Maksimov and Kolovsky, Equation (3)

Percentage Accurate: 72.2% → 99.1%

Time: 26.3s

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: 72.2% 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: 99.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_0 \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+298}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \]

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 4e+298) 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 <= 4e+298) {
		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 <= 4e+298) {
		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 <= 4e+298:
		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 <= 4e+298)
		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 <= 4e+298)
		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, 4e+298], t$95$1, U$95$m]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 6.0%

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

   3. Simplified 49.1%

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

   5. Taylor expanded in J around 0 18.7%

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

      1. neg-mul-1 18.7%

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

   7. Simplified 18.7%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 -2 J) (cos.f64 (/.f64 K 2))) (sqrt.f64 (+.f64 1 (pow.f64 (/.f64 U (*.f64 (*.f64 2 J) (cos.f64 (/.f64 K 2)))) 2)))) < 3.9999999999999998e298

   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 3.9999999999999998e298 < (*.f64 (*.f64 (*.f64 -2 J) (cos.f64 (/.f64 K 2))) (sqrt.f64 (+.f64 1 (pow.f64 (/.f64 U (*.f64 (*.f64 2 J) (cos.f64 (/.f64 K 2)))) 2))))

   1. Initial program 6.1%

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

   3. Simplified 62.5%

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

      1. expm1-log1p-u 16.0%

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

      2. expm1-undefine 16.0%

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

   6. Applied egg-rr 16.0%

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

      1. expm1-define 16.0%

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

      2. *-commutative 16.0%

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

      3. times-frac 16.0%

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

      4. *-commutative 16.0%

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

      5. associate-*r/ 16.0%

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

      6. associate-/r* 16.0%

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

      7. *-commutative 16.0%

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

      8. *-commutative 16.0%

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

      9. *-commutative 16.0%

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

   8. Simplified 16.0%

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

   9. Taylor expanded in U around -inf 57.4%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 4 \cdot 10^{+298}:\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 63.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := J \cdot \left(\left(-2 \cdot t\_0\right) \cdot \left(1 + \left(\frac{U\_m}{J} \cdot \frac{U\_m}{J}\right) \cdot 0.125\right)\right)\\ \mathbf{if}\;t\_0 \leq -0.48:\\ \;\;\;\;U\_m\\ \mathbf{elif}\;t\_0 \leq 0.03:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0.36:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_0 \leq 0.75:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \mathsf{hypot}\left(1, \frac{U\_m \cdot 0.5}{J}\right)\\ \end{array} \end{array} \]

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 -0.48)
     U_m
     (if (<= t_0 0.03)
       t_1
       (if (<= t_0 0.36)
         (- U_m)
         (if (<= t_0 0.75)
           t_1
           (* (* -2.0 J) (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 <= -0.48) {
		tmp = U_m;
	} else if (t_0 <= 0.03) {
		tmp = t_1;
	} else if (t_0 <= 0.36) {
		tmp = -U_m;
	} else if (t_0 <= 0.75) {
		tmp = t_1;
	} else {
		tmp = (-2.0 * J) * 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 <= -0.48) {
		tmp = U_m;
	} else if (t_0 <= 0.03) {
		tmp = t_1;
	} else if (t_0 <= 0.36) {
		tmp = -U_m;
	} else if (t_0 <= 0.75) {
		tmp = t_1;
	} else {
		tmp = (-2.0 * J) * 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 <= -0.48:
		tmp = U_m
	elif t_0 <= 0.03:
		tmp = t_1
	elif t_0 <= 0.36:
		tmp = -U_m
	elif t_0 <= 0.75:
		tmp = t_1
	else:
		tmp = (-2.0 * J) * 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 <= -0.48)
		tmp = U_m;
	elseif (t_0 <= 0.03)
		tmp = t_1;
	elseif (t_0 <= 0.36)
		tmp = Float64(-U_m);
	elseif (t_0 <= 0.75)
		tmp = t_1;
	else
		tmp = Float64(Float64(-2.0 * J) * hypot(1.0, Float64(Float64(U_m * 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 <= -0.48)
		tmp = U_m;
	elseif (t_0 <= 0.03)
		tmp = t_1;
	elseif (t_0 <= 0.36)
		tmp = -U_m;
	elseif (t_0 <= 0.75)
		tmp = t_1;
	else
		tmp = (-2.0 * J) * 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, -0.48], U$95$m, If[LessEqual[t$95$0, 0.03], t$95$1, If[LessEqual[t$95$0, 0.36], (-U$95$m), If[LessEqual[t$95$0, 0.75], t$95$1, N[(N[(-2.0 * J), $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[(N[(U$95$m * 0.5), $MachinePrecision] / J), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (cos.f64 (/.f64 K 2)) < -0.47999999999999998

