Maksimov and Kolovsky, Equation (3)

Percentage Accurate: 72.4% → 92.9%

Time: 11.2s

Alternatives: 16

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: 72.4% 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: 92.9% accurate, 0.3× speedup

Math

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

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))
          (* t_0 (* J_m -2.0)))))
   (*
    J_s
    (if (<= t_1 (- INFINITY))
      (- U_m)
      (if (<= t_1 INFINITY)
        t_1
        (*
         (fma
          (* (/ (* J_m J_m) (* U_m U_m)) (pow (cos (* 0.5 K)) 2.0))
          -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)) * (t_0 * (J_m * -2.0));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = fma((((J_m * J_m) / (U_m * U_m)) * pow(cos((0.5 * K)), 2.0)), -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(t_0 * Float64(J_m * -2.0)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(fma(Float64(Float64(Float64(J_m * J_m) / Float64(U_m * U_m)) * (cos(Float64(0.5 * K)) ^ 2.0)), -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[(t$95$0 * N[(J$95$m * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$1, (-Infinity)], (-U$95$m), If[LessEqual[t$95$1, Infinity], t$95$1, N[(N[(N[(N[(N[(J$95$m * J$95$m), $MachinePrecision] / N[(U$95$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]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 5.4%

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

   3. Taylor expanded in U around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 49.0

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

   5. Applied rewrites 49.0%

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

   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))))) < +inf.0

   1. Initial program 83.6%

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

   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. Initial program 72.9%

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

   3. Taylor expanded in U around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(U \cdot \left(-2 \cdot \frac{{J}^{2} \cdot {\cos \left(\frac{1}{2} \cdot K\right)}^{2}}{{U}^{2}} - 1\right)\right)} \]
   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 23.3%

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

4. Final simplification 78.8%

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

Alternative 2: 80.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := t\_0 \cdot \left(J\_m \cdot -2\right)\\ t_2 := \sqrt{{\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2} + 1} \cdot t\_1\\ t_3 := \frac{0.5 \cdot U\_m}{1 \cdot J\_m}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+306}:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{-8}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{U\_m}{J\_m \cdot J\_m} \cdot U\_m, 0.25, 1\right)} \cdot t\_1\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{-272}:\\ \;\;\;\;\left(J\_m \cdot -2\right) \cdot \sqrt{\mathsf{fma}\left(t\_3, t\_3, 1\right)}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\cos \left(0.5 \cdot K\right) \cdot \left(J\_m \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{U\_m}{J\_m} \cdot J\_m\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1 (* t_0 (* J_m -2.0)))
        (t_2 (* (sqrt (+ (pow (/ U_m (* (* 2.0 J_m) t_0)) 2.0) 1.0)) t_1))
        (t_3 (/ (* 0.5 U_m) (* 1.0 J_m))))
   (*
    J_s
    (if (<= t_2 -2e+306)
      (- U_m)
      (if (<= t_2 -1e-8)
        (* (sqrt (fma (* (/ U_m (* J_m J_m)) U_m) 0.25 1.0)) t_1)
        (if (<= t_2 -2e-272)
          (* (* J_m -2.0) (sqrt (fma t_3 t_3 1.0)))
          (if (<= t_2 INFINITY)
            (* (cos (* 0.5 K)) (* J_m -2.0))
            (* (/ U_m J_m) J_m))))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = t_0 * (J_m * -2.0);
	double t_2 = sqrt((pow((U_m / ((2.0 * J_m) * t_0)), 2.0) + 1.0)) * t_1;
	double t_3 = (0.5 * U_m) / (1.0 * J_m);
	double tmp;
	if (t_2 <= -2e+306) {
		tmp = -U_m;
	} else if (t_2 <= -1e-8) {
		tmp = sqrt(fma(((U_m / (J_m * J_m)) * U_m), 0.25, 1.0)) * t_1;
	} else if (t_2 <= -2e-272) {
		tmp = (J_m * -2.0) * sqrt(fma(t_3, t_3, 1.0));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = cos((0.5 * K)) * (J_m * -2.0);
	} else {
		tmp = (U_m / J_m) * J_m;
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(t_0 * Float64(J_m * -2.0))
	t_2 = Float64(sqrt(Float64((Float64(U_m / Float64(Float64(2.0 * J_m) * t_0)) ^ 2.0) + 1.0)) * t_1)
	t_3 = Float64(Float64(0.5 * U_m) / Float64(1.0 * J_m))
	tmp = 0.0
	if (t_2 <= -2e+306)
		tmp = Float64(-U_m);
	elseif (t_2 <= -1e-8)
		tmp = Float64(sqrt(fma(Float64(Float64(U_m / Float64(J_m * J_m)) * U_m), 0.25, 1.0)) * t_1);
	elseif (t_2 <= -2e-272)
		tmp = Float64(Float64(J_m * -2.0) * sqrt(fma(t_3, t_3, 1.0)));
	elseif (t_2 <= Inf)
		tmp = Float64(cos(Float64(0.5 * K)) * Float64(J_m * -2.0));
	else
		tmp = Float64(Float64(U_m / J_m) * J_m);
	end
	return Float64(J_s * tmp)
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(J$95$m * -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$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(0.5 * U$95$m), $MachinePrecision] / N[(1.0 * J$95$m), $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$2, -2e+306], (-U$95$m), If[LessEqual[t$95$2, -1e-8], N[(N[Sqrt[N[(N[(N[(U$95$m / N[(J$95$m * J$95$m), $MachinePrecision]), $MachinePrecision] * U$95$m), $MachinePrecision] * 0.25 + 1.0), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$2, -2e-272], N[(N[(J$95$m * -2.0), $MachinePrecision] * N[Sqrt[N[(t$95$3 * t$95$3 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] * N[(J$95$m * -2.0), $MachinePrecision]), $MachinePrecision], N[(N[(U$95$m / J$95$m), $MachinePrecision] * J$95$m), $MachinePrecision]]]]]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -2.00000000000000003e306

   1. Initial program 10.5%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   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 46.3

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

   5. Applied rewrites 46.3%

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

   if -2.00000000000000003e306 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -1e-8

   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. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 83.5

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

   5. Applied rewrites 83.5%

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

   if -1e-8 < (*.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))))) < -1.99999999999999986e-272

   1. Initial program 99.6%

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

   3. Taylor expanded in K around 0

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

   5. Applied rewrites 81.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-fma.f64 81.4

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

   7. Applied rewrites 81.4%

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

   8. Taylor expanded in K around 0

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

      1. lower-*.f64 65.3

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

   10. Applied rewrites 65.3%

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

   if -1.99999999999999986e-272 < (*.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 70.6%

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

   3. Taylor expanded in 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 49.3

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

   5. Applied rewrites 49.3%

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

   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. Initial program 72.9%

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

   3. Taylor expanded in K around 0

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

   5. Applied rewrites 62.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. div-inv N/A

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

      10. metadata-eval N/A

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

      11. lift-*.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

   7. Applied rewrites 47.9%

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

   8. Taylor expanded in U around -inf

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

      1. lower-/.f64 23.3

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

   10. Applied rewrites 23.3%

       \[\leadsto \color{blue}{\frac{U}{J}} \cdot J \]
3. Recombined 5 regimes into one program.

