Maksimov and Kolovsky, Equation (3)

Percentage Accurate: 73.3% → 99.8%

Time: 22.9s

Alternatives: 14

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: 73.3% 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: 99.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_0 \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq 10^{+308}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \]

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+308) 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+308) {
		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+308) {
		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+308:
		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+308)
		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+308)
		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+308], 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 5.5%

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

   3. Simplified 65.3%

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

   5. Taylor expanded in J around 0 50.3%

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

      1. neg-mul-1 50.3%

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

   7. Simplified 50.3%

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

   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))))) < 1e308

   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 1e308 < (*.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 64.2%

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

   5. Taylor expanded in U around -inf 59.0%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 10^{+308}:\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 68.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := J \cdot \left(\left(-2 \cdot t\_0\right) \cdot \left(1 + \left(\frac{U\_m}{J} \cdot \frac{U\_m}{J}\right) \cdot 0.125\right)\right)\\ \mathbf{if}\;t\_0 \leq -0.84:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq -0.75:\\ \;\;\;\;U\_m\\ \mathbf{elif}\;t\_0 \leq -0.4:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{elif}\;t\_0 \leq -0.246:\\ \;\;\;\;U\_m\\ \mathbf{elif}\;t\_0 \leq 0.77:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(-2 \cdot \mathsf{hypot}\left(1, U\_m \cdot \frac{0.5}{J}\right)\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1 (* J (* (* -2.0 t_0) (+ 1.0 (* (* (/ U_m J) (/ U_m J)) 0.125))))))
   (if (<= t_0 -0.84)
     t_1
     (if (<= t_0 -0.75)
       U_m
       (if (<= t_0 -0.4)
         (* (* -2.0 J) (cos (* K 0.5)))
         (if (<= t_0 -0.246)
           U_m
           (if (<= t_0 0.77)
             t_1
             (* J (* -2.0 (hypot 1.0 (* U_m (/ 0.5 J))))))))))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = J * ((-2.0 * t_0) * (1.0 + (((U_m / J) * (U_m / J)) * 0.125)));
	double tmp;
	if (t_0 <= -0.84) {
		tmp = t_1;
	} else if (t_0 <= -0.75) {
		tmp = U_m;
	} else if (t_0 <= -0.4) {
		tmp = (-2.0 * J) * cos((K * 0.5));
	} else if (t_0 <= -0.246) {
		tmp = U_m;
	} else if (t_0 <= 0.77) {
		tmp = t_1;
	} else {
		tmp = J * (-2.0 * hypot(1.0, (U_m * (0.5 / J))));
	}
	return tmp;
}

Java

U_m = Math.abs(U);
public static double code(double J, double K, double U_m) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = J * ((-2.0 * t_0) * (1.0 + (((U_m / J) * (U_m / J)) * 0.125)));
	double tmp;
	if (t_0 <= -0.84) {
		tmp = t_1;
	} else if (t_0 <= -0.75) {
		tmp = U_m;
	} else if (t_0 <= -0.4) {
		tmp = (-2.0 * J) * Math.cos((K * 0.5));
	} else if (t_0 <= -0.246) {
		tmp = U_m;
	} else if (t_0 <= 0.77) {
		tmp = t_1;
	} else {
		tmp = J * (-2.0 * Math.hypot(1.0, (U_m * (0.5 / J))));
	}
	return tmp;
}

Python

U_m = math.fabs(U)
def code(J, K, U_m):
	t_0 = math.cos((K / 2.0))
	t_1 = J * ((-2.0 * t_0) * (1.0 + (((U_m / J) * (U_m / J)) * 0.125)))
	tmp = 0
	if t_0 <= -0.84:
		tmp = t_1
	elif t_0 <= -0.75:
		tmp = U_m
	elif t_0 <= -0.4:
		tmp = (-2.0 * J) * math.cos((K * 0.5))
	elif t_0 <= -0.246:
		tmp = U_m
	elif t_0 <= 0.77:
		tmp = t_1
	else:
		tmp = J * (-2.0 * math.hypot(1.0, (U_m * (0.5 / J))))
	return tmp

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(J * Float64(Float64(-2.0 * t_0) * Float64(1.0 + Float64(Float64(Float64(U_m / J) * Float64(U_m / J)) * 0.125))))
	tmp = 0.0
	if (t_0 <= -0.84)
		tmp = t_1;
	elseif (t_0 <= -0.75)
		tmp = U_m;
	elseif (t_0 <= -0.4)
		tmp = Float64(Float64(-2.0 * J) * cos(Float64(K * 0.5)));
	elseif (t_0 <= -0.246)
		tmp = U_m;
	elseif (t_0 <= 0.77)
		tmp = t_1;
	else
		tmp = Float64(J * Float64(-2.0 * hypot(1.0, Float64(U_m * Float64(0.5 / J)))));
	end
	return tmp
end

MATLAB

U_m = abs(U);
function tmp_2 = code(J, K, U_m)
	t_0 = cos((K / 2.0));
	t_1 = J * ((-2.0 * t_0) * (1.0 + (((U_m / J) * (U_m / J)) * 0.125)));
	tmp = 0.0;
	if (t_0 <= -0.84)
		tmp = t_1;
	elseif (t_0 <= -0.75)
		tmp = U_m;
	elseif (t_0 <= -0.4)
		tmp = (-2.0 * J) * cos((K * 0.5));
	elseif (t_0 <= -0.246)
		tmp = U_m;
	elseif (t_0 <= 0.77)
		tmp = t_1;
	else
		tmp = J * (-2.0 * hypot(1.0, (U_m * (0.5 / J))));
	end
	tmp_2 = tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(J * N[(N[(-2.0 * t$95$0), $MachinePrecision] * N[(1.0 + N[(N[(N[(U$95$m / J), $MachinePrecision] * N[(U$95$m / J), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.84], t$95$1, If[LessEqual[t$95$0, -0.75], U$95$m, If[LessEqual[t$95$0, -0.4], N[(N[(-2.0 * J), $MachinePrecision] * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.246], U$95$m, If[LessEqual[t$95$0, 0.77], t$95$1, N[(J * N[(-2.0 * N[Sqrt[1.0 ^ 2 + N[(U$95$m * N[(0.5 / J), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.839999999999999969 or -0.246 < (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.77000000000000002