   1. Initial program 63.1%

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

   3. Simplified 81.5%

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

      1. expm1-log1p-u 77.6%

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

      2. expm1-undefine 77.5%

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

   6. Applied egg-rr 77.5%

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

      1. expm1-define 77.6%

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

      2. *-commutative 77.6%

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

      3. times-frac 77.6%

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

      4. *-commutative 77.6%

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

      5. associate-*r/ 77.6%

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

      6. associate-/r* 77.6%

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

      7. *-commutative 77.6%

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

      8. *-commutative 77.6%

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

      9. *-commutative 77.6%

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

   8. Simplified 77.6%

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

   9. Taylor expanded in U around -inf 40.8%

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

   if -0.47999999999999998 < (cos.f64 (/.f64 K 2)) < 0.029999999999999999 or 0.35999999999999999 < (cos.f64 (/.f64 K 2)) < 0.75

   1. Initial program 76.1%

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

   3. Simplified 82.7%

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

   5. Taylor expanded in K around 0 71.5%

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

      1. *-commutative 71.5%

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

   7. Simplified 71.5%

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

   8. Taylor expanded in U around 0 57.9%

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

      1. *-commutative 57.9%

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

   10. Simplified 57.9%

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

       1. unpow2 57.9%

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

       2. unpow2 57.9%

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

       3. frac-times 69.6%

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

   12. Applied egg-rr 69.6%

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

   if 0.029999999999999999 < (cos.f64 (/.f64 K 2)) < 0.35999999999999999

   1. Initial program 52.6%

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

   3. Simplified 89.9%

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

   5. Taylor expanded in J around 0 15.2%

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

      1. neg-mul-1 15.2%

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

   7. Simplified 15.2%

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

   if 0.75 < (cos.f64 (/.f64 K 2))

