Maksimov and Kolovsky, Equation (3)

Percentage Accurate: 74.1% → 98.9%

Time: 13.1s

Alternatives: 14

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: 74.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_0 \cdot \left(J\_m \cdot 2\right)}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;0 - U\_m\\ \mathbf{elif}\;t\_1 \leq 10^{+284}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* (* -2.0 J_m) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* t_0 (* J_m 2.0))) 2.0))))))
   (*
    J_s
    (if (<= t_1 (- INFINITY)) (- 0.0 U_m) (if (<= t_1 1e+284) t_1 U_m)))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * J_m) * t_0) * sqrt((1.0 + pow((U_m / (t_0 * (J_m * 2.0))), 2.0)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = 0.0 - U_m;
	} else if (t_1 <= 1e+284) {
		tmp = t_1;
	} else {
		tmp = U_m;
	}
	return J_s * tmp;
}

Java

U_m = Math.abs(U);
J\_m = Math.abs(J);
J\_s = Math.copySign(1.0, J);
public static double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = ((-2.0 * J_m) * t_0) * Math.sqrt((1.0 + Math.pow((U_m / (t_0 * (J_m * 2.0))), 2.0)));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = 0.0 - U_m;
	} else if (t_1 <= 1e+284) {
		tmp = t_1;
	} else {
		tmp = U_m;
	}
	return J_s * tmp;
}

Python

U_m = math.fabs(U)
J\_m = math.fabs(J)
J\_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	t_0 = math.cos((K / 2.0))
	t_1 = ((-2.0 * J_m) * t_0) * math.sqrt((1.0 + math.pow((U_m / (t_0 * (J_m * 2.0))), 2.0)))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = 0.0 - U_m
	elif t_1 <= 1e+284:
		tmp = t_1
	else:
		tmp = U_m
	return J_s * tmp

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * J_m) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_0 * Float64(J_m * 2.0))) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(0.0 - U_m);
	elseif (t_1 <= 1e+284)
		tmp = t_1;
	else
		tmp = U_m;
	end
	return Float64(J_s * tmp)
end

MATLAB

U_m = abs(U);
J\_m = abs(J);
J\_s = sign(J) * abs(1.0);
function tmp_2 = code(J_s, J_m, K, U_m)
	t_0 = cos((K / 2.0));
	t_1 = ((-2.0 * J_m) * t_0) * sqrt((1.0 + ((U_m / (t_0 * (J_m * 2.0))) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = 0.0 - U_m;
	elseif (t_1 <= 1e+284)
		tmp = t_1;
	else
		tmp = U_m;
	end
	tmp_2 = J_s * tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(t$95$0 * N[(J$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$1, (-Infinity)], N[(0.0 - U$95$m), $MachinePrecision], If[LessEqual[t$95$1, 1e+284], t$95$1, U$95$m]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 6.1%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - U} \]

      3. --lowering--.f64 52.5

         \[\leadsto \color{blue}{0 - U} \]

   5. Simplified 52.5%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 52.5

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

   7. Applied egg-rr 52.5%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 1.00000000000000008e284

   1. Initial program 99.8%

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

   if 1.00000000000000008e284 < (*.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 15.6%

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

   3. Taylor expanded in U around -inf

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

   5. Simplified 44.0%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -\infty:\\ \;\;\;\;0 - U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 10^{+284}:\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \left(-2 \cdot J\_m\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U\_m \cdot 0.25}{J\_m}, \frac{U\_m}{J\_m}, 1\right)}\\ t_1 := \cos \left(\frac{K}{2}\right)\\ t_2 := \left(-2 \cdot J\_m\right) \cdot t\_1\\ t_3 := t\_2 \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_1 \cdot \left(J\_m \cdot 2\right)}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;0 - U\_m\\ \mathbf{elif}\;t\_3 \leq -2 \cdot 10^{+248}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_3 \leq -5 \cdot 10^{-12}:\\ \;\;\;\;t\_2 \cdot \sqrt{\mathsf{fma}\left(0.25, \frac{U\_m \cdot U\_m}{J\_m \cdot J\_m}, 1\right)}\\ \mathbf{elif}\;t\_3 \leq -5 \cdot 10^{-261}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_3 \leq 10^{+284}:\\ \;\;\;\;\left(-2 \cdot J\_m\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0
         (* (* -2.0 J_m) (sqrt (fma (/ (* U_m 0.25) J_m) (/ U_m J_m) 1.0))))
        (t_1 (cos (/ K 2.0)))
        (t_2 (* (* -2.0 J_m) t_1))
        (t_3 (* t_2 (sqrt (+ 1.0 (pow (/ U_m (* t_1 (* J_m 2.0))) 2.0))))))
   (*
    J_s
    (if (<= t_3 (- INFINITY))
      (- 0.0 U_m)
      (if (<= t_3 -2e+248)
        t_0
        (if (<= t_3 -5e-12)
          (* t_2 (sqrt (fma 0.25 (/ (* U_m U_m) (* J_m J_m)) 1.0)))
          (if (<= t_3 -5e-261)
            t_0
            (if (<= t_3 1e+284) (* (* -2.0 J_m) (cos (* K 0.5))) 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 = (-2.0 * J_m) * sqrt(fma(((U_m * 0.25) / J_m), (U_m / J_m), 1.0));
	double t_1 = cos((K / 2.0));
	double t_2 = (-2.0 * J_m) * t_1;
	double t_3 = t_2 * sqrt((1.0 + pow((U_m / (t_1 * (J_m * 2.0))), 2.0)));
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = 0.0 - U_m;
	} else if (t_3 <= -2e+248) {
		tmp = t_0;
	} else if (t_3 <= -5e-12) {
		tmp = t_2 * sqrt(fma(0.25, ((U_m * U_m) / (J_m * J_m)), 1.0));
	} else if (t_3 <= -5e-261) {
		tmp = t_0;
	} else if (t_3 <= 1e+284) {
		tmp = (-2.0 * J_m) * cos((K * 0.5));
	} else {
		tmp = U_m;
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = Float64(Float64(-2.0 * J_m) * sqrt(fma(Float64(Float64(U_m * 0.25) / J_m), Float64(U_m / J_m), 1.0)))
	t_1 = cos(Float64(K / 2.0))
	t_2 = Float64(Float64(-2.0 * J_m) * t_1)
	t_3 = Float64(t_2 * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_1 * Float64(J_m * 2.0))) ^ 2.0))))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(0.0 - U_m);
	elseif (t_3 <= -2e+248)
		tmp = t_0;
	elseif (t_3 <= -5e-12)
		tmp = Float64(t_2 * sqrt(fma(0.25, Float64(Float64(U_m * U_m) / Float64(J_m * J_m)), 1.0)));
	elseif (t_3 <= -5e-261)
		tmp = t_0;
	elseif (t_3 <= 1e+284)
		tmp = Float64(Float64(-2.0 * J_m) * cos(Float64(K * 0.5)));
	else
		tmp = 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[(-2.0 * J$95$m), $MachinePrecision] * N[Sqrt[N[(N[(N[(U$95$m * 0.25), $MachinePrecision] / J$95$m), $MachinePrecision] * N[(U$95$m / J$95$m), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(t$95$1 * N[(J$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$3, (-Infinity)], N[(0.0 - U$95$m), $MachinePrecision], If[LessEqual[t$95$3, -2e+248], t$95$0, If[LessEqual[t$95$3, -5e-12], N[(t$95$2 * N[Sqrt[N[(0.25 * N[(N[(U$95$m * U$95$m), $MachinePrecision] / N[(J$95$m * J$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -5e-261], t$95$0, If[LessEqual[t$95$3, 1e+284], N[(N[(-2.0 * J$95$m), $MachinePrecision] * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], U$95$m]]]]]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 6.1%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - U} \]

      3. --lowering--.f64 52.5

         \[\leadsto \color{blue}{0 - U} \]