   1. Initial program 82.0%

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

   3. Simplified 94.2%

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

   5. Taylor expanded in K around 0 67.5%

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

   6. Taylor expanded in U around 0 59.6%

      \[\leadsto J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\left(1 + 0.125 \cdot \frac{{U}^{2}}{{J}^{2}}\right)}\right) \]
   7. Step-by-step derivation

      1. *-commutative 59.6%

         \[\leadsto J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(1 + \color{blue}{\frac{{U}^{2}}{{J}^{2}} \cdot 0.125}\right)\right) \]

   8. Simplified 59.6%

      \[\leadsto J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\left(1 + \frac{{U}^{2}}{{J}^{2}} \cdot 0.125\right)}\right) \]
   9. Step-by-step derivation

      1. add-sqr-sqrt 59.6%

         \[\leadsto J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(1 + \color{blue}{\left(\sqrt{\frac{{U}^{2}}{{J}^{2}}} \cdot \sqrt{\frac{{U}^{2}}{{J}^{2}}}\right)} \cdot 0.125\right)\right) \]

      2. pow2 59.6%

         \[\leadsto J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(1 + \color{blue}{{\left(\sqrt{\frac{{U}^{2}}{{J}^{2}}}\right)}^{2}} \cdot 0.125\right)\right) \]

      3. sqrt-div 59.6%

         \[\leadsto J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(1 + {\color{blue}{\left(\frac{\sqrt{{U}^{2}}}{\sqrt{{J}^{2}}}\right)}}^{2} \cdot 0.125\right)\right) \]

      4. sqrt-pow1 61.1%

         \[\leadsto J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(1 + {\left(\frac{\color{blue}{{U}^{\left(\frac{2}{2}\right)}}}{\sqrt{{J}^{2}}}\right)}^{2} \cdot 0.125\right)\right) \]

      5. metadata-eval 61.1%

         \[\leadsto J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(1 + {\left(\frac{{U}^{\color{blue}{1}}}{\sqrt{{J}^{2}}}\right)}^{2} \cdot 0.125\right)\right) \]

      6. pow1 61.1%

         \[\leadsto J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(1 + {\left(\frac{\color{blue}{U}}{\sqrt{{J}^{2}}}\right)}^{2} \cdot 0.125\right)\right) \]

      7. unpow2 61.1%

         \[\leadsto J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(1 + {\left(\frac{U}{\sqrt{\color{blue}{J \cdot J}}}\right)}^{2} \cdot 0.125\right)\right) \]

      8. sqrt-prod 27.6%

         \[\leadsto J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(1 + {\left(\frac{U}{\color{blue}{\sqrt{J} \cdot \sqrt{J}}}\right)}^{2} \cdot 0.125\right)\right) \]

      9. add-sqr-sqrt 62.7%

         \[\leadsto J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(1 + {\left(\frac{U}{\color{blue}{J}}\right)}^{2} \cdot 0.125\right)\right) \]

      10. pow2 62.7%

          \[\leadsto J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(1 + \color{blue}{\left(\frac{U}{J} \cdot \frac{U}{J}\right)} \cdot 0.125\right)\right) \]

   10. Applied egg-rr 62.7%

       \[\leadsto J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(1 + \color{blue}{\left(\frac{U}{J} \cdot \frac{U}{J}\right)} \cdot 0.125\right)\right) \]

   if -0.839999999999999969 < (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.75 or -0.40000000000000002 < (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.246

   1. Initial program 45.3%

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

   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 U around -inf 65.3%

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

   if -0.75 < (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.40000000000000002

   1. Initial program 94.2%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in J around inf 83.8%

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

      1. associate-*r* 83.8%

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

      2. *-commutative 83.8%

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

      3. *-commutative 83.8%

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

      4. *-commutative 83.8%

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

      5. *-commutative 83.8%

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

      6. *-commutative 83.8%

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

   7. Simplified 83.8%

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

   if 0.77000000000000002 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 77.4%

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

   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 K around 0 82.5%

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

   6. Taylor expanded in K around 0 51.8%

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

      1. metadata-eval 51.8%

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

      2. metadata-eval 51.8%

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

      3. unpow2 51.8%

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

      4. unpow2 51.8%

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

      5. times-frac 74.3%

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

      6. swap-sqr 74.3%

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

      7. associate-*r/ 74.3%

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

      8. *-commutative 74.3%

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

      9. associate-*r/ 74.2%

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

      10. associate-*r/ 74.2%

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

      11. *-commutative 74.2%

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

      12. associate-*r/ 74.2%

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

      13. hypot-undefine 86.3%

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

   8. Simplified 86.3%

      \[\leadsto J \cdot \color{blue}{\left(-2 \cdot \mathsf{hypot}\left(1, U \cdot \frac{0.5}{J}\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.84:\\ \;\;\;\;J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(1 + \left(\frac{U}{J} \cdot \frac{U}{J}\right) \cdot 0.125\right)\right)\\ \mathbf{elif}\;\cos \left(\frac{K}{2}\right) \leq -0.75:\\ \;\;\;\;U\\ \mathbf{elif}\;\cos \left(\frac{K}{2}\right) \leq -0.4:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{elif}\;\cos \left(\frac{K}{2}\right) \leq -0.246:\\ \;\;\;\;U\\ \mathbf{elif}\;\cos \left(\frac{K}{2}\right) \leq 0.77:\\ \;\;\;\;J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(1 + \left(\frac{U}{J} \cdot \frac{U}{J}\right) \cdot 0.125\right)\right)\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(-2 \cdot \mathsf{hypot}\left(1, U \cdot \frac{0.5}{J}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.2% accurate, 1.3× speedup

Math

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

FPCore

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

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = cos((K * 0.5));
	return t_0 * ((-2.0 * J) * hypot(1.0, (U_m / (t_0 * (J * 2.0)))));
}

