Maksimov and Kolovsky, Equation (3)

Percentage Accurate: 74.2% → 99.3%

Time: 16.0s

Alternatives: 11

Speedup: 1.8×

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: 74.2% 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: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_0 \cdot \left(J\_m \cdot 2\right)}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq 10^{+295}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \end{array} \]

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)) (- U_m) (if (<= t_1 1e+295) 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 = -U_m;
	} else if (t_1 <= 1e+295) {
		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 = -U_m;
	} else if (t_1 <= 1e+295) {
		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 = -U_m
	elif t_1 <= 1e+295:
		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(-U_m);
	elseif (t_1 <= 1e+295)
		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 = -U_m;
	elseif (t_1 <= 1e+295)
		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)], (-U$95$m), If[LessEqual[t$95$1, 1e+295], 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%

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

   3. Simplified 50.3%

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

   5. Taylor expanded in J around 0 55.6%

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

      1. neg-mul-1 55.6%

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

   7. Simplified 55.6%

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

   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))))) < 9.9999999999999998e294

   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 9.9999999999999998e294 < (*.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 9.9%

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

   3. Simplified 65.5%

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

   5. Taylor expanded in U around -inf 61.3%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 10^{+295}:\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 90.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;J\_m \leq 1.75 \cdot 10^{-200}:\\ \;\;\;\;-U\_m\\ \mathbf{else}:\\ \;\;\;\;J\_m \cdot \left(\left(-2 \cdot t\_0\right) \cdot \mathsf{hypot}\left(1, \frac{\frac{U\_m}{2}}{J\_m \cdot t\_0}\right)\right)\\ \end{array} \end{array} \end{array} \]

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 (<= J_m 1.75e-200)
      (- U_m)
      (* J_m (* (* -2.0 t_0) (hypot 1.0 (/ (/ U_m 2.0) (* J_m 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 <= 1.75e-200) {
		tmp = -U_m;
	} else {
		tmp = J_m * ((-2.0 * t_0) * hypot(1.0, ((U_m / 2.0) / (J_m * 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 <= 1.75e-200) {
		tmp = -U_m;
	} else {
		tmp = J_m * ((-2.0 * t_0) * Math.hypot(1.0, ((U_m / 2.0) / (J_m * 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 <= 1.75e-200:
		tmp = -U_m
	else:
		tmp = J_m * ((-2.0 * t_0) * math.hypot(1.0, ((U_m / 2.0) / (J_m * 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 <= 1.75e-200)
		tmp = Float64(-U_m);
	else
		tmp = Float64(J_m * Float64(Float64(-2.0 * t_0) * hypot(1.0, Float64(Float64(U_m / 2.0) / Float64(J_m * 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 <= 1.75e-200)
		tmp = -U_m;
	else
		tmp = J_m * ((-2.0 * t_0) * hypot(1.0, ((U_m / 2.0) / (J_m * 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, 1.75e-200], (-U$95$m), N[(J$95$m * N[(N[(-2.0 * t$95$0), $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[(N[(U$95$m / 2.0), $MachinePrecision] / N[(J$95$m * t$95$0), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if J < 1.75000000000000011e-200

   1. Initial program 65.1%

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

   3. Simplified 80.5%

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

   5. Taylor expanded in J around 0 25.9%

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

      1. neg-mul-1 25.9%

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

   7. Simplified 25.9%

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

   if 1.75000000000000011e-200 < J

   1. Initial program 80.0%

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

   3. Simplified 95.5%

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

4. Final simplification 55.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq 1.75 \cdot 10^{-200}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{\frac{U}{2}}{J \cdot \cos \left(\frac{K}{2}\right)}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ J\_s \cdot \begin{array}{l} \mathbf{if}\;J\_m \leq 6.2 \cdot 10^{-199}:\\ \;\;\;\;-U\_m\\ \mathbf{else}:\\ \;\;\;\;J\_m \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{\frac{U\_m}{\cos \left(K \cdot 0.5\right)}}{-2 \cdot J\_m}\right)\right)\\ \end{array} \end{array} \]

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 6.2e-199)
    (- U_m)
    (*
     J_m
     (*
      (* -2.0 (cos (/ K 2.0)))
      (hypot 1.0 (/ (/ 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 <= 6.2e-199) {
		tmp = -U_m;
	} else {
		tmp = J_m * ((-2.0 * cos((K / 2.0))) * hypot(1.0, ((U_m / cos((K * 0.5))) / (-2.0 * J_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 (J_m <= 6.2e-199) {
		tmp = -U_m;
	} else {
		tmp = J_m * ((-2.0 * Math.cos((K / 2.0))) * Math.hypot(1.0, ((U_m / 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 <= 6.2e-199:
		tmp = -U_m
	else:
		tmp = J_m * ((-2.0 * math.cos((K / 2.0))) * math.hypot(1.0, ((U_m / 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 <= 6.2e-199)
		tmp = Float64(-U_m);
	else
		tmp = Float64(J_m * Float64(Float64(-2.0 * cos(Float64(K / 2.0))) * hypot(1.0, Float64(Float64(U_m / 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 <= 6.2e-199)
		tmp = -U_m;
	else
		tmp = J_m * ((-2.0 * cos((K / 2.0))) * hypot(1.0, ((U_m / 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, 6.2e-199], (-U$95$m), N[(J$95$m * N[(N[(-2.0 * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[(N[(U$95$m / N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(-2.0 * J$95$m), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if J < 6.20000000000000024e-199

