Maksimov and Kolovsky, Equation (3)

Percentage Accurate: 72.7% → 99.2%

Time: 10.5s

Alternatives: 7

Speedup: 1.3×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 72.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_0 \cdot \left(J\_m \cdot 2\right)}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;0 - U\_m\\ \mathbf{elif}\;t\_1 \leq 10^{+292}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* (* -2.0 J_m) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* t_0 (* J_m 2.0))) 2.0))))))
   (*
    J_s
    (if (<= t_1 (- INFINITY)) (- 0.0 U_m) (if (<= t_1 1e+292) t_1 U_m)))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * J_m) * t_0) * sqrt((1.0 + pow((U_m / (t_0 * (J_m * 2.0))), 2.0)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = 0.0 - U_m;
	} else if (t_1 <= 1e+292) {
		tmp = t_1;
	} else {
		tmp = U_m;
	}
	return J_s * tmp;
}

Java

U_m = Math.abs(U);
J\_m = Math.abs(J);
J\_s = Math.copySign(1.0, J);
public static double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = ((-2.0 * J_m) * t_0) * Math.sqrt((1.0 + Math.pow((U_m / (t_0 * (J_m * 2.0))), 2.0)));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = 0.0 - U_m;
	} else if (t_1 <= 1e+292) {
		tmp = t_1;
	} else {
		tmp = U_m;
	}
	return J_s * tmp;
}

Python

U_m = math.fabs(U)
J\_m = math.fabs(J)
J\_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	t_0 = math.cos((K / 2.0))
	t_1 = ((-2.0 * J_m) * t_0) * math.sqrt((1.0 + math.pow((U_m / (t_0 * (J_m * 2.0))), 2.0)))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = 0.0 - U_m
	elif t_1 <= 1e+292:
		tmp = t_1
	else:
		tmp = U_m
	return J_s * tmp

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * J_m) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_0 * Float64(J_m * 2.0))) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(0.0 - U_m);
	elseif (t_1 <= 1e+292)
		tmp = t_1;
	else
		tmp = U_m;
	end
	return Float64(J_s * tmp)
end

MATLAB

U_m = abs(U);
J\_m = abs(J);
J\_s = sign(J) * abs(1.0);
function tmp_2 = code(J_s, J_m, K, U_m)
	t_0 = cos((K / 2.0));
	t_1 = ((-2.0 * J_m) * t_0) * sqrt((1.0 + ((U_m / (t_0 * (J_m * 2.0))) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = 0.0 - U_m;
	elseif (t_1 <= 1e+292)
		tmp = t_1;
	else
		tmp = U_m;
	end
	tmp_2 = J_s * tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(t$95$0 * N[(J$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$1, (-Infinity)], N[(0.0 - U$95$m), $MachinePrecision], If[LessEqual[t$95$1, 1e+292], 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 5.6%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

      3. --lowering--.f64 43.6%

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{U}\right) \]

   5. Simplified 43.6%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 43.6%

         \[\leadsto \mathsf{neg.f64}\left(U\right) \]

   7. Applied egg-rr 43.6%

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

   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))))) < 1e292

   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 1e292 < (*.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 14.6%

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

   3. Taylor expanded in U around -inf

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

   5. Simplified 37.6%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -\infty:\\ \;\;\;\;0 - U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 10^{+292}:\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 69.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -0.05:\\ \;\;\;\;U\_m\\ \mathbf{elif}\;t\_0 \leq 0.2:\\ \;\;\;\;\frac{1}{\frac{1}{\left(-2 \cdot J\_m\right) \cdot t\_0}}\\ \mathbf{elif}\;t\_0 \leq 0.925:\\ \;\;\;\;0 - U\_m\\ \mathbf{elif}\;t\_0 \leq 0.99996:\\ \;\;\;\;\left(-2 \cdot J\_m\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(1, \frac{\frac{U\_m}{J\_m \cdot 2}}{t\_0}\right) \cdot \left(-2 \cdot J\_m + 0.25 \cdot \left(J\_m \cdot \left(K \cdot K\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (*
    J_s
    (if (<= t_0 -0.05)
      U_m
      (if (<= t_0 0.2)
        (/ 1.0 (/ 1.0 (* (* -2.0 J_m) t_0)))
        (if (<= t_0 0.925)
          (- 0.0 U_m)
          (if (<= t_0 0.99996)
            (* (* -2.0 J_m) (cos (* K 0.5)))
            (*
             (hypot 1.0 (/ (/ U_m (* J_m 2.0)) t_0))
             (+ (* -2.0 J_m) (* 0.25 (* J_m (* K K))))))))))))

