Maksimov and Kolovsky, Equation (3)

Percentage Accurate: 73.9% → 99.5%

Time: 15.9s

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: 73.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_0 \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ t_2 := J \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+304}:\\ \;\;\;\;-2 \cdot \left(t\_2 \cdot \mathsf{hypot}\left(1, \frac{\frac{U\_m}{2}}{t\_2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \]

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+304)
       (* -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+304) {
		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+304) {
		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+304:
		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+304)
		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+304)
		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+304], 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 6.0%

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

   3. Simplified 6.0%

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

   5. Taylor expanded in J around 0 66.9%

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

      1. mul-1-neg 66.9%

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

   7. Simplified 66.9%

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

   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)))) < 4.9999999999999997e304

   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. associate-*l* 99.8%

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

      2. 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. unpow2 99.8%

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

      4. sqr-neg 99.8%

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

      5. distribute-frac-neg 99.8%

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

      6. distribute-frac-neg 99.8%

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

      7. unpow2 99.8%

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

   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 4.9999999999999997e304 < (*.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 5.9%

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

   3. Simplified 5.9%

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

   5. Taylor expanded in U around -inf 40.4%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 5 \cdot 10^{+304}:\\ \;\;\;\;-2 \cdot \left(\left(J \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{\frac{U}{2}}{J \cdot \cos \left(\frac{K}{2}\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 65.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \cos \left(K \cdot 0.5\right)\\ \mathbf{if}\;t\_0 \leq -0.795:\\ \;\;\;\;-2 \cdot \left(J \cdot t\_1\right) + -0.25 \cdot \left(\frac{U\_m}{t\_1} \cdot \frac{U\_m}{J}\right)\\ \mathbf{elif}\;t\_0 \leq -0.02:\\ \;\;\;\;U\_m\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(J \cdot \mathsf{hypot}\left(1, \frac{\frac{U\_m}{2}}{J}\right)\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))) (t_1 (cos (* K 0.5))))
   (if (<= t_0 -0.795)
     (+ (* -2.0 (* J t_1)) (* -0.25 (* (/ U_m t_1) (/ U_m J))))
     (if (<= t_0 -0.02) U_m (* -2.0 (* J (hypot 1.0 (/ (/ U_m 2.0) J))))))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = cos((K * 0.5));
	double tmp;
	if (t_0 <= -0.795) {
		tmp = (-2.0 * (J * t_1)) + (-0.25 * ((U_m / t_1) * (U_m / J)));
	} else if (t_0 <= -0.02) {
		tmp = U_m;
	} else {
		tmp = -2.0 * (J * hypot(1.0, ((U_m / 2.0) / J)));
	}
	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 = Math.cos((K * 0.5));
	double tmp;
	if (t_0 <= -0.795) {
		tmp = (-2.0 * (J * t_1)) + (-0.25 * ((U_m / t_1) * (U_m / J)));
	} else if (t_0 <= -0.02) {
		tmp = U_m;
	} else {
		tmp = -2.0 * (J * Math.hypot(1.0, ((U_m / 2.0) / J)));
	}
	return tmp;
}

Python

U_m = math.fabs(U)
def code(J, K, U_m):
	t_0 = math.cos((K / 2.0))
	t_1 = math.cos((K * 0.5))
	tmp = 0
	if t_0 <= -0.795:
		tmp = (-2.0 * (J * t_1)) + (-0.25 * ((U_m / t_1) * (U_m / J)))
	elif t_0 <= -0.02:
		tmp = U_m
	else:
		tmp = -2.0 * (J * math.hypot(1.0, ((U_m / 2.0) / J)))
	return tmp