   1. Initial program 65.1%

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

   3. Simplified 80.5%

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

   5. Taylor expanded in J around 0 25.9%

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

      1. neg-mul-1 25.9%

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

   7. Simplified 25.9%

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

   if 6.20000000000000024e-199 < J

   1. Initial program 80.0%

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

   3. Simplified 95.5%

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

      1. associate-/r* 95.5%

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

      2. associate-*l* 95.5%

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

      3. add-cube-cbrt 95.0%

         \[\leadsto J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{\color{blue}{\left(\sqrt[3]{U} \cdot \sqrt[3]{U}\right) \cdot \sqrt[3]{U}}}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)\right) \]

      4. associate-/l* 94.9%

         \[\leadsto J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \color{blue}{\left(\sqrt[3]{U} \cdot \sqrt[3]{U}\right) \cdot \frac{\sqrt[3]{U}}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}}\right)\right) \]

      5. pow2 94.9%

         \[\leadsto J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \color{blue}{{\left(\sqrt[3]{U}\right)}^{2}} \cdot \frac{\sqrt[3]{U}}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)\right) \]

      6. *-commutative 94.9%

         \[\leadsto J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, {\left(\sqrt[3]{U}\right)}^{2} \cdot \frac{\sqrt[3]{U}}{\color{blue}{\cos \left(\frac{K}{2}\right) \cdot \left(2 \cdot J\right)}}\right)\right) \]

      7. *-commutative 94.9%

         \[\leadsto J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, {\left(\sqrt[3]{U}\right)}^{2} \cdot \frac{\sqrt[3]{U}}{\cos \left(\frac{K}{2}\right) \cdot \color{blue}{\left(J \cdot 2\right)}}\right)\right) \]

      8. add-sqr-sqrt 94.9%

         \[\leadsto J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, {\left(\sqrt[3]{U}\right)}^{2} \cdot \frac{\sqrt[3]{U}}{\cos \left(\frac{K}{2}\right) \cdot \color{blue}{\left(\sqrt{J \cdot 2} \cdot \sqrt{J \cdot 2}\right)}}\right)\right) \]

      9. div-inv 94.9%

         \[\leadsto J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, {\left(\sqrt[3]{U}\right)}^{2} \cdot \frac{\sqrt[3]{U}}{\cos \color{blue}{\left(K \cdot \frac{1}{2}\right)} \cdot \left(\sqrt{J \cdot 2} \cdot \sqrt{J \cdot 2}\right)}\right)\right) \]

      10. metadata-eval 94.9%

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

      11. sqrt-unprod 83.1%

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

      12. swap-sqr 83.1%

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

      13. metadata-eval 83.1%

          \[\leadsto J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, {\left(\sqrt[3]{U}\right)}^{2} \cdot \frac{\sqrt[3]{U}}{\cos \left(K \cdot 0.5\right) \cdot \sqrt{\left(J \cdot J\right) \cdot \color{blue}{4}}}\right)\right) \]

      14. metadata-eval 83.1%

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

      15. swap-sqr 83.1%

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

      16. *-commutative 83.1%

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

      17. *-commutative 83.1%

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

      18. sqrt-unprod 0.0%

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

      19. add-sqr-sqrt 94.9%

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

      20. *-commutative 94.9%

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

   6. Applied egg-rr 94.9%

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

      1. associate-/r* 94.9%

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

      2. associate-*r/ 94.9%

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

      3. associate-*r/ 94.9%

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

      4. unpow2 94.9%

         \[\leadsto J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{\frac{\color{blue}{\left(\sqrt[3]{U} \cdot \sqrt[3]{U}\right)} \cdot \sqrt[3]{U}}{\cos \left(K \cdot 0.5\right)}}{J \cdot -2}\right)\right) \]

