Maksimov and Kolovsky, Equation (3)

Percentage Accurate: 73.9% → 98.8%

Time: 8.4s

Alternatives: 13

Speedup: 0.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.9% 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: 98.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+280}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \end{array} \]

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 (* (* 2.0 J_m) t_0)) 2.0))))))
   (* J_s (if (<= t_1 (- INFINITY)) (- U_m) (if (<= t_1 2e+280) 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 / ((2.0 * J_m) * t_0)), 2.0)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_1 <= 2e+280) {
		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 / ((2.0 * J_m) * t_0)), 2.0)));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = -U_m;
	} else if (t_1 <= 2e+280) {
		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 / ((2.0 * J_m) * t_0)), 2.0)))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = -U_m
	elif t_1 <= 2e+280:
		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(Float64(2.0 * J_m) * t_0)) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_1 <= 2e+280)
		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 / ((2.0 * J_m) * t_0)) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = -U_m;
	elseif (t_1 <= 2e+280)
		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[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$1, (-Infinity)], (-U$95$m), If[LessEqual[t$95$1, 2e+280], 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 5.4%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 36.6

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

   5. Applied rewrites 36.6%

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

   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))))) < 2.0000000000000001e280

   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 2.0000000000000001e280 < (*.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 26.8%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 27.7

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

   5. Applied rewrites 27.7%

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

   7. Applied rewrites 14.2%

      \[\leadsto \frac{0 - U \cdot U}{\color{blue}{0 + U}} \]
   8. Step-by-step derivation

   9. Applied rewrites 56.9%

      \[\leadsto U \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 91.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \left(-2 \cdot J\_m\right) \cdot \cos \left(0.5 \cdot K\right)\\ t_1 := \cos \left(\frac{K}{2}\right)\\ t_2 := \left(\left(-2 \cdot J\_m\right) \cdot t\_1\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_1}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-167}:\\ \;\;\;\;t\_0 \cdot \sqrt{\mathsf{fma}\left(0.25, \frac{U\_m}{J\_m} \cdot \frac{U\_m}{J\_m}, 1\right)}\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+280}:\\ \;\;\;\;t\_0 \cdot \sqrt{\mathsf{fma}\left(0.25, {\left(\cos \left(-0.5 \cdot K\right) \cdot J\_m\right)}^{-2} \cdot \left(U\_m \cdot U\_m\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \end{array} \]

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 (* (* -2.0 J_m) (cos (* 0.5 K))))
        (t_1 (cos (/ K 2.0)))
        (t_2
         (*
          (* (* -2.0 J_m) t_1)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_1)) 2.0))))))
   (*
    J_s
    (if (<= t_2 (- INFINITY))
      (- U_m)
      (if (<= t_2 5e-167)
        (* t_0 (sqrt (fma 0.25 (* (/ U_m J_m) (/ U_m J_m)) 1.0)))
        (if (<= t_2 2e+280)
          (*
           t_0
           (sqrt
            (fma
             0.25
             (* (pow (* (cos (* -0.5 K)) J_m) -2.0) (* U_m U_m))
             1.0)))
          U_m))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = (-2.0 * J_m) * cos((0.5 * K));
	double t_1 = cos((K / 2.0));
	double t_2 = ((-2.0 * J_m) * t_1) * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_1)), 2.0)));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_2 <= 5e-167) {
		tmp = t_0 * sqrt(fma(0.25, ((U_m / J_m) * (U_m / J_m)), 1.0));
	} else if (t_2 <= 2e+280) {
		tmp = t_0 * sqrt(fma(0.25, (pow((cos((-0.5 * K)) * J_m), -2.0) * (U_m * U_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 = Float64(Float64(-2.0 * J_m) * cos(Float64(0.5 * K)))
	t_1 = cos(Float64(K / 2.0))
	t_2 = Float64(Float64(Float64(-2.0 * J_m) * t_1) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * t_1)) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_2 <= 5e-167)
		tmp = Float64(t_0 * sqrt(fma(0.25, Float64(Float64(U_m / J_m) * Float64(U_m / J_m)), 1.0)));
	elseif (t_2 <= 2e+280)
		tmp = Float64(t_0 * sqrt(fma(0.25, Float64((Float64(cos(Float64(-0.5 * K)) * J_m) ^ -2.0) * Float64(U_m * U_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[(N[(-2.0 * J$95$m), $MachinePrecision] * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$1), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$2, (-Infinity)], (-U$95$m), If[LessEqual[t$95$2, 5e-167], N[(t$95$0 * N[Sqrt[N[(0.25 * N[(N[(U$95$m / J$95$m), $MachinePrecision] * N[(U$95$m / J$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+280], N[(t$95$0 * N[Sqrt[N[(0.25 * N[(N[Power[N[(N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision] * J$95$m), $MachinePrecision], -2.0], $MachinePrecision] * N[(U$95$m * U$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $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 5.4%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 36.6