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 (t_0 <= -0.05) {
		tmp = U_m;
	} else if (t_0 <= 0.2) {
		tmp = 1.0 / (1.0 / ((-2.0 * J_m) * t_0));
	} else if (t_0 <= 0.925) {
		tmp = 0.0 - U_m;
	} else if (t_0 <= 0.99996) {
		tmp = (-2.0 * J_m) * cos((K * 0.5));
	} else {
		tmp = hypot(1.0, ((U_m / (J_m * 2.0)) / t_0)) * ((-2.0 * J_m) + (0.25 * (J_m * (K * K))));
	}
	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 tmp;
	if (t_0 <= -0.05) {
		tmp = U_m;
	} else if (t_0 <= 0.2) {
		tmp = 1.0 / (1.0 / ((-2.0 * J_m) * t_0));
	} else if (t_0 <= 0.925) {
		tmp = 0.0 - U_m;
	} else if (t_0 <= 0.99996) {
		tmp = (-2.0 * J_m) * Math.cos((K * 0.5));
	} else {
		tmp = Math.hypot(1.0, ((U_m / (J_m * 2.0)) / t_0)) * ((-2.0 * J_m) + (0.25 * (J_m * (K * K))));
	}
	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 t_0 <= -0.05:
		tmp = U_m
	elif t_0 <= 0.2:
		tmp = 1.0 / (1.0 / ((-2.0 * J_m) * t_0))
	elif t_0 <= 0.925:
		tmp = 0.0 - U_m
	elif t_0 <= 0.99996:
		tmp = (-2.0 * J_m) * math.cos((K * 0.5))
	else:
		tmp = math.hypot(1.0, ((U_m / (J_m * 2.0)) / t_0)) * ((-2.0 * J_m) + (0.25 * (J_m * (K * K))))
	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 (t_0 <= -0.05)
		tmp = U_m;
	elseif (t_0 <= 0.2)
		tmp = Float64(1.0 / Float64(1.0 / Float64(Float64(-2.0 * J_m) * t_0)));
	elseif (t_0 <= 0.925)
		tmp = Float64(0.0 - U_m);
	elseif (t_0 <= 0.99996)
		tmp = Float64(Float64(-2.0 * J_m) * cos(Float64(K * 0.5)));
	else
		tmp = Float64(hypot(1.0, Float64(Float64(U_m / Float64(J_m * 2.0)) / t_0)) * Float64(Float64(-2.0 * J_m) + Float64(0.25 * Float64(J_m * Float64(K * K)))));
	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 (t_0 <= -0.05)
		tmp = U_m;
	elseif (t_0 <= 0.2)
		tmp = 1.0 / (1.0 / ((-2.0 * J_m) * t_0));
	elseif (t_0 <= 0.925)
		tmp = 0.0 - U_m;
	elseif (t_0 <= 0.99996)
		tmp = (-2.0 * J_m) * cos((K * 0.5));
	else
		tmp = hypot(1.0, ((U_m / (J_m * 2.0)) / t_0)) * ((-2.0 * J_m) + (0.25 * (J_m * (K * K))));
	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[t$95$0, -0.05], U$95$m, If[LessEqual[t$95$0, 0.2], N[(1.0 / N[(1.0 / N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.925], N[(0.0 - U$95$m), $MachinePrecision], If[LessEqual[t$95$0, 0.99996], N[(N[(-2.0 * J$95$m), $MachinePrecision] * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[1.0 ^ 2 + N[(N[(U$95$m / N[(J$95$m * 2.0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] ^ 2], $MachinePrecision] * N[(N[(-2.0 * J$95$m), $MachinePrecision] + N[(0.25 * N[(J$95$m * N[(K * K), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]]

