Maksimov and Kolovsky, Equation (3)

Percentage Accurate: 73.2% → 99.6%

Time: 14.2s

Alternatives: 8

Speedup: 0.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.2% 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: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_0 \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ t_2 := J \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+305}:\\ \;\;\;\;-2 \cdot \left(t\_2 \cdot \mathsf{hypot}\left(1, \frac{\frac{U\_m}{2}}{t\_2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* (* -2.0 J) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* t_0 (* J 2.0))) 2.0)))))
        (t_2 (* J t_0)))
   (if (<= t_1 (- INFINITY))
     (- U_m)
     (if (<= t_1 5e+305)
       (* -2.0 (* t_2 (hypot 1.0 (/ (/ U_m 2.0) t_2))))
       U_m))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * J) * t_0) * sqrt((1.0 + pow((U_m / (t_0 * (J * 2.0))), 2.0)));
	double t_2 = J * t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_1 <= 5e+305) {
		tmp = -2.0 * (t_2 * hypot(1.0, ((U_m / 2.0) / t_2)));
	} else {
		tmp = U_m;
	}
	return tmp;
}

Java

U_m = Math.abs(U);
public static double code(double J, double K, double U_m) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = ((-2.0 * J) * t_0) * Math.sqrt((1.0 + Math.pow((U_m / (t_0 * (J * 2.0))), 2.0)));
	double t_2 = J * t_0;
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = -U_m;
	} else if (t_1 <= 5e+305) {
		tmp = -2.0 * (t_2 * Math.hypot(1.0, ((U_m / 2.0) / t_2)));
	} else {
		tmp = U_m;
	}
	return tmp;
}

Python

U_m = math.fabs(U)
def code(J, K, U_m):
	t_0 = math.cos((K / 2.0))
	t_1 = ((-2.0 * J) * t_0) * math.sqrt((1.0 + math.pow((U_m / (t_0 * (J * 2.0))), 2.0)))
	t_2 = J * t_0
	tmp = 0
	if t_1 <= -math.inf:
		tmp = -U_m
	elif t_1 <= 5e+305:
		tmp = -2.0 * (t_2 * math.hypot(1.0, ((U_m / 2.0) / t_2)))
	else:
		tmp = U_m
	return tmp

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * J) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_0 * Float64(J * 2.0))) ^ 2.0))))
	t_2 = Float64(J * t_0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_1 <= 5e+305)
		tmp = Float64(-2.0 * Float64(t_2 * hypot(1.0, Float64(Float64(U_m / 2.0) / t_2))));
	else
		tmp = U_m;
	end
	return tmp
end

MATLAB

U_m = abs(U);
function tmp_2 = code(J, K, U_m)
	t_0 = cos((K / 2.0));
	t_1 = ((-2.0 * J) * t_0) * sqrt((1.0 + ((U_m / (t_0 * (J * 2.0))) ^ 2.0)));
	t_2 = J * t_0;
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = -U_m;
	elseif (t_1 <= 5e+305)
		tmp = -2.0 * (t_2 * hypot(1.0, ((U_m / 2.0) / t_2)));
	else
		tmp = U_m;
	end
	tmp_2 = tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, 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), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(t$95$0 * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(J * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], (-U$95$m), If[LessEqual[t$95$1, 5e+305], N[(-2.0 * N[(t$95$2 * N[Sqrt[1.0 ^ 2 + N[(N[(U$95$m / 2.0), $MachinePrecision] / t$95$2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], U$95$m]]]]]

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 6.2%

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

   3. Simplified 6.2%

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

   5. Taylor expanded in J around 0 39.1%

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

      1. mul-1-neg 39.1%

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

   7. Simplified 39.1%

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

   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)))) < 5.00000000000000009e305

   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. Step-by-step derivation

      1. unpow2 99.8%

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

      2. sqr-neg 99.8%

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

      3. distribute-frac-neg 99.8%

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

      4. distribute-frac-neg 99.8%

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

      5. unpow2 99.8%

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

      6. associate-*l* 99.8%

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

      7. associate-*l* 99.8%

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

   3. Simplified 99.8%

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

   if 5.00000000000000009e305 < (*.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 7.9%