   5. Simplified 52.5%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 52.5

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

   7. Applied egg-rr 52.5%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -2.00000000000000009e248 or -4.9999999999999997e-12 < (*.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))))) < -4.99999999999999981e-261

   1. Initial program 99.9%

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

   3. Taylor expanded in K around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. +-commutative N/A

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

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

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

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

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

      8. unpow2 N/A

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

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

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

      10. unpow2 N/A

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

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

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

      12. *-lowering-*.f64 28.7

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

   5. Simplified 28.7%

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

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

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

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

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

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

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

      7. /-lowering-/.f64 65.6

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

   7. Applied egg-rr 65.6%

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

   if -2.00000000000000009e248 < (*.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))))) < -4.9999999999999997e-12

   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

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

      1. sqrt-lowering-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. accelerator-lowering-fma.f64 N/A

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

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

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

      5. unpow2 N/A

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 86.7

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

   5. Simplified 86.7%

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

   if -4.99999999999999981e-261 < (*.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))))) < 1.00000000000000008e284

   1. Initial program 99.8%

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

   3. Taylor expanded in J around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

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

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

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

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

      6. *-lowering-*.f64 67.3

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

   5. Simplified 67.3%

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

   if 1.00000000000000008e284 < (*.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 15.6%

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

   3. Taylor expanded in U around -inf

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

   5. Simplified 44.0%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -\infty:\\ \;\;\;\;0 - U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -2 \cdot 10^{+248}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U \cdot 0.25}{J}, \frac{U}{J}, 1\right)}\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -5 \cdot 10^{-12}:\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\mathsf{fma}\left(0.25, \frac{U \cdot U}{J \cdot J}, 1\right)}\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -5 \cdot 10^{-261}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U \cdot 0.25}{J}, \frac{U}{J}, 1\right)}\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 10^{+284}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(-2 \cdot J\_m\right) \cdot t\_0\\ t_2 := t\_1 \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_0 \cdot \left(J\_m \cdot 2\right)}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;0 - U\_m\\ \mathbf{elif}\;t\_2 \leq 10^{-158}:\\ \;\;\;\;t\_1 \cdot \sqrt{\mathsf{fma}\left(\frac{U\_m \cdot 0.25}{J\_m}, \frac{U\_m}{J\_m}, 1\right)}\\ \mathbf{elif}\;t\_2 \leq 10^{+284}:\\ \;\;\;\;\left(-2 \cdot \sqrt{\mathsf{fma}\left(U\_m, \frac{U\_m}{\left(J\_m \cdot 2\right) \cdot \mathsf{fma}\left(J\_m, \cos K, J\_m\right)}, 1\right)}\right) \cdot \left(J\_m \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1 (* (* -2.0 J_m) t_0))
        (t_2 (* t_1 (sqrt (+ 1.0 (pow (/ U_m (* t_0 (* J_m 2.0))) 2.0))))))
   (*
    J_s
    (if (<= t_2 (- INFINITY))
      (- 0.0 U_m)
      (if (<= t_2 1e-158)
        (* t_1 (sqrt (fma (/ (* U_m 0.25) J_m) (/ U_m J_m) 1.0)))
        (if (<= t_2 1e+284)
          (*
           (*
            -2.0
            (sqrt (fma U_m (/ U_m (* (* J_m 2.0) (fma J_m (cos K) J_m))) 1.0)))
           (* J_m (cos (* K 0.5))))
          U_m))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = (-2.0 * J_m) * t_0;
	double t_2 = t_1 * sqrt((1.0 + pow((U_m / (t_0 * (J_m * 2.0))), 2.0)));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = 0.0 - U_m;
	} else if (t_2 <= 1e-158) {
		tmp = t_1 * sqrt(fma(((U_m * 0.25) / J_m), (U_m / J_m), 1.0));
	} else if (t_2 <= 1e+284) {
		tmp = (-2.0 * sqrt(fma(U_m, (U_m / ((J_m * 2.0) * fma(J_m, cos(K), J_m))), 1.0))) * (J_m * cos((K * 0.5)));
	} else {
		tmp = U_m;
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(-2.0 * J_m) * t_0)
	t_2 = Float64(t_1 * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_0 * Float64(J_m * 2.0))) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(0.0 - U_m);
	elseif (t_2 <= 1e-158)
		tmp = Float64(t_1 * sqrt(fma(Float64(Float64(U_m * 0.25) / J_m), Float64(U_m / J_m), 1.0)));
	elseif (t_2 <= 1e+284)
		tmp = Float64(Float64(-2.0 * sqrt(fma(U_m, Float64(U_m / Float64(Float64(J_m * 2.0) * fma(J_m, cos(K), J_m))), 1.0))) * Float64(J_m * cos(Float64(K * 0.5))));
	else
		tmp = 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[(-2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(t$95$0 * N[(J$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$2, (-Infinity)], N[(0.0 - U$95$m), $MachinePrecision], If[LessEqual[t$95$2, 1e-158], N[(t$95$1 * N[Sqrt[N[(N[(N[(U$95$m * 0.25), $MachinePrecision] / J$95$m), $MachinePrecision] * N[(U$95$m / J$95$m), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+284], N[(N[(-2.0 * N[Sqrt[N[(U$95$m * N[(U$95$m / N[(N[(J$95$m * 2.0), $MachinePrecision] * N[(J$95$m * N[Cos[K], $MachinePrecision] + J$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(J$95$m * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], U$95$m]]]), $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 6.1%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - U} \]

      3. --lowering--.f64 52.5

         \[\leadsto \color{blue}{0 - U} \]

   5. Simplified 52.5%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 52.5

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

   7. Applied egg-rr 52.5%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 1.00000000000000006e-158

   1. Initial program 99.8%

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

   3. Taylor expanded in K around 0

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

      1. sqrt-lowering-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. accelerator-lowering-fma.f64 N/A

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

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

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

      5. unpow2 N/A

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 66.0

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

   5. Simplified 66.0%

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

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

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

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

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

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

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

      7. /-lowering-/.f64 89.1

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

   7. Applied egg-rr 89.1%

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

   if 1.00000000000000006e-158 < (*.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))))) < 1.00000000000000008e284

   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. unpow2 N/A

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

      2. associate-/r* N/A

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

      3. frac-times N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

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

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

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

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

      8. *-commutative N/A

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

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

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

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

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

      11. associate-*l* N/A

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

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

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

      13. *-commutative N/A

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

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

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

      15. sqr-cos-a N/A

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

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

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

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

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

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

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

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

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

   4. Applied egg-rr 99.6%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. associate-*r* N/A

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

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

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

   6. Applied egg-rr 87.5%

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

   if 1.00000000000000008e284 < (*.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 15.6%

