Maksimov and Kolovsky, Equation (3)

Percentage Accurate: 74.3% → 99.1%

Time: 26.9s

Alternatives: 9

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 74.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J_m = \left|J\right| \\ J_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(t\_0 \cdot \left(-2 \cdot J\_m\right)\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_0 \cdot \left(J\_m \cdot 2\right)}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+292}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(U\_m \cdot -0.5\right)\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J_m = (fabs.f64 J)
J_s = (copysign.f64 1 J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* t_0 (* -2.0 J_m))
          (sqrt (+ 1.0 (pow (/ U_m (* t_0 (* J_m 2.0))) 2.0))))))
   (*
    J_s
    (if (<= t_1 (- INFINITY))
      (- U_m)
      (if (<= t_1 2e+292) t_1 (* -2.0 (* U_m -0.5)))))))

C

U_m = fabs(U);
J_m = fabs(J);
J_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = (t_0 * (-2.0 * J_m)) * sqrt((1.0 + pow((U_m / (t_0 * (J_m * 2.0))), 2.0)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_1 <= 2e+292) {
		tmp = t_1;
	} else {
		tmp = -2.0 * (U_m * -0.5);
	}
	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 = (t_0 * (-2.0 * J_m)) * 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 = -U_m;
	} else if (t_1 <= 2e+292) {
		tmp = t_1;
	} else {
		tmp = -2.0 * (U_m * -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):
	t_0 = math.cos((K / 2.0))
	t_1 = (t_0 * (-2.0 * J_m)) * math.sqrt((1.0 + math.pow((U_m / (t_0 * (J_m * 2.0))), 2.0)))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = -U_m
	elif t_1 <= 2e+292:
		tmp = t_1
	else:
		tmp = -2.0 * (U_m * -0.5)
	return J_s * tmp

Julia

U_m = abs(U)
J_m = abs(J)
J_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(t_0 * Float64(-2.0 * J_m)) * 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(-U_m);
	elseif (t_1 <= 2e+292)
		tmp = t_1;
	else
		tmp = Float64(-2.0 * Float64(U_m * -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)
	t_0 = cos((K / 2.0));
	t_1 = (t_0 * (-2.0 * J_m)) * sqrt((1.0 + ((U_m / (t_0 * (J_m * 2.0))) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = -U_m;
	elseif (t_1 <= 2e+292)
		tmp = t_1;
	else
		tmp = -2.0 * (U_m * -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_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * N[(-2.0 * J$95$m), $MachinePrecision]), $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)], (-U$95$m), If[LessEqual[t$95$1, 2e+292], t$95$1, N[(-2.0 * N[(U$95$m * -0.5), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 5.3%

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

   3. Taylor expanded in J around 0 51.6%

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

      1. neg-mul-1 51.6%

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

   5. Simplified 51.6%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 -2 J) (cos.f64 (/.f64 K 2))) (sqrt.f64 (+.f64 1 (pow.f64 (/.f64 U (*.f64 (*.f64 2 J) (cos.f64 (/.f64 K 2)))) 2)))) < 2e292

   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

   if 2e292 < (*.f64 (*.f64 (*.f64 -2 J) (cos.f64 (/.f64 K 2))) (sqrt.f64 (+.f64 1 (pow.f64 (/.f64 U (*.f64 (*.f64 2 J) (cos.f64 (/.f64 K 2)))) 2))))

   1. Initial program 10.5%

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

   3. Simplified 46.5%

      \[\leadsto \color{blue}{-2 \cdot \left(J \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\cos \left(\frac{K}{2}\right)} \cdot \frac{0.5}{J}\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in U around -inf 45.0%

