Maksimov and Kolovsky, Equation (3)

Percentage Accurate: 73.3% → 99.7%

Time: 15.6s

Alternatives: 5

Speedup: 0.3×

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: 73.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.3× speedup

Math

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

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 5e+305) 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 <= 5e+305) {
		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 <= 5e+305) {
		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 <= 5e+305:
		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 <= 5e+305)
		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 <= 5e+305)
		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, 5e+305], 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 7.2%

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

   3. Simplified 45.6%

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

   5. Taylor expanded in J around 0 49.1%

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

      1. mul-1-neg 49.1%

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

   7. Simplified 49.1%

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

   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))))) < 5.00000000000000009e305

   1. Initial program 99.8%

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

   if 5.00000000000000009e305 < (*.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 5.3%

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

   3. Simplified 80.4%

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

   5. Taylor expanded in U around -inf 44.7%

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

4. Final simplification 84.4%

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

Alternative 2: 51.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;J \leq 4.2 \cdot 10^{-233}:\\ \;\;\;\;-U\_m\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(\left(-2 \cdot t\_0\right) \cdot \mathsf{hypot}\left(1, \frac{\frac{U\_m}{2}}{J \cdot t\_0}\right)\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= J 4.2e-233)
     (- U_m)
     (* J (* (* -2.0 t_0) (hypot 1.0 (/ (/ U_m 2.0) (* J t_0))))))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (J <= 4.2e-233) {
		tmp = -U_m;
	} else {
		tmp = J * ((-2.0 * t_0) * hypot(1.0, ((U_m / 2.0) / (J * t_0))));
	}
	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 (J <= 4.2e-233) {
		tmp = -U_m;
	} else {
		tmp = J * ((-2.0 * t_0) * Math.hypot(1.0, ((U_m / 2.0) / (J * t_0))));
	}
	return tmp;
}

Python

U_m = math.fabs(U)
def code(J, K, U_m):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if J <= 4.2e-233:
		tmp = -U_m
	else:
		tmp = J * ((-2.0 * t_0) * math.hypot(1.0, ((U_m / 2.0) / (J * t_0))))
	return tmp

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (J <= 4.2e-233)
		tmp = Float64(-U_m);
	else
		tmp = Float64(J * Float64(Float64(-2.0 * t_0) * hypot(1.0, Float64(Float64(U_m / 2.0) / Float64(J * t_0)))));
	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 (J <= 4.2e-233)
		tmp = -U_m;
	else
		tmp = J * ((-2.0 * t_0) * hypot(1.0, ((U_m / 2.0) / (J * t_0))));
	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[J, 4.2e-233], (-U$95$m), N[(J * N[(N[(-2.0 * t$95$0), $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[(N[(U$95$m / 2.0), $MachinePrecision] / N[(J * t$95$0), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if J < 4.1999999999999997e-233

   1. Initial program 71.0%

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

   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 26.1%

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

      1. mul-1-neg 26.1%

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

   7. Simplified 26.1%

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

   if 4.1999999999999997e-233 < J

   1. Initial program 74.8%

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

   3. Simplified 91.1%

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

Alternative 3: 39.8% accurate, 3.7× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} \mathbf{if}\;J \leq 5 \cdot 10^{+20}:\\ \;\;\;\;-U\_m\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (if (<= J 5e+20) (- U_m) (* -2.0 (* J (cos (* K 0.5))))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double tmp;
	if (J <= 5e+20) {
		tmp = -U_m;
	} else {
		tmp = -2.0 * (J * cos((K * 0.5)));
	}
	return tmp;
}

Fortran

U_m = abs(u)
real(8) function code(j, k, u_m)
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u_m
    real(8) :: tmp
    if (j <= 5d+20) then
        tmp = -u_m
    else
        tmp = (-2.0d0) * (j * cos((k * 0.5d0)))
    end if
    code = tmp
end function

