Maksimov and Kolovsky, Equation (3)

Percentage Accurate: 73.6% → 99.5%

Time: 21.5s

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: 73.6% 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, 0.3× speedup

Math

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

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+304) 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+304) {
		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+304) {
		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+304:
		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+304)
		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+304)
		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+304], 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 6.0%

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

   3. Simplified 50.4%

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

   5. Taylor expanded in J around 0 63.6%

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

      1. neg-mul-1 63.6%

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

   7. Simplified 63.6%

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

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

   if 4.9999999999999997e304 < (*.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 5.7%

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

   3. Simplified 57.8%

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

   5. Taylor expanded in U around -inf 60.3%

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

4. Final simplification 89.4%

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

Alternative 2: 85.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;U\_m \leq 6.4 \cdot 10^{+198}:\\ \;\;\;\;J \cdot \left(\left(-2 \cdot t\_0\right) \cdot \mathsf{hypot}\left(1, \frac{\frac{U\_m}{2}}{J \cdot t\_0}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= U_m 6.4e+198)
     (* J (* (* -2.0 t_0) (hypot 1.0 (/ (/ U_m 2.0) (* J t_0)))))
     (- U_m))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (U_m <= 6.4e+198) {
		tmp = J * ((-2.0 * t_0) * hypot(1.0, ((U_m / 2.0) / (J * t_0))));
	} 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 tmp;
	if (U_m <= 6.4e+198) {
		tmp = J * ((-2.0 * t_0) * Math.hypot(1.0, ((U_m / 2.0) / (J * t_0))));
	} 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))
	tmp = 0
	if U_m <= 6.4e+198:
		tmp = J * ((-2.0 * t_0) * math.hypot(1.0, ((U_m / 2.0) / (J * t_0))))
	else:
		tmp = -U_m
	return tmp

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (U_m <= 6.4e+198)
		tmp = Float64(J * Float64(Float64(-2.0 * t_0) * hypot(1.0, Float64(Float64(U_m / 2.0) / Float64(J * t_0)))));
	else
		tmp = Float64(-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));
	tmp = 0.0;
	if (U_m <= 6.4e+198)
		tmp = J * ((-2.0 * t_0) * hypot(1.0, ((U_m / 2.0) / (J * t_0))));
	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]}, If[LessEqual[U$95$m, 6.4e+198], 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], (-U$95$m)]]

Derivation

1. Split input into 2 regimes

2. if U < 6.3999999999999997e198

   1. Initial program 78.6%

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

   3. Simplified 91.5%

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

   if 6.3999999999999997e198 < U

   1. Initial program 37.4%

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

   3. Simplified 53.7%

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

   5. Taylor expanded in J around 0 50.7%

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

      1. neg-mul-1 50.7%

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

   7. Simplified 50.7%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 6.4 \cdot 10^{+198}:\\ \;\;\;\;J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{\frac{U}{2}}{J \cdot \cos \left(\frac{K}{2}\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 70.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{\frac{U\_m}{2}}{J}\right)\right)\\ \mathbf{if}\;U\_m \leq 1.8 \cdot 10^{+35}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;U\_m \leq 1.15 \cdot 10^{+82}:\\ \;\;\;\;\frac{J \cdot U\_m}{-J}\\ \mathbf{elif}\;U\_m \leq 1.2 \cdot 10^{+184}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;-U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (* J (* (* -2.0 (cos (/ K 2.0))) (hypot 1.0 (/ (/ U_m 2.0) J))))))
   (if (<= U_m 1.8e+35)
     t_0
     (if (<= U_m 1.15e+82)
       (/ (* J U_m) (- J))
       (if (<= U_m 1.2e+184) t_0 (- U_m))))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = J * ((-2.0 * cos((K / 2.0))) * hypot(1.0, ((U_m / 2.0) / J)));
	double tmp;
	if (U_m <= 1.8e+35) {
		tmp = t_0;
	} else if (U_m <= 1.15e+82) {
		tmp = (J * U_m) / -J;
	} else if (U_m <= 1.2e+184) {
		tmp = t_0;
	} 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 = J * ((-2.0 * Math.cos((K / 2.0))) * Math.hypot(1.0, ((U_m / 2.0) / J)));
	double tmp;
	if (U_m <= 1.8e+35) {
		tmp = t_0;
	} else if (U_m <= 1.15e+82) {
		tmp = (J * U_m) / -J;
	} else if (U_m <= 1.2e+184) {
		tmp = t_0;
	} else {
		tmp = -U_m;
	}
	return tmp;
}