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

      1. *-commutative 45.0%

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

   7. Simplified 45.0%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;\left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 2 \cdot 10^{+292}:\\ \;\;\;\;\left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J_m = \left|J\right| \\ J_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;U\_m \leq 1.9 \cdot 10^{+236}:\\ \;\;\;\;-2 \cdot \left(J\_m \cdot \left(t\_0 \cdot \mathsf{hypot}\left(1, \frac{U\_m}{t\_0} \cdot \frac{0.5}{J\_m}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(J\_m \cdot \frac{J\_m}{U\_m}\right) - U\_m\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J_m = (fabs.f64 J)
J_s = (copysign.f64 1 J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (*
    J_s
    (if (<= U_m 1.9e+236)
      (* -2.0 (* J_m (* t_0 (hypot 1.0 (* (/ U_m t_0) (/ 0.5 J_m))))))
      (- (* -2.0 (* J_m (/ J_m 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 (U_m <= 1.9e+236) {
		tmp = -2.0 * (J_m * (t_0 * hypot(1.0, ((U_m / t_0) * (0.5 / J_m)))));
	} else {
		tmp = (-2.0 * (J_m * (J_m / U_m))) - 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 tmp;
	if (U_m <= 1.9e+236) {
		tmp = -2.0 * (J_m * (t_0 * Math.hypot(1.0, ((U_m / t_0) * (0.5 / J_m)))));
	} else {
		tmp = (-2.0 * (J_m * (J_m / U_m))) - 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 U_m <= 1.9e+236:
		tmp = -2.0 * (J_m * (t_0 * math.hypot(1.0, ((U_m / t_0) * (0.5 / J_m)))))
	else:
		tmp = (-2.0 * (J_m * (J_m / U_m))) - 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 (U_m <= 1.9e+236)
		tmp = Float64(-2.0 * Float64(J_m * Float64(t_0 * hypot(1.0, Float64(Float64(U_m / t_0) * Float64(0.5 / J_m))))));
	else
		tmp = Float64(Float64(-2.0 * Float64(J_m * Float64(J_m / U_m))) - 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 (U_m <= 1.9e+236)
		tmp = -2.0 * (J_m * (t_0 * hypot(1.0, ((U_m / t_0) * (0.5 / J_m)))));
	else
		tmp = (-2.0 * (J_m * (J_m / U_m))) - 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[U$95$m, 1.9e+236], N[(-2.0 * N[(J$95$m * N[(t$95$0 * N[Sqrt[1.0 ^ 2 + N[(N[(U$95$m / t$95$0), $MachinePrecision] * N[(0.5 / J$95$m), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 * N[(J$95$m * N[(J$95$m / U$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - U$95$m), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U < 1.89999999999999993e236

   1. Initial program 72.2%

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

   3. Simplified 87.4%

      \[\leadsto \color{blue}{-2 \cdot \left(J \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\cos \left(\frac{K}{2}\right)} \cdot \frac{0.5}{J}\right)\right)\right)} \]
   4. Add Preprocessing

   if 1.89999999999999993e236 < U

   1. Initial program 38.3%

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

   3. Taylor expanded in J around 0 27.7%

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

      1. neg-mul-1 27.7%

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

      2. unsub-neg 27.7%

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

      3. *-commutative 27.7%

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

      4. unpow2 27.7%

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

      5. *-commutative 27.7%

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

      6. unpow2 27.7%

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

      7. swap-sqr 27.7%

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

      8. unpow2 27.7%

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

      9. *-commutative 27.7%

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

   5. Simplified 27.7%

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

   6. Taylor expanded in K around 0 27.7%

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

      1. unpow2 27.7%

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

      2. associate-/l* 42.2%

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

   8. Applied egg-rr 42.2%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 1.9 \cdot 10^{+236}:\\ \;\;\;\;-2 \cdot \left(J \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\cos \left(\frac{K}{2}\right)} \cdot \frac{0.5}{J}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(J \cdot \frac{J}{U}\right) - U\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J_m = \left|J\right| \\ J_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;U\_m \leq 9 \cdot 10^{+235}:\\ \;\;\;\;-2 \cdot \left(\left(J\_m \cdot t\_0\right) \cdot \mathsf{hypot}\left(1, \frac{\frac{U\_m}{J\_m \cdot 2}}{t\_0}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(J\_m \cdot \frac{J\_m}{U\_m}\right) - U\_m\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J_m = (fabs.f64 J)
J_s = (copysign.f64 1 J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (*
    J_s
    (if (<= U_m 9e+235)
      (* -2.0 (* (* J_m t_0) (hypot 1.0 (/ (/ U_m (* J_m 2.0)) t_0))))
      (- (* -2.0 (* J_m (/ J_m 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 (U_m <= 9e+235) {
		tmp = -2.0 * ((J_m * t_0) * hypot(1.0, ((U_m / (J_m * 2.0)) / t_0)));
	} else {
		tmp = (-2.0 * (J_m * (J_m / U_m))) - 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 tmp;
	if (U_m <= 9e+235) {
		tmp = -2.0 * ((J_m * t_0) * Math.hypot(1.0, ((U_m / (J_m * 2.0)) / t_0)));
	} else {
		tmp = (-2.0 * (J_m * (J_m / U_m))) - 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 U_m <= 9e+235:
		tmp = -2.0 * ((J_m * t_0) * math.hypot(1.0, ((U_m / (J_m * 2.0)) / t_0)))
	else:
		tmp = (-2.0 * (J_m * (J_m / U_m))) - 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 (U_m <= 9e+235)
		tmp = Float64(-2.0 * Float64(Float64(J_m * t_0) * hypot(1.0, Float64(Float64(U_m / Float64(J_m * 2.0)) / t_0))));
	else
		tmp = Float64(Float64(-2.0 * Float64(J_m * Float64(J_m / U_m))) - 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 (U_m <= 9e+235)
		tmp = -2.0 * ((J_m * t_0) * hypot(1.0, ((U_m / (J_m * 2.0)) / t_0)));
	else
		tmp = (-2.0 * (J_m * (J_m / U_m))) - 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[U$95$m, 9e+235], N[(-2.0 * N[(N[(J$95$m * 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], N[(N[(-2.0 * N[(J$95$m * N[(J$95$m / U$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - U$95$m), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U < 8.9999999999999999e235