      5. rem-3cbrt-lft 95.5%

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

      6. *-commutative 95.5%

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

      7. *-commutative 95.5%

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

   8. Simplified 95.5%

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

4. Final simplification 55.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq 6.2 \cdot 10^{-199}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{\frac{U}{\cos \left(K \cdot 0.5\right)}}{-2 \cdot J}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ J\_s \cdot \begin{array}{l} \mathbf{if}\;J\_m \leq 2.5 \cdot 10^{-197}:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;J\_m \leq 1.55 \cdot 10^{-148} \lor \neg \left(J\_m \leq 1.5 \cdot 10^{-62}\right):\\ \;\;\;\;J\_m \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{\frac{U\_m}{2}}{J\_m}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot {\left(J\_m \cdot \cos \left(K \cdot 0.5\right)\right)}^{2}}{U\_m} - U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (*
  J_s
  (if (<= J_m 2.5e-197)
    (- U_m)
    (if (or (<= J_m 1.55e-148) (not (<= J_m 1.5e-62)))
      (* J_m (* (* -2.0 (cos (/ K 2.0))) (hypot 1.0 (/ (/ U_m 2.0) J_m))))
      (- (/ (* -2.0 (pow (* J_m (cos (* K 0.5))) 2.0)) U_m) U_m)))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double tmp;
	if (J_m <= 2.5e-197) {
		tmp = -U_m;
	} else if ((J_m <= 1.55e-148) || !(J_m <= 1.5e-62)) {
		tmp = J_m * ((-2.0 * cos((K / 2.0))) * hypot(1.0, ((U_m / 2.0) / J_m)));
	} else {
		tmp = ((-2.0 * pow((J_m * cos((K * 0.5))), 2.0)) / 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 (J_m <= 2.5e-197) {
		tmp = -U_m;
	} else if ((J_m <= 1.55e-148) || !(J_m <= 1.5e-62)) {
		tmp = J_m * ((-2.0 * Math.cos((K / 2.0))) * Math.hypot(1.0, ((U_m / 2.0) / J_m)));
	} else {
		tmp = ((-2.0 * Math.pow((J_m * Math.cos((K * 0.5))), 2.0)) / 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 J_m <= 2.5e-197:
		tmp = -U_m
	elif (J_m <= 1.55e-148) or not (J_m <= 1.5e-62):
		tmp = J_m * ((-2.0 * math.cos((K / 2.0))) * math.hypot(1.0, ((U_m / 2.0) / J_m)))
	else:
		tmp = ((-2.0 * math.pow((J_m * math.cos((K * 0.5))), 2.0)) / 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 (J_m <= 2.5e-197)
		tmp = Float64(-U_m);
	elseif ((J_m <= 1.55e-148) || !(J_m <= 1.5e-62))
		tmp = Float64(J_m * Float64(Float64(-2.0 * cos(Float64(K / 2.0))) * hypot(1.0, Float64(Float64(U_m / 2.0) / J_m))));
	else
		tmp = Float64(Float64(Float64(-2.0 * (Float64(J_m * cos(Float64(K * 0.5))) ^ 2.0)) / 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 (J_m <= 2.5e-197)
		tmp = -U_m;
	elseif ((J_m <= 1.55e-148) || ~((J_m <= 1.5e-62)))
		tmp = J_m * ((-2.0 * cos((K / 2.0))) * hypot(1.0, ((U_m / 2.0) / J_m)));
	else
		tmp = ((-2.0 * ((J_m * cos((K * 0.5))) ^ 2.0)) / 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[J$95$m, 2.5e-197], (-U$95$m), If[Or[LessEqual[J$95$m, 1.55e-148], N[Not[LessEqual[J$95$m, 1.5e-62]], $MachinePrecision]], N[(J$95$m * N[(N[(-2.0 * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[(N[(U$95$m / 2.0), $MachinePrecision] / J$95$m), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-2.0 * N[Power[N[(J$95$m * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / U$95$m), $MachinePrecision] - U$95$m), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if J < 2.5000000000000001e-197

   1. Initial program 65.1%

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

   3. Simplified 80.5%

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

   5. Taylor expanded in J around 0 25.9%

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

      1. neg-mul-1 25.9%

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

   7. Simplified 25.9%

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

   if 2.5000000000000001e-197 < J < 1.5500000000000001e-148 or 1.5000000000000001e-62 < J