Python

U_m = math.fabs(U)
def code(J, K, U_m):
	t_0 = J * ((-2.0 * math.cos((K / 2.0))) * math.hypot(1.0, ((U_m / 2.0) / J)))
	tmp = 0
	if U_m <= 1.8e+35:
		tmp = t_0
	elif U_m <= 1.15e+82:
		tmp = (J * U_m) / -J
	elif U_m <= 1.2e+184:
		tmp = t_0
	else:
		tmp = -U_m
	return tmp

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = Float64(J * Float64(Float64(-2.0 * cos(Float64(K / 2.0))) * hypot(1.0, Float64(Float64(U_m / 2.0) / J))))
	tmp = 0.0
	if (U_m <= 1.8e+35)
		tmp = t_0;
	elseif (U_m <= 1.15e+82)
		tmp = Float64(Float64(J * U_m) / Float64(-J));
	elseif (U_m <= 1.2e+184)
		tmp = t_0;
	else
		tmp = Float64(-U_m);
	end
	return tmp
end

MATLAB

U_m = abs(U);
function tmp_2 = code(J, K, U_m)
	t_0 = J * ((-2.0 * cos((K / 2.0))) * hypot(1.0, ((U_m / 2.0) / J)));
	tmp = 0.0;
	if (U_m <= 1.8e+35)
		tmp = t_0;
	elseif (U_m <= 1.15e+82)
		tmp = (J * U_m) / -J;
	elseif (U_m <= 1.2e+184)
		tmp = t_0;
	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[(J * N[(N[(-2.0 * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[(N[(U$95$m / 2.0), $MachinePrecision] / J), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[U$95$m, 1.8e+35], t$95$0, If[LessEqual[U$95$m, 1.15e+82], N[(N[(J * U$95$m), $MachinePrecision] / (-J)), $MachinePrecision], If[LessEqual[U$95$m, 1.2e+184], t$95$0, (-U$95$m)]]]]

Derivation

1. Split input into 3 regimes

2. if U < 1.8e35 or 1.14999999999999994e82 < U < 1.19999999999999998e184

   1. Initial program 79.7%

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

   3. Simplified 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

   5. Taylor expanded in K around 0 81.3%

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

   if 1.8e35 < U < 1.14999999999999994e82

   1. Initial program 72.0%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in U around inf 29.4%

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

      1. associate-*r/ 29.4%

         \[\leadsto J \cdot \color{blue}{\frac{-1 \cdot U}{J}} \]