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = cos(Float64(K * 0.5))
	tmp = 0.0
	if (t_0 <= -0.795)
		tmp = Float64(Float64(-2.0 * Float64(J * t_1)) + Float64(-0.25 * Float64(Float64(U_m / t_1) * Float64(U_m / J))));
	elseif (t_0 <= -0.02)
		tmp = U_m;
	else
		tmp = Float64(-2.0 * Float64(J * hypot(1.0, Float64(Float64(U_m / 2.0) / J))));
	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 = cos((K * 0.5));
	tmp = 0.0;
	if (t_0 <= -0.795)
		tmp = (-2.0 * (J * t_1)) + (-0.25 * ((U_m / t_1) * (U_m / J)));
	elseif (t_0 <= -0.02)
		tmp = U_m;
	else
		tmp = -2.0 * (J * hypot(1.0, ((U_m / 2.0) / 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 / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -0.795], N[(N[(-2.0 * N[(J * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(-0.25 * N[(N[(U$95$m / t$95$1), $MachinePrecision] * N[(U$95$m / J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.02], U$95$m, N[(-2.0 * N[(J * N[Sqrt[1.0 ^ 2 + N[(N[(U$95$m / 2.0), $MachinePrecision] / J), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 (/.f64 K 2)) < -0.79500000000000004

   1. Initial program 75.1%

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

   3. Simplified 75.1%

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

   5. Taylor expanded in J around inf 61.6%

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

      1. unpow2 61.6%

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

      2. *-commutative 61.6%

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

      3. times-frac 61.6%

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

   7. Applied egg-rr 61.6%

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

   if -0.79500000000000004 < (cos.f64 (/.f64 K 2)) < -0.0200000000000000004

   1. Initial program 75.8%

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

   3. Simplified 75.8%

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

   5. Taylor expanded in U around -inf 27.8%

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

   if -0.0200000000000000004 < (cos.f64 (/.f64 K 2))

   1. Initial program 73.7%

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

      1. associate-*l* 73.7%

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

      2. associate-*l* 73.7%

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

      3. unpow2 73.7%

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

      4. sqr-neg 73.7%

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

      5. distribute-frac-neg 73.7%

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

      6. distribute-frac-neg 73.7%

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

      7. unpow2 73.7%

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

   3. Simplified 90.5%

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

   5. Taylor expanded in K around 0 69.1%

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

   6. Taylor expanded in K around 0 80.6%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.795:\\ \;\;\;\;-2 \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right) + -0.25 \cdot \left(\frac{U}{\cos \left(K \cdot 0.5\right)} \cdot \frac{U}{J}\right)\\ \mathbf{elif}\;\cos \left(\frac{K}{2}\right) \leq -0.02:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(J \cdot \mathsf{hypot}\left(1, \frac{\frac{U}{2}}{J}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 65.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq -0.795:\\ \;\;\;\;-2 \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{elif}\;t\_0 \leq -0.02:\\ \;\;\;\;U\_m\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(J \cdot \mathsf{hypot}\left(1, \frac{\frac{U\_m}{2}}{J}\right)\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 -0.795)
     (* -2.0 (* J (cos (* K 0.5))))
     (if (<= t_0 -0.02) U_m (* -2.0 (* J (hypot 1.0 (/ (/ U_m 2.0) J))))))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= -0.795) {
		tmp = -2.0 * (J * cos((K * 0.5)));
	} else if (t_0 <= -0.02) {
		tmp = U_m;
	} else {
		tmp = -2.0 * (J * hypot(1.0, ((U_m / 2.0) / J)));
	}
	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 tmp;
	if (t_0 <= -0.795) {
		tmp = -2.0 * (J * Math.cos((K * 0.5)));
	} else if (t_0 <= -0.02) {
		tmp = U_m;
	} else {
		tmp = -2.0 * (J * Math.hypot(1.0, ((U_m / 2.0) / J)));
	}
	return tmp;
}

Python

U_m = math.fabs(U)
def code(J, K, U_m):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if t_0 <= -0.795:
		tmp = -2.0 * (J * math.cos((K * 0.5)))
	elif t_0 <= -0.02:
		tmp = U_m
	else:
		tmp = -2.0 * (J * math.hypot(1.0, ((U_m / 2.0) / J)))
	return tmp