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

   3. Simplified 7.9%

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

   5. Taylor expanded in U around -inf 43.5%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 5 \cdot 10^{+305}:\\ \;\;\;\;-2 \cdot \left(\left(J \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{\frac{U}{2}}{J \cdot \cos \left(\frac{K}{2}\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 41.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(K \cdot 0.5\right)\\ \mathbf{if}\;J \leq 1.5 \cdot 10^{-95}:\\ \;\;\;\;-U\_m\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(J \cdot t\_0\right) + -0.25 \cdot \left(\frac{U\_m}{t\_0} \cdot \frac{U\_m}{J}\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (* K 0.5))))
   (if (<= J 1.5e-95)
     (- U_m)
     (+ (* -2.0 (* J t_0)) (* -0.25 (* (/ U_m t_0) (/ U_m J)))))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = cos((K * 0.5));
	double tmp;
	if (J <= 1.5e-95) {
		tmp = -U_m;
	} else {
		tmp = (-2.0 * (J * t_0)) + (-0.25 * ((U_m / t_0) * (U_m / J)));
	}
	return tmp;
}

Fortran

U_m = abs(U)
real(8) function code(j, k, u_m)
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((k * 0.5d0))
    if (j <= 1.5d-95) then
        tmp = -u_m
    else
        tmp = ((-2.0d0) * (j * t_0)) + ((-0.25d0) * ((u_m / t_0) * (u_m / j)))
    end if
    code = tmp
end function

Java

U_m = Math.abs(U);
public static double code(double J, double K, double U_m) {
	double t_0 = Math.cos((K * 0.5));
	double tmp;
	if (J <= 1.5e-95) {
		tmp = -U_m;
	} else {
		tmp = (-2.0 * (J * t_0)) + (-0.25 * ((U_m / t_0) * (U_m / J)));
	}
	return tmp;
}

Python

U_m = math.fabs(U)
def code(J, K, U_m):
	t_0 = math.cos((K * 0.5))
	tmp = 0
	if J <= 1.5e-95:
		tmp = -U_m
	else:
		tmp = (-2.0 * (J * t_0)) + (-0.25 * ((U_m / t_0) * (U_m / J)))
	return tmp

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(K * 0.5))
	tmp = 0.0
	if (J <= 1.5e-95)
		tmp = Float64(-U_m);
	else
		tmp = Float64(Float64(-2.0 * Float64(J * t_0)) + Float64(-0.25 * Float64(Float64(U_m / t_0) * Float64(U_m / J))));
	end
	return tmp
end

MATLAB

U_m = abs(U);
function tmp_2 = code(J, K, U_m)
	t_0 = cos((K * 0.5));
	tmp = 0.0;
	if (J <= 1.5e-95)
		tmp = -U_m;
	else
		tmp = (-2.0 * (J * t_0)) + (-0.25 * ((U_m / t_0) * (U_m / J)));
	end
	tmp_2 = tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[J, 1.5e-95], (-U$95$m), N[(N[(-2.0 * N[(J * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(-0.25 * N[(N[(U$95$m / t$95$0), $MachinePrecision] * N[(U$95$m / J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if J < 1.5e-95