Java

U_m = Math.abs(U);
public static double code(double J, double K, double U_m) {
	double t_0 = Math.cos((K * 0.5));
	return t_0 * ((-2.0 * J) * Math.hypot(1.0, (U_m / (t_0 * (J * 2.0)))));
}

Python

U_m = math.fabs(U)
def code(J, K, U_m):
	t_0 = math.cos((K * 0.5))
	return t_0 * ((-2.0 * J) * math.hypot(1.0, (U_m / (t_0 * (J * 2.0)))))

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(K * 0.5))
	return Float64(t_0 * Float64(Float64(-2.0 * J) * hypot(1.0, Float64(U_m / Float64(t_0 * Float64(J * 2.0))))))
end

MATLAB

U_m = abs(U);
function tmp = code(J, K, U_m)
	t_0 = cos((K * 0.5));
	tmp = t_0 * ((-2.0 * J) * hypot(1.0, (U_m / (t_0 * (J * 2.0)))));
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]}, N[(t$95$0 * N[(N[(-2.0 * J), $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[(U$95$m / N[(t$95$0 * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 78.1%

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

3. Simplified 91.7%

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

6. Applied egg-rr 46.0%

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

   1. unpow2 46.0%

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

   2. add-sqr-sqrt 91.6%

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

   3. associate-*l* 91.6%

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

   4. metadata-eval 91.6%

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

   5. div-inv 91.6%

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

   6. associate-*r* 91.6%

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

   7. *-commutative 91.6%

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

   8. associate-*l* 91.6%

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

   9. *-commutative 91.6%

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

   10. associate-*r* 91.6%

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

8. Applied egg-rr 91.7%

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

9. Final simplification 91.7%

   \[\leadsto \cos \left(K \cdot 0.5\right) \cdot \left(\left(-2 \cdot J\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot 2\right)}\right)\right) \]
10. Add Preprocessing

Alternative 4: 88.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \left(J \cdot \left(-2 \cdot t\_0\right)\right) \cdot \mathsf{hypot}\left(1, \frac{\frac{U\_m}{J \cdot 2}}{t\_0}\right) \end{array} \end{array} \]

FPCore

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

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	return (J * (-2.0 * t_0)) * hypot(1.0, ((U_m / (J * 2.0)) / t_0));
}

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));
	return (J * (-2.0 * t_0)) * Math.hypot(1.0, ((U_m / (J * 2.0)) / t_0));
}

Python

U_m = math.fabs(U)
def code(J, K, U_m):
	t_0 = math.cos((K / 2.0))
	return (J * (-2.0 * t_0)) * math.hypot(1.0, ((U_m / (J * 2.0)) / t_0))

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	return Float64(Float64(J * Float64(-2.0 * t_0)) * hypot(1.0, Float64(Float64(U_m / Float64(J * 2.0)) / t_0)))
end

MATLAB

U_m = abs(U);
function tmp = code(J, K, U_m)
	t_0 = cos((K / 2.0));
	tmp = (J * (-2.0 * t_0)) * hypot(1.0, ((U_m / (J * 2.0)) / t_0));
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]}, N[(N[(J * N[(-2.0 * t$95$0), $MachinePrecision]), $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[(N[(U$95$m / N[(J * 2.0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 78.1%

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

   1. *-commutative 78.1%

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

   2. associate-*l* 78.1%

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

   3. unpow2 78.1%

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

   4. hypot-1-def 91.8%

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

   5. associate-/r* 91.7%

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

   6. cos-neg 91.7%

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

   7. distribute-frac-neg 91.7%

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

   8. associate-/r* 91.8%

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

   9. associate-/r* 91.7%

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

   10. *-commutative 91.7%

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

   11. distribute-frac-neg 91.7%

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

   12. cos-neg 91.7%

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

3. Simplified 91.7%

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

Alternative 5: 88.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ J \cdot \left(\left(-2 \cdot t\_0\right) \cdot \mathsf{hypot}\left(1, \frac{\frac{U\_m}{2}}{J \cdot t\_0}\right)\right) \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (* 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));
	return J * ((-2.0 * t_0) * hypot(1.0, ((U_m / 2.0) / (J * t_0))));
}

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));
	return J * ((-2.0 * t_0) * Math.hypot(1.0, ((U_m / 2.0) / (J * t_0))));
}

Python

U_m = math.fabs(U)
def code(J, K, U_m):
	t_0 = math.cos((K / 2.0))
	return J * ((-2.0 * t_0) * math.hypot(1.0, ((U_m / 2.0) / (J * t_0))))

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	return Float64(J * Float64(Float64(-2.0 * t_0) * hypot(1.0, Float64(Float64(U_m / 2.0) / Float64(J * t_0)))))
end

MATLAB

U_m = abs(U);
function tmp = code(J, K, U_m)
	t_0 = cos((K / 2.0));
	tmp = J * ((-2.0 * t_0) * hypot(1.0, ((U_m / 2.0) / (J * t_0))));
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]}, 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. Initial program 78.1%

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

3. Simplified 91.7%

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

Alternative 6: 45.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} \mathbf{if}\;J \leq 1.9 \cdot 10^{-124}:\\ \;\;\;\;\frac{-2 \cdot {\left(J \cdot \cos \left(K \cdot 0.5\right)\right)}^{2}}{U\_m} - U\_m\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{\frac{U\_m}{2}}{J}\right)\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (if (<= J 1.9e-124)
   (- (/ (* -2.0 (pow (* J (cos (* K 0.5))) 2.0)) U_m) U_m)
   (* J (* (* -2.0 (cos (/ K 2.0))) (hypot 1.0 (/ (/ U_m 2.0) J))))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double tmp;
	if (J <= 1.9e-124) {
		tmp = ((-2.0 * pow((J * cos((K * 0.5))), 2.0)) / U_m) - U_m;
	} else {
		tmp = J * ((-2.0 * cos((K / 2.0))) * hypot(1.0, ((U_m / 2.0) / J)));
	}
	return tmp;
}