Derivation

1. Split input into 5 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.050000000000000003

   1. Initial program 64.6%

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

   3. Taylor expanded in U around -inf

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

   5. Simplified 34.8%

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

   if -0.050000000000000003 < (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.20000000000000001

   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 J around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\cos \left(\frac{1}{2} \cdot K\right), \color{blue}{\left(-2 \cdot J\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot K\right)\right), \left(\color{blue}{-2} \cdot J\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, K\right)\right), \left(-2 \cdot J\right)\right) \]

      6. *-lowering-*.f64 99.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, K\right)\right), \mathsf{*.f64}\left(-2, \color{blue}{J}\right)\right) \]

   5. Simplified 99.7%

      \[\leadsto \color{blue}{\cos \left(0.5 \cdot K\right) \cdot \left(-2 \cdot J\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. div-inv N/A

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

      5. remove-double-div N/A

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

      6. un-div-inv N/A

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

      7. clear-num N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\frac{1}{\cos \left(\frac{K}{2}\right)}}{-2 \cdot J}\right)}\right) \]

      9. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{1}{\color{blue}{\frac{-2 \cdot J}{\frac{1}{\cos \left(\frac{K}{2}\right)}}}}\right)\right) \]

      10. un-div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{1}{\left(-2 \cdot J\right) \cdot \color{blue}{\frac{1}{\frac{1}{\cos \left(\frac{K}{2}\right)}}}}\right)\right) \]

      11. remove-double-div N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{1}{\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \color{blue}{\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right)}\right)\right) \]

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

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

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \cos \color{blue}{\left(\frac{K}{2}\right)}\right)\right)\right) \]

      15. div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \cos \left(K \cdot \frac{1}{2}\right)\right)\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \cos \left(K \cdot \frac{1}{2}\right)\right)\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot K\right)\right)\right)\right)\right) \]

      19. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\left(K \cdot \frac{1}{2}\right)\right)\right)\right)\right) \]

      20. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\left(K \cdot \frac{1}{2}\right)\right)\right)\right)\right) \]

      21. div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\left(\frac{K}{2}\right)\right)\right)\right)\right) \]

      22. /-lowering-/.f64 100.0%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 100.0%

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

   if 0.20000000000000001 < (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.92500000000000004

   1. Initial program 69.7%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

      3. --lowering--.f64 44.0%

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{U}\right) \]

   5. Simplified 44.0%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 44.0%

         \[\leadsto \mathsf{neg.f64}\left(U\right) \]

   7. Applied egg-rr 44.0%

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

   if 0.92500000000000004 < (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.99995999999999996

   1. Initial program 74.9%

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

   3. Taylor expanded in J around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\cos \left(\frac{1}{2} \cdot K\right), \color{blue}{\left(-2 \cdot J\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot K\right)\right), \left(\color{blue}{-2} \cdot J\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, K\right)\right), \left(-2 \cdot J\right)\right) \]

      6. *-lowering-*.f64 66.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, K\right)\right), \mathsf{*.f64}\left(-2, \color{blue}{J}\right)\right) \]

   5. Simplified 66.7%

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

   if 0.99995999999999996 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 73.6%

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right), \color{blue}{\left(\sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(-2 \cdot J\right), \cos \left(\frac{K}{2}\right)\right), \left(\sqrt{\color{blue}{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \cos \left(\frac{K}{2}\right)\right), \left(\sqrt{\color{blue}{1} + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\left(\frac{K}{2}\right)\right)\right), \left(\sqrt{1 + \color{blue}{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \left(\sqrt{1 + {\color{blue}{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}}^{2}}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \left(\sqrt{1 + \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}}\right)\right) \]

      7. hypot-1-def N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \left(\mathsf{hypot}\left(1, \color{blue}{\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{hypot.f64}\left(1, \color{blue}{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}\right)\right) \]

