Maksimov and Kolovsky, Equation (3)

Percentage Accurate: 73.5% → 98.4%

Time: 13.1s

Alternatives: 9

Speedup: 1.3×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.5% 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: 98.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_0 \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq 10^{+291}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* (* -2.0 J) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* t_0 (* J 2.0))) 2.0))))))
   (if (<= t_1 (- INFINITY)) (- U_m) (if (<= t_1 1e+291) t_1 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 tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_1 <= 1e+291) {
		tmp = t_1;
	} 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 tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = -U_m;
	} else if (t_1 <= 1e+291) {
		tmp = t_1;
	} 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)))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = -U_m
	elif t_1 <= 1e+291:
		tmp = t_1
	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))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_1 <= 1e+291)
		tmp = t_1;
	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)));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = -U_m;
	elseif (t_1 <= 1e+291)
		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[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]}, If[LessEqual[t$95$1, (-Infinity)], (-U$95$m), If[LessEqual[t$95$1, 1e+291], t$95$1, U$95$m]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -inf.0

   1. Initial program 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 56.3%

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

   5. Taylor expanded in J around 0 41.1%

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

      1. neg-mul-1 41.1%

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

   7. Simplified 41.1%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 9.9999999999999996e290

   1. Initial program 99.8%

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

   if 9.9999999999999996e290 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 5.6%

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

   3. Simplified 44.2%

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

   5. Taylor expanded in U around -inf 37.8%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 10^{+291}:\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} \mathbf{if}\;U\_m \leq 1.56 \cdot 10^{+206}:\\ \;\;\;\;J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U\_m}{J} \cdot \frac{0.5}{\cos \left(K \cdot 0.5\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (if (<= U_m 1.56e+206)
   (*
    J
    (*
     (* -2.0 (cos (/ K 2.0)))
     (hypot 1.0 (* (/ U_m J) (/ 0.5 (cos (* K 0.5)))))))
   (- U_m)))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double tmp;
	if (U_m <= 1.56e+206) {
		tmp = J * ((-2.0 * cos((K / 2.0))) * hypot(1.0, ((U_m / J) * (0.5 / cos((K * 0.5))))));
	} else {
		tmp = -U_m;
	}
	return tmp;
}

Java

U_m = Math.abs(U);
public static double code(double J, double K, double U_m) {
	double tmp;
	if (U_m <= 1.56e+206) {
		tmp = J * ((-2.0 * Math.cos((K / 2.0))) * Math.hypot(1.0, ((U_m / J) * (0.5 / Math.cos((K * 0.5))))));
	} else {
		tmp = -U_m;
	}
	return tmp;
}

Python

U_m = math.fabs(U)
def code(J, K, U_m):
	tmp = 0
	if U_m <= 1.56e+206:
		tmp = J * ((-2.0 * math.cos((K / 2.0))) * math.hypot(1.0, ((U_m / J) * (0.5 / math.cos((K * 0.5))))))
	else:
		tmp = -U_m
	return tmp

Julia

U_m = abs(U)
function code(J, K, U_m)
	tmp = 0.0
	if (U_m <= 1.56e+206)
		tmp = Float64(J * Float64(Float64(-2.0 * cos(Float64(K / 2.0))) * hypot(1.0, Float64(Float64(U_m / J) * Float64(0.5 / cos(Float64(K * 0.5)))))));
	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 <= 1.56e+206)
		tmp = J * ((-2.0 * cos((K / 2.0))) * hypot(1.0, ((U_m / J) * (0.5 / cos((K * 0.5))))));
	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, 1.56e+206], N[(J * N[(N[(-2.0 * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[(N[(U$95$m / J), $MachinePrecision] * N[(0.5 / N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-U$95$m)]