Java

U_m = Math.abs(U);
public static double code(double J, double K, double U_m) {
	double tmp;
	if (J <= 1.9e-124) {
		tmp = ((-2.0 * Math.pow((J * Math.cos((K * 0.5))), 2.0)) / U_m) - U_m;
	} else {
		tmp = J * ((-2.0 * Math.cos((K / 2.0))) * Math.hypot(1.0, ((U_m / 2.0) / J)));
	}
	return tmp;
}

Python

U_m = math.fabs(U)
def code(J, K, U_m):
	tmp = 0
	if J <= 1.9e-124:
		tmp = ((-2.0 * math.pow((J * math.cos((K * 0.5))), 2.0)) / U_m) - U_m
	else:
		tmp = J * ((-2.0 * math.cos((K / 2.0))) * math.hypot(1.0, ((U_m / 2.0) / J)))
	return tmp

Julia

U_m = abs(U)
function code(J, K, U_m)
	tmp = 0.0
	if (J <= 1.9e-124)
		tmp = Float64(Float64(Float64(-2.0 * (Float64(J * cos(Float64(K * 0.5))) ^ 2.0)) / U_m) - U_m);
	else
		tmp = Float64(J * Float64(Float64(-2.0 * cos(Float64(K / 2.0))) * hypot(1.0, Float64(Float64(U_m / 2.0) / J))));
	end
	return tmp
end

MATLAB

U_m = abs(U);
function tmp_2 = code(J, K, U_m)
	tmp = 0.0;
	if (J <= 1.9e-124)
		tmp = ((-2.0 * ((J * cos((K * 0.5))) ^ 2.0)) / U_m) - U_m;
	else
		tmp = J * ((-2.0 * cos((K / 2.0))) * hypot(1.0, ((U_m / 2.0) / J)));
	end
	tmp_2 = tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := If[LessEqual[J, 1.9e-124], N[(N[(N[(-2.0 * N[Power[N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / U$95$m), $MachinePrecision] - U$95$m), $MachinePrecision], 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]]

Derivation

1. Split input into 2 regimes

2. if J < 1.90000000000000006e-124

   1. Initial program 72.5%

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

   3. Simplified 88.1%

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

   5. Taylor expanded in J around 0 25.7%

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

      1. neg-mul-1 25.7%

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

      2. unsub-neg 25.7%

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

      3. associate-*r/ 25.7%

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

      4. unpow2 25.7%

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

      5. *-commutative 25.7%

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

      6. unpow2 25.7%

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

      7. swap-sqr 25.7%

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

      8. unpow2 25.7%

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

      9. *-commutative 25.7%

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

   7. Simplified 25.7%

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

   if 1.90000000000000006e-124 < J

   1. Initial program 90.4%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in K around 0 81.1%

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

4. Final simplification 43.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq 1.9 \cdot 10^{-124}:\\ \;\;\;\;\frac{-2 \cdot {\left(J \cdot \cos \left(K \cdot 0.5\right)\right)}^{2}}{U} - U\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{\frac{U}{2}}{J}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 58.7% accurate, 3.4× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{if}\;U\_m \leq 2.4 \cdot 10^{-99}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;U\_m \leq 1.18 \cdot 10^{-92}:\\ \;\;\;\;J \cdot \left(\frac{J - U\_m}{J} + -1\right)\\ \mathbf{elif}\;U\_m \leq 1.52 \cdot 10^{+63}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;-U\_m\\ \end{array} \end{array} \]

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 2.4e-99)
     t_0
     (if (<= U_m 1.18e-92)
       (* J (+ (/ (- J U_m) J) -1.0))
       (if (<= U_m 1.52e+63) 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 <= 2.4e-99) {
		tmp = t_0;
	} else if (U_m <= 1.18e-92) {
		tmp = J * (((J - U_m) / J) + -1.0);
	} else if (U_m <= 1.52e+63) {
		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 <= 2.4d-99) then
        tmp = t_0
    else if (u_m <= 1.18d-92) then
        tmp = j * (((j - u_m) / j) + (-1.0d0))
    else if (u_m <= 1.52d+63) 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 <= 2.4e-99) {
		tmp = t_0;
	} else if (U_m <= 1.18e-92) {
		tmp = J * (((J - U_m) / J) + -1.0);
	} else if (U_m <= 1.52e+63) {
		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 <= 2.4e-99:
		tmp = t_0
	elif U_m <= 1.18e-92:
		tmp = J * (((J - U_m) / J) + -1.0)
	elif U_m <= 1.52e+63:
		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 <= 2.4e-99)
		tmp = t_0;
	elseif (U_m <= 1.18e-92)
		tmp = Float64(J * Float64(Float64(Float64(J - U_m) / J) + -1.0));
	elseif (U_m <= 1.52e+63)
		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 <= 2.4e-99)
		tmp = t_0;
	elseif (U_m <= 1.18e-92)
		tmp = J * (((J - U_m) / J) + -1.0);
	elseif (U_m <= 1.52e+63)
		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, 2.4e-99], t$95$0, If[LessEqual[U$95$m, 1.18e-92], N[(J * N[(N[(N[(J - U$95$m), $MachinePrecision] / J), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[U$95$m, 1.52e+63], t$95$0, (-U$95$m)]]]]

Derivation

1. Split input into 3 regimes

2. if U < 2.4e-99 or 1.18e-92 < U < 1.51999999999999993e63

   1. Initial program 84.7%

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

   3. Simplified 95.5%

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

   5. Taylor expanded in J around inf 63.2%

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

      1. associate-*r* 63.2%

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

      2. *-commutative 63.2%

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

      3. *-commutative 63.2%

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

      4. *-commutative 63.2%

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

      5. *-commutative 63.2%

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

      6. *-commutative 63.2%

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

   7. Simplified 63.2%

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

   if 2.4e-99 < U < 1.18e-92

   1. Initial program 67.5%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 0.0%

      \[\leadsto J \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(-2 \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \mathsf{hypot}\left(1, U \cdot \frac{\frac{0.5}{J}}{\cos \left(K \cdot 0.5\right)}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. expm1-undefine 0.0%

         \[\leadsto J \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\left(-2 \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \mathsf{hypot}\left(1, U \cdot \frac{\frac{0.5}{J}}{\cos \left(K \cdot 0.5\right)}\right)\right)} - 1\right)} \]