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

   5. Applied rewrites 36.6%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 5.0000000000000002e-167

   1. Initial program 99.8%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. inv-pow N/A

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

      5. pow-pow N/A

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

      6. metadata-eval N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 99.7

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

      14. lift-cos.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. metadata-eval N/A

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

      17. distribute-neg-frac2 N/A

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

      18. cos-neg N/A

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

      19. lower-cos.f64 N/A

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

      20. div-inv N/A

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

      21. lower-*.f64 N/A

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

      22. metadata-eval 99.7

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

   4. Applied rewrites 99.7%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 99.7

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

   6. Applied rewrites 99.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/l* N/A

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

      8. associate-*l* N/A

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

      9. unpow-prod-down N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 99.7

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 99.7

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

   8. Applied rewrites 99.7%

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

   9. Taylor expanded in K around 0

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

       1. unpow2 N/A

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

       2. unpow2 N/A

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

       3. times-frac N/A

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

       4. lower-*.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. lower-/.f64 89.0

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

   11. Applied rewrites 89.0%

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

   if 5.0000000000000002e-167 < (*.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))))) < 2.0000000000000001e280

   1. Initial program 99.8%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. inv-pow N/A

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

      5. pow-pow N/A

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

      6. metadata-eval N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 99.7

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

      14. lift-cos.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. metadata-eval N/A

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

      17. distribute-neg-frac2 N/A

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

      18. cos-neg N/A

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

      19. lower-cos.f64 N/A

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

      20. div-inv N/A

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

      21. lower-*.f64 N/A

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

      22. metadata-eval 99.7

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

   4. Applied rewrites 99.7%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 99.7

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

   6. Applied rewrites 99.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/l* N/A

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

      8. associate-*l* N/A

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

      9. unpow-prod-down N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 99.7

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 99.7

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

   8. Applied rewrites 99.7%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. div-inv N/A

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

      6. unpow-prod-down N/A

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

      7. inv-pow N/A

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

      8. metadata-eval N/A

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

      9. pow-pow N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

   10. Applied rewrites 81.4%

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

   if 2.0000000000000001e280 < (*.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 26.8%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 27.7

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

   5. Applied rewrites 27.7%

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

   7. Applied rewrites 14.2%

      \[\leadsto \frac{0 - U \cdot U}{\color{blue}{0 + U}} \]
   8. Step-by-step derivation

   9. Applied rewrites 56.9%

      \[\leadsto U \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 3: 84.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{+195}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{0.25 \cdot U\_m}{J\_m}, \frac{U\_m}{J\_m}, 1\right)} \cdot \left(-2 \cdot J\_m\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+280}:\\ \;\;\;\;\left(\left(-2 \cdot J\_m\right) \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{0.25}{J\_m}, \frac{U\_m \cdot U\_m}{J\_m}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \end{array} \]

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)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_0)) 2.0))))))
   (*
    J_s
    (if (<= t_1 (- INFINITY))
      (- U_m)
      (if (<= t_1 -2e+195)
        (* (sqrt (fma (/ (* 0.25 U_m) J_m) (/ U_m J_m) 1.0)) (* -2.0 J_m))
        (if (<= t_1 2e+280)
          (*
           (* (* -2.0 J_m) (cos (* 0.5 K)))
           (sqrt (fma (/ 0.25 J_m) (/ (* U_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) * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_0)), 2.0)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_1 <= -2e+195) {
		tmp = sqrt(fma(((0.25 * U_m) / J_m), (U_m / J_m), 1.0)) * (-2.0 * J_m);
	} else if (t_1 <= 2e+280) {
		tmp = ((-2.0 * J_m) * cos((0.5 * K))) * sqrt(fma((0.25 / J_m), ((U_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(Float64(-2.0 * J_m) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * t_0)) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_1 <= -2e+195)
		tmp = Float64(sqrt(fma(Float64(Float64(0.25 * U_m) / J_m), Float64(U_m / J_m), 1.0)) * Float64(-2.0 * J_m));
	elseif (t_1 <= 2e+280)
		tmp = Float64(Float64(Float64(-2.0 * J_m) * cos(Float64(0.5 * K))) * sqrt(fma(Float64(0.25 / J_m), Float64(Float64(U_m * 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[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$1, (-Infinity)], (-U$95$m), If[LessEqual[t$95$1, -2e+195], N[(N[Sqrt[N[(N[(N[(0.25 * U$95$m), $MachinePrecision] / J$95$m), $MachinePrecision] * N[(U$95$m / J$95$m), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(-2.0 * J$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+280], N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(0.25 / J$95$m), $MachinePrecision] * N[(N[(U$95$m * U$95$m), $MachinePrecision] / J$95$m), $MachinePrecision] + 1.0), $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 5.4%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 36.6