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= -0.795)
		tmp = Float64(-2.0 * Float64(J * cos(Float64(K * 0.5))));
	elseif (t_0 <= -0.02)
		tmp = U_m;
	else
		tmp = Float64(-2.0 * Float64(J * hypot(1.0, Float64(Float64(U_m / 2.0) / J))));
	end
	return tmp
end

MATLAB

U_m = abs(U);
function tmp_2 = code(J, K, U_m)
	t_0 = cos((K / 2.0));
	tmp = 0.0;
	if (t_0 <= -0.795)
		tmp = -2.0 * (J * cos((K * 0.5)));
	elseif (t_0 <= -0.02)
		tmp = U_m;
	else
		tmp = -2.0 * (J * hypot(1.0, ((U_m / 2.0) / 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 / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -0.795], N[(-2.0 * N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.02], U$95$m, N[(-2.0 * N[(J * N[Sqrt[1.0 ^ 2 + N[(N[(U$95$m / 2.0), $MachinePrecision] / J), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 (/.f64 K 2)) < -0.79500000000000004

   1. Initial program 75.1%

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

      1. associate-*l* 75.1%

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

      2. associate-*l* 75.1%

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

      3. unpow2 75.1%

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

      4. sqr-neg 75.1%

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

      5. distribute-frac-neg 75.1%

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

      6. distribute-frac-neg 75.1%

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

      7. unpow2 75.1%

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

   3. Simplified 89.2%

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

   5. Taylor expanded in U around 0 61.1%

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

   if -0.79500000000000004 < (cos.f64 (/.f64 K 2)) < -0.0200000000000000004

   1. Initial program 75.8%

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

   3. Simplified 75.8%

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

   5. Taylor expanded in U around -inf 27.8%

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

   if -0.0200000000000000004 < (cos.f64 (/.f64 K 2))

   1. Initial program 73.7%

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

      1. associate-*l* 73.7%

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

      2. associate-*l* 73.7%

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

      3. unpow2 73.7%

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

      4. sqr-neg 73.7%

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

      5. distribute-frac-neg 73.7%

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

      6. distribute-frac-neg 73.7%

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

      7. unpow2 73.7%

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

   3. Simplified 90.5%

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

   5. Taylor expanded in K around 0 69.1%

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

   6. Taylor expanded in K around 0 80.6%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.795:\\ \;\;\;\;-2 \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{elif}\;\cos \left(\frac{K}{2}\right) \leq -0.02:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(J \cdot \mathsf{hypot}\left(1, \frac{\frac{U}{2}}{J}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 57.5% accurate, 3.2× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := -2 \cdot \left(U\_m \cdot 0.5 + J \cdot \frac{J}{U\_m}\right)\\ t_1 := -2 \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{if}\;U\_m \leq 70000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;U\_m \leq 1.15 \cdot 10^{+57}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;U\_m \leq 2.4 \cdot 10^{+82}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;U\_m \leq 5.6 \cdot 10^{+133}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;U\_m \leq 1.25 \cdot 10^{+160}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;-U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (* -2.0 (+ (* U_m 0.5) (* J (/ J U_m)))))
        (t_1 (* -2.0 (* J (cos (* K 0.5))))))
   (if (<= U_m 70000000.0)
     t_1
     (if (<= U_m 1.15e+57)
       t_0
       (if (<= U_m 2.4e+82)
         t_1
         (if (<= U_m 5.6e+133) t_0 (if (<= U_m 1.25e+160) t_1 (- U_m))))))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = -2.0 * ((U_m * 0.5) + (J * (J / U_m)));
	double t_1 = -2.0 * (J * cos((K * 0.5)));
	double tmp;
	if (U_m <= 70000000.0) {
		tmp = t_1;
	} else if (U_m <= 1.15e+57) {
		tmp = t_0;
	} else if (U_m <= 2.4e+82) {
		tmp = t_1;
	} else if (U_m <= 5.6e+133) {
		tmp = t_0;
	} else if (U_m <= 1.25e+160) {
		tmp = t_1;
	} 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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (-2.0d0) * ((u_m * 0.5d0) + (j * (j / u_m)))
    t_1 = (-2.0d0) * (j * cos((k * 0.5d0)))
    if (u_m <= 70000000.0d0) then
        tmp = t_1
    else if (u_m <= 1.15d+57) then
        tmp = t_0
    else if (u_m <= 2.4d+82) then
        tmp = t_1
    else if (u_m <= 5.6d+133) then
        tmp = t_0
    else if (u_m <= 1.25d+160) then
        tmp = t_1
    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 t_0 = -2.0 * ((U_m * 0.5) + (J * (J / U_m)));
	double t_1 = -2.0 * (J * Math.cos((K * 0.5)));
	double tmp;
	if (U_m <= 70000000.0) {
		tmp = t_1;
	} else if (U_m <= 1.15e+57) {
		tmp = t_0;
	} else if (U_m <= 2.4e+82) {
		tmp = t_1;
	} else if (U_m <= 5.6e+133) {
		tmp = t_0;
	} else if (U_m <= 1.25e+160) {
		tmp = t_1;
	} else {
		tmp = -U_m;
	}
	return tmp;
}

