50.1%

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

   6. Taylor expanded in U around inf

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

   8. Applied rewrites 49.6%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{J}{U} \cdot \frac{J}{U}\right) \cdot {\cos \left(0.5 \cdot K\right)}^{2}, -2, -1\right) \cdot U\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \leq 10^{+287}:\\ \;\;\;\;\left(\left(\cos \left(-0.5 \cdot K\right) \cdot J\right) \cdot -2\right) \cdot \sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(-U\right) \cdot -1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \sqrt{{\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2} + 1} \cdot \left(\left(J\_m \cdot -2\right) \cdot t\_0\right)\\ t_2 := \cos \left(-0.5 \cdot K\right)\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{J\_m}{U\_m} \cdot \frac{J\_m}{U\_m}\right) \cdot {\cos \left(0.5 \cdot K\right)}^{2}, -2, -1\right) \cdot U\_m\\ \mathbf{elif}\;t\_1 \leq 10^{+287}:\\ \;\;\;\;\left(\left(\sqrt{\mathsf{fma}\left({\left(t\_2 \cdot \frac{J\_m}{U\_m}\right)}^{-2}, 0.25, 1\right)} \cdot t\_2\right) \cdot J\_m\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\left(-U\_m\right) \cdot -1\\ \end{array} \end{array} \end{array} \]

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 (cos (* -0.5 K))))
   (*
    J_s
    (if (<= t_1 (- INFINITY))
      (*
       (fma
        (* (* (/ J_m U_m) (/ J_m U_m)) (pow (cos (* 0.5 K)) 2.0))
        -2.0
        -1.0)
       U_m)
      (if (<= t_1 1e+287)
        (*
         (* (* (sqrt (fma (pow (* t_2 (/ J_m U_m)) -2.0) 0.25 1.0)) t_2) J_m)
         -2.0)
        (* (- U_m) -1.0))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = sqrt((pow((U_m / ((2.0 * J_m) * t_0)), 2.0) + 1.0)) * ((J_m * -2.0) * t_0);
	double t_2 = cos((-0.5 * K));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma((((J_m / U_m) * (J_m / U_m)) * pow(cos((0.5 * K)), 2.0)), -2.0, -1.0) * U_m;
	} else if (t_1 <= 1e+287) {
		tmp = ((sqrt(fma(pow((t_2 * (J_m / U_m)), -2.0), 0.25, 1.0)) * t_2) * J_m) * -2.0;
	} else {
		tmp = -U_m * -1.0;
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(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 = cos(Float64(-0.5 * K))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(Float64(Float64(Float64(J_m / U_m) * Float64(J_m / U_m)) * (cos(Float64(0.5 * K)) ^ 2.0)), -2.0, -1.0) * U_m);
	elseif (t_1 <= 1e+287)
		tmp = Float64(Float64(Float64(sqrt(fma((Float64(t_2 * Float64(J_m / U_m)) ^ -2.0), 0.25, 1.0)) * t_2) * J_m) * -2.0);
	else
		tmp = Float64(Float64(-U_m) * -1.0);
	end
	return Float64(J_s * tmp)
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[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[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(N[(N[(J$95$m / U$95$m), $MachinePrecision] * N[(J$95$m / U$95$m), $MachinePrecision]), $MachinePrecision] * N[Power[N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * -2.0 + -1.0), $MachinePrecision] * U$95$m), $MachinePrecision], If[LessEqual[t$95$1, 1e+287], N[(N[(N[(N[Sqrt[N[(N[Power[N[(t$95$2 * N[(J$95$m / U$95$m), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision] * 0.25 + 1.0), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision] * J$95$m), $MachinePrecision] * -2.0), $MachinePrecision], N[((-U$95$m) * -1.0), $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 5.2%

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

   3. Taylor expanded in U around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 39.7%

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

   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))))) < 1.0000000000000001e287

   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

   4. Applied rewrites 99.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   6. Applied rewrites 99.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

   8. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\left(\left(\sqrt{\mathsf{fma}\left({\left(\cos \left(K \cdot -0.5\right) \cdot \frac{J}{U}\right)}^{-2}, 0.25, 1\right)} \cdot \cos \left(K \cdot -0.5\right)\right) \cdot J\right) \cdot -2} \]

   if 1.0000000000000001e287 < (*.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 22.1%