   8. Applied egg-rr 100.0%

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

   9. Taylor expanded in J around 0 67.5%

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

       1. neg-mul-1 67.5%

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

       2. unsub-neg 67.5%

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

   11. Simplified 67.5%

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

   if 1.51999999999999993e63 < U

   1. Initial program 52.3%

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

   3. Simplified 75.9%

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

   5. Taylor expanded in J around 0 32.0%

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

      1. neg-mul-1 32.0%

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

   7. Simplified 32.0%

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

4. Final simplification 57.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 2.4 \cdot 10^{-99}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{elif}\;U \leq 1.18 \cdot 10^{-92}:\\ \;\;\;\;J \cdot \left(\frac{J - U}{J} + -1\right)\\ \mathbf{elif}\;U \leq 1.52 \cdot 10^{+63}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 62.8% accurate, 3.7× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} \mathbf{if}\;K \leq 1.6:\\ \;\;\;\;J \cdot \left(-2 \cdot \mathsf{hypot}\left(1, U\_m \cdot \frac{0.5}{J}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (if (<= K 1.6)
   (* J (* -2.0 (hypot 1.0 (* U_m (/ 0.5 J)))))
   (* (* -2.0 J) (cos (* K 0.5)))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double tmp;
	if (K <= 1.6) {
		tmp = J * (-2.0 * hypot(1.0, (U_m * (0.5 / J))));
	} else {
		tmp = (-2.0 * J) * 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 <= 1.6) {
		tmp = J * (-2.0 * Math.hypot(1.0, (U_m * (0.5 / J))));
	} 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 K <= 1.6:
		tmp = J * (-2.0 * math.hypot(1.0, (U_m * (0.5 / J))))
	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 (K <= 1.6)
		tmp = Float64(J * Float64(-2.0 * hypot(1.0, Float64(U_m * Float64(0.5 / J)))));
	else
		tmp = Float64(Float64(-2.0 * J) * cos(Float64(K * 0.5)));
	end
	return tmp
end

MATLAB

U_m = abs(U);
function tmp_2 = code(J, K, U_m)
	tmp = 0.0;
	if (K <= 1.6)
		tmp = J * (-2.0 * hypot(1.0, (U_m * (0.5 / J))));
	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[K, 1.6], N[(J * N[(-2.0 * N[Sqrt[1.0 ^ 2 + N[(U$95$m * N[(0.5 / J), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 * J), $MachinePrecision] * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if K < 1.6000000000000001

   1. Initial program 80.0%

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

   3. Simplified 91.3%

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

   5. Taylor expanded in K around 0 81.7%

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

   6. Taylor expanded in K around 0 41.0%

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

      1. metadata-eval 41.0%

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

      2. metadata-eval 41.0%

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

      3. unpow2 41.0%

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

      4. unpow2 41.0%

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

      5. times-frac 57.1%

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

      6. swap-sqr 57.1%

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

      7. associate-*r/ 57.1%

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

      8. *-commutative 57.1%

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

      9. associate-*r/ 57.1%

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

      10. associate-*r/ 57.1%

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

      11. *-commutative 57.1%

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

      12. associate-*r/ 57.1%

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

      13. hypot-undefine 66.5%

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

   8. Simplified 66.5%

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

   if 1.6000000000000001 < K

   1. Initial program 71.6%

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

   3. Simplified 92.8%

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

   5. Taylor expanded in J around inf 46.4%

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

      1. associate-*r* 46.4%

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

      2. *-commutative 46.4%

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

      3. *-commutative 46.4%

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

      4. *-commutative 46.4%

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

      5. *-commutative 46.4%

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

      6. *-commutative 46.4%

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

   7. Simplified 46.4%

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

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;K \leq 1.6:\\ \;\;\;\;J \cdot \left(-2 \cdot \mathsf{hypot}\left(1, U \cdot \frac{0.5}{J}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 39.9% accurate, 18.2× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} \mathbf{if}\;U\_m \leq 1.15 \cdot 10^{-109}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;U\_m \leq 1.2 \cdot 10^{-92}:\\ \;\;\;\;J \cdot \left(U\_m \cdot \left(\frac{1}{U\_m} + \frac{-1}{J}\right) + -1\right)\\ \mathbf{elif}\;U\_m \leq 1.06 \cdot 10^{-58}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{else}:\\ \;\;\;\;-U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (if (<= U_m 1.15e-109)
   (* -2.0 J)
   (if (<= U_m 1.2e-92)
     (* J (+ (* U_m (+ (/ 1.0 U_m) (/ -1.0 J))) -1.0))
     (if (<= U_m 1.06e-58) (* -2.0 J) (- U_m)))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double tmp;
	if (U_m <= 1.15e-109) {
		tmp = -2.0 * J;
	} else if (U_m <= 1.2e-92) {
		tmp = J * ((U_m * ((1.0 / U_m) + (-1.0 / J))) + -1.0);
	} else if (U_m <= 1.06e-58) {
		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 <= 1.15d-109) then
        tmp = (-2.0d0) * j
    else if (u_m <= 1.2d-92) then
        tmp = j * ((u_m * ((1.0d0 / u_m) + ((-1.0d0) / j))) + (-1.0d0))
    else if (u_m <= 1.06d-58) 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 <= 1.15e-109) {
		tmp = -2.0 * J;
	} else if (U_m <= 1.2e-92) {
		tmp = J * ((U_m * ((1.0 / U_m) + (-1.0 / J))) + -1.0);
	} else if (U_m <= 1.06e-58) {
		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 <= 1.15e-109:
		tmp = -2.0 * J
	elif U_m <= 1.2e-92:
		tmp = J * ((U_m * ((1.0 / U_m) + (-1.0 / J))) + -1.0)
	elif U_m <= 1.06e-58:
		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 <= 1.15e-109)
		tmp = Float64(-2.0 * J);
	elseif (U_m <= 1.2e-92)
		tmp = Float64(J * Float64(Float64(U_m * Float64(Float64(1.0 / U_m) + Float64(-1.0 / J))) + -1.0));
	elseif (U_m <= 1.06e-58)
		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 <= 1.15e-109)
		tmp = -2.0 * J;
	elseif (U_m <= 1.2e-92)
		tmp = J * ((U_m * ((1.0 / U_m) + (-1.0 / J))) + -1.0);
	elseif (U_m <= 1.06e-58)
		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, 1.15e-109], N[(-2.0 * J), $MachinePrecision], If[LessEqual[U$95$m, 1.2e-92], N[(J * N[(N[(U$95$m * N[(N[(1.0 / U$95$m), $MachinePrecision] + N[(-1.0 / J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[U$95$m, 1.06e-58], N[(-2.0 * J), $MachinePrecision], (-U$95$m)]]]