   1. Initial program 86.1%

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

   3. Simplified 96.7%

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

   5. Taylor expanded in K around 0 86.7%

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

   if 1.5500000000000001e-148 < J < 1.5000000000000001e-62

   1. Initial program 54.3%

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

   3. Simplified 90.8%

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

   5. Taylor expanded in J around 0 27.5%

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

      1. neg-mul-1 27.5%

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

      2. unsub-neg 27.5%

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

      3. associate-*r/ 27.5%

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

      4. unpow2 27.5%

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

      5. *-commutative 27.5%

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

      6. unpow2 27.5%

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

      7. swap-sqr 27.5%

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

      8. unpow2 27.5%

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

      9. *-commutative 27.5%

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

   7. Simplified 27.5%

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

4. Final simplification 46.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq 2.5 \cdot 10^{-197}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 1.55 \cdot 10^{-148} \lor \neg \left(J \leq 1.5 \cdot 10^{-62}\right):\\ \;\;\;\;J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{\frac{U}{2}}{J}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot {\left(J \cdot \cos \left(K \cdot 0.5\right)\right)}^{2}}{U} - U\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(K \cdot 0.5\right)\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;J\_m \leq 9.5 \cdot 10^{-197}:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;J\_m \leq 7.5 \cdot 10^{-148}:\\ \;\;\;\;J\_m \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{\frac{U\_m}{2}}{J\_m}\right)\right)\\ \mathbf{elif}\;J\_m \leq 2 \cdot 10^{-62}:\\ \;\;\;\;\frac{-2 \cdot {\left(J\_m \cdot t\_0\right)}^{2}}{U\_m} - U\_m\\ \mathbf{else}:\\ \;\;\;\;\left(J\_m \cdot \left(-2 \cdot t\_0\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U\_m \cdot 0.5}{J\_m}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (* K 0.5))))
   (*
    J_s
    (if (<= J_m 9.5e-197)
      (- U_m)
      (if (<= J_m 7.5e-148)
        (* J_m (* (* -2.0 (cos (/ K 2.0))) (hypot 1.0 (/ (/ U_m 2.0) J_m))))
        (if (<= J_m 2e-62)
          (- (/ (* -2.0 (pow (* J_m t_0) 2.0)) U_m) U_m)
          (* (* J_m (* -2.0 t_0)) (hypot 1.0 (/ (* U_m 0.5) J_m)))))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K * 0.5));
	double tmp;
	if (J_m <= 9.5e-197) {
		tmp = -U_m;
	} else if (J_m <= 7.5e-148) {
		tmp = J_m * ((-2.0 * cos((K / 2.0))) * hypot(1.0, ((U_m / 2.0) / J_m)));
	} else if (J_m <= 2e-62) {
		tmp = ((-2.0 * pow((J_m * t_0), 2.0)) / U_m) - U_m;
	} else {
		tmp = (J_m * (-2.0 * t_0)) * hypot(1.0, ((U_m * 0.5) / J_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 * 0.5));
	double tmp;
	if (J_m <= 9.5e-197) {
		tmp = -U_m;
	} else if (J_m <= 7.5e-148) {
		tmp = J_m * ((-2.0 * Math.cos((K / 2.0))) * Math.hypot(1.0, ((U_m / 2.0) / J_m)));
	} else if (J_m <= 2e-62) {
		tmp = ((-2.0 * Math.pow((J_m * t_0), 2.0)) / U_m) - U_m;
	} else {
		tmp = (J_m * (-2.0 * t_0)) * Math.hypot(1.0, ((U_m * 0.5) / J_m));
	}
	return J_s * tmp;
}

Python

U_m = math.fabs(U)
J\_m = math.fabs(J)
J\_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	t_0 = math.cos((K * 0.5))
	tmp = 0
	if J_m <= 9.5e-197:
		tmp = -U_m
	elif J_m <= 7.5e-148:
		tmp = J_m * ((-2.0 * math.cos((K / 2.0))) * math.hypot(1.0, ((U_m / 2.0) / J_m)))
	elif J_m <= 2e-62:
		tmp = ((-2.0 * math.pow((J_m * t_0), 2.0)) / U_m) - U_m
	else:
		tmp = (J_m * (-2.0 * t_0)) * math.hypot(1.0, ((U_m * 0.5) / J_m))
	return J_s * tmp

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K * 0.5))
	tmp = 0.0
	if (J_m <= 9.5e-197)
		tmp = Float64(-U_m);
	elseif (J_m <= 7.5e-148)
		tmp = Float64(J_m * Float64(Float64(-2.0 * cos(Float64(K / 2.0))) * hypot(1.0, Float64(Float64(U_m / 2.0) / J_m))));
	elseif (J_m <= 2e-62)
		tmp = Float64(Float64(Float64(-2.0 * (Float64(J_m * t_0) ^ 2.0)) / U_m) - U_m);
	else
		tmp = Float64(Float64(J_m * Float64(-2.0 * t_0)) * hypot(1.0, Float64(Float64(U_m * 0.5) / J_m)));
	end
	return Float64(J_s * tmp)
end