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

   5. Applied rewrites 36.6%

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

   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))))) < -1.99999999999999995e195

   1. Initial program 99.7%

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

   3. Taylor expanded in K around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-*r/ N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. times-frac N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 49.8

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

   5. Applied rewrites 49.8%

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

   if -1.99999999999999995e195 < (*.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))))) < 2.0000000000000001e280

   1. Initial program 99.8%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. inv-pow N/A

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

      5. pow-pow N/A

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

      6. metadata-eval N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 99.7

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

      14. lift-cos.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. metadata-eval N/A

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

      17. distribute-neg-frac2 N/A

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

      18. cos-neg N/A

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

      19. lower-cos.f64 N/A

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

      20. div-inv N/A

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

      21. lower-*.f64 N/A

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

      22. metadata-eval 99.7

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

   4. Applied rewrites 99.7%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 99.7

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in K around 0

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

      1. lower-sqrt.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. unpow2 N/A

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

      5. times-frac N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 79.9

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

   9. Applied rewrites 79.9%

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

   if 2.0000000000000001e280 < (*.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 26.8%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 27.7

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

   5. Applied rewrites 27.7%

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

   7. Applied rewrites 14.2%

      \[\leadsto \frac{0 - U \cdot U}{\color{blue}{0 + U}} \]
   8. Step-by-step derivation

   9. Applied rewrites 56.9%

      \[\leadsto U \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 4: 83.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-279}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{0.25 \cdot U\_m}{J\_m}, \frac{U\_m}{J\_m}, 1\right)} \cdot \left(-2 \cdot J\_m\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+280}:\\ \;\;\;\;\left(\left(-2 \cdot J\_m\right) \cdot \cos \left(0.5 \cdot K\right)\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \end{array} \]

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 (* (* 2.0 J_m) t_0)) 2.0))))))
   (*
    J_s
    (if (<= t_1 (- INFINITY))
      (- U_m)
      (if (<= t_1 -5e-279)
        (* (sqrt (fma (/ (* 0.25 U_m) J_m) (/ U_m J_m) 1.0)) (* -2.0 J_m))
        (if (<= t_1 2e+280) (* (* (* -2.0 J_m) (cos (* 0.5 K))) 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 / ((2.0 * J_m) * t_0)), 2.0)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_1 <= -5e-279) {
		tmp = sqrt(fma(((0.25 * U_m) / J_m), (U_m / J_m), 1.0)) * (-2.0 * J_m);
	} else if (t_1 <= 2e+280) {
		tmp = ((-2.0 * J_m) * cos((0.5 * K))) * 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(Float64(2.0 * J_m) * t_0)) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_1 <= -5e-279)
		tmp = Float64(sqrt(fma(Float64(Float64(0.25 * U_m) / J_m), Float64(U_m / J_m), 1.0)) * Float64(-2.0 * J_m));
	elseif (t_1 <= 2e+280)
		tmp = Float64(Float64(Float64(-2.0 * J_m) * cos(Float64(0.5 * K))) * 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[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$1, (-Infinity)], (-U$95$m), If[LessEqual[t$95$1, -5e-279], N[(N[Sqrt[N[(N[(N[(0.25 * U$95$m), $MachinePrecision] / J$95$m), $MachinePrecision] * N[(U$95$m / J$95$m), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(-2.0 * J$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+280], N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $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 5.4%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 36.6

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

   5. Applied rewrites 36.6%

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

   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))))) < -4.99999999999999969e-279

   1. Initial program 99.8%

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

   3. Taylor expanded in K around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-*r/ N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. times-frac N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 60.6