      9. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{hypot.f64}\left(1, \left(\frac{\frac{U}{2 \cdot J}}{\color{blue}{\cos \left(\frac{K}{2}\right)}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{U}{2 \cdot J}\right), \color{blue}{\cos \left(\frac{K}{2}\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(U, \left(2 \cdot J\right)\right), \cos \color{blue}{\left(\frac{K}{2}\right)}\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(U, \left(J \cdot 2\right)\right), \cos \left(\frac{K}{\color{blue}{2}}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(U, \mathsf{*.f64}\left(J, 2\right)\right), \cos \left(\frac{K}{\color{blue}{2}}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(U, \mathsf{*.f64}\left(J, 2\right)\right), \mathsf{cos.f64}\left(\left(\frac{K}{2}\right)\right)\right)\right)\right) \]

      15. /-lowering-/.f64 85.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(U, \mathsf{*.f64}\left(J, 2\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right)\right)\right) \]

   3. Simplified 85.9%

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

   5. Taylor expanded in K around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(-2 \cdot J + \frac{1}{4} \cdot \left(J \cdot {K}^{2}\right)\right)}, \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(U, \mathsf{*.f64}\left(J, 2\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right)\right)\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(-2 \cdot J\right), \left(\frac{1}{4} \cdot \left(J \cdot {K}^{2}\right)\right)\right), \mathsf{hypot.f64}\left(\color{blue}{1}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(U, \mathsf{*.f64}\left(J, 2\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, J\right), \left(\frac{1}{4} \cdot \left(J \cdot {K}^{2}\right)\right)\right), \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(U, \mathsf{*.f64}\left(J, 2\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{*.f64}\left(\frac{1}{4}, \left(J \cdot {K}^{2}\right)\right)\right), \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(U, \mathsf{*.f64}\left(J, 2\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(J, \left({K}^{2}\right)\right)\right)\right), \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(U, \mathsf{*.f64}\left(J, 2\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(J, \left(K \cdot K\right)\right)\right)\right), \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(U, \mathsf{*.f64}\left(J, 2\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 86.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(J, \mathsf{*.f64}\left(K, K\right)\right)\right)\right), \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(U, \mathsf{*.f64}\left(J, 2\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right)\right)\right) \]

   7. Simplified 86.0%

      \[\leadsto \color{blue}{\left(-2 \cdot J + 0.25 \cdot \left(J \cdot \left(K \cdot K\right)\right)\right)} \cdot \mathsf{hypot}\left(1, \frac{\frac{U}{J \cdot 2}}{\cos \left(\frac{K}{2}\right)}\right) \]
3. Recombined 5 regimes into one program.

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.05:\\ \;\;\;\;U\\ \mathbf{elif}\;\cos \left(\frac{K}{2}\right) \leq 0.2:\\ \;\;\;\;\frac{1}{\frac{1}{\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}}\\ \mathbf{elif}\;\cos \left(\frac{K}{2}\right) \leq 0.925:\\ \;\;\;\;0 - U\\ \mathbf{elif}\;\cos \left(\frac{K}{2}\right) \leq 0.99996:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(1, \frac{\frac{U}{J \cdot 2}}{\cos \left(\frac{K}{2}\right)}\right) \cdot \left(-2 \cdot J + 0.25 \cdot \left(J \cdot \left(K \cdot K\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;J\_m \leq 1.32 \cdot 10^{-256}:\\ \;\;\;\;0 - U\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \mathsf{hypot}\left(1, \frac{\frac{U\_m}{J\_m \cdot 2}}{t\_0}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (*
    J_s
    (if (<= J_m 1.32e-256)
      (- 0.0 U_m)
      (* (* (* -2.0 J_m) t_0) (hypot 1.0 (/ (/ U_m (* J_m 2.0)) t_0)))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (J_m <= 1.32e-256) {
		tmp = 0.0 - U_m;
	} else {
		tmp = ((-2.0 * J_m) * t_0) * hypot(1.0, ((U_m / (J_m * 2.0)) / t_0));
	}
	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 tmp;
	if (J_m <= 1.32e-256) {
		tmp = 0.0 - U_m;
	} else {
		tmp = ((-2.0 * J_m) * t_0) * Math.hypot(1.0, ((U_m / (J_m * 2.0)) / t_0));
	}
	return J_s * tmp;
}