Python

U_m = math.fabs(U)
def code(J, K, U_m):
	t_0 = -2.0 * ((U_m * 0.5) + (J * (J / U_m)))
	t_1 = -2.0 * (J * math.cos((K * 0.5)))
	tmp = 0
	if U_m <= 70000000.0:
		tmp = t_1
	elif U_m <= 1.15e+57:
		tmp = t_0
	elif U_m <= 2.4e+82:
		tmp = t_1
	elif U_m <= 5.6e+133:
		tmp = t_0
	elif U_m <= 1.25e+160:
		tmp = t_1
	else:
		tmp = -U_m
	return tmp

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = Float64(-2.0 * Float64(Float64(U_m * 0.5) + Float64(J * Float64(J / U_m))))
	t_1 = Float64(-2.0 * Float64(J * cos(Float64(K * 0.5))))
	tmp = 0.0
	if (U_m <= 70000000.0)
		tmp = t_1;
	elseif (U_m <= 1.15e+57)
		tmp = t_0;
	elseif (U_m <= 2.4e+82)
		tmp = t_1;
	elseif (U_m <= 5.6e+133)
		tmp = t_0;
	elseif (U_m <= 1.25e+160)
		tmp = t_1;
	else
		tmp = Float64(-U_m);
	end
	return tmp
end

MATLAB

U_m = abs(U);
function tmp_2 = code(J, K, U_m)
	t_0 = -2.0 * ((U_m * 0.5) + (J * (J / U_m)));
	t_1 = -2.0 * (J * cos((K * 0.5)));
	tmp = 0.0;
	if (U_m <= 70000000.0)
		tmp = t_1;
	elseif (U_m <= 1.15e+57)
		tmp = t_0;
	elseif (U_m <= 2.4e+82)
		tmp = t_1;
	elseif (U_m <= 5.6e+133)
		tmp = t_0;
	elseif (U_m <= 1.25e+160)
		tmp = t_1;
	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[(-2.0 * N[(N[(U$95$m * 0.5), $MachinePrecision] + N[(J * N[(J / U$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-2.0 * N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[U$95$m, 70000000.0], t$95$1, If[LessEqual[U$95$m, 1.15e+57], t$95$0, If[LessEqual[U$95$m, 2.4e+82], t$95$1, If[LessEqual[U$95$m, 5.6e+133], t$95$0, If[LessEqual[U$95$m, 1.25e+160], t$95$1, (-U$95$m)]]]]]]]