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

   3. Taylor expanded in U around -inf

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

   5. Simplified 44.0%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -\infty:\\ \;\;\;\;0 - U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 10^{-158}:\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U \cdot 0.25}{J}, \frac{U}{J}, 1\right)}\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 10^{+284}:\\ \;\;\;\;\left(-2 \cdot \sqrt{\mathsf{fma}\left(U, \frac{U}{\left(J \cdot 2\right) \cdot \mathsf{fma}\left(J, \cos K, J\right)}, 1\right)}\right) \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_0 \cdot \left(J\_m \cdot 2\right)}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;0 - U\_m\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-261}:\\ \;\;\;\;\left(-2 \cdot J\_m\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U\_m \cdot 0.25}{J\_m}, \frac{U\_m}{J\_m}, 1\right)}\\ \mathbf{elif}\;t\_1 \leq 10^{+284}:\\ \;\;\;\;\left(-2 \cdot J\_m\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* (* -2.0 J_m) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* t_0 (* J_m 2.0))) 2.0))))))
   (*
    J_s
    (if (<= t_1 (- INFINITY))
      (- 0.0 U_m)
      (if (<= t_1 -5e-261)
        (* (* -2.0 J_m) (sqrt (fma (/ (* U_m 0.25) J_m) (/ U_m J_m) 1.0)))
        (if (<= t_1 1e+284) (* (* -2.0 J_m) (cos (* K 0.5))) U_m))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * J_m) * t_0) * sqrt((1.0 + pow((U_m / (t_0 * (J_m * 2.0))), 2.0)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = 0.0 - U_m;
	} else if (t_1 <= -5e-261) {
		tmp = (-2.0 * J_m) * sqrt(fma(((U_m * 0.25) / J_m), (U_m / J_m), 1.0));
	} else if (t_1 <= 1e+284) {
		tmp = (-2.0 * J_m) * cos((K * 0.5));
	} else {
		tmp = U_m;
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * J_m) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_0 * Float64(J_m * 2.0))) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(0.0 - U_m);
	elseif (t_1 <= -5e-261)
		tmp = Float64(Float64(-2.0 * J_m) * sqrt(fma(Float64(Float64(U_m * 0.25) / J_m), Float64(U_m / J_m), 1.0)));
	elseif (t_1 <= 1e+284)
		tmp = Float64(Float64(-2.0 * J_m) * cos(Float64(K * 0.5)));
	else
		tmp = 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[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(t$95$0 * N[(J$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$1, (-Infinity)], N[(0.0 - U$95$m), $MachinePrecision], If[LessEqual[t$95$1, -5e-261], N[(N[(-2.0 * J$95$m), $MachinePrecision] * N[Sqrt[N[(N[(N[(U$95$m * 0.25), $MachinePrecision] / J$95$m), $MachinePrecision] * N[(U$95$m / J$95$m), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+284], N[(N[(-2.0 * J$95$m), $MachinePrecision] * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], U$95$m]]]), $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 6.1%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - U} \]

      3. --lowering--.f64 52.5

         \[\leadsto \color{blue}{0 - U} \]

   5. Simplified 52.5%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 52.5

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

   7. Applied egg-rr 52.5%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -4.99999999999999981e-261

   1. Initial program 99.7%

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

   3. Taylor expanded in K around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. +-commutative N/A

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

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

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

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

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

      8. unpow2 N/A

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

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

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

      10. unpow2 N/A

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

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

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

      12. *-lowering-*.f64 39.7

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

   5. Simplified 39.7%

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

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

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

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

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

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

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

      7. /-lowering-/.f64 56.3

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

   7. Applied egg-rr 56.3%

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

   if -4.99999999999999981e-261 < (*.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))))) < 1.00000000000000008e284

   1. Initial program 99.8%

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

   3. Taylor expanded in J around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

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

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

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

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

      6. *-lowering-*.f64 67.3

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

   5. Simplified 67.3%

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

   if 1.00000000000000008e284 < (*.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 15.6%

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

   3. Taylor expanded in U around -inf

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

   5. Simplified 44.0%

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

4. Final simplification 57.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -\infty:\\ \;\;\;\;0 - U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -5 \cdot 10^{-261}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U \cdot 0.25}{J}, \frac{U}{J}, 1\right)}\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 10^{+284}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 66.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_0 \cdot \left(J\_m \cdot 2\right)}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+300}:\\ \;\;\;\;0 - U\_m\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-48}:\\ \;\;\;\;\left(-2 \cdot J\_m\right) \cdot \sqrt{\mathsf{fma}\left(0.25, \frac{U\_m \cdot U\_m}{J\_m \cdot J\_m}, 1\right)}\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-261}:\\ \;\;\;\;0 - U\_m\\ \mathbf{else}:\\ \;\;\;\;\left(0 - U\_m\right) \cdot \mathsf{fma}\left(\frac{J\_m}{U\_m}, -2 \cdot \frac{J\_m}{U\_m}, -1\right)\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* (* -2.0 J_m) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* t_0 (* J_m 2.0))) 2.0))))))
   (*
    J_s
    (if (<= t_1 -5e+300)
      (- 0.0 U_m)
      (if (<= t_1 -5e-48)
        (* (* -2.0 J_m) (sqrt (fma 0.25 (/ (* U_m U_m) (* J_m J_m)) 1.0)))
        (if (<= t_1 -5e-261)
          (- 0.0 U_m)
          (* (- 0.0 U_m) (fma (/ J_m U_m) (* -2.0 (/ J_m U_m)) -1.0))))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * J_m) * t_0) * sqrt((1.0 + pow((U_m / (t_0 * (J_m * 2.0))), 2.0)));
	double tmp;
	if (t_1 <= -5e+300) {
		tmp = 0.0 - U_m;
	} else if (t_1 <= -5e-48) {
		tmp = (-2.0 * J_m) * sqrt(fma(0.25, ((U_m * U_m) / (J_m * J_m)), 1.0));
	} else if (t_1 <= -5e-261) {
		tmp = 0.0 - U_m;
	} else {
		tmp = (0.0 - U_m) * fma((J_m / U_m), (-2.0 * (J_m / U_m)), -1.0);
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * J_m) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_0 * Float64(J_m * 2.0))) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -5e+300)
		tmp = Float64(0.0 - U_m);
	elseif (t_1 <= -5e-48)
		tmp = Float64(Float64(-2.0 * J_m) * sqrt(fma(0.25, Float64(Float64(U_m * U_m) / Float64(J_m * J_m)), 1.0)));
	elseif (t_1 <= -5e-261)
		tmp = Float64(0.0 - U_m);
	else
		tmp = Float64(Float64(0.0 - U_m) * fma(Float64(J_m / U_m), Float64(-2.0 * Float64(J_m / U_m)), -1.0));
	end
	return Float64(J_s * tmp)
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(t$95$0 * N[(J$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$1, -5e+300], N[(0.0 - U$95$m), $MachinePrecision], If[LessEqual[t$95$1, -5e-48], N[(N[(-2.0 * J$95$m), $MachinePrecision] * N[Sqrt[N[(0.25 * N[(N[(U$95$m * U$95$m), $MachinePrecision] / N[(J$95$m * J$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -5e-261], N[(0.0 - U$95$m), $MachinePrecision], N[(N[(0.0 - U$95$m), $MachinePrecision] * N[(N[(J$95$m / U$95$m), $MachinePrecision] * N[(-2.0 * N[(J$95$m / U$95$m), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -5.00000000000000026e300 or -4.9999999999999999e-48 < (*.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))))) < -4.99999999999999981e-261

   1. Initial program 34.4%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - U} \]

      3. --lowering--.f64 43.9

         \[\leadsto \color{blue}{0 - U} \]

   5. Simplified 43.9%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 43.9

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

   7. Applied egg-rr 43.9%

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

   if -5.00000000000000026e300 < (*.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))))) < -4.9999999999999999e-48

   1. Initial program 99.7%

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

   3. Taylor expanded in K around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. +-commutative N/A

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

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

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

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

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

      8. unpow2 N/A

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

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

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

      10. unpow2 N/A

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

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

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

      12. *-lowering-*.f64 45.3

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

   5. Simplified 45.3%

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

   if -4.99999999999999981e-261 < (*.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 72.0%

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

   3. Taylor expanded in K around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. +-commutative N/A

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

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

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

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

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

      8. unpow2 N/A

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

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

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

      10. unpow2 N/A

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

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

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

      12. *-lowering-*.f64 32.4

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

   5. Simplified 32.4%

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

   6. Taylor expanded in U around -inf

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

      1. associate-*r* N/A

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

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

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

      3. mul-1-neg N/A

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

      4. neg-sub0 N/A

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

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

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

         \[\leadsto \left(0 - U\right) \cdot \left(-2 \cdot \frac{{J}^{2}}{{U}^{2}} + \color{blue}{-1}\right) \]