Derivation

1. Split input into 3 regimes

2. if U < 1.1500000000000001e-109 or 1.2000000000000001e-92 < U < 1.0600000000000001e-58

   1. Initial program 84.5%

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

   3. Simplified 94.9%

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

   5. Taylor expanded in K around 0 81.3%

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

   6. Taylor expanded in K around 0 36.7%

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

      1. metadata-eval 36.7%

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

      2. metadata-eval 36.7%

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

      3. unpow2 36.7%

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

      4. unpow2 36.7%

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

      5. times-frac 49.5%

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

      6. swap-sqr 49.5%

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

      7. associate-*r/ 49.5%

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

      8. *-commutative 49.5%

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

      9. associate-*r/ 49.5%

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

      10. associate-*r/ 49.5%

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

      11. *-commutative 49.5%

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

      12. associate-*r/ 49.5%

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

      13. hypot-undefine 57.7%

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

   8. Simplified 57.7%

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

   9. Taylor expanded in J around inf 37.2%

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

   if 1.1500000000000001e-109 < U < 1.2000000000000001e-92

   1. Initial program 86.1%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 0.0%

      \[\leadsto J \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(-2 \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \mathsf{hypot}\left(1, U \cdot \frac{\frac{0.5}{J}}{\cos \left(K \cdot 0.5\right)}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. expm1-undefine 0.0%

         \[\leadsto J \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\left(-2 \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \mathsf{hypot}\left(1, U \cdot \frac{\frac{0.5}{J}}{\cos \left(K \cdot 0.5\right)}\right)\right)} - 1\right)} \]

   8. Applied egg-rr 100.0%

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

   9. Taylor expanded in U around inf 59.0%

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

   if 1.0600000000000001e-58 < U

   1. Initial program 62.8%

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

   3. Simplified 83.7%

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

   5. Taylor expanded in J around 0 32.1%

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

      1. neg-mul-1 32.1%

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

   7. Simplified 32.1%

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

4. Final simplification 36.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 1.15 \cdot 10^{-109}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;U \leq 1.2 \cdot 10^{-92}:\\ \;\;\;\;J \cdot \left(U \cdot \left(\frac{1}{U} + \frac{-1}{J}\right) + -1\right)\\ \mathbf{elif}\;U \leq 1.06 \cdot 10^{-58}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 39.9% accurate, 23.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} \mathbf{if}\;U\_m \leq 3.8 \cdot 10^{-109} \lor \neg \left(U\_m \leq 1.18 \cdot 10^{-92}\right) \land U\_m \leq 1.3 \cdot 10^{-58}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{else}:\\ \;\;\;\;-U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (if (or (<= U_m 3.8e-109) (and (not (<= U_m 1.18e-92)) (<= U_m 1.3e-58)))
   (* -2.0 J)
   (- U_m)))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double tmp;
	if ((U_m <= 3.8e-109) || (!(U_m <= 1.18e-92) && (U_m <= 1.3e-58))) {
		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 <= 3.8d-109) .or. (.not. (u_m <= 1.18d-92)) .and. (u_m <= 1.3d-58)) 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 <= 3.8e-109) || (!(U_m <= 1.18e-92) && (U_m <= 1.3e-58))) {
		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 <= 3.8e-109) or (not (U_m <= 1.18e-92) and (U_m <= 1.3e-58)):
		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 <= 3.8e-109) || (!(U_m <= 1.18e-92) && (U_m <= 1.3e-58)))
		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 <= 3.8e-109) || (~((U_m <= 1.18e-92)) && (U_m <= 1.3e-58)))
		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[Or[LessEqual[U$95$m, 3.8e-109], And[N[Not[LessEqual[U$95$m, 1.18e-92]], $MachinePrecision], LessEqual[U$95$m, 1.3e-58]]], N[(-2.0 * J), $MachinePrecision], (-U$95$m)]