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

   5. Applied rewrites 60.6%

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

   if -4.99999999999999969e-279 < (*.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))))) < 2.0000000000000001e280

   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

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

   5. Applied rewrites 69.2%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 69.2

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

   7. Applied rewrites 69.2%

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

   if 2.0000000000000001e280 < (*.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 26.8%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 27.7

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

   5. Applied rewrites 27.7%

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

   7. Applied rewrites 14.2%

      \[\leadsto \frac{0 - U \cdot U}{\color{blue}{0 + U}} \]
   8. Step-by-step derivation

   9. Applied rewrites 56.9%

      \[\leadsto U \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 5: 98.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+280}:\\ \;\;\;\;\left(\left(-2 \cdot J\_m\right) \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \sqrt{1 + {\left(\frac{2 \cdot J\_m}{U\_m} \cdot \cos \left(K \cdot -0.5\right)\right)}^{-2}}\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \end{array} \]

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 (* (* 2.0 J_m) t_0)) 2.0))))))
   (*
    J_s
    (if (<= t_1 (- INFINITY))
      (- U_m)
      (if (<= t_1 2e+280)
        (*
         (* (* -2.0 J_m) (cos (* 0.5 K)))
         (sqrt (+ 1.0 (pow (* (/ (* 2.0 J_m) U_m) (cos (* K -0.5))) -2.0))))
        U_m)))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * J_m) * t_0) * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_0)), 2.0)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_1 <= 2e+280) {
		tmp = ((-2.0 * J_m) * cos((0.5 * K))) * sqrt((1.0 + pow((((2.0 * J_m) / U_m) * cos((K * -0.5))), -2.0)));
	} 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 / ((2.0 * J_m) * t_0)), 2.0)));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = -U_m;
	} else if (t_1 <= 2e+280) {
		tmp = ((-2.0 * J_m) * Math.cos((0.5 * K))) * Math.sqrt((1.0 + Math.pow((((2.0 * J_m) / U_m) * Math.cos((K * -0.5))), -2.0)));
	} else {
		tmp = U_m;
	}
	return J_s * tmp;
}

Python

U_m = math.fabs(U)
J\_m = math.fabs(J)
J\_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	t_0 = math.cos((K / 2.0))
	t_1 = ((-2.0 * J_m) * t_0) * math.sqrt((1.0 + math.pow((U_m / ((2.0 * J_m) * t_0)), 2.0)))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = -U_m
	elif t_1 <= 2e+280:
		tmp = ((-2.0 * J_m) * math.cos((0.5 * K))) * math.sqrt((1.0 + math.pow((((2.0 * J_m) / U_m) * math.cos((K * -0.5))), -2.0)))
	else:
		tmp = U_m
	return J_s * tmp

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	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(Float64(2.0 * J_m) * t_0)) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_1 <= 2e+280)
		tmp = Float64(Float64(Float64(-2.0 * J_m) * cos(Float64(0.5 * K))) * sqrt(Float64(1.0 + (Float64(Float64(Float64(2.0 * J_m) / U_m) * cos(Float64(K * -0.5))) ^ -2.0))));
	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 / ((2.0 * J_m) * t_0)) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = -U_m;
	elseif (t_1 <= 2e+280)
		tmp = ((-2.0 * J_m) * cos((0.5 * K))) * sqrt((1.0 + ((((2.0 * J_m) / U_m) * cos((K * -0.5))) ^ -2.0)));
	else
		tmp = U_m;
	end
	tmp_2 = J_s * tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := 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[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$1, (-Infinity)], (-U$95$m), If[LessEqual[t$95$1, 2e+280], N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(N[(N[(2.0 * J$95$m), $MachinePrecision] / U$95$m), $MachinePrecision] * N[Cos[N[(K * -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -2.0], $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 5.4%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 36.6

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

   5. Applied rewrites 36.6%

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

   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))))) < 2.0000000000000001e280

   1. Initial program 99.8%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. inv-pow N/A

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

      5. pow-pow N/A

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

      6. metadata-eval N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 99.7

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

      14. lift-cos.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. metadata-eval N/A

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

      17. distribute-neg-frac2 N/A

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

      18. cos-neg N/A

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

      19. lower-cos.f64 N/A

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

      20. div-inv N/A

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

      21. lower-*.f64 N/A

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

      22. metadata-eval 99.7

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

   4. Applied rewrites 99.7%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 99.7

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

   6. Applied rewrites 99.7%

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

   if 2.0000000000000001e280 < (*.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 26.8%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 27.7