MATLAB

U_m = abs(U);
J\_m = abs(J);
J\_s = sign(J) * abs(1.0);
function tmp_2 = code(J_s, J_m, K, U_m)
	t_0 = cos((K * 0.5));
	tmp = 0.0;
	if (J_m <= 9.5e-197)
		tmp = -U_m;
	elseif (J_m <= 7.5e-148)
		tmp = J_m * ((-2.0 * cos((K / 2.0))) * hypot(1.0, ((U_m / 2.0) / J_m)));
	elseif (J_m <= 2e-62)
		tmp = ((-2.0 * ((J_m * t_0) ^ 2.0)) / U_m) - U_m;
	else
		tmp = (J_m * (-2.0 * t_0)) * hypot(1.0, ((U_m * 0.5) / J_m));
	end
	tmp_2 = J_s * tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]}, N[(J$95$s * If[LessEqual[J$95$m, 9.5e-197], (-U$95$m), If[LessEqual[J$95$m, 7.5e-148], N[(J$95$m * N[(N[(-2.0 * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[(N[(U$95$m / 2.0), $MachinePrecision] / J$95$m), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[J$95$m, 2e-62], N[(N[(N[(-2.0 * N[Power[N[(J$95$m * t$95$0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / U$95$m), $MachinePrecision] - U$95$m), $MachinePrecision], N[(N[(J$95$m * N[(-2.0 * t$95$0), $MachinePrecision]), $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[(N[(U$95$m * 0.5), $MachinePrecision] / J$95$m), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if J < 9.5000000000000003e-197

   1. Initial program 65.1%

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

   3. Simplified 80.5%

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

   5. Taylor expanded in J around 0 25.9%

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

      1. neg-mul-1 25.9%

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

   7. Simplified 25.9%

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

   if 9.5000000000000003e-197 < J < 7.5000000000000005e-148

   1. Initial program 45.0%

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

   3. Simplified 85.1%

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

   5. Taylor expanded in K around 0 65.4%

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

   if 7.5000000000000005e-148 < J < 2.0000000000000001e-62

   1. Initial program 54.3%

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

   3. Simplified 90.8%

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

   5. Taylor expanded in J around 0 27.5%

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

      1. neg-mul-1 27.5%

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

      2. unsub-neg 27.5%

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

      3. associate-*r/ 27.5%

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

      4. unpow2 27.5%

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

      5. *-commutative 27.5%

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

      6. unpow2 27.5%

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

      7. swap-sqr 27.5%

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

      8. unpow2 27.5%

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

      9. *-commutative 27.5%

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

   7. Simplified 27.5%

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

   if 2.0000000000000001e-62 < J

   1. Initial program 92.6%

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

   3. Simplified 98.5%

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

   5. Taylor expanded in K around 0 90.1%

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

   6. Taylor expanded in K around inf 65.9%

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

      1. associate-*r* 65.9%

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

      2. *-commutative 65.9%

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

      3. *-commutative 65.9%

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

      4. associate-*l* 65.9%

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

      5. *-commutative 65.9%

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

      6. *-commutative 65.9%

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

      7. metadata-eval 65.9%

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

      8. unpow2 65.9%

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

      9. unpow2 65.9%

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

      10. times-frac 86.1%

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

      11. swap-sqr 86.1%

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

      12. metadata-eval 86.1%

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

      13. hypot-undefine 90.1%

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

      14. associate-*r/ 90.1%

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

      15. *-commutative 90.1%

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

   8. Simplified 90.1%

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

4. Final simplification 46.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq 9.5 \cdot 10^{-197}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 7.5 \cdot 10^{-148}:\\ \;\;\;\;J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{\frac{U}{2}}{J}\right)\right)\\ \mathbf{elif}\;J \leq 2 \cdot 10^{-62}:\\ \;\;\;\;\frac{-2 \cdot {\left(J \cdot \cos \left(K \cdot 0.5\right)\right)}^{2}}{U} - U\\ \mathbf{else}:\\ \;\;\;\;\left(J \cdot \left(-2 \cdot \cos \left(K \cdot 0.5\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U \cdot 0.5}{J}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ J\_s \cdot \begin{array}{l} \mathbf{if}\;J\_m \leq 4.35 \cdot 10^{-44}:\\ \;\;\;\;\frac{-2 \cdot {\left(J\_m \cdot \cos \left(K \cdot 0.5\right)\right)}^{2}}{U\_m} - U\_m\\ \mathbf{else}:\\ \;\;\;\;J\_m \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(1 + \left(\frac{U\_m}{J\_m} \cdot \frac{U\_m}{J\_m}\right) \cdot 0.125\right)\right)\\ \end{array} \end{array} \]

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 4.35e-44)
    (- (/ (* -2.0 (pow (* J_m (cos (* K 0.5))) 2.0)) U_m) U_m)
    (*
     J_m
     (*
      (* -2.0 (cos (/ K 2.0)))
      (+ 1.0 (* (* (/ U_m J_m) (/ U_m J_m)) 0.125)))))))