   1. Initial program 75.2%

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

   3. Simplified 89.7%

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

      1. expm1-log1p-u 0.0%

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

      2. expm1-undefine 0.0%

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

   6. Applied egg-rr 0.0%

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

      1. expm1-define 0.0%

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

      2. *-commutative 0.0%

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

      3. times-frac 0.0%

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

      4. *-commutative 0.0%

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

      5. associate-*r/ 0.0%

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

      6. associate-/r* 0.0%

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

      7. *-commutative 0.0%

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

      8. *-commutative 0.0%

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

      9. *-commutative 0.0%

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

   8. Simplified 0.0%

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

      1. expm1-undefine 0.0%

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

      2. sub-neg 0.0%

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

   10. Applied egg-rr 89.7%

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

       1. associate-+l+ 89.7%

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

       2. metadata-eval 89.7%

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

       3. sub-neg 89.7%

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

       4. associate-*r* 89.7%

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

       5. *-commutative 89.7%

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

       6. fma-neg 89.7%

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

       7. associate-/l/ 89.7%

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

       8. *-commutative 89.7%

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

       9. metadata-eval 89.7%

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

   12. Simplified 89.7%

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

   13. Taylor expanded in K around 0 50.3%

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

       1. associate-*r* 50.3%

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

       2. metadata-eval 50.3%

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

       3. unpow2 50.3%

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

       4. unpow2 50.3%

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

       5. times-frac 68.0%

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

       6. swap-sqr 68.0%

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

       7. unpow2 68.0%

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

       8. metadata-eval 68.0%

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

       9. unpow2 68.0%

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

       10. hypot-undefine 82.5%

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

       11. associate-*r/ 82.5%

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

   15. Simplified 82.5%

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

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.48:\\ \;\;\;\;U\\ \mathbf{elif}\;\cos \left(\frac{K}{2}\right) \leq 0.03:\\ \;\;\;\;J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(1 + \left(\frac{U}{J} \cdot \frac{U}{J}\right) \cdot 0.125\right)\right)\\ \mathbf{elif}\;\cos \left(\frac{K}{2}\right) \leq 0.36:\\ \;\;\;\;-U\\ \mathbf{elif}\;\cos \left(\frac{K}{2}\right) \leq 0.75:\\ \;\;\;\;J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(1 + \left(\frac{U}{J} \cdot \frac{U}{J}\right) \cdot 0.125\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \mathsf{hypot}\left(1, \frac{U \cdot 0.5}{J}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 63.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := J \cdot \left(-2 \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{if}\;t\_0 \leq -0.48:\\ \;\;\;\;U\_m\\ \mathbf{elif}\;t\_0 \leq 0.03:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0.36:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_0 \leq 0.75:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \mathsf{hypot}\left(1, \frac{U\_m \cdot 0.5}{J}\right)\\ \end{array} \end{array} \]

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 (cos (* K 0.5))))))
   (if (<= t_0 -0.48)
     U_m
     (if (<= t_0 0.03)
       t_1
       (if (<= t_0 0.36)
         (- U_m)
         (if (<= t_0 0.75)
           t_1
           (* (* -2.0 J) (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 * cos((K * 0.5)));
	double tmp;
	if (t_0 <= -0.48) {
		tmp = U_m;
	} else if (t_0 <= 0.03) {
		tmp = t_1;
	} else if (t_0 <= 0.36) {
		tmp = -U_m;
	} else if (t_0 <= 0.75) {
		tmp = t_1;
	} else {
		tmp = (-2.0 * J) * 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 * Math.cos((K * 0.5)));
	double tmp;
	if (t_0 <= -0.48) {
		tmp = U_m;
	} else if (t_0 <= 0.03) {
		tmp = t_1;
	} else if (t_0 <= 0.36) {
		tmp = -U_m;
	} else if (t_0 <= 0.75) {
		tmp = t_1;
	} else {
		tmp = (-2.0 * J) * 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 * math.cos((K * 0.5)))
	tmp = 0
	if t_0 <= -0.48:
		tmp = U_m
	elif t_0 <= 0.03:
		tmp = t_1
	elif t_0 <= 0.36:
		tmp = -U_m
	elif t_0 <= 0.75:
		tmp = t_1
	else:
		tmp = (-2.0 * J) * 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(-2.0 * cos(Float64(K * 0.5))))
	tmp = 0.0
	if (t_0 <= -0.48)
		tmp = U_m;
	elseif (t_0 <= 0.03)
		tmp = t_1;
	elseif (t_0 <= 0.36)
		tmp = Float64(-U_m);
	elseif (t_0 <= 0.75)
		tmp = t_1;
	else
		tmp = Float64(Float64(-2.0 * J) * hypot(1.0, Float64(Float64(U_m * 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 * cos((K * 0.5)));
	tmp = 0.0;
	if (t_0 <= -0.48)
		tmp = U_m;
	elseif (t_0 <= 0.03)
		tmp = t_1;
	elseif (t_0 <= 0.36)
		tmp = -U_m;
	elseif (t_0 <= 0.75)
		tmp = t_1;
	else
		tmp = (-2.0 * J) * 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[(-2.0 * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.48], U$95$m, If[LessEqual[t$95$0, 0.03], t$95$1, If[LessEqual[t$95$0, 0.36], (-U$95$m), If[LessEqual[t$95$0, 0.75], t$95$1, N[(N[(-2.0 * J), $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[(N[(U$95$m * 0.5), $MachinePrecision] / J), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (cos.f64 (/.f64 K 2)) < -0.47999999999999998