4. Final simplification 59.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq -2 \cdot 10^{+306}:\\ \;\;\;\;-U\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq -1 \cdot 10^{-8}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{U}{J \cdot J} \cdot U, 0.25, 1\right)} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right)\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq -2 \cdot 10^{-272}:\\ \;\;\;\;\left(J \cdot -2\right) \cdot \sqrt{\mathsf{fma}\left(\frac{0.5 \cdot U}{1 \cdot J}, \frac{0.5 \cdot U}{1 \cdot J}, 1\right)}\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq \infty:\\ \;\;\;\;\cos \left(0.5 \cdot K\right) \cdot \left(J \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{U}{J} \cdot J\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := t\_0 \cdot \left(J\_m \cdot -2\right)\\ t_2 := \sqrt{{\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2} + 1} \cdot t\_1\\ t_3 := \cos \left(\left(-0.5 \cdot K\right) \cdot 2\right) \cdot 0.5 + 0.5\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{+247}:\\ \;\;\;\;\sqrt{\frac{U\_m}{\left(\frac{2 \cdot J\_m}{U\_m} \cdot t\_3\right) \cdot \left(2 \cdot J\_m\right)} + 1} \cdot t\_1\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\sqrt{\frac{\frac{0.5}{J\_m} \cdot \left(\frac{U\_m}{2 \cdot J\_m} \cdot U\_m\right)}{t\_3} + 1} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{J\_m \cdot J\_m}{U\_m \cdot U\_m} \cdot {\cos \left(0.5 \cdot K\right)}^{2}, -2, -1\right) \cdot \left(-U\_m\right)\\ \end{array} \end{array} \end{array} \]

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 (* t_0 (* J_m -2.0)))
        (t_2 (* (sqrt (+ (pow (/ U_m (* (* 2.0 J_m) t_0)) 2.0) 1.0)) t_1))
        (t_3 (+ (* (cos (* (* -0.5 K) 2.0)) 0.5) 0.5)))
   (*
    J_s
    (if (<= t_2 (- INFINITY))
      (- U_m)
      (if (<= t_2 -1e+247)
        (*
         (sqrt (+ (/ U_m (* (* (/ (* 2.0 J_m) U_m) t_3) (* 2.0 J_m))) 1.0))
         t_1)
        (if (<= t_2 INFINITY)
          (*
           (sqrt (+ (/ (* (/ 0.5 J_m) (* (/ U_m (* 2.0 J_m)) U_m)) t_3) 1.0))
           t_1)
          (*
           (fma
            (* (/ (* J_m J_m) (* U_m U_m)) (pow (cos (* 0.5 K)) 2.0))
            -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 = t_0 * (J_m * -2.0);
	double t_2 = sqrt((pow((U_m / ((2.0 * J_m) * t_0)), 2.0) + 1.0)) * t_1;
	double t_3 = (cos(((-0.5 * K) * 2.0)) * 0.5) + 0.5;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_2 <= -1e+247) {
		tmp = sqrt(((U_m / ((((2.0 * J_m) / U_m) * t_3) * (2.0 * J_m))) + 1.0)) * t_1;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt(((((0.5 / J_m) * ((U_m / (2.0 * J_m)) * U_m)) / t_3) + 1.0)) * t_1;
	} else {
		tmp = fma((((J_m * J_m) / (U_m * U_m)) * pow(cos((0.5 * K)), 2.0)), -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(t_0 * Float64(J_m * -2.0))
	t_2 = Float64(sqrt(Float64((Float64(U_m / Float64(Float64(2.0 * J_m) * t_0)) ^ 2.0) + 1.0)) * t_1)
	t_3 = Float64(Float64(cos(Float64(Float64(-0.5 * K) * 2.0)) * 0.5) + 0.5)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_2 <= -1e+247)
		tmp = Float64(sqrt(Float64(Float64(U_m / Float64(Float64(Float64(Float64(2.0 * J_m) / U_m) * t_3) * Float64(2.0 * J_m))) + 1.0)) * t_1);
	elseif (t_2 <= Inf)
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(0.5 / J_m) * Float64(Float64(U_m / Float64(2.0 * J_m)) * U_m)) / t_3) + 1.0)) * t_1);
	else
		tmp = Float64(fma(Float64(Float64(Float64(J_m * J_m) / Float64(U_m * U_m)) * (cos(Float64(0.5 * K)) ^ 2.0)), -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[(t$95$0 * N[(J$95$m * -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$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Cos[N[(N[(-0.5 * K), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision] + 0.5), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$2, (-Infinity)], (-U$95$m), If[LessEqual[t$95$2, -1e+247], N[(N[Sqrt[N[(N[(U$95$m / N[(N[(N[(N[(2.0 * J$95$m), $MachinePrecision] / U$95$m), $MachinePrecision] * t$95$3), $MachinePrecision] * N[(2.0 * J$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[Sqrt[N[(N[(N[(N[(0.5 / J$95$m), $MachinePrecision] * N[(N[(U$95$m / N[(2.0 * J$95$m), $MachinePrecision]), $MachinePrecision] * U$95$m), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[(N[(N[(N[(J$95$m * J$95$m), $MachinePrecision] / N[(U$95$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]]]]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 5.4%

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

   3. Taylor expanded in U around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 49.0

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

   5. Applied rewrites 49.0%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -9.99999999999999952e246

   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. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

   4. Applied rewrites 99.5%

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

   if -9.99999999999999952e246 < (*.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 81.8%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. frac-times N/A

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

      10. lower-/.f64 N/A

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

   4. Applied rewrites 80.3%

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

   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. Initial program 72.9%

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

   3. Taylor expanded in U around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(U \cdot \left(-2 \cdot \frac{{J}^{2} \cdot {\cos \left(\frac{1}{2} \cdot K\right)}^{2}}{{U}^{2}} - 1\right)\right)} \]
   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 23.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\cos \left(K \cdot 0.5\right)}^{2} \cdot \frac{J \cdot J}{U \cdot U}, -2, -1\right) \cdot \left(-U\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq -1 \cdot 10^{+247}:\\ \;\;\;\;\sqrt{\frac{U}{\left(\frac{2 \cdot J}{U} \cdot \left(\cos \left(\left(-0.5 \cdot K\right) \cdot 2\right) \cdot 0.5 + 0.5\right)\right) \cdot \left(2 \cdot J\right)} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right)\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq \infty:\\ \;\;\;\;\sqrt{\frac{\frac{0.5}{J} \cdot \left(\frac{U}{2 \cdot J} \cdot U\right)}{\cos \left(\left(-0.5 \cdot K\right) \cdot 2\right) \cdot 0.5 + 0.5} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{J \cdot J}{U \cdot U} \cdot {\cos \left(0.5 \cdot K\right)}^{2}, -2, -1\right) \cdot \left(-U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := t\_0 \cdot \left(J\_m \cdot -2\right)\\ t_2 := \sqrt{{\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2} + 1} \cdot t\_1\\ t_3 := \cos \left(0.5 \cdot K\right)\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{+63}:\\ \;\;\;\;\sqrt{\frac{U\_m}{\left(\frac{2 \cdot J\_m}{U\_m} \cdot \left(\cos \left(\left(-0.5 \cdot K\right) \cdot 2\right) \cdot 0.5 + 0.5\right)\right) \cdot \left(2 \cdot J\_m\right)} + 1} \cdot t\_1\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\left(\left(t\_3 \cdot -2\right) \cdot \sqrt{\mathsf{fma}\left(\frac{0.5}{\mathsf{fma}\left(\cos K, 0.5, 0.5\right) \cdot J\_m}, \frac{U\_m}{2 \cdot J\_m} \cdot U\_m, 1\right)}\right) \cdot J\_m\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{J\_m \cdot J\_m}{U\_m \cdot U\_m} \cdot {t\_3}^{2}, -2, -1\right) \cdot \left(-U\_m\right)\\ \end{array} \end{array} \end{array} \]