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 <= 4.35e-44) {
		tmp = ((-2.0 * pow((J_m * cos((K * 0.5))), 2.0)) / U_m) - U_m;
	} else {
		tmp = J_m * ((-2.0 * cos((K / 2.0))) * (1.0 + (((U_m / J_m) * (U_m / J_m)) * 0.125)));
	}
	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 <= 4.35d-44) then
        tmp = (((-2.0d0) * ((j_m * cos((k * 0.5d0))) ** 2.0d0)) / u_m) - u_m
    else
        tmp = j_m * (((-2.0d0) * cos((k / 2.0d0))) * (1.0d0 + (((u_m / j_m) * (u_m / j_m)) * 0.125d0)))
    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 <= 4.35e-44) {
		tmp = ((-2.0 * Math.pow((J_m * Math.cos((K * 0.5))), 2.0)) / U_m) - U_m;
	} else {
		tmp = J_m * ((-2.0 * Math.cos((K / 2.0))) * (1.0 + (((U_m / J_m) * (U_m / J_m)) * 0.125)));
	}
	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 <= 4.35e-44:
		tmp = ((-2.0 * math.pow((J_m * math.cos((K * 0.5))), 2.0)) / U_m) - U_m
	else:
		tmp = J_m * ((-2.0 * math.cos((K / 2.0))) * (1.0 + (((U_m / J_m) * (U_m / J_m)) * 0.125)))
	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 <= 4.35e-44)
		tmp = Float64(Float64(Float64(-2.0 * (Float64(J_m * cos(Float64(K * 0.5))) ^ 2.0)) / U_m) - U_m);
	else
		tmp = Float64(J_m * Float64(Float64(-2.0 * cos(Float64(K / 2.0))) * Float64(1.0 + Float64(Float64(Float64(U_m / J_m) * Float64(U_m / J_m)) * 0.125))));
	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 <= 4.35e-44)
		tmp = ((-2.0 * ((J_m * cos((K * 0.5))) ^ 2.0)) / U_m) - U_m;
	else
		tmp = J_m * ((-2.0 * cos((K / 2.0))) * (1.0 + (((U_m / J_m) * (U_m / J_m)) * 0.125)));
	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, 4.35e-44], N[(N[(N[(-2.0 * N[Power[N[(J$95$m * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / U$95$m), $MachinePrecision] - U$95$m), $MachinePrecision], N[(J$95$m * N[(N[(-2.0 * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(N[(U$95$m / J$95$m), $MachinePrecision] * N[(U$95$m / J$95$m), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if J < 4.3500000000000002e-44

   1. Initial program 62.7%

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

   3. Simplified 82.1%

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

   5. Taylor expanded in J around 0 26.2%

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

      1. neg-mul-1 26.2%

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

      2. unsub-neg 26.2%

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

      3. associate-*r/ 26.2%

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

      4. unpow2 26.2%

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

      5. *-commutative 26.2%

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

      6. unpow2 26.2%

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

      7. swap-sqr 26.2%

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

      8. unpow2 26.2%

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

      9. *-commutative 26.2%

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

   7. Simplified 26.2%

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

   if 4.3500000000000002e-44 < J

   1. Initial program 92.5%

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

   3. Simplified 98.5%

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

   5. Taylor expanded in K around 0 89.9%

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

   6. Taylor expanded in U around 0 63.9%

      \[\leadsto J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\left(1 + 0.125 \cdot \frac{{U}^{2}}{{J}^{2}}\right)}\right) \]
   7. Step-by-step derivation

      1. *-commutative 63.9%

         \[\leadsto J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(1 + \color{blue}{\frac{{U}^{2}}{{J}^{2}} \cdot 0.125}\right)\right) \]

   8. Simplified 63.9%

      \[\leadsto J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\left(1 + \frac{{U}^{2}}{{J}^{2}} \cdot 0.125\right)}\right) \]
   9. Step-by-step derivation

      1. unpow2 63.9%

         \[\leadsto J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(1 + \frac{\color{blue}{U \cdot U}}{{J}^{2}} \cdot 0.125\right)\right) \]

      2. unpow2 63.9%

         \[\leadsto J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(1 + \frac{U \cdot U}{\color{blue}{J \cdot J}} \cdot 0.125\right)\right) \]

      3. frac-times 75.0%

         \[\leadsto J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(1 + \color{blue}{\left(\frac{U}{J} \cdot \frac{U}{J}\right)} \cdot 0.125\right)\right) \]

   10. Applied egg-rr 75.0%

       \[\leadsto J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(1 + \color{blue}{\left(\frac{U}{J} \cdot \frac{U}{J}\right)} \cdot 0.125\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq 4.35 \cdot 10^{-44}:\\ \;\;\;\;\frac{-2 \cdot {\left(J \cdot \cos \left(K \cdot 0.5\right)\right)}^{2}}{U} - U\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(1 + \left(\frac{U}{J} \cdot \frac{U}{J}\right) \cdot 0.125\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.7% accurate, 3.4× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ J\_s \cdot \begin{array}{l} \mathbf{if}\;J\_m \leq 3.9 \cdot 10^{-44}:\\ \;\;\;\;\frac{-2 \cdot {J\_m}^{2}}{U\_m} - U\_m\\ \mathbf{else}:\\ \;\;\;\;J\_m \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(1 + \left(\frac{U\_m}{J\_m} \cdot \frac{U\_m}{J\_m}\right) \cdot 0.125\right)\right)\\ \end{array} \end{array} \]