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

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

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

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

      10. unpow2 N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 20.1

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

   8. Simplified 20.1%

      \[\leadsto \color{blue}{\left(0 - U\right) \cdot \mathsf{fma}\left(-2, \frac{J \cdot J}{U \cdot U}, -1\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(0 - U\right) \cdot \left(\color{blue}{\frac{J \cdot J}{U \cdot U} \cdot -2} + -1\right) \]

      2. times-frac N/A

         \[\leadsto \left(0 - U\right) \cdot \left(\color{blue}{\left(\frac{J}{U} \cdot \frac{J}{U}\right)} \cdot -2 + -1\right) \]

      3. associate-*l* N/A

         \[\leadsto \left(0 - U\right) \cdot \left(\color{blue}{\frac{J}{U} \cdot \left(\frac{J}{U} \cdot -2\right)} + -1\right) \]

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

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

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

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

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

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

      7. /-lowering-/.f64 24.7

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

   10. Applied egg-rr 24.7%

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

4. Final simplification 35.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -5 \cdot 10^{+300}:\\ \;\;\;\;0 - U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -5 \cdot 10^{-48}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \sqrt{\mathsf{fma}\left(0.25, \frac{U \cdot U}{J \cdot J}, 1\right)}\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -5 \cdot 10^{-261}:\\ \;\;\;\;0 - U\\ \mathbf{else}:\\ \;\;\;\;\left(0 - U\right) \cdot \mathsf{fma}\left(\frac{J}{U}, -2 \cdot \frac{J}{U}, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 62.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_0 \cdot \left(J\_m \cdot 2\right)}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+306}:\\ \;\;\;\;0 - U\_m\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-48}:\\ \;\;\;\;\mathsf{fma}\left(U\_m, \frac{U\_m \cdot -0.25}{J\_m}, -2 \cdot J\_m\right)\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-261}:\\ \;\;\;\;0 - U\_m\\ \mathbf{else}:\\ \;\;\;\;\left(0 - U\_m\right) \cdot \mathsf{fma}\left(\frac{J\_m}{U\_m}, -2 \cdot \frac{J\_m}{U\_m}, -1\right)\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* (* -2.0 J_m) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* t_0 (* J_m 2.0))) 2.0))))))
   (*
    J_s
    (if (<= t_1 -1e+306)
      (- 0.0 U_m)
      (if (<= t_1 -5e-48)
        (fma U_m (/ (* U_m -0.25) J_m) (* -2.0 J_m))
        (if (<= t_1 -5e-261)
          (- 0.0 U_m)
          (* (- 0.0 U_m) (fma (/ J_m U_m) (* -2.0 (/ J_m U_m)) -1.0))))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * J_m) * t_0) * sqrt((1.0 + pow((U_m / (t_0 * (J_m * 2.0))), 2.0)));
	double tmp;
	if (t_1 <= -1e+306) {
		tmp = 0.0 - U_m;
	} else if (t_1 <= -5e-48) {
		tmp = fma(U_m, ((U_m * -0.25) / J_m), (-2.0 * J_m));
	} else if (t_1 <= -5e-261) {
		tmp = 0.0 - U_m;
	} else {
		tmp = (0.0 - U_m) * fma((J_m / U_m), (-2.0 * (J_m / U_m)), -1.0);
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * J_m) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_0 * Float64(J_m * 2.0))) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -1e+306)
		tmp = Float64(0.0 - U_m);
	elseif (t_1 <= -5e-48)
		tmp = fma(U_m, Float64(Float64(U_m * -0.25) / J_m), Float64(-2.0 * J_m));
	elseif (t_1 <= -5e-261)
		tmp = Float64(0.0 - U_m);
	else
		tmp = Float64(Float64(0.0 - U_m) * fma(Float64(J_m / U_m), Float64(-2.0 * Float64(J_m / U_m)), -1.0));
	end
	return Float64(J_s * tmp)
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(t$95$0 * N[(J$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$1, -1e+306], N[(0.0 - U$95$m), $MachinePrecision], If[LessEqual[t$95$1, -5e-48], N[(U$95$m * N[(N[(U$95$m * -0.25), $MachinePrecision] / J$95$m), $MachinePrecision] + N[(-2.0 * J$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -5e-261], N[(0.0 - U$95$m), $MachinePrecision], N[(N[(0.0 - U$95$m), $MachinePrecision] * N[(N[(J$95$m / U$95$m), $MachinePrecision] * N[(-2.0 * N[(J$95$m / U$95$m), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]]]), $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))))) < -1.00000000000000002e306 or -4.9999999999999999e-48 < (*.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))))) < -4.99999999999999981e-261

   1. Initial program 30.0%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - U} \]

      3. --lowering--.f64 45.0

         \[\leadsto \color{blue}{0 - U} \]

   5. Simplified 45.0%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 45.0

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

   7. Applied egg-rr 45.0%

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

   if -1.00000000000000002e306 < (*.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))))) < -4.9999999999999999e-48

   1. Initial program 99.7%

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

   3. Taylor expanded in U around 0

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

      1. accelerator-lowering-fma.f64 N/A

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

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

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

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

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

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

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

      5. associate-*r/ N/A

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

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

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

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

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

      8. unpow2 N/A

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

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

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

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

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

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

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

      12. *-lowering-*.f64 77.3

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

   5. Simplified 77.3%

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

   6. Taylor expanded in K around 0

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{J \cdot -2} + \frac{-1}{4} \cdot \frac{{U}^{2}}{J} \]

      2. accelerator-lowering-fma.f64 N/A

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

      3. associate-*r/ N/A

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

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

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

      5. *-commutative N/A

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 41.7

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

   8. Simplified 41.7%

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

      1. +-commutative N/A

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

      2. associate-*l* N/A

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

      3. associate-/l* N/A

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

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

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

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

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

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

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

      7. *-lowering-*.f64 42.2

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

   10. Applied egg-rr 42.2%

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

   if -4.99999999999999981e-261 < (*.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 72.0%

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

   3. Taylor expanded in K around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. +-commutative N/A

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

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

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

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

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

      8. unpow2 N/A

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

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

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

      10. unpow2 N/A

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

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

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

      12. *-lowering-*.f64 32.4

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

   5. Simplified 32.4%

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

   6. Taylor expanded in U around -inf

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

      1. associate-*r* N/A

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

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

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

      3. mul-1-neg N/A

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

      4. neg-sub0 N/A

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

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

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

         \[\leadsto \left(0 - U\right) \cdot \left(-2 \cdot \frac{{J}^{2}}{{U}^{2}} + \color{blue}{-1}\right) \]

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

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

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

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

      10. unpow2 N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 20.1

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

   8. Simplified 20.1%

      \[\leadsto \color{blue}{\left(0 - U\right) \cdot \mathsf{fma}\left(-2, \frac{J \cdot J}{U \cdot U}, -1\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(0 - U\right) \cdot \left(\color{blue}{\frac{J \cdot J}{U \cdot U} \cdot -2} + -1\right) \]

      2. times-frac N/A

         \[\leadsto \left(0 - U\right) \cdot \left(\color{blue}{\left(\frac{J}{U} \cdot \frac{J}{U}\right)} \cdot -2 + -1\right) \]

      3. associate-*l* N/A

         \[\leadsto \left(0 - U\right) \cdot \left(\color{blue}{\frac{J}{U} \cdot \left(\frac{J}{U} \cdot -2\right)} + -1\right) \]