      2. neg-mul-1 29.4%

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

   7. Simplified 29.4%

      \[\leadsto J \cdot \color{blue}{\frac{-U}{J}} \]
   8. Step-by-step derivation

      1. distribute-frac-neg 29.4%

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

      2. distribute-frac-neg2 29.4%

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

      3. associate-*r/ 29.2%

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

   9. Applied egg-rr 29.2%

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

   if 1.19999999999999998e184 < U

   1. Initial program 38.2%

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

   3. Simplified 60.7%

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

   5. Taylor expanded in J around 0 49.3%

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

      1. neg-mul-1 49.3%

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

   7. Simplified 49.3%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 1.8 \cdot 10^{+35}:\\ \;\;\;\;J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{\frac{U}{2}}{J}\right)\right)\\ \mathbf{elif}\;U \leq 1.15 \cdot 10^{+82}:\\ \;\;\;\;\frac{J \cdot U}{-J}\\ \mathbf{elif}\;U \leq 1.2 \cdot 10^{+184}:\\ \;\;\;\;J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{\frac{U}{2}}{J}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} \mathbf{if}\;U\_m \leq 1.8 \cdot 10^{+35}:\\ \;\;\;\;\left(-2 \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \left(J \cdot \mathsf{hypot}\left(1, 0.5 \cdot \frac{U\_m}{J}\right)\right)\\ \mathbf{elif}\;U\_m \leq 1.2 \cdot 10^{+75}:\\ \;\;\;\;\frac{J \cdot U\_m}{-J}\\ \mathbf{elif}\;U\_m \leq 1.08 \cdot 10^{+184}:\\ \;\;\;\;J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{\frac{U\_m}{2}}{J}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (if (<= U_m 1.8e+35)
   (* (* -2.0 (cos (* K 0.5))) (* J (hypot 1.0 (* 0.5 (/ U_m J)))))
   (if (<= U_m 1.2e+75)
     (/ (* J U_m) (- J))
     (if (<= U_m 1.08e+184)
       (* J (* (* -2.0 (cos (/ K 2.0))) (hypot 1.0 (/ (/ U_m 2.0) J))))
       (- U_m)))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double tmp;
	if (U_m <= 1.8e+35) {
		tmp = (-2.0 * cos((K * 0.5))) * (J * hypot(1.0, (0.5 * (U_m / J))));
	} else if (U_m <= 1.2e+75) {
		tmp = (J * U_m) / -J;
	} else if (U_m <= 1.08e+184) {
		tmp = J * ((-2.0 * cos((K / 2.0))) * hypot(1.0, ((U_m / 2.0) / J)));
	} 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 <= 1.8e+35) {
		tmp = (-2.0 * Math.cos((K * 0.5))) * (J * Math.hypot(1.0, (0.5 * (U_m / J))));
	} else if (U_m <= 1.2e+75) {
		tmp = (J * U_m) / -J;
	} else if (U_m <= 1.08e+184) {
		tmp = J * ((-2.0 * Math.cos((K / 2.0))) * Math.hypot(1.0, ((U_m / 2.0) / J)));
	} else {
		tmp = -U_m;
	}
	return tmp;
}

Python

U_m = math.fabs(U)
def code(J, K, U_m):
	tmp = 0
	if U_m <= 1.8e+35:
		tmp = (-2.0 * math.cos((K * 0.5))) * (J * math.hypot(1.0, (0.5 * (U_m / J))))
	elif U_m <= 1.2e+75:
		tmp = (J * U_m) / -J
	elif U_m <= 1.08e+184:
		tmp = J * ((-2.0 * math.cos((K / 2.0))) * math.hypot(1.0, ((U_m / 2.0) / J)))
	else:
		tmp = -U_m
	return tmp

Julia

U_m = abs(U)
function code(J, K, U_m)
	tmp = 0.0
	if (U_m <= 1.8e+35)
		tmp = Float64(Float64(-2.0 * cos(Float64(K * 0.5))) * Float64(J * hypot(1.0, Float64(0.5 * Float64(U_m / J)))));
	elseif (U_m <= 1.2e+75)
		tmp = Float64(Float64(J * U_m) / Float64(-J));
	elseif (U_m <= 1.08e+184)
		tmp = Float64(J * Float64(Float64(-2.0 * cos(Float64(K / 2.0))) * hypot(1.0, Float64(Float64(U_m / 2.0) / J))));
	else
		tmp = Float64(-U_m);
	end
	return tmp
end

MATLAB

U_m = abs(U);
function tmp_2 = code(J, K, U_m)
	tmp = 0.0;
	if (U_m <= 1.8e+35)
		tmp = (-2.0 * cos((K * 0.5))) * (J * hypot(1.0, (0.5 * (U_m / J))));
	elseif (U_m <= 1.2e+75)
		tmp = (J * U_m) / -J;
	elseif (U_m <= 1.08e+184)
		tmp = J * ((-2.0 * cos((K / 2.0))) * hypot(1.0, ((U_m / 2.0) / J)));
	else
		tmp = -U_m;
	end
	tmp_2 = tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := If[LessEqual[U$95$m, 1.8e+35], N[(N[(-2.0 * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(J * N[Sqrt[1.0 ^ 2 + N[(0.5 * N[(U$95$m / J), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[U$95$m, 1.2e+75], N[(N[(J * U$95$m), $MachinePrecision] / (-J)), $MachinePrecision], If[LessEqual[U$95$m, 1.08e+184], 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 / 2.0), $MachinePrecision] / J), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-U$95$m)]]]