Derivation

1. Split input into 2 regimes

2. if U < 1.5599999999999999e206

   1. Initial program 78.0%

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

   3. Simplified 89.3%

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

      1. div-inv 89.3%

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

      2. metadata-eval 89.3%

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

      3. times-frac 89.3%

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

      4. div-inv 89.3%

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

      5. metadata-eval 89.3%

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

   6. Applied egg-rr 89.3%

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

   if 1.5599999999999999e206 < U

   1. Initial program 50.2%

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

   3. Simplified 56.0%

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

   5. Taylor expanded in J around 0 60.5%

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

      1. neg-mul-1 60.5%

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

   7. Simplified 60.5%

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

Alternative 3: 62.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} \mathbf{if}\;K \leq 0.00021:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \mathsf{hypot}\left(1, \frac{1}{\frac{-2 \cdot J}{U\_m}}\right)\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(-2 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(K \cdot 0.5\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (if (<= K 0.00021)
   (* (* -2.0 J) (hypot 1.0 (/ 1.0 (/ (* -2.0 J) U_m))))
   (* J (* -2.0 (expm1 (log1p (cos (* K 0.5))))))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double tmp;
	if (K <= 0.00021) {
		tmp = (-2.0 * J) * hypot(1.0, (1.0 / ((-2.0 * J) / U_m)));
	} else {
		tmp = J * (-2.0 * expm1(log1p(cos((K * 0.5)))));
	}
	return tmp;
}

Java

U_m = Math.abs(U);
public static double code(double J, double K, double U_m) {
	double tmp;
	if (K <= 0.00021) {
		tmp = (-2.0 * J) * Math.hypot(1.0, (1.0 / ((-2.0 * J) / U_m)));
	} else {
		tmp = J * (-2.0 * Math.expm1(Math.log1p(Math.cos((K * 0.5)))));
	}
	return tmp;
}

Python

U_m = math.fabs(U)
def code(J, K, U_m):
	tmp = 0
	if K <= 0.00021:
		tmp = (-2.0 * J) * math.hypot(1.0, (1.0 / ((-2.0 * J) / U_m)))
	else:
		tmp = J * (-2.0 * math.expm1(math.log1p(math.cos((K * 0.5)))))
	return tmp

Julia

U_m = abs(U)
function code(J, K, U_m)
	tmp = 0.0
	if (K <= 0.00021)
		tmp = Float64(Float64(-2.0 * J) * hypot(1.0, Float64(1.0 / Float64(Float64(-2.0 * J) / U_m))));
	else
		tmp = Float64(J * Float64(-2.0 * expm1(log1p(cos(Float64(K * 0.5))))));
	end
	return tmp
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := If[LessEqual[K, 0.00021], N[(N[(-2.0 * J), $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[(1.0 / N[(N[(-2.0 * J), $MachinePrecision] / U$95$m), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(J * N[(-2.0 * N[(Exp[N[Log[1 + N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if K < 2.1000000000000001e-4

   1. Initial program 74.3%

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

   3. Simplified 86.4%

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

   5. Taylor expanded in K around 0 43.1%

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

      1. associate-*r* 43.1%

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

      2. *-commutative 43.1%

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

      3. unpow2 43.1%

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

      4. unpow2 43.1%

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

      5. times-frac 56.8%

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

      6. metadata-eval 56.8%

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

      7. swap-sqr 56.8%

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

      8. associate-*l/ 56.8%

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

      9. associate-*l/ 56.8%

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

      10. unpow2 56.8%

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

   7. Simplified 56.8%

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

      1. unpow2 56.8%

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

      2. hypot-1-def 67.0%

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

      3. associate-/l* 67.0%

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

   9. Applied egg-rr 67.0%

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

       1. clear-num 67.0%

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

       2. div-inv 67.0%

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

       3. metadata-eval 67.0%

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

       4. *-commutative 67.0%

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

       5. div-inv 67.0%

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

       6. clear-num 67.0%

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

       7. add-sqr-sqrt 35.3%

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

       8. sqrt-unprod 52.7%

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

       9. swap-sqr 52.7%

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

       10. metadata-eval 52.7%

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

       11. metadata-eval 52.7%

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

       12. swap-sqr 52.7%

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

       13. add-sqr-sqrt 25.2%

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

       14. associate-*l* 25.2%

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

       15. add-sqr-sqrt 25.2%

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

       16. associate-*l* 25.2%

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

       17. sqrt-unprod 0.0%

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

       18. add-sqr-sqrt 35.3%

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

       19. associate-*l* 35.3%

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

       20. add-sqr-sqrt 67.0%

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

   11. Applied egg-rr 67.0%

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

   if 2.1000000000000001e-4 < K

   1. Initial program 82.7%

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

   3. Simplified 90.5%

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

   5. Taylor expanded in U around 0 59.4%

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

      1. *-commutative 59.4%

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

      2. expm1-log1p-u 59.5%

         \[\leadsto J \cdot \left(-2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(K \cdot 0.5\right)\right)\right)}\right) \]