   1. Initial program 72.2%

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

   3. Simplified 87.4%

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

   if 8.9999999999999999e235 < U

   1. Initial program 38.3%

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

   3. Taylor expanded in J around 0 27.7%

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

      1. neg-mul-1 27.7%

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

      2. unsub-neg 27.7%

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

      3. *-commutative 27.7%

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

      4. unpow2 27.7%

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

      5. *-commutative 27.7%

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

      6. unpow2 27.7%

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

      7. swap-sqr 27.7%

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

      8. unpow2 27.7%

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

      9. *-commutative 27.7%

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

   5. Simplified 27.7%

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

   6. Taylor expanded in K around 0 27.7%

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

      1. unpow2 27.7%

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

      2. associate-/l* 42.2%

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

   8. Applied egg-rr 42.2%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 9 \cdot 10^{+235}:\\ \;\;\;\;-2 \cdot \left(\left(J \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{\frac{U}{J \cdot 2}}{\cos \left(\frac{K}{2}\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(J \cdot \frac{J}{U}\right) - U\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J_m = \left|J\right| \\ J_s = \mathsf{copysign}\left(1, J\right) \\ J\_s \cdot \begin{array}{l} \mathbf{if}\;U\_m \leq 3.3 \cdot 10^{+178}:\\ \;\;\;\;-2 \cdot \left(J\_m \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \mathsf{hypot}\left(1, U\_m \cdot \frac{0.5}{J\_m}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(J\_m \cdot \frac{J\_m}{U\_m}\right) - U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J_m = (fabs.f64 J)
J_s = (copysign.f64 1 J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (*
  J_s
  (if (<= U_m 3.3e+178)
    (* -2.0 (* J_m (* (cos (/ K 2.0)) (hypot 1.0 (* U_m (/ 0.5 J_m))))))
    (- (* -2.0 (* J_m (/ J_m 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 tmp;
	if (U_m <= 3.3e+178) {
		tmp = -2.0 * (J_m * (cos((K / 2.0)) * hypot(1.0, (U_m * (0.5 / J_m)))));
	} else {
		tmp = (-2.0 * (J_m * (J_m / U_m))) - 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 tmp;
	if (U_m <= 3.3e+178) {
		tmp = -2.0 * (J_m * (Math.cos((K / 2.0)) * Math.hypot(1.0, (U_m * (0.5 / J_m)))));
	} else {
		tmp = (-2.0 * (J_m * (J_m / U_m))) - U_m;
	}
	return J_s * tmp;
}

Python

U_m = math.fabs(U)
J_m = math.fabs(J)
J_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	tmp = 0
	if U_m <= 3.3e+178:
		tmp = -2.0 * (J_m * (math.cos((K / 2.0)) * math.hypot(1.0, (U_m * (0.5 / J_m)))))
	else:
		tmp = (-2.0 * (J_m * (J_m / U_m))) - U_m
	return J_s * tmp