Derivation

1. Split input into 2 regimes

2. if U < 3.80000000000000002e-109 or 1.18e-92 < U < 1.30000000000000003e-58

   1. Initial program 84.5%

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

   3. Simplified 94.9%

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

   5. Taylor expanded in K around 0 81.3%

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

   6. Taylor expanded in K around 0 36.7%

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

      1. metadata-eval 36.7%

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

      2. metadata-eval 36.7%

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

      3. unpow2 36.7%

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

      4. unpow2 36.7%

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

      5. times-frac 49.5%

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

      6. swap-sqr 49.5%

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

      7. associate-*r/ 49.5%

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

      8. *-commutative 49.5%

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

      9. associate-*r/ 49.5%

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

      10. associate-*r/ 49.5%

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

      11. *-commutative 49.5%

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

      12. associate-*r/ 49.5%

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

      13. hypot-undefine 57.7%

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

   8. Simplified 57.7%

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

   9. Taylor expanded in J around inf 37.2%

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

   if 3.80000000000000002e-109 < U < 1.18e-92 or 1.30000000000000003e-58 < U

   1. Initial program 64.8%

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

   3. Simplified 85.0%

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

   5. Taylor expanded in J around 0 34.3%

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

      1. neg-mul-1 34.3%

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

   7. Simplified 34.3%

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

4. Final simplification 36.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 3.8 \cdot 10^{-109} \lor \neg \left(U \leq 1.18 \cdot 10^{-92}\right) \land U \leq 1.3 \cdot 10^{-58}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 40.1% accurate, 23.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} \mathbf{if}\;U\_m \leq 2.5 \cdot 10^{-109}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;U\_m \leq 1.18 \cdot 10^{-92}:\\ \;\;\;\;J \cdot \left(-1 - \frac{U\_m}{J}\right)\\ \mathbf{elif}\;U\_m \leq 5 \cdot 10^{-59}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{else}:\\ \;\;\;\;-U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (if (<= U_m 2.5e-109)
   (* -2.0 J)
   (if (<= U_m 1.18e-92)
     (* J (- -1.0 (/ U_m J)))
     (if (<= U_m 5e-59) (* -2.0 J) (- U_m)))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double tmp;
	if (U_m <= 2.5e-109) {
		tmp = -2.0 * J;
	} else if (U_m <= 1.18e-92) {
		tmp = J * (-1.0 - (U_m / J));
	} else if (U_m <= 5e-59) {
		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 <= 2.5d-109) then
        tmp = (-2.0d0) * j
    else if (u_m <= 1.18d-92) then
        tmp = j * ((-1.0d0) - (u_m / j))
    else if (u_m <= 5d-59) 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 <= 2.5e-109) {
		tmp = -2.0 * J;
	} else if (U_m <= 1.18e-92) {
		tmp = J * (-1.0 - (U_m / J));
	} else if (U_m <= 5e-59) {
		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 <= 2.5e-109:
		tmp = -2.0 * J
	elif U_m <= 1.18e-92:
		tmp = J * (-1.0 - (U_m / J))
	elif U_m <= 5e-59:
		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 <= 2.5e-109)
		tmp = Float64(-2.0 * J);
	elseif (U_m <= 1.18e-92)
		tmp = Float64(J * Float64(-1.0 - Float64(U_m / J)));
	elseif (U_m <= 5e-59)
		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 <= 2.5e-109)
		tmp = -2.0 * J;
	elseif (U_m <= 1.18e-92)
		tmp = J * (-1.0 - (U_m / J));
	elseif (U_m <= 5e-59)
		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, 2.5e-109], N[(-2.0 * J), $MachinePrecision], If[LessEqual[U$95$m, 1.18e-92], N[(J * N[(-1.0 - N[(U$95$m / J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[U$95$m, 5e-59], N[(-2.0 * J), $MachinePrecision], (-U$95$m)]]]

Derivation

1. Split input into 3 regimes

2. if U < 2.5000000000000001e-109 or 1.18e-92 < U < 5.0000000000000001e-59

   1. Initial program 84.5%

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

   3. Simplified 94.9%

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

   5. Taylor expanded in K around 0 81.3%

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

   6. Taylor expanded in K around 0 36.7%

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

      1. metadata-eval 36.7%

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

      2. metadata-eval 36.7%

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

      3. unpow2 36.7%

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

      4. unpow2 36.7%

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

      5. times-frac 49.5%

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

      6. swap-sqr 49.5%

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

      7. associate-*r/ 49.5%

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

      8. *-commutative 49.5%

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

      9. associate-*r/ 49.5%

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

      10. associate-*r/ 49.5%

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

      11. *-commutative 49.5%

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

      12. associate-*r/ 49.5%

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

      13. hypot-undefine 57.7%

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

   8. Simplified 57.7%

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

   9. Taylor expanded in J around inf 37.2%

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

   if 2.5000000000000001e-109 < U < 1.18e-92

   1. Initial program 86.1%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 0.0%

      \[\leadsto J \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(-2 \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \mathsf{hypot}\left(1, U \cdot \frac{\frac{0.5}{J}}{\cos \left(K \cdot 0.5\right)}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. expm1-undefine 0.0%

         \[\leadsto J \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\left(-2 \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \mathsf{hypot}\left(1, U \cdot \frac{\frac{0.5}{J}}{\cos \left(K \cdot 0.5\right)}\right)\right)} - 1\right)} \]

   8. Applied egg-rr 100.0%

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

      1. expm1-log1p-u 28.6%

         \[\leadsto J \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 + -2 \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot 2\right)}\right)\right)\right)\right)} - 1\right) \]

      2. expm1-undefine 28.6%

         \[\leadsto J \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left(1 + -2 \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot 2\right)}\right)\right)\right)} - 1\right)} - 1\right) \]

   10. Applied egg-rr 100.0%

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

       1. +-commutative 100.0%

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

       2. rem-exp-log 28.6%

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

       3. +-commutative 28.6%

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

       4. log1p-undefine 28.6%

          \[\leadsto J \cdot \left(\left(e^{\color{blue}{\mathsf{log1p}\left(\mathsf{fma}\left(-2, \cos \left(K \cdot 0.5\right) \cdot \mathsf{hypot}\left(1, U \cdot \frac{0.5}{\cos \left(K \cdot 0.5\right) \cdot J}\right), 1\right)\right)}} - 1\right) - 1\right) \]

       5. expm1-undefine 28.6%

          \[\leadsto J \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(-2, \cos \left(K \cdot 0.5\right) \cdot \mathsf{hypot}\left(1, U \cdot \frac{0.5}{\cos \left(K \cdot 0.5\right) \cdot J}\right), 1\right)\right)\right)} - 1\right) \]

       6. fma-undefine 28.6%

          \[\leadsto J \cdot \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{-2 \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \mathsf{hypot}\left(1, U \cdot \frac{0.5}{\cos \left(K \cdot 0.5\right) \cdot J}\right)\right) + 1}\right)\right) - 1\right) \]

   12. Simplified 28.6%

       \[\leadsto J \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(-2 \cdot \cos \left(0.5 \cdot K\right), \mathsf{hypot}\left(1, 0.5 \cdot \frac{\frac{U}{J}}{\cos \left(0.5 \cdot K\right)}\right), 1\right)\right)\right)} - 1\right) \]