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 (* t_0 (* J_m -2.0)))
        (t_2 (* (sqrt (+ (pow (/ U_m (* (* 2.0 J_m) t_0)) 2.0) 1.0)) t_1))
        (t_3 (cos (* 0.5 K))))
   (*
    J_s
    (if (<= t_2 (- INFINITY))
      (- U_m)
      (if (<= t_2 -2e+63)
        (*
         (sqrt
          (+
           (/
            U_m
            (*
             (* (/ (* 2.0 J_m) U_m) (+ (* (cos (* (* -0.5 K) 2.0)) 0.5) 0.5))
             (* 2.0 J_m)))
           1.0))
         t_1)
        (if (<= t_2 INFINITY)
          (*
           (*
            (* t_3 -2.0)
            (sqrt
             (fma
              (/ 0.5 (* (fma (cos K) 0.5 0.5) J_m))
              (* (/ U_m (* 2.0 J_m)) U_m)
              1.0)))
           J_m)
          (*
           (fma (* (/ (* J_m J_m) (* U_m U_m)) (pow t_3 2.0)) -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 = t_0 * (J_m * -2.0);
	double t_2 = sqrt((pow((U_m / ((2.0 * J_m) * t_0)), 2.0) + 1.0)) * t_1;
	double t_3 = cos((0.5 * K));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_2 <= -2e+63) {
		tmp = sqrt(((U_m / ((((2.0 * J_m) / U_m) * ((cos(((-0.5 * K) * 2.0)) * 0.5) + 0.5)) * (2.0 * J_m))) + 1.0)) * t_1;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = ((t_3 * -2.0) * sqrt(fma((0.5 / (fma(cos(K), 0.5, 0.5) * J_m)), ((U_m / (2.0 * J_m)) * U_m), 1.0))) * J_m;
	} else {
		tmp = fma((((J_m * J_m) / (U_m * U_m)) * pow(t_3, 2.0)), -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(t_0 * Float64(J_m * -2.0))
	t_2 = Float64(sqrt(Float64((Float64(U_m / Float64(Float64(2.0 * J_m) * t_0)) ^ 2.0) + 1.0)) * t_1)
	t_3 = cos(Float64(0.5 * K))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_2 <= -2e+63)
		tmp = Float64(sqrt(Float64(Float64(U_m / Float64(Float64(Float64(Float64(2.0 * J_m) / U_m) * Float64(Float64(cos(Float64(Float64(-0.5 * K) * 2.0)) * 0.5) + 0.5)) * Float64(2.0 * J_m))) + 1.0)) * t_1);
	elseif (t_2 <= Inf)
		tmp = Float64(Float64(Float64(t_3 * -2.0) * sqrt(fma(Float64(0.5 / Float64(fma(cos(K), 0.5, 0.5) * J_m)), Float64(Float64(U_m / Float64(2.0 * J_m)) * U_m), 1.0))) * J_m);
	else
		tmp = Float64(fma(Float64(Float64(Float64(J_m * J_m) / Float64(U_m * U_m)) * (t_3 ^ 2.0)), -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[(t$95$0 * N[(J$95$m * -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$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$2, (-Infinity)], (-U$95$m), If[LessEqual[t$95$2, -2e+63], N[(N[Sqrt[N[(N[(U$95$m / N[(N[(N[(N[(2.0 * J$95$m), $MachinePrecision] / U$95$m), $MachinePrecision] * N[(N[(N[Cos[N[(N[(-0.5 * K), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] * N[(2.0 * J$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[(N[(t$95$3 * -2.0), $MachinePrecision] * N[Sqrt[N[(N[(0.5 / N[(N[(N[Cos[K], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision] * J$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(U$95$m / N[(2.0 * J$95$m), $MachinePrecision]), $MachinePrecision] * U$95$m), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * J$95$m), $MachinePrecision], N[(N[(N[(N[(N[(J$95$m * J$95$m), $MachinePrecision] / N[(U$95$m * U$95$m), $MachinePrecision]), $MachinePrecision] * N[Power[t$95$3, 2.0], $MachinePrecision]), $MachinePrecision] * -2.0 + -1.0), $MachinePrecision] * (-U$95$m)), $MachinePrecision]]]]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 5.4%

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

   3. Taylor expanded in U around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 49.0

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

   5. Applied rewrites 49.0%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -2.00000000000000012e63

   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-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

   4. Applied rewrites 99.6%

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

   if -2.00000000000000012e63 < (*.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 77.8%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. frac-times N/A

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

      10. lower-/.f64 N/A

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

   4. Applied rewrites 75.9%

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

   6. Applied rewrites 75.9%

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

   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. Initial program 72.9%

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

   3. Taylor expanded in U around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(U \cdot \left(-2 \cdot \frac{{J}^{2} \cdot {\cos \left(\frac{1}{2} \cdot K\right)}^{2}}{{U}^{2}} - 1\right)\right)} \]
   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 23.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\cos \left(K \cdot 0.5\right)}^{2} \cdot \frac{J \cdot J}{U \cdot U}, -2, -1\right) \cdot \left(-U\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq -2 \cdot 10^{+63}:\\ \;\;\;\;\sqrt{\frac{U}{\left(\frac{2 \cdot J}{U} \cdot \left(\cos \left(\left(-0.5 \cdot K\right) \cdot 2\right) \cdot 0.5 + 0.5\right)\right) \cdot \left(2 \cdot J\right)} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right)\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq \infty:\\ \;\;\;\;\left(\left(\cos \left(0.5 \cdot K\right) \cdot -2\right) \cdot \sqrt{\mathsf{fma}\left(\frac{0.5}{\mathsf{fma}\left(\cos K, 0.5, 0.5\right) \cdot J}, \frac{U}{2 \cdot J} \cdot U, 1\right)}\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{J \cdot J}{U \cdot U} \cdot {\cos \left(0.5 \cdot K\right)}^{2}, -2, -1\right) \cdot \left(-U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \sqrt{{\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2} + 1} \cdot \left(t\_0 \cdot \left(J\_m \cdot -2\right)\right)\\ t_2 := \frac{0.5 \cdot U\_m}{1 \cdot J\_m}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-272}:\\ \;\;\;\;\left(J\_m \cdot -2\right) \cdot \sqrt{\mathsf{fma}\left(t\_2, t\_2, 1\right)}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\cos \left(0.5 \cdot K\right) \cdot \left(J\_m \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{U\_m}{J\_m} \cdot J\_m\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (sqrt (+ (pow (/ U_m (* (* 2.0 J_m) t_0)) 2.0) 1.0))
          (* t_0 (* J_m -2.0))))
        (t_2 (/ (* 0.5 U_m) (* 1.0 J_m))))
   (*
    J_s
    (if (<= t_1 (- INFINITY))
      (- U_m)
      (if (<= t_1 -2e-272)
        (* (* J_m -2.0) (sqrt (fma t_2 t_2 1.0)))
        (if (<= t_1 INFINITY)
          (* (cos (* 0.5 K)) (* J_m -2.0))
          (* (/ U_m J_m) J_m)))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = sqrt((pow((U_m / ((2.0 * J_m) * t_0)), 2.0) + 1.0)) * (t_0 * (J_m * -2.0));
	double t_2 = (0.5 * U_m) / (1.0 * J_m);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_1 <= -2e-272) {
		tmp = (J_m * -2.0) * sqrt(fma(t_2, t_2, 1.0));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = cos((0.5 * K)) * (J_m * -2.0);
	} else {
		tmp = (U_m / J_m) * J_m;
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(sqrt(Float64((Float64(U_m / Float64(Float64(2.0 * J_m) * t_0)) ^ 2.0) + 1.0)) * Float64(t_0 * Float64(J_m * -2.0)))
	t_2 = Float64(Float64(0.5 * U_m) / Float64(1.0 * J_m))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_1 <= -2e-272)
		tmp = Float64(Float64(J_m * -2.0) * sqrt(fma(t_2, t_2, 1.0)));
	elseif (t_1 <= Inf)
		tmp = Float64(cos(Float64(0.5 * K)) * Float64(J_m * -2.0));
	else
		tmp = Float64(Float64(U_m / J_m) * J_m);
	end
	return Float64(J_s * tmp)
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[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[(t$95$0 * N[(J$95$m * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(0.5 * U$95$m), $MachinePrecision] / N[(1.0 * J$95$m), $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$1, (-Infinity)], (-U$95$m), If[LessEqual[t$95$1, -2e-272], N[(N[(J$95$m * -2.0), $MachinePrecision] * N[Sqrt[N[(t$95$2 * t$95$2 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] * N[(J$95$m * -2.0), $MachinePrecision]), $MachinePrecision], N[(N[(U$95$m / J$95$m), $MachinePrecision] * J$95$m), $MachinePrecision]]]]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 5.4%