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

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

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

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

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

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

      7. /-lowering-/.f64 24.7

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

   10. Applied egg-rr 24.7%

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

4. Final simplification 34.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -1 \cdot 10^{+306}:\\ \;\;\;\;0 - U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -5 \cdot 10^{-48}:\\ \;\;\;\;\mathsf{fma}\left(U, \frac{U \cdot -0.25}{J}, -2 \cdot J\right)\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -5 \cdot 10^{-261}:\\ \;\;\;\;0 - U\\ \mathbf{else}:\\ \;\;\;\;\left(0 - U\right) \cdot \mathsf{fma}\left(\frac{J}{U}, -2 \cdot \frac{J}{U}, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 62.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_0 \cdot \left(J\_m \cdot 2\right)}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+306}:\\ \;\;\;\;0 - U\_m\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-48}:\\ \;\;\;\;\mathsf{fma}\left(U\_m, \frac{U\_m \cdot -0.25}{J\_m}, -2 \cdot J\_m\right)\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-261}:\\ \;\;\;\;0 - U\_m\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* (* -2.0 J_m) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* t_0 (* J_m 2.0))) 2.0))))))
   (*
    J_s
    (if (<= t_1 -1e+306)
      (- 0.0 U_m)
      (if (<= t_1 -5e-48)
        (fma U_m (/ (* U_m -0.25) J_m) (* -2.0 J_m))
        (if (<= t_1 -5e-261) (- 0.0 U_m) U_m))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * J_m) * t_0) * sqrt((1.0 + pow((U_m / (t_0 * (J_m * 2.0))), 2.0)));
	double tmp;
	if (t_1 <= -1e+306) {
		tmp = 0.0 - U_m;
	} else if (t_1 <= -5e-48) {
		tmp = fma(U_m, ((U_m * -0.25) / J_m), (-2.0 * J_m));
	} else if (t_1 <= -5e-261) {
		tmp = 0.0 - U_m;
	} else {
		tmp = U_m;
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * J_m) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_0 * Float64(J_m * 2.0))) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -1e+306)
		tmp = Float64(0.0 - U_m);
	elseif (t_1 <= -5e-48)
		tmp = fma(U_m, Float64(Float64(U_m * -0.25) / J_m), Float64(-2.0 * J_m));
	elseif (t_1 <= -5e-261)
		tmp = Float64(0.0 - U_m);
	else
		tmp = 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[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(t$95$0 * N[(J$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$1, -1e+306], N[(0.0 - U$95$m), $MachinePrecision], If[LessEqual[t$95$1, -5e-48], N[(U$95$m * N[(N[(U$95$m * -0.25), $MachinePrecision] / J$95$m), $MachinePrecision] + N[(-2.0 * J$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -5e-261], N[(0.0 - U$95$m), $MachinePrecision], U$95$m]]]), $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))))) < -1.00000000000000002e306 or -4.9999999999999999e-48 < (*.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))))) < -4.99999999999999981e-261

   1. Initial program 30.0%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - U} \]

      3. --lowering--.f64 45.0

         \[\leadsto \color{blue}{0 - U} \]

   5. Simplified 45.0%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 45.0

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

   7. Applied egg-rr 45.0%

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

   if -1.00000000000000002e306 < (*.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))))) < -4.9999999999999999e-48

   1. Initial program 99.7%

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

   3. Taylor expanded in U around 0

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

      1. accelerator-lowering-fma.f64 N/A

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

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

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

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

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

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

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

      5. associate-*r/ N/A

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

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

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

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

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

      8. unpow2 N/A

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

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

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

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

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

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

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

      12. *-lowering-*.f64 77.3

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

   5. Simplified 77.3%

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

   6. Taylor expanded in K around 0

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{J \cdot -2} + \frac{-1}{4} \cdot \frac{{U}^{2}}{J} \]

      2. accelerator-lowering-fma.f64 N/A

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

      3. associate-*r/ N/A

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

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

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

      5. *-commutative N/A

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 41.7

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

   8. Simplified 41.7%

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

      1. +-commutative N/A

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

      2. associate-*l* N/A

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

      3. associate-/l* N/A

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

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

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

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

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

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

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

      7. *-lowering-*.f64 42.2

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

   10. Applied egg-rr 42.2%

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

   if -4.99999999999999981e-261 < (*.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 72.0%

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

   3. Taylor expanded in U around -inf

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

   5. Simplified 24.6%

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

4. Final simplification 34.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -1 \cdot 10^{+306}:\\ \;\;\;\;0 - U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -5 \cdot 10^{-48}:\\ \;\;\;\;\mathsf{fma}\left(U, \frac{U \cdot -0.25}{J}, -2 \cdot J\right)\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -5 \cdot 10^{-261}:\\ \;\;\;\;0 - U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 62.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_0 \cdot \left(J\_m \cdot 2\right)}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+306}:\\ \;\;\;\;0 - U\_m\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-48}:\\ \;\;\;\;-2 \cdot J\_m\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-261}:\\ \;\;\;\;0 - U\_m\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* (* -2.0 J_m) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* t_0 (* J_m 2.0))) 2.0))))))
   (*
    J_s
    (if (<= t_1 -1e+306)
      (- 0.0 U_m)
      (if (<= t_1 -5e-48)
        (* -2.0 J_m)
        (if (<= t_1 -5e-261) (- 0.0 U_m) U_m))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * J_m) * t_0) * sqrt((1.0 + pow((U_m / (t_0 * (J_m * 2.0))), 2.0)));
	double tmp;
	if (t_1 <= -1e+306) {
		tmp = 0.0 - U_m;
	} else if (t_1 <= -5e-48) {
		tmp = -2.0 * J_m;
	} else if (t_1 <= -5e-261) {
		tmp = 0.0 - U_m;
	} else {
		tmp = U_m;
	}
	return J_s * tmp;
}

Fortran

U_m = abs(u)
J\_m = abs(j)
J\_s = copysign(1.0d0, j)
real(8) function code(j_s, j_m, k, u_m)
    real(8), intent (in) :: j_s
    real(8), intent (in) :: j_m
    real(8), intent (in) :: k
    real(8), intent (in) :: u_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    t_1 = (((-2.0d0) * j_m) * t_0) * sqrt((1.0d0 + ((u_m / (t_0 * (j_m * 2.0d0))) ** 2.0d0)))
    if (t_1 <= (-1d+306)) then
        tmp = 0.0d0 - u_m
    else if (t_1 <= (-5d-48)) then
        tmp = (-2.0d0) * j_m
    else if (t_1 <= (-5d-261)) then
        tmp = 0.0d0 - u_m
    else
        tmp = u_m
    end if
    code = j_s * tmp
end function

Java

U_m = Math.abs(U);
J\_m = Math.abs(J);
J\_s = Math.copySign(1.0, J);
public static double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = ((-2.0 * J_m) * t_0) * Math.sqrt((1.0 + Math.pow((U_m / (t_0 * (J_m * 2.0))), 2.0)));
	double tmp;
	if (t_1 <= -1e+306) {
		tmp = 0.0 - U_m;
	} else if (t_1 <= -5e-48) {
		tmp = -2.0 * J_m;
	} else if (t_1 <= -5e-261) {
		tmp = 0.0 - U_m;
	} else {
		tmp = U_m;
	}
	return J_s * tmp;
}

Python

U_m = math.fabs(U)
J\_m = math.fabs(J)
J\_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	t_0 = math.cos((K / 2.0))
	t_1 = ((-2.0 * J_m) * t_0) * math.sqrt((1.0 + math.pow((U_m / (t_0 * (J_m * 2.0))), 2.0)))
	tmp = 0
	if t_1 <= -1e+306:
		tmp = 0.0 - U_m
	elif t_1 <= -5e-48:
		tmp = -2.0 * J_m
	elif t_1 <= -5e-261:
		tmp = 0.0 - U_m
	else:
		tmp = U_m
	return J_s * tmp

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * J_m) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_0 * Float64(J_m * 2.0))) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -1e+306)
		tmp = Float64(0.0 - U_m);
	elseif (t_1 <= -5e-48)
		tmp = Float64(-2.0 * J_m);
	elseif (t_1 <= -5e-261)
		tmp = Float64(0.0 - U_m);
	else
		tmp = U_m;
	end
	return Float64(J_s * tmp)
end