   1. Initial program 64.8%

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

   3. Simplified 64.8%

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

   5. Taylor expanded in J around 0 27.0%

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

      1. mul-1-neg 27.0%

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

   7. Simplified 27.0%

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

   if 1.5e-95 < J

   1. Initial program 89.5%

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

   3. Simplified 89.5%

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

   5. Taylor expanded in J around inf 64.5%

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

      1. unpow2 64.5%

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

      2. *-commutative 64.5%

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

      3. times-frac 71.4%

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

      4. *-commutative 71.4%

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

   7. Applied egg-rr 71.4%

      \[\leadsto -2 \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right) + -0.25 \cdot \color{blue}{\left(\frac{U}{\cos \left(K \cdot 0.5\right)} \cdot \frac{U}{J}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq 1.5 \cdot 10^{-95}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right) + -0.25 \cdot \left(\frac{U}{\cos \left(K \cdot 0.5\right)} \cdot \frac{U}{J}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 41.4% accurate, 3.5× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} \mathbf{if}\;J \leq 1.75 \cdot 10^{-95}:\\ \;\;\;\;-U\_m\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right) + -0.25 \cdot \left(U\_m \cdot \frac{U\_m}{J}\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (if (<= J 1.75e-95)
   (- U_m)
   (+ (* -2.0 (* J (cos (* K 0.5)))) (* -0.25 (* U_m (/ U_m J))))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double tmp;
	if (J <= 1.75e-95) {
		tmp = -U_m;
	} else {
		tmp = (-2.0 * (J * cos((K * 0.5)))) + (-0.25 * (U_m * (U_m / J)));
	}
	return tmp;
}

Fortran

U_m = abs(U)
real(8) function code(j, k, u_m)
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u_m
    real(8) :: tmp
    if (j <= 1.75d-95) then
        tmp = -u_m
    else
        tmp = ((-2.0d0) * (j * cos((k * 0.5d0)))) + ((-0.25d0) * (u_m * (u_m / j)))
    end if
    code = tmp
end function

Java

U_m = Math.abs(U);
public static double code(double J, double K, double U_m) {
	double tmp;
	if (J <= 1.75e-95) {
		tmp = -U_m;
	} else {
		tmp = (-2.0 * (J * Math.cos((K * 0.5)))) + (-0.25 * (U_m * (U_m / J)));
	}
	return tmp;
}

Python

U_m = math.fabs(U)
def code(J, K, U_m):
	tmp = 0
	if J <= 1.75e-95:
		tmp = -U_m
	else:
		tmp = (-2.0 * (J * math.cos((K * 0.5)))) + (-0.25 * (U_m * (U_m / J)))
	return tmp

Julia

U_m = abs(U)
function code(J, K, U_m)
	tmp = 0.0
	if (J <= 1.75e-95)
		tmp = Float64(-U_m);
	else
		tmp = Float64(Float64(-2.0 * Float64(J * cos(Float64(K * 0.5)))) + Float64(-0.25 * Float64(U_m * Float64(U_m / J))));
	end
	return tmp
end

MATLAB

U_m = abs(U);
function tmp_2 = code(J, K, U_m)
	tmp = 0.0;
	if (J <= 1.75e-95)
		tmp = -U_m;
	else
		tmp = (-2.0 * (J * cos((K * 0.5)))) + (-0.25 * (U_m * (U_m / J)));
	end
	tmp_2 = tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := If[LessEqual[J, 1.75e-95], (-U$95$m), N[(N[(-2.0 * N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.25 * N[(U$95$m * N[(U$95$m / J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if J < 1.7499999999999999e-95

   1. Initial program 64.8%

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

   3. Simplified 64.8%

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

   5. Taylor expanded in J around 0 27.0%

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

      1. mul-1-neg 27.0%

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

   7. Simplified 27.0%

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

   if 1.7499999999999999e-95 < J

   1. Initial program 89.5%

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

   3. Simplified 89.5%

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

   5. Taylor expanded in J around inf 64.5%

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

   6. Taylor expanded in K around 0 63.5%

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

      1. *-commutative 63.5%

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

   8. Simplified 63.5%

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

      1. unpow2 63.5%

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

      2. *-un-lft-identity 63.5%

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

      3. times-frac 70.5%

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

   10. Applied egg-rr 70.5%

       \[\leadsto -2 \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right) + \color{blue}{\left(\frac{U}{1} \cdot \frac{U}{J}\right)} \cdot -0.25 \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq 1.75 \cdot 10^{-95}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right) + -0.25 \cdot \left(U \cdot \frac{U}{J}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 41.3% accurate, 3.7× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} \mathbf{if}\;J \leq 1.9 \cdot 10^{-95}:\\ \;\;\;\;-U\_m\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (if (<= J 1.9e-95) (- U_m) (* (* -2.0 J) (cos (* K 0.5)))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double tmp;
	if (J <= 1.9e-95) {
		tmp = -U_m;
	} else {
		tmp = (-2.0 * J) * cos((K * 0.5));
	}
	return tmp;
}