   1. Initial program 63.1%

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

   3. Simplified 81.5%

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

      1. expm1-log1p-u 77.6%

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

      2. expm1-undefine 77.5%

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

   6. Applied egg-rr 77.5%

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

      1. expm1-define 77.6%

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

      2. *-commutative 77.6%

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

      3. times-frac 77.6%

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

      4. *-commutative 77.6%

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

      5. associate-*r/ 77.6%

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

      6. associate-/r* 77.6%

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

      7. *-commutative 77.6%

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

      8. *-commutative 77.6%

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

      9. *-commutative 77.6%

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

   8. Simplified 77.6%

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

   9. Taylor expanded in U around -inf 40.8%

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

   if -0.47999999999999998 < (cos.f64 (/.f64 K 2)) < 0.029999999999999999 or 0.35999999999999999 < (cos.f64 (/.f64 K 2)) < 0.75

   1. Initial program 76.1%

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

   3. Simplified 82.7%

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

   5. Taylor expanded in J around inf 68.4%

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

      1. *-commutative 68.4%

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

      2. *-commutative 68.4%

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

      3. metadata-eval 68.4%

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

      4. distribute-rgt-neg-in 68.4%

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

      5. associate-*r* 68.4%

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

      6. distribute-rgt-neg-in 68.4%

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

      7. *-commutative 68.4%

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

      8. distribute-lft-neg-in 68.4%

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

      9. metadata-eval 68.4%

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

      10. *-commutative 68.4%

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

   7. Simplified 68.4%

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

   if 0.029999999999999999 < (cos.f64 (/.f64 K 2)) < 0.35999999999999999

   1. Initial program 52.6%

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

   3. Simplified 89.9%

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

   5. Taylor expanded in J around 0 15.2%

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

      1. neg-mul-1 15.2%

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

   7. Simplified 15.2%

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

   if 0.75 < (cos.f64 (/.f64 K 2))