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

   5. Applied rewrites 27.7%

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

   7. Applied rewrites 14.2%

      \[\leadsto \frac{0 - U \cdot U}{\color{blue}{0 + U}} \]
   8. Step-by-step derivation

   9. Applied rewrites 56.9%

      \[\leadsto U \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 98.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+280}:\\ \;\;\;\;\left(\left(-2 \cdot J\_m\right) \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \sqrt{\mathsf{fma}\left(0.25, {\left(\frac{J\_m}{U\_m} \cdot \cos \left(-0.5 \cdot K\right)\right)}^{-2}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \end{array} \]

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 (* (* 2.0 J_m) t_0)) 2.0))))))
   (*
    J_s
    (if (<= t_1 (- INFINITY))
      (- U_m)
      (if (<= t_1 2e+280)
        (*
         (* (* -2.0 J_m) (cos (* 0.5 K)))
         (sqrt (fma 0.25 (pow (* (/ J_m U_m) (cos (* -0.5 K))) -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 / ((2.0 * J_m) * t_0)), 2.0)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_1 <= 2e+280) {
		tmp = ((-2.0 * J_m) * cos((0.5 * K))) * sqrt(fma(0.25, pow(((J_m / U_m) * cos((-0.5 * K))), -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(Float64(2.0 * J_m) * t_0)) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_1 <= 2e+280)
		tmp = Float64(Float64(Float64(-2.0 * J_m) * cos(Float64(0.5 * K))) * sqrt(fma(0.25, (Float64(Float64(J_m / U_m) * cos(Float64(-0.5 * K))) ^ -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[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$1, (-Infinity)], (-U$95$m), If[LessEqual[t$95$1, 2e+280], N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(0.25 * N[Power[N[(N[(J$95$m / U$95$m), $MachinePrecision] * N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -2.0], $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 5.4%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 36.6

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

   5. Applied rewrites 36.6%

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

   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))))) < 2.0000000000000001e280

   1. Initial program 99.8%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. inv-pow N/A

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

      5. pow-pow N/A

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

      6. metadata-eval N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 99.7

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

      14. lift-cos.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. metadata-eval N/A

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

      17. distribute-neg-frac2 N/A

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

      18. cos-neg N/A

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

      19. lower-cos.f64 N/A

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

      20. div-inv N/A

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

      21. lower-*.f64 N/A

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

      22. metadata-eval 99.7

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

   4. Applied rewrites 99.7%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 99.7

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

   6. Applied rewrites 99.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/l* N/A

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

      8. associate-*l* N/A

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

      9. unpow-prod-down N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 99.7

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 99.7

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

   8. Applied rewrites 99.7%

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

   if 2.0000000000000001e280 < (*.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 26.8%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 27.7

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

   5. Applied rewrites 27.7%

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

   7. Applied rewrites 14.2%

      \[\leadsto \frac{0 - U \cdot U}{\color{blue}{0 + U}} \]
   8. Step-by-step derivation

   9. Applied rewrites 56.9%

      \[\leadsto U \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 90.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+280}:\\ \;\;\;\;\left(\left(-2 \cdot J\_m\right) \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \sqrt{\mathsf{fma}\left(0.25, \frac{U\_m}{J\_m} \cdot \frac{U\_m}{J\_m}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \end{array} \]

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)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_0)) 2.0))))))
   (*
    J_s
    (if (<= t_1 (- INFINITY))
      (- U_m)
      (if (<= t_1 2e+280)
        (*
         (* (* -2.0 J_m) (cos (* 0.5 K)))
         (sqrt (fma 0.25 (* (/ U_m 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) * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_0)), 2.0)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_1 <= 2e+280) {
		tmp = ((-2.0 * J_m) * cos((0.5 * K))) * sqrt(fma(0.25, ((U_m / 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(Float64(-2.0 * J_m) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * t_0)) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_1 <= 2e+280)
		tmp = Float64(Float64(Float64(-2.0 * J_m) * cos(Float64(0.5 * K))) * sqrt(fma(0.25, Float64(Float64(U_m / 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[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$1, (-Infinity)], (-U$95$m), If[LessEqual[t$95$1, 2e+280], N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(0.25 * N[(N[(U$95$m / J$95$m), $MachinePrecision] * N[(U$95$m / J$95$m), $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 5.4%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 36.6