   7. Applied egg-rr 59.5%

      \[\leadsto J \cdot \left(-2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(K \cdot 0.5\right)\right)\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 63.0% accurate, 3.6× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} \mathbf{if}\;K \leq 0.00185:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \mathsf{hypot}\left(1, \frac{1}{\frac{-2 \cdot J}{U\_m}}\right)\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(-2 \cdot \cos \left(K \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (if (<= K 0.00185)
   (* (* -2.0 J) (hypot 1.0 (/ 1.0 (/ (* -2.0 J) U_m))))
   (* J (* -2.0 (cos (* K 0.5))))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double tmp;
	if (K <= 0.00185) {
		tmp = (-2.0 * J) * hypot(1.0, (1.0 / ((-2.0 * J) / U_m)));
	} else {
		tmp = J * (-2.0 * cos((K * 0.5)));
	}
	return tmp;
}

Java

U_m = Math.abs(U);
public static double code(double J, double K, double U_m) {
	double tmp;
	if (K <= 0.00185) {
		tmp = (-2.0 * J) * Math.hypot(1.0, (1.0 / ((-2.0 * J) / U_m)));
	} else {
		tmp = J * (-2.0 * Math.cos((K * 0.5)));
	}
	return tmp;
}

Python

U_m = math.fabs(U)
def code(J, K, U_m):
	tmp = 0
	if K <= 0.00185:
		tmp = (-2.0 * J) * math.hypot(1.0, (1.0 / ((-2.0 * J) / U_m)))
	else:
		tmp = J * (-2.0 * math.cos((K * 0.5)))
	return tmp

Julia

U_m = abs(U)
function code(J, K, U_m)
	tmp = 0.0
	if (K <= 0.00185)
		tmp = Float64(Float64(-2.0 * J) * hypot(1.0, Float64(1.0 / Float64(Float64(-2.0 * J) / U_m))));
	else
		tmp = Float64(J * Float64(-2.0 * 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 (K <= 0.00185)
		tmp = (-2.0 * J) * hypot(1.0, (1.0 / ((-2.0 * J) / U_m)));
	else
		tmp = J * (-2.0 * cos((K * 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := If[LessEqual[K, 0.00185], N[(N[(-2.0 * J), $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[(1.0 / N[(N[(-2.0 * J), $MachinePrecision] / U$95$m), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(J * N[(-2.0 * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if K < 0.0018500000000000001