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

   3. Taylor expanded in U around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 49.0

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

   5. Applied rewrites 49.0%

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

   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))))) < -1.99999999999999986e-272

   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

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

   5. Applied rewrites 86.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-fma.f64 86.3

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

   7. Applied rewrites 86.3%

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

   8. Taylor expanded in K around 0

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

      1. lower-*.f64 62.9

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

   10. Applied rewrites 62.9%

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

   if -1.99999999999999986e-272 < (*.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 70.6%

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

   3. Taylor expanded in 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 49.3

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

   5. Applied rewrites 49.3%

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

   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. Initial program 72.9%

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

   3. Taylor expanded in K around 0

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

   5. Applied rewrites 62.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. div-inv N/A

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

      10. metadata-eval N/A

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

      11. lift-*.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

   7. Applied rewrites 47.9%

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

   8. Taylor expanded in U around -inf

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

      1. lower-/.f64 23.3

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

   10. Applied rewrites 23.3%

       \[\leadsto \color{blue}{\frac{U}{J}} \cdot J \]
3. Recombined 4 regimes into one program.

4. Final simplification 54.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq -2 \cdot 10^{-272}:\\ \;\;\;\;\left(J \cdot -2\right) \cdot \sqrt{\mathsf{fma}\left(\frac{0.5 \cdot U}{1 \cdot J}, \frac{0.5 \cdot U}{1 \cdot J}, 1\right)}\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq \infty:\\ \;\;\;\;\cos \left(0.5 \cdot K\right) \cdot \left(J \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{U}{J} \cdot J\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \sqrt{{\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2} + 1} \cdot \left(t\_0 \cdot \left(J\_m \cdot -2\right)\right)\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+306}:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-74}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{U\_m}{J\_m \cdot J\_m} \cdot U\_m, 0.25, 1\right)} \cdot \left(J\_m \cdot -2\right)\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-272}:\\ \;\;\;\;-U\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{U\_m}{J\_m} \cdot J\_m\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (sqrt (+ (pow (/ U_m (* (* 2.0 J_m) t_0)) 2.0) 1.0))
          (* t_0 (* J_m -2.0)))))
   (*
    J_s
    (if (<= t_1 -2e+306)
      (- U_m)
      (if (<= t_1 -5e-74)
        (* (sqrt (fma (* (/ U_m (* J_m J_m)) U_m) 0.25 1.0)) (* J_m -2.0))
        (if (<= t_1 -2e-272) (- U_m) (* (/ U_m J_m) J_m)))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = sqrt((pow((U_m / ((2.0 * J_m) * t_0)), 2.0) + 1.0)) * (t_0 * (J_m * -2.0));
	double tmp;
	if (t_1 <= -2e+306) {
		tmp = -U_m;
	} else if (t_1 <= -5e-74) {
		tmp = sqrt(fma(((U_m / (J_m * J_m)) * U_m), 0.25, 1.0)) * (J_m * -2.0);
	} else if (t_1 <= -2e-272) {
		tmp = -U_m;
	} else {
		tmp = (U_m / J_m) * J_m;
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(sqrt(Float64((Float64(U_m / Float64(Float64(2.0 * J_m) * t_0)) ^ 2.0) + 1.0)) * Float64(t_0 * Float64(J_m * -2.0)))
	tmp = 0.0
	if (t_1 <= -2e+306)
		tmp = Float64(-U_m);
	elseif (t_1 <= -5e-74)
		tmp = Float64(sqrt(fma(Float64(Float64(U_m / Float64(J_m * J_m)) * U_m), 0.25, 1.0)) * Float64(J_m * -2.0));
	elseif (t_1 <= -2e-272)
		tmp = Float64(-U_m);
	else
		tmp = Float64(Float64(U_m / J_m) * J_m);
	end
	return Float64(J_s * tmp)
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[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[(t$95$0 * N[(J$95$m * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$1, -2e+306], (-U$95$m), If[LessEqual[t$95$1, -5e-74], N[(N[Sqrt[N[(N[(N[(U$95$m / N[(J$95$m * J$95$m), $MachinePrecision]), $MachinePrecision] * U$95$m), $MachinePrecision] * 0.25 + 1.0), $MachinePrecision]], $MachinePrecision] * N[(J$95$m * -2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -2e-272], (-U$95$m), N[(N[(U$95$m / J$95$m), $MachinePrecision] * J$95$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))))) < -2.00000000000000003e306 or -4.99999999999999998e-74 < (*.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))))) < -1.99999999999999986e-272

   1. Initial program 41.8%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   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 35.6

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

   5. Applied rewrites 35.6%

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

   if -2.00000000000000003e306 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -4.99999999999999998e-74

   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. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 59.0

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

   5. Applied rewrites 59.0%

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

   if -1.99999999999999986e-272 < (*.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 70.6%

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

   3. Taylor expanded in K around 0

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

   5. Applied rewrites 59.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. div-inv N/A

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

      10. metadata-eval N/A

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

      11. lift-*.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

   7. Applied rewrites 49.0%

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

   8. Taylor expanded in U around -inf

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

      1. lower-/.f64 30.1

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

   10. Applied rewrites 30.1%

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

4. Final simplification 39.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq -2 \cdot 10^{+306}:\\ \;\;\;\;-U\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq -5 \cdot 10^{-74}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{U}{J \cdot J} \cdot U, 0.25, 1\right)} \cdot \left(J \cdot -2\right)\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq -2 \cdot 10^{-272}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;\frac{U}{J} \cdot J\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 58.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \sqrt{{\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2} + 1} \cdot \left(t\_0 \cdot \left(J\_m \cdot -2\right)\right)\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+306}:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-74}:\\ \;\;\;\;\mathsf{fma}\left(\frac{U\_m \cdot U\_m}{J\_m}, -0.25, J\_m \cdot -2\right)\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-272}:\\ \;\;\;\;-U\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{U\_m}{J\_m} \cdot J\_m\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (sqrt (+ (pow (/ U_m (* (* 2.0 J_m) t_0)) 2.0) 1.0))
          (* t_0 (* J_m -2.0)))))
   (*
    J_s
    (if (<= t_1 -2e+306)
      (- U_m)
      (if (<= t_1 -5e-74)
        (fma (/ (* U_m U_m) J_m) -0.25 (* J_m -2.0))
        (if (<= t_1 -2e-272) (- U_m) (* (/ U_m J_m) J_m)))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = sqrt((pow((U_m / ((2.0 * J_m) * t_0)), 2.0) + 1.0)) * (t_0 * (J_m * -2.0));
	double tmp;
	if (t_1 <= -2e+306) {
		tmp = -U_m;
	} else if (t_1 <= -5e-74) {
		tmp = fma(((U_m * U_m) / J_m), -0.25, (J_m * -2.0));
	} else if (t_1 <= -2e-272) {
		tmp = -U_m;
	} else {
		tmp = (U_m / J_m) * J_m;
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(sqrt(Float64((Float64(U_m / Float64(Float64(2.0 * J_m) * t_0)) ^ 2.0) + 1.0)) * Float64(t_0 * Float64(J_m * -2.0)))
	tmp = 0.0
	if (t_1 <= -2e+306)
		tmp = Float64(-U_m);
	elseif (t_1 <= -5e-74)
		tmp = fma(Float64(Float64(U_m * U_m) / J_m), -0.25, Float64(J_m * -2.0));
	elseif (t_1 <= -2e-272)
		tmp = Float64(-U_m);
	else
		tmp = Float64(Float64(U_m / J_m) * J_m);
	end
	return Float64(J_s * tmp)
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[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[(t$95$0 * N[(J$95$m * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$1, -2e+306], (-U$95$m), If[LessEqual[t$95$1, -5e-74], N[(N[(N[(U$95$m * U$95$m), $MachinePrecision] / J$95$m), $MachinePrecision] * -0.25 + N[(J$95$m * -2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -2e-272], (-U$95$m), N[(N[(U$95$m / J$95$m), $MachinePrecision] * J$95$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))))) < -2.00000000000000003e306 or -4.99999999999999998e-74 < (*.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))))) < -1.99999999999999986e-272