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

   5. Applied rewrites 36.6%

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

   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))))) < 2.0000000000000001e280

   1. Initial program 99.8%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. inv-pow N/A

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

      5. pow-pow N/A

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

      6. metadata-eval N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 99.7

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

      14. lift-cos.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. metadata-eval N/A

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

      17. distribute-neg-frac2 N/A

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

      18. cos-neg N/A

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

      19. lower-cos.f64 N/A

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

      20. div-inv N/A

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

      21. lower-*.f64 N/A

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

      22. metadata-eval 99.7

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

   4. Applied rewrites 99.7%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 99.7

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

   6. Applied rewrites 99.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/l* N/A

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

      8. associate-*l* N/A

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

      9. unpow-prod-down N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 99.7

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 99.7

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

   8. Applied rewrites 99.7%

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

   9. Taylor expanded in K around 0

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

       1. unpow2 N/A

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

       2. unpow2 N/A

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

       3. times-frac N/A

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

       4. lower-*.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. lower-/.f64 88.7

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

   11. Applied rewrites 88.7%

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

   if 2.0000000000000001e280 < (*.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 26.8%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 27.7

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

   5. Applied rewrites 27.7%

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

   7. Applied rewrites 14.2%

      \[\leadsto \frac{0 - U \cdot U}{\color{blue}{0 + U}} \]
   8. Step-by-step derivation

   9. Applied rewrites 56.9%

      \[\leadsto U \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 76.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-286}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{0.25 \cdot U\_m}{J\_m}, \frac{U\_m}{J\_m}, 1\right)} \cdot \left(-2 \cdot J\_m\right)\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \end{array} \]

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 (* (* 2.0 J_m) t_0)) 2.0))))))
   (*
    J_s
    (if (<= t_1 (- INFINITY))
      (- U_m)
      (if (<= t_1 -5e-286)
        (* (sqrt (fma (/ (* 0.25 U_m) J_m) (/ U_m J_m) 1.0)) (* -2.0 J_m))
        U_m)))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * J_m) * t_0) * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_0)), 2.0)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_1 <= -5e-286) {
		tmp = sqrt(fma(((0.25 * U_m) / J_m), (U_m / J_m), 1.0)) * (-2.0 * J_m);
	} else {
		tmp = U_m;
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * J_m) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * t_0)) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_1 <= -5e-286)
		tmp = Float64(sqrt(fma(Float64(Float64(0.25 * U_m) / J_m), Float64(U_m / J_m), 1.0)) * Float64(-2.0 * J_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[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$1, (-Infinity)], (-U$95$m), If[LessEqual[t$95$1, -5e-286], N[(N[Sqrt[N[(N[(N[(0.25 * U$95$m), $MachinePrecision] / J$95$m), $MachinePrecision] * N[(U$95$m / J$95$m), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(-2.0 * J$95$m), $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 5.4%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 36.6

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

   5. Applied rewrites 36.6%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -5.00000000000000037e-286

   1. Initial program 99.8%

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

   3. Taylor expanded in K around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-*r/ N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. times-frac N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 61.1

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

   5. Applied rewrites 61.1%

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

   if -5.00000000000000037e-286 < (*.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 73.3%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 20.3

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

   5. Applied rewrites 20.3%

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

   7. Applied rewrites 9.6%

      \[\leadsto \frac{0 - U \cdot U}{\color{blue}{0 + U}} \]
   8. Step-by-step derivation

   9. Applied rewrites 34.3%

      \[\leadsto U \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 68.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+293}:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-286}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{0.25}{J\_m}, \frac{U\_m \cdot U\_m}{J\_m}, 1\right)} \cdot \left(-2 \cdot J\_m\right)\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \end{array} \]