Python

U_m = math.fabs(U)
J\_m = math.fabs(J)
J\_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if J_m <= 1.32e-256:
		tmp = 0.0 - U_m
	else:
		tmp = ((-2.0 * J_m) * t_0) * math.hypot(1.0, ((U_m / (J_m * 2.0)) / t_0))
	return J_s * tmp

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (J_m <= 1.32e-256)
		tmp = Float64(0.0 - U_m);
	else
		tmp = Float64(Float64(Float64(-2.0 * J_m) * t_0) * hypot(1.0, Float64(Float64(U_m / Float64(J_m * 2.0)) / t_0)));
	end
	return Float64(J_s * tmp)
end

MATLAB

U_m = abs(U);
J\_m = abs(J);
J\_s = sign(J) * abs(1.0);
function tmp_2 = code(J_s, J_m, K, U_m)
	t_0 = cos((K / 2.0));
	tmp = 0.0;
	if (J_m <= 1.32e-256)
		tmp = 0.0 - U_m;
	else
		tmp = ((-2.0 * J_m) * t_0) * hypot(1.0, ((U_m / (J_m * 2.0)) / t_0));
	end
	tmp_2 = J_s * tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, N[(J$95$s * If[LessEqual[J$95$m, 1.32e-256], N[(0.0 - U$95$m), $MachinePrecision], N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[(N[(U$95$m / N[(J$95$m * 2.0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if J < 1.32e-256

   1. Initial program 62.8%

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

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

      2. neg-sub0 N/A

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

      3. --lowering--.f64 32.8%

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{U}\right) \]

   5. Simplified 32.8%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 32.8%

         \[\leadsto \mathsf{neg.f64}\left(U\right) \]

   7. Applied egg-rr 32.8%

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

   if 1.32e-256 < J

   1. Initial program 82.3%

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right), \color{blue}{\left(\sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(-2 \cdot J\right), \cos \left(\frac{K}{2}\right)\right), \left(\sqrt{\color{blue}{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \cos \left(\frac{K}{2}\right)\right), \left(\sqrt{\color{blue}{1} + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\left(\frac{K}{2}\right)\right)\right), \left(\sqrt{1 + \color{blue}{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \left(\sqrt{1 + {\color{blue}{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}}^{2}}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \left(\sqrt{1 + \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}}\right)\right) \]

      7. hypot-1-def N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \left(\mathsf{hypot}\left(1, \color{blue}{\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{hypot.f64}\left(1, \color{blue}{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}\right)\right) \]

      9. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{hypot.f64}\left(1, \left(\frac{\frac{U}{2 \cdot J}}{\color{blue}{\cos \left(\frac{K}{2}\right)}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{U}{2 \cdot J}\right), \color{blue}{\cos \left(\frac{K}{2}\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(U, \left(2 \cdot J\right)\right), \cos \color{blue}{\left(\frac{K}{2}\right)}\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(U, \left(J \cdot 2\right)\right), \cos \left(\frac{K}{\color{blue}{2}}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(U, \mathsf{*.f64}\left(J, 2\right)\right), \cos \left(\frac{K}{\color{blue}{2}}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(U, \mathsf{*.f64}\left(J, 2\right)\right), \mathsf{cos.f64}\left(\left(\frac{K}{2}\right)\right)\right)\right)\right) \]

      15. /-lowering-/.f64 94.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(U, \mathsf{*.f64}\left(J, 2\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right)\right)\right) \]

   3. Simplified 94.2%

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

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq 1.32 \cdot 10^{-256}:\\ \;\;\;\;0 - U\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{\frac{U}{J \cdot 2}}{\cos \left(\frac{K}{2}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 66.6% accurate, 3.7× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ J\_s \cdot \begin{array}{l} \mathbf{if}\;J\_m \leq 7.5 \cdot 10^{-61}:\\ \;\;\;\;0 - U\_m\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot J\_m\right) \cdot \cos \left(K \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (* J_s (if (<= J_m 7.5e-61) (- 0.0 U_m) (* (* -2.0 J_m) (cos (* K 0.5))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double tmp;
	if (J_m <= 7.5e-61) {
		tmp = 0.0 - U_m;
	} else {
		tmp = (-2.0 * J_m) * cos((K * 0.5));
	}
	return J_s * tmp;
}