Fortran

U_m = abs(U)
real(8) function code(j, k, u_m)
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u_m
    real(8) :: tmp
    if (j <= 1.9d-95) then
        tmp = -u_m
    else
        tmp = ((-2.0d0) * j) * cos((k * 0.5d0))
    end if
    code = tmp
end function

Java

U_m = Math.abs(U);
public static double code(double J, double K, double U_m) {
	double tmp;
	if (J <= 1.9e-95) {
		tmp = -U_m;
	} else {
		tmp = (-2.0 * J) * Math.cos((K * 0.5));
	}
	return tmp;
}

Python

U_m = math.fabs(U)
def code(J, K, U_m):
	tmp = 0
	if J <= 1.9e-95:
		tmp = -U_m
	else:
		tmp = (-2.0 * J) * math.cos((K * 0.5))
	return tmp

Julia

U_m = abs(U)
function code(J, K, U_m)
	tmp = 0.0
	if (J <= 1.9e-95)
		tmp = Float64(-U_m);
	else
		tmp = Float64(Float64(-2.0 * J) * cos(Float64(K * 0.5)));
	end
	return tmp
end

MATLAB

U_m = abs(U);
function tmp_2 = code(J, K, U_m)
	tmp = 0.0;
	if (J <= 1.9e-95)
		tmp = -U_m;
	else
		tmp = (-2.0 * J) * cos((K * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := If[LessEqual[J, 1.9e-95], (-U$95$m), N[(N[(-2.0 * J), $MachinePrecision] * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if J < 1.8999999999999999e-95

   1. Initial program 64.8%

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

   3. Simplified 64.8%

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

   5. Taylor expanded in J around 0 27.0%

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

      1. mul-1-neg 27.0%

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

   7. Simplified 27.0%

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

   if 1.8999999999999999e-95 < J

   1. Initial program 89.5%

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

   3. Simplified 89.5%

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

   5. Taylor expanded in J around inf 70.8%

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

      1. associate-*r* 70.8%

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

   7. Simplified 70.8%

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

4. Final simplification 40.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq 1.9 \cdot 10^{-95}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 40.4% accurate, 26.2× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} \mathbf{if}\;U\_m \leq 5.4 \cdot 10^{-45}:\\ \;\;\;\;-2 \cdot J + -0.25 \cdot \frac{U\_m \cdot U\_m}{J}\\ \mathbf{else}:\\ \;\;\;\;-U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (if (<= U_m 5.4e-45) (+ (* -2.0 J) (* -0.25 (/ (* U_m U_m) J))) (- U_m)))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double tmp;
	if (U_m <= 5.4e-45) {
		tmp = (-2.0 * J) + (-0.25 * ((U_m * U_m) / J));
	} else {
		tmp = -U_m;
	}
	return tmp;
}

Fortran

U_m = abs(U)
real(8) function code(j, k, u_m)
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u_m
    real(8) :: tmp
    if (u_m <= 5.4d-45) then
        tmp = ((-2.0d0) * j) + ((-0.25d0) * ((u_m * u_m) / j))
    else
        tmp = -u_m
    end if
    code = tmp
end function

Java

U_m = Math.abs(U);
public static double code(double J, double K, double U_m) {
	double tmp;
	if (U_m <= 5.4e-45) {
		tmp = (-2.0 * J) + (-0.25 * ((U_m * U_m) / J));
	} else {
		tmp = -U_m;
	}
	return tmp;
}

Python

U_m = math.fabs(U)
def code(J, K, U_m):
	tmp = 0
	if U_m <= 5.4e-45:
		tmp = (-2.0 * J) + (-0.25 * ((U_m * U_m) / J))
	else:
		tmp = -U_m
	return tmp

Julia

U_m = abs(U)
function code(J, K, U_m)
	tmp = 0.0
	if (U_m <= 5.4e-45)
		tmp = Float64(Float64(-2.0 * J) + Float64(-0.25 * Float64(Float64(U_m * U_m) / J)));
	else
		tmp = Float64(-U_m);
	end
	return tmp
end