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 (* (* 2.0 J_m) t_0)) 2.0))))))
   (*
    J_s
    (if (<= t_1 -5e+293)
      (- U_m)
      (if (<= t_1 -5e-286)
        (* (sqrt (fma (/ 0.25 J_m) (/ (* U_m U_m) J_m) 1.0)) (* -2.0 J_m))
        U_m)))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * J_m) * t_0) * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_0)), 2.0)));
	double tmp;
	if (t_1 <= -5e+293) {
		tmp = -U_m;
	} else if (t_1 <= -5e-286) {
		tmp = sqrt(fma((0.25 / J_m), ((U_m * U_m) / J_m), 1.0)) * (-2.0 * J_m);
	} else {
		tmp = U_m;
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * J_m) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * t_0)) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -5e+293)
		tmp = Float64(-U_m);
	elseif (t_1 <= -5e-286)
		tmp = Float64(sqrt(fma(Float64(0.25 / J_m), Float64(Float64(U_m * U_m) / J_m), 1.0)) * Float64(-2.0 * J_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[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$1, -5e+293], (-U$95$m), If[LessEqual[t$95$1, -5e-286], N[(N[Sqrt[N[(N[(0.25 / J$95$m), $MachinePrecision] * N[(N[(U$95$m * U$95$m), $MachinePrecision] / J$95$m), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(-2.0 * J$95$m), $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))))) < -5.00000000000000033e293

   1. Initial program 10.4%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   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. lower-neg.f64 34.8

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

   5. Applied rewrites 34.8%

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

   if -5.00000000000000033e293 < (*.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))))) < -5.00000000000000037e-286

   1. Initial program 99.8%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. inv-pow N/A

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

      5. pow-pow N/A

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

      6. metadata-eval N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 99.7

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

      14. lift-cos.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. metadata-eval N/A

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

      17. distribute-neg-frac2 N/A

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

      18. cos-neg N/A

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

      19. lower-cos.f64 N/A

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

      20. div-inv N/A

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

      21. lower-*.f64 N/A

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

      22. metadata-eval 99.7

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

   4. Applied rewrites 99.7%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 99.7

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r/ N/A

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

      9. unpow2 N/A

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

      10. times-frac N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 57.8

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

   9. Applied rewrites 57.8%

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

   if -5.00000000000000037e-286 < (*.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 73.3%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 20.3

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

   5. Applied rewrites 20.3%

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

   7. Applied rewrites 9.6%

      \[\leadsto \frac{0 - U \cdot U}{\color{blue}{0 + U}} \]
   8. Step-by-step derivation

   9. Applied rewrites 34.3%

      \[\leadsto U \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 59.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+293}:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-286}:\\ \;\;\;\;\mathsf{fma}\left(\frac{U\_m \cdot U\_m}{J\_m}, -0.25, -2 \cdot J\_m\right)\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \end{array} \]

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 (* (* 2.0 J_m) t_0)) 2.0))))))
   (*
    J_s
    (if (<= t_1 -5e+293)
      (- U_m)
      (if (<= t_1 -5e-286)
        (fma (/ (* U_m U_m) J_m) -0.25 (* -2.0 J_m))
        U_m)))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * J_m) * t_0) * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_0)), 2.0)));
	double tmp;
	if (t_1 <= -5e+293) {
		tmp = -U_m;
	} else if (t_1 <= -5e-286) {
		tmp = fma(((U_m * U_m) / J_m), -0.25, (-2.0 * J_m));
	} else {
		tmp = U_m;
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * J_m) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * t_0)) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -5e+293)
		tmp = Float64(-U_m);
	elseif (t_1 <= -5e-286)
		tmp = fma(Float64(Float64(U_m * U_m) / J_m), -0.25, Float64(-2.0 * J_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[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$1, -5e+293], (-U$95$m), If[LessEqual[t$95$1, -5e-286], N[(N[(N[(U$95$m * U$95$m), $MachinePrecision] / J$95$m), $MachinePrecision] * -0.25 + N[(-2.0 * J$95$m), $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))))) < -5.00000000000000033e293

   1. Initial program 10.4%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   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. lower-neg.f64 34.8

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

   5. Applied rewrites 34.8%

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

   if -5.00000000000000033e293 < (*.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))))) < -5.00000000000000037e-286

   1. Initial program 99.8%

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

   3. Taylor expanded in U around 0

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

      1. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-*.f64 71.4

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

   5. Applied rewrites 71.4%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 41.1%

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

   if -5.00000000000000037e-286 < (*.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 73.3%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 20.3

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

   5. Applied rewrites 20.3%

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

   7. Applied rewrites 9.6%

      \[\leadsto \frac{0 - U \cdot U}{\color{blue}{0 + U}} \]
   8. Step-by-step derivation

   9. Applied rewrites 34.3%

      \[\leadsto U \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 58.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+293}:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-286}:\\ \;\;\;\;\left(\left(-2 \cdot J\_m\right) \cdot \mathsf{fma}\left(-0.125, K \cdot K, 1\right)\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \end{array} \]