Julia

U_m = abs(U)
J_m = abs(J)
J_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	tmp = 0.0
	if (U_m <= 3.3e+178)
		tmp = Float64(-2.0 * Float64(J_m * Float64(cos(Float64(K / 2.0)) * hypot(1.0, Float64(U_m * Float64(0.5 / J_m))))));
	else
		tmp = Float64(Float64(-2.0 * Float64(J_m * Float64(J_m / U_m))) - U_m);
	end
	return Float64(J_s * tmp)
end

MATLAB

U_m = abs(U);
J_m = abs(J);
J_s = sign(J) * abs(1.0);
function tmp_2 = code(J_s, J_m, K, U_m)
	tmp = 0.0;
	if (U_m <= 3.3e+178)
		tmp = -2.0 * (J_m * (cos((K / 2.0)) * hypot(1.0, (U_m * (0.5 / J_m)))));
	else
		tmp = (-2.0 * (J_m * (J_m / U_m))) - U_m;
	end
	tmp_2 = J_s * tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J_m = N[Abs[J], $MachinePrecision]
J_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := N[(J$95$s * If[LessEqual[U$95$m, 3.3e+178], N[(-2.0 * N[(J$95$m * N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[(U$95$m * N[(0.5 / J$95$m), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 * N[(J$95$m * N[(J$95$m / U$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - U$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if U < 3.2999999999999998e178

   1. Initial program 74.3%

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

   3. Simplified 89.8%

      \[\leadsto \color{blue}{-2 \cdot \left(J \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\cos \left(\frac{K}{2}\right)} \cdot \frac{0.5}{J}\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in K around 0 76.8%

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

      1. associate-*r/ 76.8%

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

      2. *-commutative 76.8%

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

      3. associate-/l* 76.8%

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

   7. Simplified 76.8%

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

   if 3.2999999999999998e178 < U

   1. Initial program 33.5%

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

   3. Taylor expanded in J around 0 28.6%

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

      1. neg-mul-1 28.6%

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

      2. unsub-neg 28.6%

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

      3. *-commutative 28.6%

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

      4. unpow2 28.6%

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

      5. *-commutative 28.6%

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

      6. unpow2 28.6%

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

      7. swap-sqr 28.6%

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

      8. unpow2 28.6%

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

      9. *-commutative 28.6%

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

   5. Simplified 28.6%

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

   6. Taylor expanded in K around 0 28.6%

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

      1. unpow2 28.6%

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

      2. associate-/l* 38.0%

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

   8. Applied egg-rr 38.0%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 3.3 \cdot 10^{+178}:\\ \;\;\;\;-2 \cdot \left(J \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \mathsf{hypot}\left(1, U \cdot \frac{0.5}{J}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(J \cdot \frac{J}{U}\right) - U\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J_m = \left|J\right| \\ J_s = \mathsf{copysign}\left(1, J\right) \\ J\_s \cdot \begin{array}{l} \mathbf{if}\;U\_m \leq 3.8 \cdot 10^{+178}:\\ \;\;\;\;-2 \cdot \left(\left(J\_m \cdot \mathsf{hypot}\left(1, 0.5 \cdot \frac{U\_m}{J\_m}\right)\right) \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(J\_m \cdot \frac{J\_m}{U\_m}\right) - U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J_m = (fabs.f64 J)
J_s = (copysign.f64 1 J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (*
  J_s
  (if (<= U_m 3.8e+178)
    (* -2.0 (* (* J_m (hypot 1.0 (* 0.5 (/ U_m J_m)))) (cos (* K 0.5))))
    (- (* -2.0 (* J_m (/ J_m 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 tmp;
	if (U_m <= 3.8e+178) {
		tmp = -2.0 * ((J_m * hypot(1.0, (0.5 * (U_m / J_m)))) * cos((K * 0.5)));
	} else {
		tmp = (-2.0 * (J_m * (J_m / U_m))) - 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 tmp;
	if (U_m <= 3.8e+178) {
		tmp = -2.0 * ((J_m * Math.hypot(1.0, (0.5 * (U_m / J_m)))) * Math.cos((K * 0.5)));
	} else {
		tmp = (-2.0 * (J_m * (J_m / U_m))) - U_m;
	}
	return J_s * tmp;
}

Python

U_m = math.fabs(U)
J_m = math.fabs(J)
J_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	tmp = 0
	if U_m <= 3.8e+178:
		tmp = -2.0 * ((J_m * math.hypot(1.0, (0.5 * (U_m / J_m)))) * math.cos((K * 0.5)))
	else:
		tmp = (-2.0 * (J_m * (J_m / U_m))) - U_m
	return J_s * tmp