MATLAB

U_m = abs(U);
J\_m = abs(J);
J\_s = sign(J) * abs(1.0);
function tmp_2 = code(J_s, J_m, K, U_m)
	t_0 = cos((K / 2.0));
	t_1 = ((-2.0 * J_m) * t_0) * sqrt((1.0 + ((U_m / (t_0 * (J_m * 2.0))) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= -1e+306)
		tmp = 0.0 - U_m;
	elseif (t_1 <= -5e-48)
		tmp = -2.0 * J_m;
	elseif (t_1 <= -5e-261)
		tmp = 0.0 - U_m;
	else
		tmp = U_m;
	end
	tmp_2 = J_s * tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(t$95$0 * N[(J$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$1, -1e+306], N[(0.0 - U$95$m), $MachinePrecision], If[LessEqual[t$95$1, -5e-48], N[(-2.0 * J$95$m), $MachinePrecision], If[LessEqual[t$95$1, -5e-261], N[(0.0 - U$95$m), $MachinePrecision], U$95$m]]]), $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))))) < -1.00000000000000002e306 or -4.9999999999999999e-48 < (*.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))))) < -4.99999999999999981e-261

   1. Initial program 30.0%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - U} \]

      3. --lowering--.f64 45.0

         \[\leadsto \color{blue}{0 - U} \]

   5. Simplified 45.0%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 45.0

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

   7. Applied egg-rr 45.0%

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

   if -1.00000000000000002e306 < (*.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))))) < -4.9999999999999999e-48

   1. Initial program 99.7%

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

   3. Taylor expanded in K around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. +-commutative N/A

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

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

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

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

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

      8. unpow2 N/A

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

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

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

      10. unpow2 N/A

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

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

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

      12. *-lowering-*.f64 44.3

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

   5. Simplified 44.3%

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

   6. Taylor expanded in U around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 42.1

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

   8. Simplified 42.1%

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

   if -4.99999999999999981e-261 < (*.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 72.0%

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

   3. Taylor expanded in U around -inf

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

   5. Simplified 24.6%

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

4. Final simplification 34.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -1 \cdot 10^{+306}:\\ \;\;\;\;0 - U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -5 \cdot 10^{-48}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -5 \cdot 10^{-261}:\\ \;\;\;\;0 - U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(-2 \cdot J\_m\right) \cdot t\_0\\ t_2 := t\_1 \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_0 \cdot \left(J\_m \cdot 2\right)}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;0 - U\_m\\ \mathbf{elif}\;t\_2 \leq 10^{+284}:\\ \;\;\;\;t\_1 \cdot \sqrt{1 + \frac{U\_m}{J\_m \cdot 2} \cdot \frac{U\_m}{\left(J\_m \cdot 2\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(K \cdot 0.5\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1 (* (* -2.0 J_m) t_0))
        (t_2 (* t_1 (sqrt (+ 1.0 (pow (/ U_m (* t_0 (* J_m 2.0))) 2.0))))))
   (*
    J_s
    (if (<= t_2 (- INFINITY))
      (- 0.0 U_m)
      (if (<= t_2 1e+284)
        (*
         t_1
         (sqrt
          (+
           1.0
           (*
            (/ U_m (* J_m 2.0))
            (/ U_m (* (* J_m 2.0) (+ 0.5 (* 0.5 (cos (* 2.0 (* K 0.5)))))))))))
        U_m)))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = (-2.0 * J_m) * t_0;
	double t_2 = t_1 * sqrt((1.0 + pow((U_m / (t_0 * (J_m * 2.0))), 2.0)));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = 0.0 - U_m;
	} else if (t_2 <= 1e+284) {
		tmp = t_1 * sqrt((1.0 + ((U_m / (J_m * 2.0)) * (U_m / ((J_m * 2.0) * (0.5 + (0.5 * cos((2.0 * (K * 0.5))))))))));
	} else {
		tmp = U_m;
	}
	return J_s * tmp;
}

Java

U_m = Math.abs(U);
J\_m = Math.abs(J);
J\_s = Math.copySign(1.0, J);
public static double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = (-2.0 * J_m) * t_0;
	double t_2 = t_1 * Math.sqrt((1.0 + Math.pow((U_m / (t_0 * (J_m * 2.0))), 2.0)));
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = 0.0 - U_m;
	} else if (t_2 <= 1e+284) {
		tmp = t_1 * Math.sqrt((1.0 + ((U_m / (J_m * 2.0)) * (U_m / ((J_m * 2.0) * (0.5 + (0.5 * Math.cos((2.0 * (K * 0.5))))))))));
	} else {
		tmp = U_m;
	}
	return J_s * tmp;
}

Python

U_m = math.fabs(U)
J\_m = math.fabs(J)
J\_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	t_0 = math.cos((K / 2.0))
	t_1 = (-2.0 * J_m) * t_0
	t_2 = t_1 * math.sqrt((1.0 + math.pow((U_m / (t_0 * (J_m * 2.0))), 2.0)))
	tmp = 0
	if t_2 <= -math.inf:
		tmp = 0.0 - U_m
	elif t_2 <= 1e+284:
		tmp = t_1 * math.sqrt((1.0 + ((U_m / (J_m * 2.0)) * (U_m / ((J_m * 2.0) * (0.5 + (0.5 * math.cos((2.0 * (K * 0.5))))))))))
	else:
		tmp = U_m
	return J_s * tmp

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(-2.0 * J_m) * t_0)
	t_2 = Float64(t_1 * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_0 * Float64(J_m * 2.0))) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(0.0 - U_m);
	elseif (t_2 <= 1e+284)
		tmp = Float64(t_1 * sqrt(Float64(1.0 + Float64(Float64(U_m / Float64(J_m * 2.0)) * Float64(U_m / Float64(Float64(J_m * 2.0) * Float64(0.5 + Float64(0.5 * cos(Float64(2.0 * Float64(K * 0.5)))))))))));
	else
		tmp = U_m;
	end
	return Float64(J_s * tmp)
end

MATLAB

U_m = abs(U);
J\_m = abs(J);
J\_s = sign(J) * abs(1.0);
function tmp_2 = code(J_s, J_m, K, U_m)
	t_0 = cos((K / 2.0));
	t_1 = (-2.0 * J_m) * t_0;
	t_2 = t_1 * sqrt((1.0 + ((U_m / (t_0 * (J_m * 2.0))) ^ 2.0)));
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = 0.0 - U_m;
	elseif (t_2 <= 1e+284)
		tmp = t_1 * sqrt((1.0 + ((U_m / (J_m * 2.0)) * (U_m / ((J_m * 2.0) * (0.5 + (0.5 * cos((2.0 * (K * 0.5))))))))));
	else
		tmp = U_m;
	end
	tmp_2 = J_s * tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(t$95$0 * N[(J$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$2, (-Infinity)], N[(0.0 - U$95$m), $MachinePrecision], If[LessEqual[t$95$2, 1e+284], N[(t$95$1 * N[Sqrt[N[(1.0 + N[(N[(U$95$m / N[(J$95$m * 2.0), $MachinePrecision]), $MachinePrecision] * N[(U$95$m / N[(N[(J$95$m * 2.0), $MachinePrecision] * N[(0.5 + N[(0.5 * N[Cos[N[(2.0 * N[(K * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], U$95$m]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 6.1%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - U} \]

      3. --lowering--.f64 52.5

         \[\leadsto \color{blue}{0 - U} \]

   5. Simplified 52.5%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 52.5

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

   7. Applied egg-rr 52.5%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 1.00000000000000008e284

   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. unpow2 N/A

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

      2. associate-/r* N/A

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

      3. frac-times N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

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

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

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

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

      8. *-commutative N/A

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

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

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

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

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

      11. associate-*l* N/A

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

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

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

      13. *-commutative N/A

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

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

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

      15. sqr-cos-a N/A

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

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

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

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

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

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

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

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

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

   4. Applied egg-rr 99.6%

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

   if 1.00000000000000008e284 < (*.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 15.6%