   1. Initial program 74.3%

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

   3. Simplified 86.4%

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

   5. Taylor expanded in K around 0 43.1%

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

      1. associate-*r* 43.1%

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

      2. *-commutative 43.1%

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

      3. unpow2 43.1%

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

      4. unpow2 43.1%

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

      5. times-frac 56.8%

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

      6. metadata-eval 56.8%

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

      7. swap-sqr 56.8%

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

      8. associate-*l/ 56.8%

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

      9. associate-*l/ 56.8%

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

      10. unpow2 56.8%

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

   7. Simplified 56.8%

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

      1. unpow2 56.8%

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

      2. hypot-1-def 67.0%

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

      3. associate-/l* 67.0%

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

   9. Applied egg-rr 67.0%

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

       1. clear-num 67.0%

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

       2. div-inv 67.0%

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

       3. metadata-eval 67.0%

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

       4. *-commutative 67.0%

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

       5. div-inv 67.0%

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

       6. clear-num 67.0%

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

       7. add-sqr-sqrt 35.3%

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

       8. sqrt-unprod 52.7%

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

       9. swap-sqr 52.7%

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

       10. metadata-eval 52.7%

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

       11. metadata-eval 52.7%

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

       12. swap-sqr 52.7%

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

       13. add-sqr-sqrt 25.2%

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

       14. associate-*l* 25.2%

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

       15. add-sqr-sqrt 25.2%

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

       16. associate-*l* 25.2%

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

       17. sqrt-unprod 0.0%

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

       18. add-sqr-sqrt 35.3%

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

       19. associate-*l* 35.3%

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

       20. add-sqr-sqrt 67.0%

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

   11. Applied egg-rr 67.0%

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

   if 0.0018500000000000001 < K

   1. Initial program 82.7%

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

   3. Simplified 90.5%

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

   5. Taylor expanded in U around 0 59.4%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;K \leq 0.00185:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \mathsf{hypot}\left(1, \frac{1}{\frac{-2 \cdot J}{U}}\right)\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(-2 \cdot \cos \left(K \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 63.0% accurate, 3.7× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} \mathbf{if}\;K \leq 0.00036:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \mathsf{hypot}\left(1, U\_m \cdot \frac{0.5}{J}\right)\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(-2 \cdot \cos \left(K \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (if (<= K 0.00036)
   (* (* -2.0 J) (hypot 1.0 (* U_m (/ 0.5 J))))
   (* J (* -2.0 (cos (* K 0.5))))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double tmp;
	if (K <= 0.00036) {
		tmp = (-2.0 * J) * hypot(1.0, (U_m * (0.5 / J)));
	} else {
		tmp = J * (-2.0 * cos((K * 0.5)));
	}
	return tmp;
}

Java

U_m = Math.abs(U);
public static double code(double J, double K, double U_m) {
	double tmp;
	if (K <= 0.00036) {
		tmp = (-2.0 * J) * Math.hypot(1.0, (U_m * (0.5 / J)));
	} else {
		tmp = J * (-2.0 * Math.cos((K * 0.5)));
	}
	return tmp;
}

Python

U_m = math.fabs(U)
def code(J, K, U_m):
	tmp = 0
	if K <= 0.00036:
		tmp = (-2.0 * J) * math.hypot(1.0, (U_m * (0.5 / J)))
	else:
		tmp = J * (-2.0 * math.cos((K * 0.5)))
	return tmp

Julia

U_m = abs(U)
function code(J, K, U_m)
	tmp = 0.0
	if (K <= 0.00036)
		tmp = Float64(Float64(-2.0 * J) * hypot(1.0, Float64(U_m * Float64(0.5 / J))));
	else
		tmp = Float64(J * Float64(-2.0 * 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 (K <= 0.00036)
		tmp = (-2.0 * J) * hypot(1.0, (U_m * (0.5 / J)));
	else
		tmp = J * (-2.0 * cos((K * 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := If[LessEqual[K, 0.00036], N[(N[(-2.0 * J), $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[(U$95$m * N[(0.5 / J), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(J * N[(-2.0 * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if K < 3.60000000000000023e-4

   1. Initial program 74.3%

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

   3. Simplified 86.4%

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

   5. Taylor expanded in K around 0 43.1%

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

      1. associate-*r* 43.1%

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

      2. *-commutative 43.1%

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

      3. unpow2 43.1%

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

      4. unpow2 43.1%

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

      5. times-frac 56.8%

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

      6. metadata-eval 56.8%

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

      7. swap-sqr 56.8%

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

      8. associate-*l/ 56.8%

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

      9. associate-*l/ 56.8%

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

      10. unpow2 56.8%

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

   7. Simplified 56.8%

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

      1. unpow2 56.8%

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

      2. hypot-1-def 67.0%

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

      3. associate-/l* 67.0%

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

   9. Applied egg-rr 67.0%

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

   if 3.60000000000000023e-4 < K

   1. Initial program 82.7%

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

   3. Simplified 90.5%

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

   5. Taylor expanded in U around 0 59.4%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;K \leq 0.00036:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \mathsf{hypot}\left(1, U \cdot \frac{0.5}{J}\right)\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(-2 \cdot \cos \left(K \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 59.0% accurate, 3.7× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} \mathbf{if}\;U\_m \leq 6.7 \cdot 10^{+75}:\\ \;\;\;\;J \cdot \left(-2 \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (if (<= U_m 6.7e+75) (* J (* -2.0 (cos (* K 0.5)))) (- U_m)))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double tmp;
	if (U_m <= 6.7e+75) {
		tmp = J * (-2.0 * cos((K * 0.5)));
	} 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 <= 6.7d+75) then
        tmp = j * ((-2.0d0) * cos((k * 0.5d0)))
    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 <= 6.7e+75) {
		tmp = J * (-2.0 * Math.cos((K * 0.5)));
	} else {
		tmp = -U_m;
	}
	return tmp;
}