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

   3. Taylor expanded in U around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(U \cdot \left(-2 \cdot \frac{{J}^{2} \cdot {\cos \left(\frac{1}{2} \cdot K\right)}^{2}}{{U}^{2}} - 1\right)\right)} \]
   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 50.1%

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

   6. Taylor expanded in U around inf

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

   8. Applied rewrites 49.6%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{J}{U} \cdot \frac{J}{U}\right) \cdot {\cos \left(0.5 \cdot K\right)}^{2}, -2, -1\right) \cdot U\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \leq 10^{+287}:\\ \;\;\;\;\left(\left(\sqrt{\mathsf{fma}\left({\left(\cos \left(-0.5 \cdot K\right) \cdot \frac{J}{U}\right)}^{-2}, 0.25, 1\right)} \cdot \cos \left(-0.5 \cdot K\right)\right) \cdot J\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\left(-U\right) \cdot -1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 90.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \sqrt{{\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2} + 1} \cdot \left(\left(J\_m \cdot -2\right) \cdot t\_0\right)\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{J\_m}{U\_m} \cdot \frac{J\_m}{U\_m}\right) \cdot {\cos \left(0.5 \cdot K\right)}^{2}, -2, -1\right) \cdot U\_m\\ \mathbf{elif}\;t\_1 \leq 10^{+287}:\\ \;\;\;\;\left(\left(\sqrt{\mathsf{fma}\left(\frac{U\_m}{J\_m} \cdot \frac{U\_m}{J\_m}, 0.25, 1\right)} \cdot \cos \left(-0.5 \cdot K\right)\right) \cdot J\_m\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\left(-U\_m\right) \cdot -1\\ \end{array} \end{array} \end{array} \]

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))
      (*
       (fma
        (* (* (/ J_m U_m) (/ J_m U_m)) (pow (cos (* 0.5 K)) 2.0))
        -2.0
        -1.0)
       U_m)
      (if (<= t_1 1e+287)
        (*
         (*
          (*
           (sqrt (fma (* (/ U_m J_m) (/ U_m J_m)) 0.25 1.0))
           (cos (* -0.5 K)))
          J_m)
         -2.0)
        (* (- U_m) -1.0))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = 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 = fma((((J_m / U_m) * (J_m / U_m)) * pow(cos((0.5 * K)), 2.0)), -2.0, -1.0) * U_m;
	} else if (t_1 <= 1e+287) {
		tmp = ((sqrt(fma(((U_m / J_m) * (U_m / J_m)), 0.25, 1.0)) * cos((-0.5 * K))) * J_m) * -2.0;
	} else {
		tmp = -U_m * -1.0;
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(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(fma(Float64(Float64(Float64(J_m / U_m) * Float64(J_m / U_m)) * (cos(Float64(0.5 * K)) ^ 2.0)), -2.0, -1.0) * U_m);
	elseif (t_1 <= 1e+287)
		tmp = Float64(Float64(Float64(sqrt(fma(Float64(Float64(U_m / J_m) * Float64(U_m / J_m)), 0.25, 1.0)) * cos(Float64(-0.5 * K))) * J_m) * -2.0);
	else
		tmp = Float64(Float64(-U_m) * -1.0);
	end
	return Float64(J_s * tmp)
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[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)], N[(N[(N[(N[(N[(J$95$m / U$95$m), $MachinePrecision] * N[(J$95$m / U$95$m), $MachinePrecision]), $MachinePrecision] * N[Power[N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * -2.0 + -1.0), $MachinePrecision] * U$95$m), $MachinePrecision], If[LessEqual[t$95$1, 1e+287], N[(N[(N[(N[Sqrt[N[(N[(N[(U$95$m / J$95$m), $MachinePrecision] * N[(U$95$m / J$95$m), $MachinePrecision]), $MachinePrecision] * 0.25 + 1.0), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * J$95$m), $MachinePrecision] * -2.0), $MachinePrecision], N[((-U$95$m) * -1.0), $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 5.2%

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

   3. Taylor expanded in U around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 39.7%

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

   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))))) < 1.0000000000000001e287

   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

   4. Applied rewrites 99.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   6. Applied rewrites 99.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

   8. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\left(\left(\sqrt{\mathsf{fma}\left({\left(\cos \left(K \cdot -0.5\right) \cdot \frac{J}{U}\right)}^{-2}, 0.25, 1\right)} \cdot \cos \left(K \cdot -0.5\right)\right) \cdot J\right) \cdot -2} \]