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

   3. Taylor expanded in U around -inf

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

   5. Simplified 44.0%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -\infty:\\ \;\;\;\;0 - U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 10^{+284}:\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + \frac{U}{J \cdot 2} \cdot \frac{U}{\left(J \cdot 2\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(K \cdot 0.5\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 98.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_0 \cdot \left(J\_m \cdot 2\right)}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;0 - U\_m\\ \mathbf{elif}\;t\_1 \leq 10^{+284}:\\ \;\;\;\;\left(-2 \cdot \left(J\_m \cdot \cos \left(K \cdot 0.5\right)\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U\_m}{\mathsf{fma}\left(J\_m, \cos K, J\_m\right)}, \frac{U\_m}{J\_m \cdot 2}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* (* -2.0 J_m) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* t_0 (* J_m 2.0))) 2.0))))))
   (*
    J_s
    (if (<= t_1 (- INFINITY))
      (- 0.0 U_m)
      (if (<= t_1 1e+284)
        (*
         (* -2.0 (* J_m (cos (* K 0.5))))
         (sqrt (fma (/ U_m (fma J_m (cos K) J_m)) (/ U_m (* J_m 2.0)) 1.0)))
        U_m)))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * J_m) * t_0) * sqrt((1.0 + pow((U_m / (t_0 * (J_m * 2.0))), 2.0)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = 0.0 - U_m;
	} else if (t_1 <= 1e+284) {
		tmp = (-2.0 * (J_m * cos((K * 0.5)))) * sqrt(fma((U_m / fma(J_m, cos(K), J_m)), (U_m / (J_m * 2.0)), 1.0));
	} else {
		tmp = U_m;
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * J_m) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_0 * Float64(J_m * 2.0))) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(0.0 - U_m);
	elseif (t_1 <= 1e+284)
		tmp = Float64(Float64(-2.0 * Float64(J_m * cos(Float64(K * 0.5)))) * sqrt(fma(Float64(U_m / fma(J_m, cos(K), J_m)), Float64(U_m / Float64(J_m * 2.0)), 1.0)));
	else
		tmp = 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[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(t$95$0 * N[(J$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$1, (-Infinity)], N[(0.0 - U$95$m), $MachinePrecision], If[LessEqual[t$95$1, 1e+284], N[(N[(-2.0 * N[(J$95$m * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(U$95$m / N[(J$95$m * N[Cos[K], $MachinePrecision] + J$95$m), $MachinePrecision]), $MachinePrecision] * N[(U$95$m / N[(J$95$m * 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], U$95$m]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 6.1%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - U} \]

      3. --lowering--.f64 52.5

         \[\leadsto \color{blue}{0 - U} \]

   5. Simplified 52.5%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 52.5

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

   7. Applied egg-rr 52.5%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 1.00000000000000008e284

   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. unpow2 N/A

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

      2. associate-/r* N/A

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

      3. frac-times N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

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

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

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

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

      8. *-commutative N/A

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

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

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

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

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

      11. associate-*l* N/A

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

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

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

      13. *-commutative N/A

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

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

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

      15. sqr-cos-a N/A

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

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

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

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

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

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

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

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

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

   4. Applied egg-rr 99.6%

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

      1. associate-/r* N/A

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

      2. frac-times N/A

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

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

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

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

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

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

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

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

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

      7. +-commutative N/A

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

      8. distribute-rgt-in N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. associate-*l* N/A

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

      12. metadata-eval N/A

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

      13. *-rgt-identity N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

      16. metadata-eval N/A

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

      17. *-lft-identity N/A

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

   6. Applied egg-rr 95.4%

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

      1. associate-*l* N/A

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

      2. div-inv N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

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

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

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

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 95.4

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

   8. Applied egg-rr 95.4%

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

      1. +-commutative N/A

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

      2. times-frac N/A

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

      3. *-commutative N/A

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

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

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

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

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

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

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

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

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

      8. associate-/l/ N/A

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

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

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 99.6

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

   10. Applied egg-rr 99.6%

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

   if 1.00000000000000008e284 < (*.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 15.6%

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

   3. Taylor expanded in U around -inf

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

   5. Simplified 44.0%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -\infty:\\ \;\;\;\;0 - U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 10^{+284}:\\ \;\;\;\;\left(-2 \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U}{\mathsf{fma}\left(J, \cos K, J\right)}, \frac{U}{J \cdot 2}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 89.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(-2 \cdot J\_m\right) \cdot t\_0\\ t_2 := t\_1 \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_0 \cdot \left(J\_m \cdot 2\right)}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;0 - U\_m\\ \mathbf{elif}\;t\_2 \leq 10^{+284}:\\ \;\;\;\;t\_1 \cdot \sqrt{\mathsf{fma}\left(\frac{U\_m \cdot 0.25}{J\_m}, \frac{U\_m}{J\_m}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1 (* (* -2.0 J_m) t_0))
        (t_2 (* t_1 (sqrt (+ 1.0 (pow (/ U_m (* t_0 (* J_m 2.0))) 2.0))))))
   (*
    J_s
    (if (<= t_2 (- INFINITY))
      (- 0.0 U_m)
      (if (<= t_2 1e+284)
        (* t_1 (sqrt (fma (/ (* U_m 0.25) J_m) (/ U_m J_m) 1.0)))
        U_m)))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = (-2.0 * J_m) * t_0;
	double t_2 = t_1 * sqrt((1.0 + pow((U_m / (t_0 * (J_m * 2.0))), 2.0)));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = 0.0 - U_m;
	} else if (t_2 <= 1e+284) {
		tmp = t_1 * sqrt(fma(((U_m * 0.25) / J_m), (U_m / J_m), 1.0));
	} else {
		tmp = U_m;
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(-2.0 * J_m) * t_0)
	t_2 = Float64(t_1 * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_0 * Float64(J_m * 2.0))) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(0.0 - U_m);
	elseif (t_2 <= 1e+284)
		tmp = Float64(t_1 * sqrt(fma(Float64(Float64(U_m * 0.25) / J_m), Float64(U_m / J_m), 1.0)));
	else
		tmp = 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[(-2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(t$95$0 * N[(J$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$2, (-Infinity)], N[(0.0 - U$95$m), $MachinePrecision], If[LessEqual[t$95$2, 1e+284], N[(t$95$1 * N[Sqrt[N[(N[(N[(U$95$m * 0.25), $MachinePrecision] / J$95$m), $MachinePrecision] * N[(U$95$m / J$95$m), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], U$95$m]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 6.1%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - U} \]

      3. --lowering--.f64 52.5

         \[\leadsto \color{blue}{0 - U} \]

   5. Simplified 52.5%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 52.5

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

   7. Applied egg-rr 52.5%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 1.00000000000000008e284

   1. Initial program 99.8%

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

   3. Taylor expanded in K around 0

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

      1. sqrt-lowering-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. accelerator-lowering-fma.f64 N/A

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

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

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

      5. unpow2 N/A

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 70.8

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

   5. Simplified 70.8%

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

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

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

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

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

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

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

      7. /-lowering-/.f64 89.7

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

   7. Applied egg-rr 89.7%

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

   if 1.00000000000000008e284 < (*.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 15.6%