Java

U_m = Math.abs(U);
public static double code(double J, double K, double U_m) {
	double tmp;
	if (J <= 5e+20) {
		tmp = -U_m;
	} else {
		tmp = -2.0 * (J * Math.cos((K * 0.5)));
	}
	return tmp;
}

Python

U_m = math.fabs(U)
def code(J, K, U_m):
	tmp = 0
	if J <= 5e+20:
		tmp = -U_m
	else:
		tmp = -2.0 * (J * math.cos((K * 0.5)))
	return tmp

Julia

U_m = abs(U)
function code(J, K, U_m)
	tmp = 0.0
	if (J <= 5e+20)
		tmp = Float64(-U_m);
	else
		tmp = Float64(-2.0 * Float64(J * cos(Float64(K * 0.5))));
	end
	return tmp
end

MATLAB

U_m = abs(U);
function tmp_2 = code(J, K, U_m)
	tmp = 0.0;
	if (J <= 5e+20)
		tmp = -U_m;
	else
		tmp = -2.0 * (J * cos((K * 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := If[LessEqual[J, 5e+20], (-U$95$m), N[(-2.0 * N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if J < 5e20

   1. Initial program 66.4%

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

   3. Simplified 85.3%

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

   5. Taylor expanded in J around 0 28.5%

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

      1. mul-1-neg 28.5%

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

   7. Simplified 28.5%

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

   if 5e20 < J

   1. Initial program 93.6%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in J around inf 72.0%

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

      1. *-commutative 72.0%

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

   7. Simplified 72.0%

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

4. Final simplification 38.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq 5 \cdot 10^{+20}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 26.6% accurate, 24.6× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} \mathbf{if}\;K \leq 3.7 \cdot 10^{+28} \lor \neg \left(K \leq 3.6 \cdot 10^{+147}\right) \land K \leq 2.8 \cdot 10^{+206}:\\ \;\;\;\;-U\_m\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (if (or (<= K 3.7e+28) (and (not (<= K 3.6e+147)) (<= K 2.8e+206)))
   (- U_m)
   U_m))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double tmp;
	if ((K <= 3.7e+28) || (!(K <= 3.6e+147) && (K <= 2.8e+206))) {
		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 <= 3.7d+28) .or. (.not. (k <= 3.6d+147)) .and. (k <= 2.8d+206)) 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 <= 3.7e+28) || (!(K <= 3.6e+147) && (K <= 2.8e+206))) {
		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 <= 3.7e+28) or (not (K <= 3.6e+147) and (K <= 2.8e+206)):
		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 <= 3.7e+28) || (!(K <= 3.6e+147) && (K <= 2.8e+206)))
		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 <= 3.7e+28) || (~((K <= 3.6e+147)) && (K <= 2.8e+206)))
		tmp = -U_m;
	else
		tmp = U_m;
	end
	tmp_2 = tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := If[Or[LessEqual[K, 3.7e+28], And[N[Not[LessEqual[K, 3.6e+147]], $MachinePrecision], LessEqual[K, 2.8e+206]]], (-U$95$m), U$95$m]

Derivation

1. Split input into 2 regimes

2. if K < 3.6999999999999999e28 or 3.6000000000000002e147 < K < 2.7999999999999998e206

   1. Initial program 73.1%

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

   3. Simplified 88.9%

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

   5. Taylor expanded in J around 0 26.5%

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

      1. mul-1-neg 26.5%

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

   7. Simplified 26.5%

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

   if 3.6999999999999999e28 < K < 3.6000000000000002e147 or 2.7999999999999998e206 < K

   1. Initial program 68.3%

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

   3. Simplified 86.5%

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

   5. Taylor expanded in U around -inf 34.7%

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

4. Final simplification 27.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;K \leq 3.7 \cdot 10^{+28} \lor \neg \left(K \leq 3.6 \cdot 10^{+147}\right) \land K \leq 2.8 \cdot 10^{+206}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 26.7% accurate, 420.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.4%

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

3. Simplified 88.5%

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

5. Taylor expanded in U around -inf 28.3%

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

Reproduce

herbie shell --seed 2024111 
(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)))))