   9. Taylor expanded in K around 0

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

       1. unpow2 N/A

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

       2. unpow2 N/A

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

       3. times-frac N/A

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

       4. lower-*.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. lower-/.f64 87.1

          \[\leadsto \left(\left(\sqrt{\mathsf{fma}\left(\frac{U}{J} \cdot \color{blue}{\frac{U}{J}}, 0.25, 1\right)} \cdot \cos \left(K \cdot -0.5\right)\right) \cdot J\right) \cdot -2 \]

   11. Applied rewrites 87.1%

       \[\leadsto \left(\left(\sqrt{\mathsf{fma}\left(\color{blue}{\frac{U}{J} \cdot \frac{U}{J}}, 0.25, 1\right)} \cdot \cos \left(K \cdot -0.5\right)\right) \cdot J\right) \cdot -2 \]

   if 1.0000000000000001e287 < (*.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 22.1%

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

   3. Taylor expanded in U around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(U \cdot \left(-2 \cdot \frac{{J}^{2} \cdot {\cos \left(\frac{1}{2} \cdot K\right)}^{2}}{{U}^{2}} - 1\right)\right)} \]
   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 50.1%

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

   6. Taylor expanded in U around inf

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

   8. Applied rewrites 49.6%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{J}{U} \cdot \frac{J}{U}\right) \cdot {\cos \left(0.5 \cdot K\right)}^{2}, -2, -1\right) \cdot U\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \leq 10^{+287}:\\ \;\;\;\;\left(\left(\sqrt{\mathsf{fma}\left(\frac{U}{J} \cdot \frac{U}{J}, 0.25, 1\right)} \cdot \cos \left(-0.5 \cdot K\right)\right) \cdot J\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\left(-U\right) \cdot -1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 90.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \sqrt{{\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2} + 1} \cdot \left(\left(J\_m \cdot -2\right) \cdot t\_0\right)\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq 10^{+287}:\\ \;\;\;\;\left(\left(\sqrt{\mathsf{fma}\left(\frac{U\_m}{J\_m} \cdot \frac{U\_m}{J\_m}, 0.25, 1\right)} \cdot \cos \left(-0.5 \cdot K\right)\right) \cdot J\_m\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\left(-U\_m\right) \cdot -1\\ \end{array} \end{array} \end{array} \]

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 1e+287)
        (*
         (*
          (*
           (sqrt (fma (* (/ U_m J_m) (/ U_m J_m)) 0.25 1.0))
           (cos (* -0.5 K)))
          J_m)
         -2.0)
        (* (- U_m) -1.0))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = 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 <= 1e+287) {
		tmp = ((sqrt(fma(((U_m / J_m) * (U_m / J_m)), 0.25, 1.0)) * cos((-0.5 * K))) * J_m) * -2.0;
	} else {
		tmp = -U_m * -1.0;
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(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 <= 1e+287)
		tmp = Float64(Float64(Float64(sqrt(fma(Float64(Float64(U_m / J_m) * Float64(U_m / J_m)), 0.25, 1.0)) * cos(Float64(-0.5 * K))) * J_m) * -2.0);
	else
		tmp = Float64(Float64(-U_m) * -1.0);
	end
	return Float64(J_s * tmp)
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[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, 1e+287], N[(N[(N[(N[Sqrt[N[(N[(N[(U$95$m / J$95$m), $MachinePrecision] * N[(U$95$m / J$95$m), $MachinePrecision]), $MachinePrecision] * 0.25 + 1.0), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * J$95$m), $MachinePrecision] * -2.0), $MachinePrecision], N[((-U$95$m) * -1.0), $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 5.2%

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

   3. Taylor expanded in 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 39.7

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

   5. Applied rewrites 39.7%

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

   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))))) < 1.0000000000000001e287

   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

   4. Applied rewrites 99.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   6. Applied rewrites 99.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

   8. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\left(\left(\sqrt{\mathsf{fma}\left({\left(\cos \left(K \cdot -0.5\right) \cdot \frac{J}{U}\right)}^{-2}, 0.25, 1\right)} \cdot \cos \left(K \cdot -0.5\right)\right) \cdot J\right) \cdot -2} \]

   9. Taylor expanded in K around 0

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

       1. unpow2 N/A

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

       2. unpow2 N/A

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

       3. times-frac N/A

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

       4. lower-*.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. lower-/.f64 87.1

          \[\leadsto \left(\left(\sqrt{\mathsf{fma}\left(\frac{U}{J} \cdot \color{blue}{\frac{U}{J}}, 0.25, 1\right)} \cdot \cos \left(K \cdot -0.5\right)\right) \cdot J\right) \cdot -2 \]