   1. Initial program 41.8%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   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 35.6

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

   5. Applied rewrites 35.6%

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

   if -2.00000000000000003e306 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -4.99999999999999998e-74

   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

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-cos.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 66.9

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

   5. Applied rewrites 66.9%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 43.5%

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

   if -1.99999999999999986e-272 < (*.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 70.6%

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

   3. Taylor expanded in K around 0

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

   5. Applied rewrites 59.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. div-inv N/A

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

      10. metadata-eval N/A

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

      11. lift-*.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

   7. Applied rewrites 49.0%

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

   8. Taylor expanded in U around -inf

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

      1. lower-/.f64 30.1

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

   10. Applied rewrites 30.1%

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

4. Final simplification 35.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq -2 \cdot 10^{+306}:\\ \;\;\;\;-U\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq -5 \cdot 10^{-74}:\\ \;\;\;\;\mathsf{fma}\left(\frac{U \cdot U}{J}, -0.25, J \cdot -2\right)\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq -2 \cdot 10^{-272}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;\frac{U}{J} \cdot J\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 57.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \sqrt{{\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2} + 1} \cdot \left(t\_0 \cdot \left(J\_m \cdot -2\right)\right)\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+298}:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-69}:\\ \;\;\;\;\mathsf{fma}\left(\left(K \cdot K\right) \cdot J\_m, -0.125, J\_m\right) \cdot -2\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-272}:\\ \;\;\;\;-U\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{U\_m}{J\_m} \cdot J\_m\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (sqrt (+ (pow (/ U_m (* (* 2.0 J_m) t_0)) 2.0) 1.0))
          (* t_0 (* J_m -2.0)))))
   (*
    J_s
    (if (<= t_1 -4e+298)
      (- U_m)
      (if (<= t_1 -2e-69)
        (* (fma (* (* K K) J_m) -0.125 J_m) -2.0)
        (if (<= t_1 -2e-272) (- U_m) (* (/ U_m J_m) J_m)))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = sqrt((pow((U_m / ((2.0 * J_m) * t_0)), 2.0) + 1.0)) * (t_0 * (J_m * -2.0));
	double tmp;
	if (t_1 <= -4e+298) {
		tmp = -U_m;
	} else if (t_1 <= -2e-69) {
		tmp = fma(((K * K) * J_m), -0.125, J_m) * -2.0;
	} else if (t_1 <= -2e-272) {
		tmp = -U_m;
	} else {
		tmp = (U_m / J_m) * J_m;
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(sqrt(Float64((Float64(U_m / Float64(Float64(2.0 * J_m) * t_0)) ^ 2.0) + 1.0)) * Float64(t_0 * Float64(J_m * -2.0)))
	tmp = 0.0
	if (t_1 <= -4e+298)
		tmp = Float64(-U_m);
	elseif (t_1 <= -2e-69)
		tmp = Float64(fma(Float64(Float64(K * K) * J_m), -0.125, J_m) * -2.0);
	elseif (t_1 <= -2e-272)
		tmp = Float64(-U_m);
	else
		tmp = Float64(Float64(U_m / J_m) * J_m);
	end
	return Float64(J_s * tmp)
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[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[(t$95$0 * N[(J$95$m * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$1, -4e+298], (-U$95$m), If[LessEqual[t$95$1, -2e-69], N[(N[(N[(N[(K * K), $MachinePrecision] * J$95$m), $MachinePrecision] * -0.125 + J$95$m), $MachinePrecision] * -2.0), $MachinePrecision], If[LessEqual[t$95$1, -2e-272], (-U$95$m), N[(N[(U$95$m / J$95$m), $MachinePrecision] * J$95$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))))) < -3.9999999999999998e298 or -1.9999999999999999e-69 < (*.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))))) < -1.99999999999999986e-272

   1. Initial program 44.7%

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

   3. Taylor expanded in U around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 35.7

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

   5. Applied rewrites 35.7%

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

   if -3.9999999999999998e298 < (*.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))))) < -1.9999999999999999e-69

   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

      \[\leadsto \color{blue}{-2 \cdot \left(J \cdot \sqrt{1 + \frac{1}{4} \cdot \frac{{U}^{2}}{{J}^{2}}}\right) + -2 \cdot \left({K}^{2} \cdot \left(\frac{-1}{8} \cdot \left(J \cdot \sqrt{1 + \frac{1}{4} \cdot \frac{{U}^{2}}{{J}^{2}}}\right) + \frac{1}{32} \cdot \left(\frac{{U}^{2}}{J} \cdot \sqrt{\frac{1}{1 + \frac{1}{4} \cdot \frac{{U}^{2}}{{J}^{2}}}}\right)\right)\right)} \]

   5. Applied rewrites 47.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.125, K \cdot K, 1\right), \sqrt{\mathsf{fma}\left(U \cdot \frac{U}{J \cdot J}, 0.25, 1\right)} \cdot J, \left(\left(K \cdot K\right) \cdot \left(\frac{U \cdot U}{J} \cdot 0.03125\right)\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(U \cdot \frac{U}{J \cdot J}, 0.25, 1\right)}}\right) \cdot -2} \]

   6. Taylor expanded in U around 0

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

   8. Applied rewrites 47.1%

      \[\leadsto \left(\mathsf{fma}\left(K \cdot K, -0.125, 1\right) \cdot J\right) \cdot -2 \]

   9. Taylor expanded in K around 0

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

   11. Applied rewrites 47.1%

       \[\leadsto \mathsf{fma}\left(\left(K \cdot K\right) \cdot J, -0.125, J\right) \cdot -2 \]

   if -1.99999999999999986e-272 < (*.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 70.6%

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

   3. Taylor expanded in K around 0

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

   5. Applied rewrites 59.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. div-inv N/A

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

      10. metadata-eval N/A

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

      11. lift-*.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

   7. Applied rewrites 49.0%

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

   8. Taylor expanded in U around -inf

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

      1. lower-/.f64 30.1

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

   10. Applied rewrites 30.1%

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

4. Final simplification 36.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq -4 \cdot 10^{+298}:\\ \;\;\;\;-U\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq -2 \cdot 10^{-69}:\\ \;\;\;\;\mathsf{fma}\left(\left(K \cdot K\right) \cdot J, -0.125, J\right) \cdot -2\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq -2 \cdot 10^{-272}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;\frac{U}{J} \cdot J\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 89.5% accurate, 0.4× speedup