Derivation

1. Split input into 4 regimes

2. if U < 1.8e35

   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 93.0%

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

   6. Applied egg-rr 91.2%

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

   7. Taylor expanded in K around 0 80.6%

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

      1. rem-cube-cbrt 82.1%

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

      2. *-commutative 82.1%

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

      3. associate-*l* 82.1%

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

      4. rem-cube-cbrt 79.4%

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

      5. *-commutative 79.4%

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

      6. rem-cube-cbrt 82.1%

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

      7. *-commutative 82.1%

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

      8. div-inv 82.1%

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

      9. associate-*l* 82.1%

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

      10. associate-*l/ 82.1%

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

      11. *-un-lft-identity 82.1%

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

   9. Applied egg-rr 82.1%

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

   if 1.8e35 < U < 1.2e75

   1. Initial program 72.0%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in U around inf 29.4%

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

      1. associate-*r/ 29.4%

         \[\leadsto J \cdot \color{blue}{\frac{-1 \cdot U}{J}} \]

      2. neg-mul-1 29.4%

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

   7. Simplified 29.4%

      \[\leadsto J \cdot \color{blue}{\frac{-U}{J}} \]
   8. Step-by-step derivation

      1. distribute-frac-neg 29.4%

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

      2. distribute-frac-neg2 29.4%

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

      3. associate-*r/ 29.2%

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

   9. Applied egg-rr 29.2%

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

   if 1.2e75 < U < 1.07999999999999993e184

   1. Initial program 54.3%

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

   3. Simplified 75.0%

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

   5. Taylor expanded in K around 0 75.0%

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

   if 1.07999999999999993e184 < U

   1. Initial program 38.2%

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

   3. Simplified 60.7%

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

   5. Taylor expanded in J around 0 49.3%

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

      1. neg-mul-1 49.3%

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

   7. Simplified 49.3%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 1.8 \cdot 10^{+35}:\\ \;\;\;\;\left(-2 \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \left(J \cdot \mathsf{hypot}\left(1, 0.5 \cdot \frac{U}{J}\right)\right)\\ \mathbf{elif}\;U \leq 1.2 \cdot 10^{+75}:\\ \;\;\;\;\frac{J \cdot U}{-J}\\ \mathbf{elif}\;U \leq 1.08 \cdot 10^{+184}:\\ \;\;\;\;J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{\frac{U}{2}}{J}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 63.3% accurate, 3.2× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := -2 \cdot \left(J \cdot \mathsf{hypot}\left(1, 0.5 \cdot \frac{U\_m}{J}\right)\right)\\ t_1 := \left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{if}\;U\_m \leq 4.5 \cdot 10^{-139}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;U\_m \leq 7.8 \cdot 10^{-115}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;U\_m \leq 22000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;U\_m \leq 1.4 \cdot 10^{+180}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;-U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (* -2.0 (* J (hypot 1.0 (* 0.5 (/ U_m J))))))
        (t_1 (* (* -2.0 J) (cos (* K 0.5)))))
   (if (<= U_m 4.5e-139)
     t_1
     (if (<= U_m 7.8e-115)
       t_0
       (if (<= U_m 22000000000.0) t_1 (if (<= U_m 1.4e+180) t_0 (- U_m)))))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = -2.0 * (J * hypot(1.0, (0.5 * (U_m / J))));
	double t_1 = (-2.0 * J) * cos((K * 0.5));
	double tmp;
	if (U_m <= 4.5e-139) {
		tmp = t_1;
	} else if (U_m <= 7.8e-115) {
		tmp = t_0;
	} else if (U_m <= 22000000000.0) {
		tmp = t_1;
	} else if (U_m <= 1.4e+180) {
		tmp = t_0;
	} 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 = -2.0 * (J * Math.hypot(1.0, (0.5 * (U_m / J))));
	double t_1 = (-2.0 * J) * Math.cos((K * 0.5));
	double tmp;
	if (U_m <= 4.5e-139) {
		tmp = t_1;
	} else if (U_m <= 7.8e-115) {
		tmp = t_0;
	} else if (U_m <= 22000000000.0) {
		tmp = t_1;
	} else if (U_m <= 1.4e+180) {
		tmp = t_0;
	} else {
		tmp = -U_m;
	}
	return tmp;
}