MATLAB

U_m = abs(U);
function tmp_2 = code(J, K, U_m)
	tmp = 0.0;
	if (U_m <= 5.4e-45)
		tmp = (-2.0 * J) + (-0.25 * ((U_m * U_m) / J));
	else
		tmp = -U_m;
	end
	tmp_2 = tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := If[LessEqual[U$95$m, 5.4e-45], N[(N[(-2.0 * J), $MachinePrecision] + N[(-0.25 * N[(N[(U$95$m * U$95$m), $MachinePrecision] / J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-U$95$m)]

Derivation

1. Split input into 2 regimes

2. if U < 5.3999999999999997e-45

   1. Initial program 77.3%

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

   3. Simplified 77.3%

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

   5. Taylor expanded in J around inf 60.1%

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

   6. Taylor expanded in K around 0 59.5%

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

      1. *-commutative 59.5%

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

   8. Simplified 59.5%

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

   9. Taylor expanded in K around 0 34.5%

      \[\leadsto -2 \cdot \color{blue}{J} + \frac{{U}^{2}}{J} \cdot -0.25 \]
   10. Step-by-step derivation

       1. unpow2 34.5%

          \[\leadsto -2 \cdot J + \frac{\color{blue}{U \cdot U}}{J} \cdot -0.25 \]

   11. Applied egg-rr 34.5%

       \[\leadsto -2 \cdot J + \frac{\color{blue}{U \cdot U}}{J} \cdot -0.25 \]

   if 5.3999999999999997e-45 < U

   1. Initial program 59.8%

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

   3. Simplified 59.8%

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

   5. Taylor expanded in J around 0 27.8%

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

      1. mul-1-neg 27.8%

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

   7. Simplified 27.8%

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

4. Final simplification 32.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 5.4 \cdot 10^{-45}:\\ \;\;\;\;-2 \cdot J + -0.25 \cdot \frac{U \cdot U}{J}\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 27.4% accurate, 34.9× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} \mathbf{if}\;K \leq 4.7 \cdot 10^{+61} \lor \neg \left(K \leq 4.4 \cdot 10^{+169}\right):\\ \;\;\;\;-U\_m\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (if (or (<= K 4.7e+61) (not (<= K 4.4e+169))) (- U_m) U_m))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double tmp;
	if ((K <= 4.7e+61) || !(K <= 4.4e+169)) {
		tmp = -U_m;
	} else {
		tmp = U_m;
	}
	return tmp;
}

Fortran

U_m = abs(U)
real(8) function code(j, k, u_m)
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u_m
    real(8) :: tmp
    if ((k <= 4.7d+61) .or. (.not. (k <= 4.4d+169))) then
        tmp = -u_m
    else
        tmp = u_m
    end if
    code = tmp
end function

Java

U_m = Math.abs(U);
public static double code(double J, double K, double U_m) {
	double tmp;
	if ((K <= 4.7e+61) || !(K <= 4.4e+169)) {
		tmp = -U_m;
	} else {
		tmp = U_m;
	}
	return tmp;
}

Python

U_m = math.fabs(U)
def code(J, K, U_m):
	tmp = 0
	if (K <= 4.7e+61) or not (K <= 4.4e+169):
		tmp = -U_m
	else:
		tmp = U_m
	return tmp

Julia

U_m = abs(U)
function code(J, K, U_m)
	tmp = 0.0
	if ((K <= 4.7e+61) || !(K <= 4.4e+169))
		tmp = Float64(-U_m);
	else
		tmp = U_m;
	end
	return tmp
end

MATLAB

U_m = abs(U);
function tmp_2 = code(J, K, U_m)
	tmp = 0.0;
	if ((K <= 4.7e+61) || ~((K <= 4.4e+169)))
		tmp = -U_m;
	else
		tmp = U_m;
	end
	tmp_2 = tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := If[Or[LessEqual[K, 4.7e+61], N[Not[LessEqual[K, 4.4e+169]], $MachinePrecision]], (-U$95$m), U$95$m]