Math

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

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 (* 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))
          (* t_1 (* J_m -2.0)))))
   (*
    J_s
    (if (<= t_2 -2e+306)
      (- U_m)
      (if (<= t_2 INFINITY)
        (*
         (*
          (* t_0 -2.0)
          (sqrt
           (fma
            (/ 0.5 (* (fma (cos K) 0.5 0.5) J_m))
            (* (/ U_m (* 2.0 J_m)) U_m)
            1.0)))
         J_m)
        (*
         (fma (* (/ (* J_m J_m) (* U_m U_m)) (pow t_0 2.0)) -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((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)) * (t_1 * (J_m * -2.0));
	double tmp;
	if (t_2 <= -2e+306) {
		tmp = -U_m;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = ((t_0 * -2.0) * sqrt(fma((0.5 / (fma(cos(K), 0.5, 0.5) * J_m)), ((U_m / (2.0 * J_m)) * U_m), 1.0))) * J_m;
	} else {
		tmp = fma((((J_m * J_m) / (U_m * U_m)) * pow(t_0, 2.0)), -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(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(t_1 * Float64(J_m * -2.0)))
	tmp = 0.0
	if (t_2 <= -2e+306)
		tmp = Float64(-U_m);
	elseif (t_2 <= Inf)
		tmp = Float64(Float64(Float64(t_0 * -2.0) * sqrt(fma(Float64(0.5 / Float64(fma(cos(K), 0.5, 0.5) * J_m)), Float64(Float64(U_m / Float64(2.0 * J_m)) * U_m), 1.0))) * J_m);
	else
		tmp = Float64(fma(Float64(Float64(Float64(J_m * J_m) / Float64(U_m * U_m)) * (t_0 ^ 2.0)), -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[(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[(t$95$1 * N[(J$95$m * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$2, -2e+306], (-U$95$m), If[LessEqual[t$95$2, Infinity], N[(N[(N[(t$95$0 * -2.0), $MachinePrecision] * N[Sqrt[N[(N[(0.5 / N[(N[(N[Cos[K], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision] * J$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(U$95$m / N[(2.0 * J$95$m), $MachinePrecision]), $MachinePrecision] * U$95$m), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * J$95$m), $MachinePrecision], N[(N[(N[(N[(N[(J$95$m * J$95$m), $MachinePrecision] / N[(U$95$m * U$95$m), $MachinePrecision]), $MachinePrecision] * N[Power[t$95$0, 2.0], $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))))) < -2.00000000000000003e306

   1. Initial program 10.5%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   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 46.3

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

   5. Applied rewrites 46.3%

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

   if -2.00000000000000003e306 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < +inf.0

   1. Initial program 83.4%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. frac-times N/A

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

      10. lower-/.f64 N/A

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

   4. Applied rewrites 79.9%

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

   6. Applied rewrites 79.9%

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

   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. Initial program 72.9%

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

   3. Taylor expanded in U around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(U \cdot \left(-2 \cdot \frac{{J}^{2} \cdot {\cos \left(\frac{1}{2} \cdot K\right)}^{2}}{{U}^{2}} - 1\right)\right)} \]
   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 23.3%

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

4. Final simplification 75.0%

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

Alternative 10: 84.1% accurate, 0.4× speedup

Math

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

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 (/ (* 0.5 U_m) (* 1.0 J_m)))
        (t_1 (cos (/ K 2.0)))
        (t_2 (cos (* 0.5 K)))
        (t_3
         (*
          (sqrt (+ (pow (/ U_m (* (* 2.0 J_m) t_1)) 2.0) 1.0))
          (* t_1 (* J_m -2.0)))))
   (*
    J_s
    (if (<= t_3 (- INFINITY))
      (- U_m)
      (if (<= t_3 INFINITY)
        (* (sqrt (fma t_0 t_0 1.0)) (* (* t_2 J_m) -2.0))
        (*
         (fma (* (/ (* J_m J_m) (* U_m U_m)) (pow t_2 2.0)) -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 = (0.5 * U_m) / (1.0 * J_m);
	double t_1 = cos((K / 2.0));
	double t_2 = cos((0.5 * K));
	double t_3 = sqrt((pow((U_m / ((2.0 * J_m) * t_1)), 2.0) + 1.0)) * (t_1 * (J_m * -2.0));
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt(fma(t_0, t_0, 1.0)) * ((t_2 * J_m) * -2.0);
	} else {
		tmp = fma((((J_m * J_m) / (U_m * U_m)) * pow(t_2, 2.0)), -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 = Float64(Float64(0.5 * U_m) / Float64(1.0 * J_m))
	t_1 = cos(Float64(K / 2.0))
	t_2 = cos(Float64(0.5 * K))
	t_3 = Float64(sqrt(Float64((Float64(U_m / Float64(Float64(2.0 * J_m) * t_1)) ^ 2.0) + 1.0)) * Float64(t_1 * Float64(J_m * -2.0)))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_3 <= Inf)
		tmp = Float64(sqrt(fma(t_0, t_0, 1.0)) * Float64(Float64(t_2 * J_m) * -2.0));
	else
		tmp = Float64(fma(Float64(Float64(Float64(J_m * J_m) / Float64(U_m * U_m)) * (t_2 ^ 2.0)), -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[(N[(0.5 * U$95$m), $MachinePrecision] / N[(1.0 * J$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = 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[(t$95$1 * N[(J$95$m * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$3, (-Infinity)], (-U$95$m), If[LessEqual[t$95$3, Infinity], N[(N[Sqrt[N[(t$95$0 * t$95$0 + 1.0), $MachinePrecision]], $MachinePrecision] * N[(N[(t$95$2 * J$95$m), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(J$95$m * J$95$m), $MachinePrecision] / N[(U$95$m * U$95$m), $MachinePrecision]), $MachinePrecision] * N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision] * -2.0 + -1.0), $MachinePrecision] * (-U$95$m)), $MachinePrecision]]]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 5.4%

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

   3. Taylor expanded in U around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 49.0

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

   5. Applied rewrites 49.0%

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

   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))))) < +inf.0

   1. Initial program 83.6%

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

   3. Taylor expanded in K around 0

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

   5. Applied rewrites 71.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-fma.f64 71.3

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

   7. Applied rewrites 71.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 71.3

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

      8. lift-/.f64 N/A

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

      9. div-inv N/A

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

      10. metadata-eval N/A

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

      11. lift-*.f64 71.3

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

   9. Applied rewrites 71.3%

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

   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. Initial program 72.9%

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

   3. Taylor expanded in U around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(U \cdot \left(-2 \cdot \frac{{J}^{2} \cdot {\cos \left(\frac{1}{2} \cdot K\right)}^{2}}{{U}^{2}} - 1\right)\right)} \]
   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 23.3%

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

4. Final simplification 68.2%

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

Alternative 11: 84.1% accurate, 0.4× speedup

Math

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

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 (/ (* 0.5 U_m) (* 1.0 J_m)))
        (t_1 (cos (/ K 2.0)))
        (t_2 (cos (* 0.5 K)))
        (t_3
         (*
          (sqrt (+ (pow (/ U_m (* (* 2.0 J_m) t_1)) 2.0) 1.0))
          (* t_1 (* J_m -2.0)))))
   (*
    J_s
    (if (<= t_3 (- INFINITY))
      (- U_m)
      (if (<= t_3 INFINITY)
        (* (sqrt (fma t_0 t_0 1.0)) (* (* t_2 J_m) -2.0))
        (/ (* t_2 (- U_m)) (sqrt (fma (cos K) 0.5 0.5))))))))