   13. Taylor expanded in U around inf 63.4%

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

       1. associate-*r/ 63.4%

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

       2. mul-1-neg 63.4%

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

   15. Simplified 63.4%

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

   if 5.0000000000000001e-59 < U

   1. Initial program 62.8%

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

   3. Simplified 83.7%

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

   5. Taylor expanded in J around 0 32.1%

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

      1. neg-mul-1 32.1%

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

   7. Simplified 32.1%

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

4. Final simplification 36.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 2.5 \cdot 10^{-109}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;U \leq 1.18 \cdot 10^{-92}:\\ \;\;\;\;J \cdot \left(-1 - \frac{U}{J}\right)\\ \mathbf{elif}\;U \leq 5 \cdot 10^{-59}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 39.8% accurate, 23.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} \mathbf{if}\;U\_m \leq 1.1 \cdot 10^{-109}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;U\_m \leq 1.18 \cdot 10^{-92}:\\ \;\;\;\;\frac{U\_m}{J} \cdot \left(-J\right)\\ \mathbf{elif}\;U\_m \leq 2.2 \cdot 10^{-59}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{else}:\\ \;\;\;\;-U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (if (<= U_m 1.1e-109)
   (* -2.0 J)
   (if (<= U_m 1.18e-92)
     (* (/ U_m J) (- J))
     (if (<= U_m 2.2e-59) (* -2.0 J) (- U_m)))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double tmp;
	if (U_m <= 1.1e-109) {
		tmp = -2.0 * J;
	} else if (U_m <= 1.18e-92) {
		tmp = (U_m / J) * -J;
	} else if (U_m <= 2.2e-59) {
		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 <= 1.1d-109) then
        tmp = (-2.0d0) * j
    else if (u_m <= 1.18d-92) then
        tmp = (u_m / j) * -j
    else if (u_m <= 2.2d-59) 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 <= 1.1e-109) {
		tmp = -2.0 * J;
	} else if (U_m <= 1.18e-92) {
		tmp = (U_m / J) * -J;
	} else if (U_m <= 2.2e-59) {
		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 <= 1.1e-109:
		tmp = -2.0 * J
	elif U_m <= 1.18e-92:
		tmp = (U_m / J) * -J
	elif U_m <= 2.2e-59:
		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 <= 1.1e-109)
		tmp = Float64(-2.0 * J);
	elseif (U_m <= 1.18e-92)
		tmp = Float64(Float64(U_m / J) * Float64(-J));
	elseif (U_m <= 2.2e-59)
		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 <= 1.1e-109)
		tmp = -2.0 * J;
	elseif (U_m <= 1.18e-92)
		tmp = (U_m / J) * -J;
	elseif (U_m <= 2.2e-59)
		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, 1.1e-109], N[(-2.0 * J), $MachinePrecision], If[LessEqual[U$95$m, 1.18e-92], N[(N[(U$95$m / J), $MachinePrecision] * (-J)), $MachinePrecision], If[LessEqual[U$95$m, 2.2e-59], N[(-2.0 * J), $MachinePrecision], (-U$95$m)]]]

Derivation

1. Split input into 3 regimes

2. if U < 1.1e-109 or 1.18e-92 < U < 2.1999999999999999e-59

   1. Initial program 84.5%

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

   3. Simplified 94.9%

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

   5. Taylor expanded in K around 0 81.3%

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

   6. Taylor expanded in K around 0 36.7%

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

      1. metadata-eval 36.7%

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

      2. metadata-eval 36.7%

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

      3. unpow2 36.7%

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

      4. unpow2 36.7%

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

      5. times-frac 49.5%

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

      6. swap-sqr 49.5%

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

      7. associate-*r/ 49.5%

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

      8. *-commutative 49.5%

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

      9. associate-*r/ 49.5%

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

      10. associate-*r/ 49.5%

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

      11. *-commutative 49.5%

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

      12. associate-*r/ 49.5%

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

      13. hypot-undefine 57.7%

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

   8. Simplified 57.7%

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

   9. Taylor expanded in J around inf 37.2%

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

   if 1.1e-109 < U < 1.18e-92

   1. Initial program 86.1%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in U around inf 57.9%

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

      1. associate-*r/ 57.9%

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

      2. neg-mul-1 57.9%

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

   7. Simplified 57.9%

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

   if 2.1999999999999999e-59 < U

   1. Initial program 62.8%

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

   3. Simplified 83.7%

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

   5. Taylor expanded in J around 0 32.1%

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

      1. neg-mul-1 32.1%

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

   7. Simplified 32.1%

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

4. Final simplification 36.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 1.1 \cdot 10^{-109}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;U \leq 1.18 \cdot 10^{-92}:\\ \;\;\;\;\frac{U}{J} \cdot \left(-J\right)\\ \mathbf{elif}\;U \leq 2.2 \cdot 10^{-59}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 26.8% accurate, 59.9× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} \mathbf{if}\;K \leq 3.4 \cdot 10^{+26}:\\ \;\;\;\;-U\_m\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m) :precision binary64 (if (<= K 3.4e+26) (- U_m) U_m))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double tmp;
	if (K <= 3.4e+26) {
		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 <= 3.4d+26) 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 <= 3.4e+26) {
		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 <= 3.4e+26:
		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 <= 3.4e+26)
		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 <= 3.4e+26)
		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, 3.4e+26], (-U$95$m), U$95$m]

Derivation

1. Split input into 2 regimes

2. if K < 3.4000000000000003e26

   1. Initial program 80.2%

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

   3. Simplified 91.6%

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

   5. Taylor expanded in J around 0 24.8%

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

      1. neg-mul-1 24.8%

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

   7. Simplified 24.8%

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

   if 3.4000000000000003e26 < K

   1. Initial program 69.6%

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

   3. Simplified 92.0%

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

   5. Taylor expanded in U around -inf 41.2%

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

Alternative 14: 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 78.1%

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

3. Simplified 91.7%

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

5. Taylor expanded in U around -inf 28.0%

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

Reproduce

herbie shell --seed 2024090 
(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)))))