Fortran

U_m = abs(u)
J\_m = abs(j)
J\_s = copysign(1.0d0, j)
real(8) function code(j_s, j_m, k, u_m)
    real(8), intent (in) :: j_s
    real(8), intent (in) :: j_m
    real(8), intent (in) :: k
    real(8), intent (in) :: u_m
    real(8) :: tmp
    if (j_m <= 7.5d-61) then
        tmp = 0.0d0 - u_m
    else
        tmp = ((-2.0d0) * j_m) * cos((k * 0.5d0))
    end if
    code = j_s * tmp
end function

Java

U_m = Math.abs(U);
J\_m = Math.abs(J);
J\_s = Math.copySign(1.0, J);
public static double code(double J_s, double J_m, double K, double U_m) {
	double tmp;
	if (J_m <= 7.5e-61) {
		tmp = 0.0 - U_m;
	} else {
		tmp = (-2.0 * J_m) * Math.cos((K * 0.5));
	}
	return J_s * tmp;
}

Python

U_m = math.fabs(U)
J\_m = math.fabs(J)
J\_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	tmp = 0
	if J_m <= 7.5e-61:
		tmp = 0.0 - U_m
	else:
		tmp = (-2.0 * J_m) * math.cos((K * 0.5))
	return J_s * tmp

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	tmp = 0.0
	if (J_m <= 7.5e-61)
		tmp = Float64(0.0 - U_m);
	else
		tmp = Float64(Float64(-2.0 * J_m) * cos(Float64(K * 0.5)));
	end
	return Float64(J_s * tmp)
end

MATLAB

U_m = abs(U);
J\_m = abs(J);
J\_s = sign(J) * abs(1.0);
function tmp_2 = code(J_s, J_m, K, U_m)
	tmp = 0.0;
	if (J_m <= 7.5e-61)
		tmp = 0.0 - U_m;
	else
		tmp = (-2.0 * J_m) * cos((K * 0.5));
	end
	tmp_2 = J_s * tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := N[(J$95$s * If[LessEqual[J$95$m, 7.5e-61], N[(0.0 - U$95$m), $MachinePrecision], N[(N[(-2.0 * J$95$m), $MachinePrecision] * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if J < 7.50000000000000047e-61

   1. Initial program 62.2%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

      3. --lowering--.f64 32.5%

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{U}\right) \]

   5. Simplified 32.5%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 32.5%

         \[\leadsto \mathsf{neg.f64}\left(U\right) \]

   7. Applied egg-rr 32.5%

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

   if 7.50000000000000047e-61 < J

   1. Initial program 94.9%

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

   3. Taylor expanded in J around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\cos \left(\frac{1}{2} \cdot K\right), \color{blue}{\left(-2 \cdot J\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot K\right)\right), \left(\color{blue}{-2} \cdot J\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, K\right)\right), \left(-2 \cdot J\right)\right) \]

      6. *-lowering-*.f64 72.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, K\right)\right), \mathsf{*.f64}\left(-2, \color{blue}{J}\right)\right) \]

   5. Simplified 72.8%

      \[\leadsto \color{blue}{\cos \left(0.5 \cdot K\right) \cdot \left(-2 \cdot J\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq 7.5 \cdot 10^{-61}:\\ \;\;\;\;0 - U\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 50.1% accurate, 52.4× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ J\_s \cdot \begin{array}{l} \mathbf{if}\;J\_m \leq 1.55 \cdot 10^{+77}:\\ \;\;\;\;0 - U\_m\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot J\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (* J_s (if (<= J_m 1.55e+77) (- 0.0 U_m) (* -2.0 J_m))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double tmp;
	if (J_m <= 1.55e+77) {
		tmp = 0.0 - U_m;
	} else {
		tmp = -2.0 * J_m;
	}
	return J_s * tmp;
}

Fortran

U_m = abs(u)
J\_m = abs(j)
J\_s = copysign(1.0d0, j)
real(8) function code(j_s, j_m, k, u_m)
    real(8), intent (in) :: j_s
    real(8), intent (in) :: j_m
    real(8), intent (in) :: k
    real(8), intent (in) :: u_m
    real(8) :: tmp
    if (j_m <= 1.55d+77) then
        tmp = 0.0d0 - u_m
    else
        tmp = (-2.0d0) * j_m
    end if
    code = j_s * tmp
end function