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 3.9e-44)
    (- (/ (* -2.0 (pow J_m 2.0)) U_m) U_m)
    (*
     J_m
     (*
      (* -2.0 (cos (/ K 2.0)))
      (+ 1.0 (* (* (/ U_m J_m) (/ U_m J_m)) 0.125)))))))

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 <= 3.9e-44) {
		tmp = ((-2.0 * pow(J_m, 2.0)) / U_m) - U_m;
	} else {
		tmp = J_m * ((-2.0 * cos((K / 2.0))) * (1.0 + (((U_m / J_m) * (U_m / J_m)) * 0.125)));
	}
	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 <= 3.9d-44) then
        tmp = (((-2.0d0) * (j_m ** 2.0d0)) / u_m) - u_m
    else
        tmp = j_m * (((-2.0d0) * cos((k / 2.0d0))) * (1.0d0 + (((u_m / j_m) * (u_m / j_m)) * 0.125d0)))
    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 <= 3.9e-44) {
		tmp = ((-2.0 * Math.pow(J_m, 2.0)) / U_m) - U_m;
	} else {
		tmp = J_m * ((-2.0 * Math.cos((K / 2.0))) * (1.0 + (((U_m / J_m) * (U_m / J_m)) * 0.125)));
	}
	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 <= 3.9e-44:
		tmp = ((-2.0 * math.pow(J_m, 2.0)) / U_m) - U_m
	else:
		tmp = J_m * ((-2.0 * math.cos((K / 2.0))) * (1.0 + (((U_m / J_m) * (U_m / J_m)) * 0.125)))
	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 <= 3.9e-44)
		tmp = Float64(Float64(Float64(-2.0 * (J_m ^ 2.0)) / U_m) - U_m);
	else
		tmp = Float64(J_m * Float64(Float64(-2.0 * cos(Float64(K / 2.0))) * Float64(1.0 + Float64(Float64(Float64(U_m / J_m) * Float64(U_m / J_m)) * 0.125))));
	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 <= 3.9e-44)
		tmp = ((-2.0 * (J_m ^ 2.0)) / U_m) - U_m;
	else
		tmp = J_m * ((-2.0 * cos((K / 2.0))) * (1.0 + (((U_m / J_m) * (U_m / J_m)) * 0.125)));
	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, 3.9e-44], N[(N[(N[(-2.0 * N[Power[J$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / U$95$m), $MachinePrecision] - U$95$m), $MachinePrecision], N[(J$95$m * N[(N[(-2.0 * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(N[(U$95$m / J$95$m), $MachinePrecision] * N[(U$95$m / J$95$m), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if J < 3.9000000000000002e-44

   1. Initial program 62.7%

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

   3. Simplified 82.1%

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

   5. Taylor expanded in J around 0 26.2%

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

      1. neg-mul-1 26.2%

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

      2. unsub-neg 26.2%

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

      3. associate-*r/ 26.2%

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

      4. unpow2 26.2%

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

      5. *-commutative 26.2%

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

      6. unpow2 26.2%

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

      7. swap-sqr 26.2%

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

      8. unpow2 26.2%

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

      9. *-commutative 26.2%

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

   7. Simplified 26.2%

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

   8. Taylor expanded in K around 0 26.2%

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

   if 3.9000000000000002e-44 < J

   1. Initial program 92.5%

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

   3. Simplified 98.5%

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

   5. Taylor expanded in K around 0 89.9%

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

   6. Taylor expanded in U around 0 63.9%

      \[\leadsto J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\left(1 + 0.125 \cdot \frac{{U}^{2}}{{J}^{2}}\right)}\right) \]
   7. Step-by-step derivation

      1. *-commutative 63.9%

         \[\leadsto J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(1 + \color{blue}{\frac{{U}^{2}}{{J}^{2}} \cdot 0.125}\right)\right) \]

   8. Simplified 63.9%

      \[\leadsto J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\left(1 + \frac{{U}^{2}}{{J}^{2}} \cdot 0.125\right)}\right) \]
   9. Step-by-step derivation

      1. unpow2 63.9%

         \[\leadsto J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(1 + \frac{\color{blue}{U \cdot U}}{{J}^{2}} \cdot 0.125\right)\right) \]

      2. unpow2 63.9%

         \[\leadsto J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(1 + \frac{U \cdot U}{\color{blue}{J \cdot J}} \cdot 0.125\right)\right) \]

      3. frac-times 75.0%

         \[\leadsto J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(1 + \color{blue}{\left(\frac{U}{J} \cdot \frac{U}{J}\right)} \cdot 0.125\right)\right) \]