Derivation

1. Split input into 2 regimes

2. if K < 4.6999999999999998e61 or 4.4e169 < K

   1. Initial program 73.1%

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

   3. Simplified 73.1%

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

   5. Taylor expanded in J around 0 24.4%

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

      1. mul-1-neg 24.4%

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

   7. Simplified 24.4%

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

   if 4.6999999999999998e61 < K < 4.4e169

   1. Initial program 65.6%

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

   3. Simplified 65.6%

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

   5. Taylor expanded in U around -inf 44.3%

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

4. Final simplification 25.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;K \leq 4.7 \cdot 10^{+61} \lor \neg \left(K \leq 4.4 \cdot 10^{+169}\right):\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 40.4% accurate, 52.4× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} \mathbf{if}\;U\_m \leq 1.95 \cdot 10^{-44}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{else}:\\ \;\;\;\;-U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (if (<= U_m 1.95e-44) (* -2.0 J) (- U_m)))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double tmp;
	if (U_m <= 1.95e-44) {
		tmp = -2.0 * J;
	} else {
		tmp = -U_m;
	}
	return tmp;
}

Fortran

U_m = abs(U)
real(8) function code(j, k, u_m)
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u_m
    real(8) :: tmp
    if (u_m <= 1.95d-44) then
        tmp = (-2.0d0) * j
    else
        tmp = -u_m
    end if
    code = tmp
end function

Java

U_m = Math.abs(U);
public static double code(double J, double K, double U_m) {
	double tmp;
	if (U_m <= 1.95e-44) {
		tmp = -2.0 * J;
	} else {
		tmp = -U_m;
	}
	return tmp;
}

Python

U_m = math.fabs(U)
def code(J, K, U_m):
	tmp = 0
	if U_m <= 1.95e-44:
		tmp = -2.0 * J
	else:
		tmp = -U_m
	return tmp

Julia

U_m = abs(U)
function code(J, K, U_m)
	tmp = 0.0
	if (U_m <= 1.95e-44)
		tmp = Float64(-2.0 * J);
	else
		tmp = Float64(-U_m);
	end
	return tmp
end

MATLAB

U_m = abs(U);
function tmp_2 = code(J, K, U_m)
	tmp = 0.0;
	if (U_m <= 1.95e-44)
		tmp = -2.0 * J;
	else
		tmp = -U_m;
	end
	tmp_2 = tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := If[LessEqual[U$95$m, 1.95e-44], N[(-2.0 * J), $MachinePrecision], (-U$95$m)]

Derivation

1. Split input into 2 regimes

2. if U < 1.9500000000000001e-44

   1. Initial program 77.3%

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

   3. Simplified 77.3%

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

   5. Taylor expanded in J around inf 61.9%

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

      1. associate-*r* 61.9%

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

   7. Simplified 61.9%

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

   8. Taylor expanded in K around 0 35.6%

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

      1. *-commutative 35.6%

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

   10. Simplified 35.6%

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

   if 1.9500000000000001e-44 < U

   1. Initial program 59.8%

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

   3. Simplified 59.8%

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

   5. Taylor expanded in J around 0 27.8%

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

      1. mul-1-neg 27.8%

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

   7. Simplified 27.8%

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

4. Final simplification 33.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 1.95 \cdot 10^{-44}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 25.7% accurate, 420.0× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ U\_m \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m) :precision binary64 U_m)

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	return U_m;
}

Fortran

U_m = abs(U)
real(8) function code(j, k, u_m)
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u_m
    code = u_m
end function

Java

U_m = Math.abs(U);
public static double code(double J, double K, double U_m) {
	return U_m;
}

Python

U_m = math.fabs(U)
def code(J, K, U_m):
	return U_m

Julia

U_m = abs(U)
function code(J, K, U_m)
	return U_m
end

MATLAB

U_m = abs(U);
function tmp = code(J, K, U_m)
	tmp = U_m;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := U$95$m

Derivation

1. Initial program 72.6%

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

3. Simplified 72.6%

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

5. Taylor expanded in U around -inf 28.0%

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

6. Final simplification 28.0%

   \[\leadsto U \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024036 
(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)))))