Python

U_m = math.fabs(U)
def code(J, K, U_m):
	tmp = 0
	if U_m <= 6.7e+75:
		tmp = J * (-2.0 * math.cos((K * 0.5)))
	else:
		tmp = -U_m
	return tmp

Julia

U_m = abs(U)
function code(J, K, U_m)
	tmp = 0.0
	if (U_m <= 6.7e+75)
		tmp = Float64(J * Float64(-2.0 * cos(Float64(K * 0.5))));
	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 <= 6.7e+75)
		tmp = J * (-2.0 * cos((K * 0.5)));
	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, 6.7e+75], N[(J * N[(-2.0 * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-U$95$m)]

Derivation

1. Split input into 2 regimes

2. if U < 6.7000000000000001e75

   1. Initial program 79.8%

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

   3. Simplified 90.3%

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

   5. Taylor expanded in U around 0 62.0%

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

   if 6.7000000000000001e75 < U

   1. Initial program 57.4%

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

   3. Simplified 71.5%

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

   5. Taylor expanded in J around 0 47.5%

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

      1. neg-mul-1 47.5%

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

   7. Simplified 47.5%

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

4. Final simplification 59.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 6.7 \cdot 10^{+75}:\\ \;\;\;\;J \cdot \left(-2 \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 39.8% accurate, 52.4× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} \mathbf{if}\;U\_m \leq 7 \cdot 10^{+75}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{else}:\\ \;\;\;\;-U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m) :precision binary64 (if (<= U_m 7e+75) (* -2.0 J) (- U_m)))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double tmp;
	if (U_m <= 7e+75) {
		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 <= 7d+75) 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 <= 7e+75) {
		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 <= 7e+75:
		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 <= 7e+75)
		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 <= 7e+75)
		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, 7e+75], N[(-2.0 * J), $MachinePrecision], (-U$95$m)]

Derivation

1. Split input into 2 regimes

2. if U < 6.9999999999999997e75

   1. Initial program 79.8%

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

   3. Simplified 90.3%

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

   5. Taylor expanded in U around 0 62.0%

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

   6. Taylor expanded in K around 0 37.6%

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

   if 6.9999999999999997e75 < U

   1. Initial program 57.4%

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

   3. Simplified 71.5%

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

   5. Taylor expanded in J around 0 47.5%

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

      1. neg-mul-1 47.5%

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

   7. Simplified 47.5%

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

4. Final simplification 39.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 7 \cdot 10^{+75}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 26.2% accurate, 59.9× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} \mathbf{if}\;K \leq 1.6 \cdot 10^{+122}:\\ \;\;\;\;-U\_m\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m) :precision binary64 (if (<= K 1.6e+122) (- U_m) U_m))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double tmp;
	if (K <= 1.6e+122) {
		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 <= 1.6d+122) 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 <= 1.6e+122) {
		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 <= 1.6e+122:
		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 <= 1.6e+122)
		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 <= 1.6e+122)
		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[LessEqual[K, 1.6e+122], (-U$95$m), U$95$m]

Derivation

1. Split input into 2 regimes

2. if K < 1.60000000000000006e122

   1. Initial program 75.0%

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

   3. Simplified 87.0%

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

   5. Taylor expanded in J around 0 26.0%

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

      1. neg-mul-1 26.0%

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

   7. Simplified 26.0%

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

   if 1.60000000000000006e122 < K

   1. Initial program 84.4%

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

   3. Simplified 89.5%

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

   5. Taylor expanded in U around -inf 24.4%

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

Alternative 9: 27.3% 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 76.3%

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

3. Simplified 87.4%

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

5. Taylor expanded in U around -inf 25.4%

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

Reproduce

herbie shell --seed 2024150 
(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)))))