Python

U_m = math.fabs(U)
def code(J, K, U_m):
	t_0 = -2.0 * (J * math.hypot(1.0, (0.5 * (U_m / J))))
	t_1 = (-2.0 * J) * math.cos((K * 0.5))
	tmp = 0
	if U_m <= 4.5e-139:
		tmp = t_1
	elif U_m <= 7.8e-115:
		tmp = t_0
	elif U_m <= 22000000000.0:
		tmp = t_1
	elif U_m <= 1.4e+180:
		tmp = t_0
	else:
		tmp = -U_m
	return tmp

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = Float64(-2.0 * Float64(J * hypot(1.0, Float64(0.5 * Float64(U_m / J)))))
	t_1 = Float64(Float64(-2.0 * J) * cos(Float64(K * 0.5)))
	tmp = 0.0
	if (U_m <= 4.5e-139)
		tmp = t_1;
	elseif (U_m <= 7.8e-115)
		tmp = t_0;
	elseif (U_m <= 22000000000.0)
		tmp = t_1;
	elseif (U_m <= 1.4e+180)
		tmp = t_0;
	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 * (J * hypot(1.0, (0.5 * (U_m / J))));
	t_1 = (-2.0 * J) * cos((K * 0.5));
	tmp = 0.0;
	if (U_m <= 4.5e-139)
		tmp = t_1;
	elseif (U_m <= 7.8e-115)
		tmp = t_0;
	elseif (U_m <= 22000000000.0)
		tmp = t_1;
	elseif (U_m <= 1.4e+180)
		tmp = t_0;
	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[(J * N[Sqrt[1.0 ^ 2 + N[(0.5 * N[(U$95$m / J), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(-2.0 * J), $MachinePrecision] * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[U$95$m, 4.5e-139], t$95$1, If[LessEqual[U$95$m, 7.8e-115], t$95$0, If[LessEqual[U$95$m, 22000000000.0], t$95$1, If[LessEqual[U$95$m, 1.4e+180], t$95$0, (-U$95$m)]]]]]]

Derivation

1. Split input into 3 regimes

2. if U < 4.50000000000000023e-139 or 7.7999999999999997e-115 < U < 2.2e10

   1. Initial program 83.3%

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

   3. Simplified 92.6%

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

   5. Taylor expanded in J around inf 65.5%

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

      1. associate-*r* 65.5%

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

      2. *-commutative 65.5%

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

      3. *-commutative 65.5%

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

      4. *-commutative 65.5%

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

      5. *-commutative 65.5%

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

      6. *-commutative 65.5%

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

   7. Simplified 65.5%

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

   if 4.50000000000000023e-139 < U < 7.7999999999999997e-115 or 2.2e10 < U < 1.40000000000000006e180