Julia

U_m = abs(U)
J_m = abs(J)
J_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	tmp = 0.0
	if (U_m <= 3.8e+178)
		tmp = Float64(-2.0 * Float64(Float64(J_m * hypot(1.0, Float64(0.5 * Float64(U_m / J_m)))) * cos(Float64(K * 0.5))));
	else
		tmp = Float64(Float64(-2.0 * Float64(J_m * Float64(J_m / U_m))) - U_m);
	end
	return Float64(J_s * tmp)
end

MATLAB

U_m = abs(U);
J_m = abs(J);
J_s = sign(J) * abs(1.0);
function tmp_2 = code(J_s, J_m, K, U_m)
	tmp = 0.0;
	if (U_m <= 3.8e+178)
		tmp = -2.0 * ((J_m * hypot(1.0, (0.5 * (U_m / J_m)))) * cos((K * 0.5)));
	else
		tmp = (-2.0 * (J_m * (J_m / U_m))) - U_m;
	end
	tmp_2 = J_s * tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J_m = N[Abs[J], $MachinePrecision]
J_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := N[(J$95$s * If[LessEqual[U$95$m, 3.8e+178], N[(-2.0 * N[(N[(J$95$m * N[Sqrt[1.0 ^ 2 + N[(0.5 * N[(U$95$m / J$95$m), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 * N[(J$95$m * N[(J$95$m / U$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - U$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if U < 3.79999999999999998e178

   1. Initial program 74.3%

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

   3. Simplified 89.8%

      \[\leadsto \color{blue}{-2 \cdot \left(J \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\cos \left(\frac{K}{2}\right)} \cdot \frac{0.5}{J}\right)\right)\right)} \]
   4. Add Preprocessing

   6. Applied egg-rr 88.2%

      \[\leadsto -2 \cdot \color{blue}{{\left(\sqrt[3]{\left(J \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \mathsf{hypot}\left(1, U \cdot \frac{\frac{0.5}{J}}{\cos \left(K \cdot 0.5\right)}\right)}\right)}^{3}} \]
   7. Step-by-step derivation

      1. rem-cube-cbrt 89.8%

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

      2. associate-*l* 89.7%

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

      3. *-commutative 89.7%

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

      4. *-commutative 89.7%

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

      5. associate-*r* 89.8%

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

      6. associate-/l/ 89.8%

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

      7. *-commutative 89.8%

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

      8. *-commutative 89.8%

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

   8. Applied egg-rr 89.8%

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

   9. Taylor expanded in K around 0 76.8%

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

   if 3.79999999999999998e178 < U

   1. Initial program 33.5%

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

   3. Taylor expanded in J around 0 28.6%

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

      1. neg-mul-1 28.6%

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

      2. unsub-neg 28.6%

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

      3. *-commutative 28.6%

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

      4. unpow2 28.6%

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

      5. *-commutative 28.6%

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

      6. unpow2 28.6%

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

      7. swap-sqr 28.6%

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

      8. unpow2 28.6%

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

      9. *-commutative 28.6%

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

   5. Simplified 28.6%

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

   6. Taylor expanded in K around 0 28.6%

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

      1. unpow2 28.6%

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

      2. associate-/l* 38.0%

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

   8. Applied egg-rr 38.0%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 3.8 \cdot 10^{+178}:\\ \;\;\;\;-2 \cdot \left(\left(J \cdot \mathsf{hypot}\left(1, 0.5 \cdot \frac{U}{J}\right)\right) \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(J \cdot \frac{J}{U}\right) - U\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 66.1% accurate, 3.7× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J_m = \left|J\right| \\ J_s = \mathsf{copysign}\left(1, J\right) \\ J\_s \cdot \begin{array}{l} \mathbf{if}\;J\_m \leq 8 \cdot 10^{-58}:\\ \;\;\;\;-U\_m\\ \mathbf{else}:\\ \;\;\;\;\cos \left(K \cdot 0.5\right) \cdot \left(-2 \cdot J\_m\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J_m = (fabs.f64 J)
J_s = (copysign.f64 1 J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (* J_s (if (<= J_m 8e-58) (- U_m) (* (cos (* K 0.5)) (* -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 <= 8e-58) {
		tmp = -U_m;
	} else {
		tmp = cos((K * 0.5)) * (-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 <= 8d-58) then
        tmp = -u_m
    else
        tmp = cos((k * 0.5d0)) * ((-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 <= 8e-58) {
		tmp = -U_m;
	} else {
		tmp = Math.cos((K * 0.5)) * (-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 <= 8e-58:
		tmp = -U_m
	else:
		tmp = math.cos((K * 0.5)) * (-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 <= 8e-58)
		tmp = Float64(-U_m);
	else
		tmp = Float64(cos(Float64(K * 0.5)) * 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 <= 8e-58)
		tmp = -U_m;
	else
		tmp = cos((K * 0.5)) * (-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, 8e-58], (-U$95$m), N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(-2.0 * J$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if J < 8.0000000000000002e-58