Java

U_m = Math.abs(U);
J\_m = Math.abs(J);
J\_s = Math.copySign(1.0, J);
public static double code(double J_s, double J_m, double K, double U_m) {
	double tmp;
	if (J_m <= 1.55e+77) {
		tmp = 0.0 - U_m;
	} else {
		tmp = -2.0 * J_m;
	}
	return J_s * tmp;
}

Python

U_m = math.fabs(U)
J\_m = math.fabs(J)
J\_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	tmp = 0
	if J_m <= 1.55e+77:
		tmp = 0.0 - U_m
	else:
		tmp = -2.0 * J_m
	return J_s * tmp

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	tmp = 0.0
	if (J_m <= 1.55e+77)
		tmp = Float64(0.0 - U_m);
	else
		tmp = Float64(-2.0 * J_m);
	end
	return Float64(J_s * tmp)
end

MATLAB

U_m = abs(U);
J\_m = abs(J);
J\_s = sign(J) * abs(1.0);
function tmp_2 = code(J_s, J_m, K, U_m)
	tmp = 0.0;
	if (J_m <= 1.55e+77)
		tmp = 0.0 - U_m;
	else
		tmp = -2.0 * J_m;
	end
	tmp_2 = J_s * tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := N[(J$95$s * If[LessEqual[J$95$m, 1.55e+77], N[(0.0 - U$95$m), $MachinePrecision], N[(-2.0 * J$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if J < 1.54999999999999999e77

   1. Initial program 64.3%

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

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

      2. neg-sub0 N/A

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

      3. --lowering--.f64 33.2%

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{U}\right) \]

   5. Simplified 33.2%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 33.2%

         \[\leadsto \mathsf{neg.f64}\left(U\right) \]

   7. Applied egg-rr 33.2%

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

   if 1.54999999999999999e77 < J

   1. Initial program 98.3%

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

   3. Taylor expanded in J around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\cos \left(\frac{1}{2} \cdot K\right), \color{blue}{\left(-2 \cdot J\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot K\right)\right), \left(\color{blue}{-2} \cdot J\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, K\right)\right), \left(-2 \cdot J\right)\right) \]

      6. *-lowering-*.f64 85.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, K\right)\right), \mathsf{*.f64}\left(-2, \color{blue}{J}\right)\right) \]

   5. Simplified 85.1%

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

   6. Taylor expanded in K around 0

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

      1. *-lowering-*.f64 52.5%

         \[\leadsto \mathsf{*.f64}\left(-2, \color{blue}{J}\right) \]

   8. Simplified 52.5%

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

4. Final simplification 37.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq 1.55 \cdot 10^{+77}:\\ \;\;\;\;0 - U\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot J\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 39.7% accurate, 140.0× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ J\_s \cdot \left(0 - U\_m\right) \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m) :precision binary64 (* J_s (- 0.0 U_m)))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	return J_s * (0.0 - 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 * (0.0d0 - 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 * (0.0 - 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 * (0.0 - U_m)

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	return Float64(J_s * Float64(0.0 - 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 * (0.0 - 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 * N[(0.0 - U$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 71.9%

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

3. Taylor expanded in J around 0

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

   1. mul-1-neg N/A

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

   2. neg-sub0 N/A

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

   3. --lowering--.f64 28.7%

      \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{U}\right) \]

5. Simplified 28.7%

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

   1. sub0-neg N/A

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

   2. neg-lowering-neg.f64 28.7%

      \[\leadsto \mathsf{neg.f64}\left(U\right) \]

7. Applied egg-rr 28.7%

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

8. Final simplification 28.7%

   \[\leadsto 0 - U \]
9. Add Preprocessing

Alternative 7: 14.4% accurate, 420.0× 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 71.9%

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

3. Taylor expanded in U around -inf

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

5. Simplified 26.8%

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

Reproduce

herbie shell --seed 2024138 
(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)))))