   1. Initial program 62.9%

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

   3. Simplified 85.2%

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

   6. Applied egg-rr 83.4%

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

      1. rem-cube-cbrt 85.0%

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

      2. *-commutative 85.0%

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

      3. add-sqr-sqrt 38.6%

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

      4. associate-*r* 38.6%

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

   8. Applied egg-rr 38.6%

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

   9. Taylor expanded in K around 0 40.1%

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

       1. metadata-eval 40.1%

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

       2. metadata-eval 40.1%

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

       3. unpow2 40.1%

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

       4. unpow2 40.1%

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

       5. times-frac 50.2%

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

       6. swap-sqr 50.2%

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

       7. hypot-undefine 72.5%

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

   11. Simplified 72.5%

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

   if 1.40000000000000006e180 < U

   1. Initial program 37.2%

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

   3. Simplified 61.9%

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

   5. Taylor expanded in J around 0 47.9%

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

      1. neg-mul-1 47.9%

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

   7. Simplified 47.9%

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

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 4.5 \cdot 10^{-139}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{elif}\;U \leq 7.8 \cdot 10^{-115}:\\ \;\;\;\;-2 \cdot \left(J \cdot \mathsf{hypot}\left(1, 0.5 \cdot \frac{U}{J}\right)\right)\\ \mathbf{elif}\;U \leq 22000000000:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{elif}\;U \leq 1.4 \cdot 10^{+180}:\\ \;\;\;\;-2 \cdot \left(J \cdot \mathsf{hypot}\left(1, 0.5 \cdot \frac{U}{J}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 56.8% accurate, 3.4× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{if}\;U\_m \leq 1.65 \cdot 10^{+27}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;U\_m \leq 10^{+121}:\\ \;\;\;\;\frac{J \cdot U\_m}{-J}\\ \mathbf{elif}\;U\_m \leq 9 \cdot 10^{+158}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;-U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (* (* -2.0 J) (cos (* K 0.5)))))
   (if (<= U_m 1.65e+27)
     t_0
     (if (<= U_m 1e+121)
       (/ (* J U_m) (- J))
       (if (<= U_m 9e+158) t_0 (- U_m))))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = (-2.0 * J) * cos((K * 0.5));
	double tmp;
	if (U_m <= 1.65e+27) {
		tmp = t_0;
	} else if (U_m <= 1e+121) {
		tmp = (J * U_m) / -J;
	} else if (U_m <= 9e+158) {
		tmp = t_0;
	} 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) :: tmp
    t_0 = ((-2.0d0) * j) * cos((k * 0.5d0))
    if (u_m <= 1.65d+27) then
        tmp = t_0
    else if (u_m <= 1d+121) then
        tmp = (j * u_m) / -j
    else if (u_m <= 9d+158) then
        tmp = t_0
    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 * J) * Math.cos((K * 0.5));
	double tmp;
	if (U_m <= 1.65e+27) {
		tmp = t_0;
	} else if (U_m <= 1e+121) {
		tmp = (J * U_m) / -J;
	} else if (U_m <= 9e+158) {
		tmp = t_0;
	} else {
		tmp = -U_m;
	}
	return tmp;
}

Python

U_m = math.fabs(U)
def code(J, K, U_m):
	t_0 = (-2.0 * J) * math.cos((K * 0.5))
	tmp = 0
	if U_m <= 1.65e+27:
		tmp = t_0
	elif U_m <= 1e+121:
		tmp = (J * U_m) / -J
	elif U_m <= 9e+158:
		tmp = t_0
	else:
		tmp = -U_m
	return tmp

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = Float64(Float64(-2.0 * J) * cos(Float64(K * 0.5)))
	tmp = 0.0
	if (U_m <= 1.65e+27)
		tmp = t_0;
	elseif (U_m <= 1e+121)
		tmp = Float64(Float64(J * U_m) / Float64(-J));
	elseif (U_m <= 9e+158)
		tmp = t_0;
	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 * J) * cos((K * 0.5));
	tmp = 0.0;
	if (U_m <= 1.65e+27)
		tmp = t_0;
	elseif (U_m <= 1e+121)
		tmp = (J * U_m) / -J;
	elseif (U_m <= 9e+158)
		tmp = t_0;
	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[(N[(-2.0 * J), $MachinePrecision] * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[U$95$m, 1.65e+27], t$95$0, If[LessEqual[U$95$m, 1e+121], N[(N[(J * U$95$m), $MachinePrecision] / (-J)), $MachinePrecision], If[LessEqual[U$95$m, 9e+158], t$95$0, (-U$95$m)]]]]

Derivation

1. Split input into 3 regimes

2. if U < 1.6499999999999999e27 or 1.00000000000000004e121 < U < 9.00000000000000092e158

   1. Initial program 82.6%

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

   3. Simplified 92.2%

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

   5. Taylor expanded in J around inf 64.5%

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

      1. associate-*r* 64.5%

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

      2. *-commutative 64.5%

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

      3. *-commutative 64.5%

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

      4. *-commutative 64.5%

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

      5. *-commutative 64.5%

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

      6. *-commutative 64.5%

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

   7. Simplified 64.5%

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

   if 1.6499999999999999e27 < U < 1.00000000000000004e121

   1. Initial program 59.3%

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

   3. Simplified 84.7%

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

   5. Taylor expanded in U around inf 23.2%

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

      1. associate-*r/ 23.2%

         \[\leadsto J \cdot \color{blue}{\frac{-1 \cdot U}{J}} \]