Derivation

1. Split input into 3 regimes

2. if U < 7e7 or 1.1499999999999999e57 < U < 2.39999999999999998e82 or 5.60000000000000033e133 < U < 1.25e160

   1. Initial program 80.7%

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

      1. associate-*l* 80.7%

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

      2. associate-*l* 80.7%

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

      3. unpow2 80.7%

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

      4. sqr-neg 80.7%

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

      5. distribute-frac-neg 80.7%

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

      6. distribute-frac-neg 80.7%

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

      7. unpow2 80.7%

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

   3. Simplified 93.8%

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

   5. Taylor expanded in U around 0 59.8%

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

   if 7e7 < U < 1.1499999999999999e57 or 2.39999999999999998e82 < U < 5.60000000000000033e133

   1. Initial program 62.4%

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

      1. associate-*l* 62.4%

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

      2. associate-*l* 62.4%

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

      3. unpow2 62.4%

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

      4. sqr-neg 62.4%

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

      5. distribute-frac-neg 62.4%

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

      6. distribute-frac-neg 62.4%

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

      7. unpow2 62.4%

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

   3. Simplified 83.8%

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

   5. Taylor expanded in K around 0 58.6%

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

   6. Taylor expanded in K around 0 72.5%

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

   7. Taylor expanded in U around inf 39.8%

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

      1. div-inv 39.8%

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

      2. unpow2 39.8%

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

      3. associate-*l* 39.8%

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

      4. div-inv 39.8%

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

   9. Applied egg-rr 39.8%

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

   if 1.25e160 < U

   1. Initial program 42.0%

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

   3. Simplified 42.0%

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

   5. Taylor expanded in J around 0 46.1%

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

      1. mul-1-neg 46.1%

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

   7. Simplified 46.1%

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

4. Final simplification 56.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 70000000:\\ \;\;\;\;-2 \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{elif}\;U \leq 1.15 \cdot 10^{+57}:\\ \;\;\;\;-2 \cdot \left(U \cdot 0.5 + J \cdot \frac{J}{U}\right)\\ \mathbf{elif}\;U \leq 2.4 \cdot 10^{+82}:\\ \;\;\;\;-2 \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{elif}\;U \leq 5.6 \cdot 10^{+133}:\\ \;\;\;\;-2 \cdot \left(U \cdot 0.5 + J \cdot \frac{J}{U}\right)\\ \mathbf{elif}\;U \leq 1.25 \cdot 10^{+160}:\\ \;\;\;\;-2 \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 38.1% accurate, 26.2× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} \mathbf{if}\;U\_m \leq 1.2 \cdot 10^{-114}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(U\_m \cdot 0.5 + J \cdot \frac{J}{U\_m}\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (if (<= U_m 1.2e-114) (* -2.0 J) (* -2.0 (+ (* U_m 0.5) (* J (/ J U_m))))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double tmp;
	if (U_m <= 1.2e-114) {
		tmp = -2.0 * J;
	} else {
		tmp = -2.0 * ((U_m * 0.5) + (J * (J / 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 <= 1.2d-114) then
        tmp = (-2.0d0) * j
    else
        tmp = (-2.0d0) * ((u_m * 0.5d0) + (j * (j / 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 <= 1.2e-114) {
		tmp = -2.0 * J;
	} else {
		tmp = -2.0 * ((U_m * 0.5) + (J * (J / U_m)));
	}
	return tmp;
}

Python

U_m = math.fabs(U)
def code(J, K, U_m):
	tmp = 0
	if U_m <= 1.2e-114:
		tmp = -2.0 * J
	else:
		tmp = -2.0 * ((U_m * 0.5) + (J * (J / U_m)))
	return tmp