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

   3. Taylor expanded in U around -inf

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

   5. Simplified 44.0%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -\infty:\\ \;\;\;\;0 - U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 10^{+284}:\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U \cdot 0.25}{J}, \frac{U}{J}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 77.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_0 \cdot \left(J\_m \cdot 2\right)}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;0 - U\_m\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-261}:\\ \;\;\;\;\left(-2 \cdot J\_m\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U\_m \cdot 0.25}{J\_m}, \frac{U\_m}{J\_m}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(0 - U\_m\right) \cdot \mathsf{fma}\left(\frac{J\_m}{U\_m}, -2 \cdot \frac{J\_m}{U\_m}, -1\right)\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* (* -2.0 J_m) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* t_0 (* J_m 2.0))) 2.0))))))
   (*
    J_s
    (if (<= t_1 (- INFINITY))
      (- 0.0 U_m)
      (if (<= t_1 -5e-261)
        (* (* -2.0 J_m) (sqrt (fma (/ (* U_m 0.25) J_m) (/ U_m J_m) 1.0)))
        (* (- 0.0 U_m) (fma (/ J_m U_m) (* -2.0 (/ J_m U_m)) -1.0)))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * J_m) * t_0) * sqrt((1.0 + pow((U_m / (t_0 * (J_m * 2.0))), 2.0)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = 0.0 - U_m;
	} else if (t_1 <= -5e-261) {
		tmp = (-2.0 * J_m) * sqrt(fma(((U_m * 0.25) / J_m), (U_m / J_m), 1.0));
	} else {
		tmp = (0.0 - U_m) * fma((J_m / U_m), (-2.0 * (J_m / U_m)), -1.0);
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * J_m) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_0 * Float64(J_m * 2.0))) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(0.0 - U_m);
	elseif (t_1 <= -5e-261)
		tmp = Float64(Float64(-2.0 * J_m) * sqrt(fma(Float64(Float64(U_m * 0.25) / J_m), Float64(U_m / J_m), 1.0)));
	else
		tmp = Float64(Float64(0.0 - U_m) * fma(Float64(J_m / U_m), Float64(-2.0 * Float64(J_m / U_m)), -1.0));
	end
	return Float64(J_s * tmp)
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(t$95$0 * N[(J$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$1, (-Infinity)], N[(0.0 - U$95$m), $MachinePrecision], If[LessEqual[t$95$1, -5e-261], N[(N[(-2.0 * J$95$m), $MachinePrecision] * N[Sqrt[N[(N[(N[(U$95$m * 0.25), $MachinePrecision] / J$95$m), $MachinePrecision] * N[(U$95$m / J$95$m), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(0.0 - U$95$m), $MachinePrecision] * N[(N[(J$95$m / U$95$m), $MachinePrecision] * N[(-2.0 * N[(J$95$m / U$95$m), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 6.1%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - U} \]

      3. --lowering--.f64 52.5

         \[\leadsto \color{blue}{0 - U} \]

   5. Simplified 52.5%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 52.5

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

   7. Applied egg-rr 52.5%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -4.99999999999999981e-261

   1. Initial program 99.7%

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

   3. Taylor expanded in K around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. +-commutative N/A

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

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

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

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

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

      8. unpow2 N/A

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

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

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

      10. unpow2 N/A

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

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

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

      12. *-lowering-*.f64 39.7

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

   5. Simplified 39.7%

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

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

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

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

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

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

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

      7. /-lowering-/.f64 56.3

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

   7. Applied egg-rr 56.3%

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

   if -4.99999999999999981e-261 < (*.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 72.0%

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

   3. Taylor expanded in K around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. +-commutative N/A

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

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

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

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

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

      8. unpow2 N/A

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

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

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

      10. unpow2 N/A

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

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

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

      12. *-lowering-*.f64 32.4

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

   5. Simplified 32.4%

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

   6. Taylor expanded in U around -inf

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

      1. associate-*r* N/A

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

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

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

      3. mul-1-neg N/A

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

      4. neg-sub0 N/A

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

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

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

         \[\leadsto \left(0 - U\right) \cdot \left(-2 \cdot \frac{{J}^{2}}{{U}^{2}} + \color{blue}{-1}\right) \]

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

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

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

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

      10. unpow2 N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 20.1

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

   8. Simplified 20.1%

      \[\leadsto \color{blue}{\left(0 - U\right) \cdot \mathsf{fma}\left(-2, \frac{J \cdot J}{U \cdot U}, -1\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(0 - U\right) \cdot \left(\color{blue}{\frac{J \cdot J}{U \cdot U} \cdot -2} + -1\right) \]

      2. times-frac N/A

         \[\leadsto \left(0 - U\right) \cdot \left(\color{blue}{\left(\frac{J}{U} \cdot \frac{J}{U}\right)} \cdot -2 + -1\right) \]

      3. associate-*l* N/A

         \[\leadsto \left(0 - U\right) \cdot \left(\color{blue}{\frac{J}{U} \cdot \left(\frac{J}{U} \cdot -2\right)} + -1\right) \]

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

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

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

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

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

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

      7. /-lowering-/.f64 24.7

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

   10. Applied egg-rr 24.7%

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

4. Final simplification 41.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -\infty:\\ \;\;\;\;0 - U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -5 \cdot 10^{-261}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U \cdot 0.25}{J}, \frac{U}{J}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(0 - U\right) \cdot \mathsf{fma}\left(\frac{J}{U}, -2 \cdot \frac{J}{U}, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 51.8% accurate, 3.1× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ J\_s \cdot \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -1 \cdot 10^{-309}:\\ \;\;\;\;U\_m\\ \mathbf{else}:\\ \;\;\;\;0 - U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (* J_s (if (<= (cos (/ K 2.0)) -1e-309) U_m (- 0.0 U_m))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double tmp;
	if (cos((K / 2.0)) <= -1e-309) {
		tmp = U_m;
	} else {
		tmp = 0.0 - U_m;
	}
	return J_s * tmp;
}

Fortran

U_m = abs(u)
J\_m = abs(j)
J\_s = copysign(1.0d0, j)
real(8) function code(j_s, j_m, k, u_m)
    real(8), intent (in) :: j_s
    real(8), intent (in) :: j_m
    real(8), intent (in) :: k
    real(8), intent (in) :: u_m
    real(8) :: tmp
    if (cos((k / 2.0d0)) <= (-1d-309)) then
        tmp = u_m
    else
        tmp = 0.0d0 - u_m
    end if
    code = j_s * tmp
end function

Java

U_m = Math.abs(U);
J\_m = Math.abs(J);
J\_s = Math.copySign(1.0, J);
public static double code(double J_s, double J_m, double K, double U_m) {
	double tmp;
	if (Math.cos((K / 2.0)) <= -1e-309) {
		tmp = U_m;
	} else {
		tmp = 0.0 - U_m;
	}
	return J_s * tmp;
}

Python

U_m = math.fabs(U)
J\_m = math.fabs(J)
J\_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	tmp = 0
	if math.cos((K / 2.0)) <= -1e-309:
		tmp = U_m
	else:
		tmp = 0.0 - U_m
	return J_s * tmp

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -1e-309)
		tmp = U_m;
	else
		tmp = Float64(0.0 - U_m);
	end
	return Float64(J_s * tmp)
end

MATLAB

U_m = abs(U);
J\_m = abs(J);
J\_s = sign(J) * abs(1.0);
function tmp_2 = code(J_s, J_m, K, U_m)
	tmp = 0.0;
	if (cos((K / 2.0)) <= -1e-309)
		tmp = U_m;
	else
		tmp = 0.0 - U_m;
	end
	tmp_2 = J_s * tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := N[(J$95$s * If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -1e-309], U$95$m, N[(0.0 - U$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -1.000000000000002e-309

   1. Initial program 73.6%

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

   3. Taylor expanded in U around -inf

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

   5. Simplified 21.3%

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

   if -1.000000000000002e-309 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 70.0%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - U} \]

      3. --lowering--.f64 30.0

         \[\leadsto \color{blue}{0 - U} \]

   5. Simplified 30.0%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 30.0

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

   7. Applied egg-rr 30.0%

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

4. Final simplification 27.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -1 \cdot 10^{-309}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;0 - U\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 14.1% accurate, 373.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.8%

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

3. Taylor expanded in U around -inf

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

5. Simplified 24.5%

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

Reproduce

herbie shell --seed 2024199 
(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)))))