   10. Applied egg-rr 75.0%

       \[\leadsto J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(1 + \color{blue}{\left(\frac{U}{J} \cdot \frac{U}{J}\right)} \cdot 0.125\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq 3.9 \cdot 10^{-44}:\\ \;\;\;\;\frac{-2 \cdot {J}^{2}}{U} - U\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(1 + \left(\frac{U}{J} \cdot \frac{U}{J}\right) \cdot 0.125\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.8% accurate, 3.7× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ J\_s \cdot \begin{array}{l} \mathbf{if}\;J\_m \leq 4.8 \cdot 10^{-13}:\\ \;\;\;\;\frac{-2 \cdot {J\_m}^{2}}{U\_m} - U\_m\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot J\_m\right) \cdot \cos \left(K \cdot 0.5\right)\\ \end{array} \end{array} \]

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 4.8e-13)
    (- (/ (* -2.0 (pow J_m 2.0)) U_m) 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 <= 4.8e-13) {
		tmp = ((-2.0 * pow(J_m, 2.0)) / U_m) - 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 <= 4.8d-13) then
        tmp = (((-2.0d0) * (j_m ** 2.0d0)) / u_m) - 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 <= 4.8e-13) {
		tmp = ((-2.0 * Math.pow(J_m, 2.0)) / U_m) - 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 <= 4.8e-13:
		tmp = ((-2.0 * math.pow(J_m, 2.0)) / U_m) - 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 <= 4.8e-13)
		tmp = Float64(Float64(Float64(-2.0 * (J_m ^ 2.0)) / U_m) - 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 <= 4.8e-13)
		tmp = ((-2.0 * (J_m ^ 2.0)) / U_m) - 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, 4.8e-13], N[(N[(N[(-2.0 * N[Power[J$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / U$95$m), $MachinePrecision] - 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 < 4.7999999999999997e-13

   1. Initial program 61.8%

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

   3. Simplified 81.9%

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

   5. Taylor expanded in J around 0 26.3%

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

      1. neg-mul-1 26.3%

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

      2. unsub-neg 26.3%

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

      3. associate-*r/ 26.3%

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

      4. unpow2 26.3%

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

      5. *-commutative 26.3%

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

      6. unpow2 26.3%

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

      7. swap-sqr 26.3%

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

      8. unpow2 26.3%

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

      9. *-commutative 26.3%

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

   7. Simplified 26.3%

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

   8. Taylor expanded in K around 0 26.3%

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

   if 4.7999999999999997e-13 < J

   1. Initial program 96.0%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in J around inf 77.3%

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

      1. associate-*r* 77.3%

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

      2. *-commutative 77.3%

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

      3. *-commutative 77.3%

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

      4. *-commutative 77.3%

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

      5. *-commutative 77.3%

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

      6. *-commutative 77.3%

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

   7. Simplified 77.3%

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

4. Final simplification 40.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq 4.8 \cdot 10^{-13}:\\ \;\;\;\;\frac{-2 \cdot {J}^{2}}{U} - U\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 67.7% accurate, 3.7× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ J\_s \cdot \begin{array}{l} \mathbf{if}\;J\_m \leq 7.2 \cdot 10^{-16}:\\ \;\;\;\;-U\_m\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot J\_m\right) \cdot \cos \left(K \cdot 0.5\right)\\ \end{array} \end{array} \]

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 7.2e-16) (- 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 <= 7.2e-16) {
		tmp = -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 <= 7.2d-16) then
        tmp = -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 <= 7.2e-16) {
		tmp = -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 <= 7.2e-16:
		tmp = -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 <= 7.2e-16)
		tmp = Float64(-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 <= 7.2e-16)
		tmp = -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, 7.2e-16], (-U$95$m), 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 < 7.19999999999999965e-16

   1. Initial program 61.8%

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

   3. Simplified 81.9%

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

   5. Taylor expanded in J around 0 26.2%

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

      1. neg-mul-1 26.2%

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

   7. Simplified 26.2%

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

   if 7.19999999999999965e-16 < J

   1. Initial program 96.0%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in J around inf 77.3%

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

      1. associate-*r* 77.3%

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

      2. *-commutative 77.3%

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

      3. *-commutative 77.3%

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

      4. *-commutative 77.3%

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

      5. *-commutative 77.3%

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

      6. *-commutative 77.3%

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

   7. Simplified 77.3%

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

4. Final simplification 40.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq 7.2 \cdot 10^{-16}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 38.9% 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 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m) :precision binary64 (* J_s (- U_m)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.4%

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

3. Simplified 86.9%

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

5. Taylor expanded in J around 0 22.6%

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

   1. neg-mul-1 22.6%

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

7. Simplified 22.6%

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

8. Final simplification 22.6%

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

Alternative 11: 14.0% 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 71.4%

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

3. Simplified 86.9%

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

5. Taylor expanded in U around -inf 29.4%

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

6. Final simplification 29.4%

   \[\leadsto U \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024059 
(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)))))