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 = (0.5 * U_m) / (1.0 * J_m);
	double t_1 = cos((K / 2.0));
	double t_2 = cos((0.5 * K));
	double t_3 = sqrt((pow((U_m / ((2.0 * J_m) * t_1)), 2.0) + 1.0)) * (t_1 * (J_m * -2.0));
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt(fma(t_0, t_0, 1.0)) * ((t_2 * J_m) * -2.0);
	} else {
		tmp = (t_2 * -U_m) / sqrt(fma(cos(K), 0.5, 0.5));
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = Float64(Float64(0.5 * U_m) / Float64(1.0 * J_m))
	t_1 = cos(Float64(K / 2.0))
	t_2 = cos(Float64(0.5 * K))
	t_3 = Float64(sqrt(Float64((Float64(U_m / Float64(Float64(2.0 * J_m) * t_1)) ^ 2.0) + 1.0)) * Float64(t_1 * Float64(J_m * -2.0)))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_3 <= Inf)
		tmp = Float64(sqrt(fma(t_0, t_0, 1.0)) * Float64(Float64(t_2 * J_m) * -2.0));
	else
		tmp = Float64(Float64(t_2 * Float64(-U_m)) / sqrt(fma(cos(K), 0.5, 0.5)));
	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[(N[(0.5 * U$95$m), $MachinePrecision] / N[(1.0 * J$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = 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[(t$95$1 * N[(J$95$m * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$3, (-Infinity)], (-U$95$m), If[LessEqual[t$95$3, Infinity], N[(N[Sqrt[N[(t$95$0 * t$95$0 + 1.0), $MachinePrecision]], $MachinePrecision] * N[(N[(t$95$2 * J$95$m), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$2 * (-U$95$m)), $MachinePrecision] / N[Sqrt[N[(N[Cos[K], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 5.4%

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

   3. Taylor expanded in U around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 49.0

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

   5. Applied rewrites 49.0%

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

   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))))) < +inf.0

   1. Initial program 83.6%

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

   3. Taylor expanded in K around 0

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

   5. Applied rewrites 71.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-fma.f64 71.3

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

   7. Applied rewrites 71.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 71.3

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

      8. lift-/.f64 N/A

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

      9. div-inv N/A

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

      10. metadata-eval N/A

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

      11. lift-*.f64 71.3

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

   9. Applied rewrites 71.3%

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

   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. Initial program 72.9%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. frac-times N/A

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

      10. lower-/.f64 N/A

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

   4. Applied rewrites 69.5%

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

   5. Taylor expanded in U around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. mul-1-neg N/A

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

      12. cos-neg N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

   7. Applied rewrites 24.3%

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

   9. Applied rewrites 24.3%

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

   11. Applied rewrites 24.3%

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

4. Final simplification 68.2%

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

Alternative 12: 84.1% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 5.4%

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

   3. Taylor expanded in U around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 49.0

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

   5. Applied rewrites 49.0%

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

   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))))) < +inf.0

   1. Initial program 83.6%

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

   3. Taylor expanded in K around 0

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

   5. Applied rewrites 71.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-fma.f64 71.3

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

   7. Applied rewrites 71.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 71.3

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

      8. lift-/.f64 N/A

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

      9. div-inv N/A

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

      10. metadata-eval N/A

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

      11. lift-*.f64 71.3

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

   9. Applied rewrites 71.3%

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

   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. Initial program 72.9%

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

   3. Taylor expanded in K around 0

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

   5. Applied rewrites 62.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. div-inv N/A

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

      10. metadata-eval N/A

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

      11. lift-*.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

   7. Applied rewrites 47.9%

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

   8. Taylor expanded in U around -inf

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

      1. lower-/.f64 23.3

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

   10. Applied rewrites 23.3%

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

4. Final simplification 68.2%

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

Alternative 13: 82.0% accurate, 0.4× speedup

Math

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

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (sqrt (+ (pow (/ U_m (* (* 2.0 J_m) t_0)) 2.0) 1.0))
          (* t_0 (* J_m -2.0)))))
   (*
    J_s
    (if (<= t_1 -2e+306)
      (- U_m)
      (if (<= t_1 INFINITY)
        (*
         (*
          (sqrt (fma (/ 0.5 J_m) (* (/ U_m (* 2.0 J_m)) U_m) 1.0))
          (* (cos (* 0.5 K)) -2.0))
         J_m)
        (* (/ U_m J_m) J_m))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = sqrt((pow((U_m / ((2.0 * J_m) * t_0)), 2.0) + 1.0)) * (t_0 * (J_m * -2.0));
	double tmp;
	if (t_1 <= -2e+306) {
		tmp = -U_m;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (sqrt(fma((0.5 / J_m), ((U_m / (2.0 * J_m)) * U_m), 1.0)) * (cos((0.5 * K)) * -2.0)) * J_m;
	} else {
		tmp = (U_m / J_m) * J_m;
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(sqrt(Float64((Float64(U_m / Float64(Float64(2.0 * J_m) * t_0)) ^ 2.0) + 1.0)) * Float64(t_0 * Float64(J_m * -2.0)))
	tmp = 0.0
	if (t_1 <= -2e+306)
		tmp = Float64(-U_m);
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(sqrt(fma(Float64(0.5 / J_m), Float64(Float64(U_m / Float64(2.0 * J_m)) * U_m), 1.0)) * Float64(cos(Float64(0.5 * K)) * -2.0)) * J_m);
	else
		tmp = Float64(Float64(U_m / J_m) * J_m);
	end
	return Float64(J_s * tmp)
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[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[(t$95$0 * N[(J$95$m * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$1, -2e+306], (-U$95$m), If[LessEqual[t$95$1, Infinity], N[(N[(N[Sqrt[N[(N[(0.5 / J$95$m), $MachinePrecision] * N[(N[(U$95$m / N[(2.0 * J$95$m), $MachinePrecision]), $MachinePrecision] * U$95$m), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision] * J$95$m), $MachinePrecision], N[(N[(U$95$m / J$95$m), $MachinePrecision] * J$95$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))))) < -2.00000000000000003e306

   1. Initial program 10.5%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   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 46.3

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

   5. Applied rewrites 46.3%

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

   if -2.00000000000000003e306 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < +inf.0