   11. Applied rewrites 87.1%

       \[\leadsto \left(\left(\sqrt{\mathsf{fma}\left(\color{blue}{\frac{U}{J} \cdot \frac{U}{J}}, 0.25, 1\right)} \cdot \cos \left(K \cdot -0.5\right)\right) \cdot J\right) \cdot -2 \]

   if 1.0000000000000001e287 < (*.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 22.1%

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

   3. Taylor expanded in U around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(U \cdot \left(-2 \cdot \frac{{J}^{2} \cdot {\cos \left(\frac{1}{2} \cdot K\right)}^{2}}{{U}^{2}} - 1\right)\right)} \]
   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 50.1%

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

   6. Taylor expanded in U around inf

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

   8. Applied rewrites 49.6%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \leq 10^{+287}:\\ \;\;\;\;\left(\left(\sqrt{\mathsf{fma}\left(\frac{U}{J} \cdot \frac{U}{J}, 0.25, 1\right)} \cdot \cos \left(-0.5 \cdot K\right)\right) \cdot J\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\left(-U\right) \cdot -1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 90.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \sqrt{{\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2} + 1} \cdot \left(\left(J\_m \cdot -2\right) \cdot t\_0\right)\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq 10^{+287}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{0.25 \cdot U\_m}{J\_m}, \frac{U\_m}{J\_m}, 1\right)} \cdot \left(\left(J\_m \cdot -2\right) \cdot \cos \left(0.5 \cdot K\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-U\_m\right) \cdot -1\\ \end{array} \end{array} \end{array} \]

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 1e+287)
        (*
         (sqrt (fma (/ (* 0.25 U_m) J_m) (/ U_m J_m) 1.0))
         (* (* J_m -2.0) (cos (* 0.5 K))))
        (* (- U_m) -1.0))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = 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 <= 1e+287) {
		tmp = sqrt(fma(((0.25 * U_m) / J_m), (U_m / J_m), 1.0)) * ((J_m * -2.0) * cos((0.5 * K)));
	} else {
		tmp = -U_m * -1.0;
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(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 <= 1e+287)
		tmp = Float64(sqrt(fma(Float64(Float64(0.25 * U_m) / J_m), Float64(U_m / J_m), 1.0)) * Float64(Float64(J_m * -2.0) * cos(Float64(0.5 * K))));
	else
		tmp = Float64(Float64(-U_m) * -1.0);
	end
	return Float64(J_s * tmp)
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[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, 1e+287], 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] * N[(N[(J$95$m * -2.0), $MachinePrecision] * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-U$95$m) * -1.0), $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 5.2%

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

   3. Taylor expanded in 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 39.7

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

   5. Applied rewrites 39.7%

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

   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))))) < 1.0000000000000001e287

   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

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      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 87.1

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

   5. Applied rewrites 87.1%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 87.1

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \color{blue}{\left(0.5 \cdot K\right)}\right) \cdot \sqrt{\mathsf{fma}\left(\frac{0.25 \cdot U}{J}, \frac{U}{J}, 1\right)} \]

   7. Applied rewrites 87.1%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\cos \left(0.5 \cdot K\right)}\right) \cdot \sqrt{\mathsf{fma}\left(\frac{0.25 \cdot U}{J}, \frac{U}{J}, 1\right)} \]

   if 1.0000000000000001e287 < (*.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 22.1%

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

   3. Taylor expanded in U around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(U \cdot \left(-2 \cdot \frac{{J}^{2} \cdot {\cos \left(\frac{1}{2} \cdot K\right)}^{2}}{{U}^{2}} - 1\right)\right)} \]
   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 50.1%

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

   6. Taylor expanded in U around inf

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

   8. Applied rewrites 49.6%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \leq 10^{+287}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{0.25 \cdot U}{J}, \frac{U}{J}, 1\right)} \cdot \left(\left(J \cdot -2\right) \cdot \cos \left(0.5 \cdot K\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-U\right) \cdot -1\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 77.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \sqrt{{\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2} + 1} \cdot \left(\left(J\_m \cdot -2\right) \cdot t\_0\right)\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-288}:\\ \;\;\;\;\left(J\_m \cdot -2\right) \cdot \sqrt{\mathsf{fma}\left(\frac{0.25 \cdot U\_m}{J\_m}, \frac{U\_m}{J\_m}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{J\_m}{U\_m} \cdot \frac{J\_m}{U\_m}, -2, -1\right) \cdot \left(-U\_m\right)\\ \end{array} \end{array} \end{array} \]