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 (* (* 2.0 J_m) t_0)) 2.0))))))
   (*
    J_s
    (if (<= t_1 -5e+293)
      (- U_m)
      (if (<= t_1 -5e-286)
        (* (* (* -2.0 J_m) (fma -0.125 (* K K) 1.0)) 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 / ((2.0 * J_m) * t_0)), 2.0)));
	double tmp;
	if (t_1 <= -5e+293) {
		tmp = -U_m;
	} else if (t_1 <= -5e-286) {
		tmp = ((-2.0 * J_m) * fma(-0.125, (K * K), 1.0)) * 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(Float64(2.0 * J_m) * t_0)) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -5e+293)
		tmp = Float64(-U_m);
	elseif (t_1 <= -5e-286)
		tmp = Float64(Float64(Float64(-2.0 * J_m) * fma(-0.125, Float64(K * K), 1.0)) * 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[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$1, -5e+293], (-U$95$m), If[LessEqual[t$95$1, -5e-286], N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * N[(-0.125 * N[(K * K), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 1.0), $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))))) < -5.00000000000000033e293

   1. Initial program 10.4%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   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. lower-neg.f64 34.8

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

   5. Applied rewrites 34.8%

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

   if -5.00000000000000033e293 < (*.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))))) < -5.00000000000000037e-286

   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

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

   5. Applied rewrites 72.4%

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

   6. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 38.6

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

   8. Applied rewrites 38.6%

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

   if -5.00000000000000037e-286 < (*.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 73.3%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 20.3

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

   5. Applied rewrites 20.3%

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

   7. Applied rewrites 9.6%

      \[\leadsto \frac{0 - U \cdot U}{\color{blue}{0 + U}} \]
   8. Step-by-step derivation

   9. Applied rewrites 34.3%

      \[\leadsto U \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 51.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2}} \leq -5 \cdot 10^{-286}:\\ \;\;\;\;-U\_m\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \end{array} \]

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))))
   (*
    J_s
    (if (<=
         (*
          (* (* -2.0 J_m) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_0)) 2.0))))
         -5e-286)
      (- 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 tmp;
	if ((((-2.0 * J_m) * t_0) * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_0)), 2.0)))) <= -5e-286) {
		tmp = -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) :: tmp
    t_0 = cos((k / 2.0d0))
    if (((((-2.0d0) * j_m) * t_0) * sqrt((1.0d0 + ((u_m / ((2.0d0 * j_m) * t_0)) ** 2.0d0)))) <= (-5d-286)) then
        tmp = -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 tmp;
	if ((((-2.0 * J_m) * t_0) * Math.sqrt((1.0 + Math.pow((U_m / ((2.0 * J_m) * t_0)), 2.0)))) <= -5e-286) {
		tmp = -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))
	tmp = 0
	if (((-2.0 * J_m) * t_0) * math.sqrt((1.0 + math.pow((U_m / ((2.0 * J_m) * t_0)), 2.0)))) <= -5e-286:
		tmp = -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))
	tmp = 0.0
	if (Float64(Float64(Float64(-2.0 * J_m) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * t_0)) ^ 2.0)))) <= -5e-286)
		tmp = Float64(-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));
	tmp = 0.0;
	if ((((-2.0 * J_m) * t_0) * sqrt((1.0 + ((U_m / ((2.0 * J_m) * t_0)) ^ 2.0)))) <= -5e-286)
		tmp = -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]}, N[(J$95$s * If[LessEqual[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[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -5e-286], (-U$95$m), U$95$m]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -5.00000000000000037e-286

   1. Initial program 72.4%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   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. lower-neg.f64 19.7

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

   5. Applied rewrites 19.7%

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

   if -5.00000000000000037e-286 < (*.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 73.3%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 20.3

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

   5. Applied rewrites 20.3%

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

   7. Applied rewrites 9.6%

      \[\leadsto \frac{0 - U \cdot U}{\color{blue}{0 + U}} \]
   8. Step-by-step derivation

   9. Applied rewrites 34.3%

      \[\leadsto U \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 14.3% 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 72.8%

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

3. Taylor expanded in J around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 20.0

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

5. Applied rewrites 20.0%

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

7. Applied rewrites 11.6%

   \[\leadsto \frac{0 - U \cdot U}{\color{blue}{0 + U}} \]
8. Step-by-step derivation

9. Applied rewrites 32.9%

   \[\leadsto U \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024321 
(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)))))