Julia

U_m = abs(U)
function code(J, K, U_m)
	tmp = 0.0
	if (U_m <= 1.2e-114)
		tmp = Float64(-2.0 * J);
	else
		tmp = Float64(-2.0 * Float64(Float64(U_m * 0.5) + Float64(J * Float64(J / 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 <= 1.2e-114)
		tmp = -2.0 * J;
	else
		tmp = -2.0 * ((U_m * 0.5) + (J * (J / 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, 1.2e-114], N[(-2.0 * J), $MachinePrecision], N[(-2.0 * N[(N[(U$95$m * 0.5), $MachinePrecision] + N[(J * N[(J / U$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U < 1.2000000000000001e-114

   1. Initial program 80.4%

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

      1. associate-*l* 80.4%

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

      2. associate-*l* 80.4%

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

      3. unpow2 80.4%

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

      4. sqr-neg 80.4%

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

      5. distribute-frac-neg 80.4%

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

      6. distribute-frac-neg 80.4%

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

      7. unpow2 80.4%

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

   3. Simplified 93.0%

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

   5. Taylor expanded in U around 0 60.1%

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

   6. Taylor expanded in K around 0 36.8%

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

   if 1.2000000000000001e-114 < U

   1. Initial program 63.1%

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

      1. associate-*l* 63.1%

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

      2. associate-*l* 63.1%

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

      3. unpow2 63.1%

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

      4. sqr-neg 63.1%

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

      5. distribute-frac-neg 63.1%

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

      6. distribute-frac-neg 63.1%

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

      7. unpow2 63.1%

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

   3. Simplified 85.6%

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

   5. Taylor expanded in K around 0 45.5%

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

   6. Taylor expanded in K around 0 56.4%

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

   7. Taylor expanded in U around inf 35.1%

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

      1. div-inv 35.1%

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

      2. unpow2 35.1%

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

      3. associate-*l* 37.3%

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

      4. div-inv 37.3%

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

   9. Applied egg-rr 37.3%

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

4. Final simplification 37.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 1.2 \cdot 10^{-114}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(U \cdot 0.5 + J \cdot \frac{J}{U}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 37.9% accurate, 52.4× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} \mathbf{if}\;U\_m \leq 10^{-114}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{else}:\\ \;\;\;\;-U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m) :precision binary64 (if (<= U_m 1e-114) (* -2.0 J) (- U_m)))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double tmp;
	if (U_m <= 1e-114) {
		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 <= 1d-114) 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 <= 1e-114) {
		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 <= 1e-114:
		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 <= 1e-114)
		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 <= 1e-114)
		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, 1e-114], N[(-2.0 * J), $MachinePrecision], (-U$95$m)]

Derivation

1. Split input into 2 regimes

2. if U < 1.0000000000000001e-114

   1. Initial program 80.4%

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

      1. associate-*l* 80.4%

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

      2. associate-*l* 80.4%

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

      3. unpow2 80.4%

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

      4. sqr-neg 80.4%

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

      5. distribute-frac-neg 80.4%

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

      6. distribute-frac-neg 80.4%

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

      7. unpow2 80.4%

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

   3. Simplified 93.0%

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

   5. Taylor expanded in U around 0 60.1%

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

   6. Taylor expanded in K around 0 36.8%

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

   if 1.0000000000000001e-114 < U

   1. Initial program 63.1%

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

   3. Simplified 63.1%

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

   5. Taylor expanded in J around 0 37.2%

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

      1. mul-1-neg 37.2%

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

   7. Simplified 37.2%

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

4. Final simplification 37.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 10^{-114}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 26.7% accurate, 210.0× 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 Float64(-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 74.2%

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

3. Simplified 74.2%

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

5. Taylor expanded in J around 0 30.1%

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

   1. mul-1-neg 30.1%

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

7. Simplified 30.1%

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

8. Final simplification 30.1%

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

Alternative 8: 26.4% 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 74.2%

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

3. Simplified 74.2%

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

5. Taylor expanded in U around -inf 25.5%

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

6. Final simplification 25.5%

   \[\leadsto U \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024027 
(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)))))