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 -5e-288)
        (* (* 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 <= -5e-288) {
		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 <= -5e-288)
		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, -5e-288], 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 5.2%

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

   3. Taylor expanded in 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 39.7

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

   5. Applied rewrites 39.7%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -5.00000000000000011e-288

   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

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

      9. associate-*r* N/A

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

      10. unpow2 N/A

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

      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 69.5

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

   5. Applied rewrites 69.5%

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

   if -5.00000000000000011e-288 < (*.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 70.3%

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

   3. Taylor expanded in U around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(U \cdot \left(-2 \cdot \frac{{J}^{2} \cdot {\cos \left(\frac{1}{2} \cdot K\right)}^{2}}{{U}^{2}} - 1\right)\right)} \]
   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 28.4%

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

   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 28.3%

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

4. Final simplification 46.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \leq -5 \cdot 10^{-288}:\\ \;\;\;\;\left(J \cdot -2\right) \cdot \sqrt{\mathsf{fma}\left(\frac{0.25 \cdot U}{J}, \frac{U}{J}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{J}{U} \cdot \frac{J}{U}, -2, -1\right) \cdot \left(-U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 49.5% accurate, 31.0× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ J\_s \cdot \begin{array}{l} \mathbf{if}\;U\_m \leq 1.05 \cdot 10^{+37}:\\ \;\;\;\;J\_m \cdot -2\\ \mathbf{else}:\\ \;\;\;\;-U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (* J_s (if (<= U_m 1.05e+37) (* J_m -2.0) (- U_m))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double tmp;
	if (U_m <= 1.05e+37) {
		tmp = J_m * -2.0;
	} else {
		tmp = -U_m;
	}
	return J_s * tmp;
}

Fortran

U_m = abs(u)
J\_m = abs(j)
J\_s = copysign(1.0d0, j)
real(8) function code(j_s, j_m, k, u_m)
    real(8), intent (in) :: j_s
    real(8), intent (in) :: j_m
    real(8), intent (in) :: k
    real(8), intent (in) :: u_m
    real(8) :: tmp
    if (u_m <= 1.05d+37) then
        tmp = j_m * (-2.0d0)
    else
        tmp = -u_m
    end if
    code = j_s * tmp
end function

Java

U_m = Math.abs(U);
J\_m = Math.abs(J);
J\_s = Math.copySign(1.0, J);
public static double code(double J_s, double J_m, double K, double U_m) {
	double tmp;
	if (U_m <= 1.05e+37) {
		tmp = J_m * -2.0;
	} else {
		tmp = -U_m;
	}
	return J_s * tmp;
}

Python

U_m = math.fabs(U)
J\_m = math.fabs(J)
J\_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	tmp = 0
	if U_m <= 1.05e+37:
		tmp = J_m * -2.0
	else:
		tmp = -U_m
	return J_s * tmp

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	tmp = 0.0
	if (U_m <= 1.05e+37)
		tmp = Float64(J_m * -2.0);
	else
		tmp = Float64(-U_m);
	end
	return Float64(J_s * tmp)
end

MATLAB

U_m = abs(U);
J\_m = abs(J);
J\_s = sign(J) * abs(1.0);
function tmp_2 = code(J_s, J_m, K, U_m)
	tmp = 0.0;
	if (U_m <= 1.05e+37)
		tmp = J_m * -2.0;
	else
		tmp = -U_m;
	end
	tmp_2 = J_s * tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := N[(J$95$s * If[LessEqual[U$95$m, 1.05e+37], N[(J$95$m * -2.0), $MachinePrecision], (-U$95$m)]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if U < 1.0500000000000001e37

   1. Initial program 82.7%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   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 58.6

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

   5. Applied rewrites 58.6%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 36.0%

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

   if 1.0500000000000001e37 < U

   1. Initial program 49.5%

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

   3. Taylor expanded in U around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 38.1

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

   5. Applied rewrites 38.1%

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

Alternative 13: 39.3% 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 75.2%

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

3. Taylor expanded in 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 25.8

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

5. Applied rewrites 25.8%

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

Reproduce

herbie shell --seed 2024243 
(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)))))