   1. Initial program 60.8%

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

   3. Taylor expanded in J around 0 32.3%

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

      1. neg-mul-1 32.3%

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

   5. Simplified 32.3%

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

   if 8.0000000000000002e-58 < J

   1. Initial program 93.2%

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

   3. Taylor expanded in J around inf 75.8%

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

      1. associate-*r* 75.8%

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

      2. *-commutative 75.8%

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

   5. Simplified 75.8%

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

4. Final simplification 44.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq 8 \cdot 10^{-58}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;\cos \left(K \cdot 0.5\right) \cdot \left(-2 \cdot J\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 48.7% accurate, 30.0× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J_m = \left|J\right| \\ J_s = \mathsf{copysign}\left(1, J\right) \\ J\_s \cdot \begin{array}{l} \mathbf{if}\;U\_m \leq 2.8 \cdot 10^{-156}:\\ \;\;\;\;-2 \cdot J\_m\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(J\_m \cdot \frac{J\_m}{U\_m}\right) - U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J_m = (fabs.f64 J)
J_s = (copysign.f64 1 J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (*
  J_s
  (if (<= U_m 2.8e-156) (* -2.0 J_m) (- (* -2.0 (* J_m (/ J_m 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 tmp;
	if (U_m <= 2.8e-156) {
		tmp = -2.0 * J_m;
	} else {
		tmp = (-2.0 * (J_m * (J_m / U_m))) - U_m;
	}
	return J_s * tmp;
}

Fortran

U_m = abs(U)
J_m = abs(J)
J_s = copysign(1.0d0, J)
real(8) function code(j_s, j_m, k, u_m)
    real(8), intent (in) :: j_s
    real(8), intent (in) :: j_m
    real(8), intent (in) :: k
    real(8), intent (in) :: u_m
    real(8) :: tmp
    if (u_m <= 2.8d-156) then
        tmp = (-2.0d0) * j_m
    else
        tmp = ((-2.0d0) * (j_m * (j_m / u_m))) - u_m
    end if
    code = j_s * tmp
end function

Java

U_m = Math.abs(U);
J_m = Math.abs(J);
J_s = Math.copySign(1.0, J);
public static double code(double J_s, double J_m, double K, double U_m) {
	double tmp;
	if (U_m <= 2.8e-156) {
		tmp = -2.0 * J_m;
	} else {
		tmp = (-2.0 * (J_m * (J_m / U_m))) - U_m;
	}
	return J_s * tmp;
}

Python

U_m = math.fabs(U)
J_m = math.fabs(J)
J_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	tmp = 0
	if U_m <= 2.8e-156:
		tmp = -2.0 * J_m
	else:
		tmp = (-2.0 * (J_m * (J_m / U_m))) - U_m
	return J_s * tmp

Julia

U_m = abs(U)
J_m = abs(J)
J_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	tmp = 0.0
	if (U_m <= 2.8e-156)
		tmp = Float64(-2.0 * J_m);
	else
		tmp = Float64(Float64(-2.0 * Float64(J_m * Float64(J_m / U_m))) - U_m);
	end
	return Float64(J_s * tmp)
end

MATLAB

U_m = abs(U);
J_m = abs(J);
J_s = sign(J) * abs(1.0);
function tmp_2 = code(J_s, J_m, K, U_m)
	tmp = 0.0;
	if (U_m <= 2.8e-156)
		tmp = -2.0 * J_m;
	else
		tmp = (-2.0 * (J_m * (J_m / U_m))) - U_m;
	end
	tmp_2 = J_s * tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J_m = N[Abs[J], $MachinePrecision]
J_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := N[(J$95$s * If[LessEqual[U$95$m, 2.8e-156], N[(-2.0 * J$95$m), $MachinePrecision], N[(N[(-2.0 * N[(J$95$m * N[(J$95$m / U$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - U$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if U < 2.8000000000000002e-156