   1. Initial program 83.4%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. frac-times N/A

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

      10. lower-/.f64 N/A

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

   4. Applied rewrites 79.9%

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

   6. Applied rewrites 79.9%

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

   7. Taylor expanded in K around 0

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

      1. lower-/.f64 69.4

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

   9. Applied rewrites 69.4%

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

   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. Initial program 72.9%

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

   3. Taylor expanded in K around 0

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

   5. Applied rewrites 62.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. div-inv N/A

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

      10. metadata-eval N/A

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

      11. lift-*.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

   7. Applied rewrites 47.9%

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

   8. Taylor expanded in U around -inf

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

      1. lower-/.f64 23.3

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

   10. Applied rewrites 23.3%

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

4. Final simplification 66.1%

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

Alternative 14: 74.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \sqrt{{\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2} + 1} \cdot \left(t\_0 \cdot \left(J\_m \cdot -2\right)\right)\\ t_2 := \frac{0.5 \cdot U\_m}{1 \cdot J\_m}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-272}:\\ \;\;\;\;\left(J\_m \cdot -2\right) \cdot \sqrt{\mathsf{fma}\left(t\_2, t\_2, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{U\_m}{J\_m} \cdot J\_m\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (sqrt (+ (pow (/ U_m (* (* 2.0 J_m) t_0)) 2.0) 1.0))
          (* t_0 (* J_m -2.0))))
        (t_2 (/ (* 0.5 U_m) (* 1.0 J_m))))
   (*
    J_s
    (if (<= t_1 (- INFINITY))
      (- U_m)
      (if (<= t_1 -2e-272)
        (* (* J_m -2.0) (sqrt (fma t_2 t_2 1.0)))
        (* (/ U_m J_m) J_m))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = sqrt((pow((U_m / ((2.0 * J_m) * t_0)), 2.0) + 1.0)) * (t_0 * (J_m * -2.0));
	double t_2 = (0.5 * U_m) / (1.0 * J_m);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_1 <= -2e-272) {
		tmp = (J_m * -2.0) * sqrt(fma(t_2, t_2, 1.0));
	} else {
		tmp = (U_m / J_m) * J_m;
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(sqrt(Float64((Float64(U_m / Float64(Float64(2.0 * J_m) * t_0)) ^ 2.0) + 1.0)) * Float64(t_0 * Float64(J_m * -2.0)))
	t_2 = Float64(Float64(0.5 * U_m) / Float64(1.0 * J_m))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_1 <= -2e-272)
		tmp = Float64(Float64(J_m * -2.0) * sqrt(fma(t_2, t_2, 1.0)));
	else
		tmp = Float64(Float64(U_m / J_m) * J_m);
	end
	return Float64(J_s * tmp)
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[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[(t$95$0 * N[(J$95$m * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(0.5 * U$95$m), $MachinePrecision] / N[(1.0 * J$95$m), $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$1, (-Infinity)], (-U$95$m), If[LessEqual[t$95$1, -2e-272], N[(N[(J$95$m * -2.0), $MachinePrecision] * N[Sqrt[N[(t$95$2 * t$95$2 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(U$95$m / J$95$m), $MachinePrecision] * J$95$m), $MachinePrecision]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 5.4%

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

   3. Taylor expanded in U around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 49.0

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

   5. Applied rewrites 49.0%

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

   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))))) < -1.99999999999999986e-272

   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

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

   5. Applied rewrites 86.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-fma.f64 86.3

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

   7. Applied rewrites 86.3%

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

   8. Taylor expanded in K around 0

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

      1. lower-*.f64 62.9

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

   10. Applied rewrites 62.9%

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

   if -1.99999999999999986e-272 < (*.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 70.6%

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

   3. Taylor expanded in K around 0

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

   5. Applied rewrites 59.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. div-inv N/A

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

      10. metadata-eval N/A

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

      11. lift-*.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

   7. Applied rewrites 49.0%

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

   8. Taylor expanded in U around -inf

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

      1. lower-/.f64 30.1

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

   10. Applied rewrites 30.1%

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

4. Final simplification 45.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq -2 \cdot 10^{-272}:\\ \;\;\;\;\left(J \cdot -2\right) \cdot \sqrt{\mathsf{fma}\left(\frac{0.5 \cdot U}{1 \cdot J}, \frac{0.5 \cdot U}{1 \cdot J}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{U}{J} \cdot J\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 50.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;\sqrt{{\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2} + 1} \cdot \left(t\_0 \cdot \left(J\_m \cdot -2\right)\right) \leq -2 \cdot 10^{-272}:\\ \;\;\;\;-U\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{U\_m}{J\_m} \cdot J\_m\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (*
    J_s
    (if (<=
         (*
          (sqrt (+ (pow (/ U_m (* (* 2.0 J_m) t_0)) 2.0) 1.0))
          (* t_0 (* J_m -2.0)))
         -2e-272)
      (- U_m)
      (* (/ U_m J_m) J_m)))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if ((sqrt((pow((U_m / ((2.0 * J_m) * t_0)), 2.0) + 1.0)) * (t_0 * (J_m * -2.0))) <= -2e-272) {
		tmp = -U_m;
	} else {
		tmp = (U_m / J_m) * J_m;
	}
	return J_s * tmp;
}

Fortran

U_m = abs(u)
J\_m = abs(j)
J\_s = copysign(1.0d0, j)
real(8) function code(j_s, j_m, k, u_m)
    real(8), intent (in) :: j_s
    real(8), intent (in) :: j_m
    real(8), intent (in) :: k
    real(8), intent (in) :: u_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    if ((sqrt((((u_m / ((2.0d0 * j_m) * t_0)) ** 2.0d0) + 1.0d0)) * (t_0 * (j_m * (-2.0d0)))) <= (-2d-272)) then
        tmp = -u_m
    else
        tmp = (u_m / j_m) * j_m
    end if
    code = j_s * tmp
end function

Java

U_m = Math.abs(U);
J\_m = Math.abs(J);
J\_s = Math.copySign(1.0, J);
public static double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = Math.cos((K / 2.0));
	double tmp;
	if ((Math.sqrt((Math.pow((U_m / ((2.0 * J_m) * t_0)), 2.0) + 1.0)) * (t_0 * (J_m * -2.0))) <= -2e-272) {
		tmp = -U_m;
	} else {
		tmp = (U_m / J_m) * J_m;
	}
	return J_s * tmp;
}

Python

U_m = math.fabs(U)
J\_m = math.fabs(J)
J\_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if (math.sqrt((math.pow((U_m / ((2.0 * J_m) * t_0)), 2.0) + 1.0)) * (t_0 * (J_m * -2.0))) <= -2e-272:
		tmp = -U_m
	else:
		tmp = (U_m / J_m) * J_m
	return J_s * tmp

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (Float64(sqrt(Float64((Float64(U_m / Float64(Float64(2.0 * J_m) * t_0)) ^ 2.0) + 1.0)) * Float64(t_0 * Float64(J_m * -2.0))) <= -2e-272)
		tmp = Float64(-U_m);
	else
		tmp = Float64(Float64(U_m / J_m) * J_m);
	end
	return Float64(J_s * tmp)
end

MATLAB

U_m = abs(U);
J\_m = abs(J);
J\_s = sign(J) * abs(1.0);
function tmp_2 = code(J_s, J_m, K, U_m)
	t_0 = cos((K / 2.0));
	tmp = 0.0;
	if ((sqrt((((U_m / ((2.0 * J_m) * t_0)) ^ 2.0) + 1.0)) * (t_0 * (J_m * -2.0))) <= -2e-272)
		tmp = -U_m;
	else
		tmp = (U_m / J_m) * J_m;
	end
	tmp_2 = J_s * tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, N[(J$95$s * If[LessEqual[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[(t$95$0 * N[(J$95$m * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-272], (-U$95$m), N[(N[(U$95$m / J$95$m), $MachinePrecision] * J$95$m), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -1.99999999999999986e-272

   1. Initial program 74.9%

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

   3. Taylor expanded in U around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 26.7

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

   5. Applied rewrites 26.7%

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

   if -1.99999999999999986e-272 < (*.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 70.6%

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

   3. Taylor expanded in K around 0

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

   5. Applied rewrites 59.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. div-inv N/A

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

      10. metadata-eval N/A

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

      11. lift-*.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

   7. Applied rewrites 49.0%

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

   8. Taylor expanded in U around -inf

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

      1. lower-/.f64 30.1

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

   10. Applied rewrites 30.1%

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

4. Final simplification 28.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq -2 \cdot 10^{-272}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;\frac{U}{J} \cdot J\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 40.6% accurate, 124.3× speedup

Math

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

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m) :precision binary64 (* J_s (- U_m)))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	return J_s * -U_m;
}

Fortran

U_m = abs(u)
J\_m = abs(j)
J\_s = copysign(1.0d0, j)
real(8) function code(j_s, j_m, k, u_m)
    real(8), intent (in) :: j_s
    real(8), intent (in) :: j_m
    real(8), intent (in) :: k
    real(8), intent (in) :: u_m
    code = j_s * -u_m
end function

Java

U_m = Math.abs(U);
J\_m = Math.abs(J);
J\_s = Math.copySign(1.0, J);
public static double code(double J_s, double J_m, double K, double U_m) {
	return J_s * -U_m;
}

Python

U_m = math.fabs(U)
J\_m = math.fabs(J)
J\_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	return J_s * -U_m

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	return Float64(J_s * Float64(-U_m))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.9%

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

3. Taylor expanded in U around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 24.2

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

5. Applied rewrites 24.2%

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

Reproduce

herbie shell --seed 2024240 
(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)))))