      2. neg-mul-1 23.2%

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

   7. Simplified 23.2%

      \[\leadsto J \cdot \color{blue}{\frac{-U}{J}} \]
   8. Step-by-step derivation

      1. distribute-frac-neg 23.2%

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

      2. distribute-frac-neg2 23.2%

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

      3. associate-*r/ 28.2%

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

   9. Applied egg-rr 28.2%

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

   if 9.00000000000000092e158 < U

   1. Initial program 38.0%

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

   3. Simplified 64.3%

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

   5. Taylor expanded in J around 0 41.9%

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

      1. neg-mul-1 41.9%

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

   7. Simplified 41.9%

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

4. Final simplification 58.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 1.65 \cdot 10^{+27}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{elif}\;U \leq 10^{+121}:\\ \;\;\;\;\frac{J \cdot U}{-J}\\ \mathbf{elif}\;U \leq 9 \cdot 10^{+158}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 39.8% accurate, 52.4× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} \mathbf{if}\;U\_m \leq 1.7 \cdot 10^{+26}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{else}:\\ \;\;\;\;-U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m) :precision binary64 (if (<= U_m 1.7e+26) (* -2.0 J) (- U_m)))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double tmp;
	if (U_m <= 1.7e+26) {
		tmp = -2.0 * J;
	} else {
		tmp = -U_m;
	}
	return tmp;
}

Fortran

U_m = abs(u)
real(8) function code(j, k, u_m)
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u_m
    real(8) :: tmp
    if (u_m <= 1.7d+26) then
        tmp = (-2.0d0) * j
    else
        tmp = -u_m
    end if
    code = tmp
end function

Java

U_m = Math.abs(U);
public static double code(double J, double K, double U_m) {
	double tmp;
	if (U_m <= 1.7e+26) {
		tmp = -2.0 * J;
	} else {
		tmp = -U_m;
	}
	return tmp;
}

Python

U_m = math.fabs(U)
def code(J, K, U_m):
	tmp = 0
	if U_m <= 1.7e+26:
		tmp = -2.0 * J
	else:
		tmp = -U_m
	return tmp

Julia

U_m = abs(U)
function code(J, K, U_m)
	tmp = 0.0
	if (U_m <= 1.7e+26)
		tmp = Float64(-2.0 * J);
	else
		tmp = Float64(-U_m);
	end
	return tmp
end

MATLAB

U_m = abs(U);
function tmp_2 = code(J, K, U_m)
	tmp = 0.0;
	if (U_m <= 1.7e+26)
		tmp = -2.0 * J;
	else
		tmp = -U_m;
	end
	tmp_2 = tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := If[LessEqual[U$95$m, 1.7e+26], N[(-2.0 * J), $MachinePrecision], (-U$95$m)]

Derivation

1. Split input into 2 regimes

2. if U < 1.7000000000000001e26

   1. Initial program 83.1%

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

   3. Simplified 93.0%

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

   6. Applied egg-rr 91.2%

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

   7. Taylor expanded in J around inf 63.3%

      \[\leadsto {\color{blue}{\left(\sqrt[3]{J \cdot \cos \left(0.5 \cdot K\right)} \cdot \sqrt[3]{-2}\right)}}^{3} \]

   8. Taylor expanded in K around 0 38.3%

      \[\leadsto \color{blue}{J \cdot {\left(\sqrt[3]{-2}\right)}^{3}} \]
   9. Step-by-step derivation

      1. rem-cube-cbrt 39.5%

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

   10. Simplified 39.5%

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

   if 1.7000000000000001e26 < U

   1. Initial program 47.1%

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

   3. Simplified 70.7%

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

   5. Taylor expanded in J around 0 34.2%

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

      1. neg-mul-1 34.2%

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

   7. Simplified 34.2%

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

4. Final simplification 38.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 1.7 \cdot 10^{+26}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 26.6% 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 74.1%

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

3. Simplified 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 J around 0 25.7%

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

   1. neg-mul-1 25.7%

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

7. Simplified 25.7%

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

8. Final simplification 25.7%

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

Alternative 9: 26.6% 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 74.1%

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

3. Simplified 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 25.2%

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

6. Final simplification 25.2%

   \[\leadsto U \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024073 
(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)))))