   1. Initial program 75.4%

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

   3. Taylor expanded in K around 0 41.6%

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

      1. *-commutative 41.6%

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

   5. Simplified 41.6%

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

   6. Taylor expanded in J around inf 34.5%

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

      1. *-commutative 34.5%

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

   8. Simplified 34.5%

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

   if 2.8000000000000002e-156 < U

   1. Initial program 58.8%

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

   3. Taylor expanded in J around 0 35.1%

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

      1. neg-mul-1 35.1%

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

      2. unsub-neg 35.1%

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

      3. *-commutative 35.1%

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

      4. unpow2 35.1%

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

      5. *-commutative 35.1%

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

      6. unpow2 35.1%

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

      7. swap-sqr 35.1%

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

      8. unpow2 35.1%

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

      9. *-commutative 35.1%

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

   5. Simplified 35.1%

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

   6. Taylor expanded in K around 0 35.1%

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

      1. unpow2 35.1%

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

      2. associate-/l* 38.2%

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

   8. Applied egg-rr 38.2%

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

4. Final simplification 35.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 2.8 \cdot 10^{-156}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(J \cdot \frac{J}{U}\right) - U\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 48.3% accurate, 52.4× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J_m = \left|J\right| \\ J_s = \mathsf{copysign}\left(1, J\right) \\ J\_s \cdot \begin{array}{l} \mathbf{if}\;U\_m \leq 5 \cdot 10^{-157}:\\ \;\;\;\;-2 \cdot J\_m\\ \mathbf{else}:\\ \;\;\;\;-U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J_m = (fabs.f64 J)
J_s = (copysign.f64 1 J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (* J_s (if (<= U_m 5e-157) (* -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 tmp;
	if (U_m <= 5e-157) {
		tmp = -2.0 * J_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) :: tmp
    if (u_m <= 5d-157) then
        tmp = (-2.0d0) * j_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 tmp;
	if (U_m <= 5e-157) {
		tmp = -2.0 * J_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):
	tmp = 0
	if U_m <= 5e-157:
		tmp = -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)
	tmp = 0.0
	if (U_m <= 5e-157)
		tmp = Float64(-2.0 * J_m);
	else
		tmp = Float64(-U_m);
	end
	return Float64(J_s * tmp)
end

MATLAB

U_m = abs(U);
J_m = abs(J);
J_s = sign(J) * abs(1.0);
function tmp_2 = code(J_s, J_m, K, U_m)
	tmp = 0.0;
	if (U_m <= 5e-157)
		tmp = -2.0 * J_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_] := N[(J$95$s * If[LessEqual[U$95$m, 5e-157], N[(-2.0 * J$95$m), $MachinePrecision], (-U$95$m)]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if U < 5.0000000000000002e-157

   1. Initial program 75.3%

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

   3. Taylor expanded in K around 0 41.8%

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

      1. *-commutative 41.8%

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

   5. Simplified 41.8%

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

   6. Taylor expanded in J around inf 34.7%

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

      1. *-commutative 34.7%

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

   8. Simplified 34.7%

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

   if 5.0000000000000002e-157 < U

   1. Initial program 59.2%

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

   3. Taylor expanded in J around 0 37.8%

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

      1. neg-mul-1 37.8%

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

   5. Simplified 37.8%

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

4. Final simplification 35.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 5 \cdot 10^{-157}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 39.7% accurate, 210.0× speedup

Math

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

FPCore

U_m = (fabs.f64 U)
J_m = (fabs.f64 J)
J_s = (copysign.f64 1 J)
(FPCore (J_s J_m K U_m) :precision binary64 (* J_s (- U_m)))

C

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

Fortran

U_m = abs(U)
J_m = abs(J)
J_s = copysign(1.0d0, J)
real(8) function code(j_s, j_m, k, u_m)
    real(8), intent (in) :: j_s
    real(8), intent (in) :: j_m
    real(8), intent (in) :: k
    real(8), intent (in) :: u_m
    code = j_s * -u_m
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.6%

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

3. Taylor expanded in J around 0 27.6%

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

   1. neg-mul-1 27.6%

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

5. Simplified 27.6%

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

6. Final simplification 27.6%

   \[\leadsto -U \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024039 
(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)))))