   1. Initial program 75.2%

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

   3. Simplified 89.7%

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

      1. expm1-log1p-u 0.0%

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

      2. expm1-undefine 0.0%

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

   6. Applied egg-rr 0.0%

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

      1. expm1-define 0.0%

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

      2. *-commutative 0.0%

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

      3. times-frac 0.0%

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

      4. *-commutative 0.0%

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

      5. associate-*r/ 0.0%

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

      6. associate-/r* 0.0%

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

      7. *-commutative 0.0%

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

      8. *-commutative 0.0%

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

      9. *-commutative 0.0%

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

   8. Simplified 0.0%

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

      1. expm1-undefine 0.0%

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

      2. sub-neg 0.0%

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

   10. Applied egg-rr 89.7%

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

       1. associate-+l+ 89.7%

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

       2. metadata-eval 89.7%

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

       3. sub-neg 89.7%

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

       4. associate-*r* 89.7%

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

       5. *-commutative 89.7%

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

       6. fma-neg 89.7%

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

       7. associate-/l/ 89.7%

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

       8. *-commutative 89.7%

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

       9. metadata-eval 89.7%

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

   12. Simplified 89.7%

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

   13. Taylor expanded in K around 0 50.3%

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

       1. associate-*r* 50.3%

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

       2. metadata-eval 50.3%

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

       3. unpow2 50.3%

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

       4. unpow2 50.3%

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

       5. times-frac 68.0%

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

       6. swap-sqr 68.0%

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

       7. unpow2 68.0%

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

       8. metadata-eval 68.0%

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

       9. unpow2 68.0%

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

       10. hypot-undefine 82.5%

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

       11. associate-*r/ 82.5%

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

   15. Simplified 82.5%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.48:\\ \;\;\;\;U\\ \mathbf{elif}\;\cos \left(\frac{K}{2}\right) \leq 0.03:\\ \;\;\;\;J \cdot \left(-2 \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{elif}\;\cos \left(\frac{K}{2}\right) \leq 0.36:\\ \;\;\;\;-U\\ \mathbf{elif}\;\cos \left(\frac{K}{2}\right) \leq 0.75:\\ \;\;\;\;J \cdot \left(-2 \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \mathsf{hypot}\left(1, \frac{U \cdot 0.5}{J}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 70.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} \mathbf{if}\;U\_m \leq 220000000000:\\ \;\;\;\;J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U\_m}{J \cdot 2}\right)\right)\\ \mathbf{elif}\;U\_m \leq 2.1 \cdot 10^{+170}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \mathsf{hypot}\left(1, \frac{U\_m \cdot 0.5}{J}\right)\\ \mathbf{elif}\;U\_m \leq 3.6 \cdot 10^{+186}:\\ \;\;\;\;J \cdot \left(1 + \mathsf{fma}\left(-2, \cos \left(K \cdot 0.5\right) \cdot \mathsf{hypot}\left(1, 0.5 \cdot \frac{U\_m}{J}\right), -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (if (<= U_m 220000000000.0)
   (* J (* (* -2.0 (cos (/ K 2.0))) (hypot 1.0 (/ U_m (* J 2.0)))))
   (if (<= U_m 2.1e+170)
     (* (* -2.0 J) (hypot 1.0 (/ (* U_m 0.5) J)))
     (if (<= U_m 3.6e+186)
       (*
        J
        (+
         1.0
         (fma -2.0 (* (cos (* K 0.5)) (hypot 1.0 (* 0.5 (/ U_m J)))) -1.0)))
       (- U_m)))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double tmp;
	if (U_m <= 220000000000.0) {
		tmp = J * ((-2.0 * cos((K / 2.0))) * hypot(1.0, (U_m / (J * 2.0))));
	} else if (U_m <= 2.1e+170) {
		tmp = (-2.0 * J) * hypot(1.0, ((U_m * 0.5) / J));
	} else if (U_m <= 3.6e+186) {
		tmp = J * (1.0 + fma(-2.0, (cos((K * 0.5)) * hypot(1.0, (0.5 * (U_m / J)))), -1.0));
	} else {
		tmp = -U_m;
	}
	return tmp;
}

Julia

U_m = abs(U)
function code(J, K, U_m)
	tmp = 0.0
	if (U_m <= 220000000000.0)
		tmp = Float64(J * Float64(Float64(-2.0 * cos(Float64(K / 2.0))) * hypot(1.0, Float64(U_m / Float64(J * 2.0)))));
	elseif (U_m <= 2.1e+170)
		tmp = Float64(Float64(-2.0 * J) * hypot(1.0, Float64(Float64(U_m * 0.5) / J)));
	elseif (U_m <= 3.6e+186)
		tmp = Float64(J * Float64(1.0 + fma(-2.0, Float64(cos(Float64(K * 0.5)) * hypot(1.0, Float64(0.5 * Float64(U_m / J)))), -1.0)));
	else
		tmp = Float64(-U_m);
	end
	return tmp
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := If[LessEqual[U$95$m, 220000000000.0], N[(J * N[(N[(-2.0 * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[(U$95$m / N[(J * 2.0), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[U$95$m, 2.1e+170], N[(N[(-2.0 * J), $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[(N[(U$95$m * 0.5), $MachinePrecision] / J), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[U$95$m, 3.6e+186], N[(J * N[(1.0 + N[(-2.0 * N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[(0.5 * N[(U$95$m / J), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-U$95$m)]]]

Derivation

1. Split input into 4 regimes

2. if U < 2.2e11

   1. Initial program 77.6%

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

   3. Simplified 89.0%

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

   5. Taylor expanded in K around 0 75.1%

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

      1. *-commutative 75.1%

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

   7. Simplified 75.1%

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

   if 2.2e11 < U < 2.09999999999999998e170

   1. Initial program 60.0%

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

   3. Simplified 87.8%

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

      1. expm1-log1p-u 11.2%

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

      2. expm1-undefine 11.2%

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

   6. Applied egg-rr 11.2%

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

      1. expm1-define 11.2%

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

      2. *-commutative 11.2%

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

      3. times-frac 11.2%

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

      4. *-commutative 11.2%

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

      5. associate-*r/ 11.2%

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

      6. associate-/r* 11.2%

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

      7. *-commutative 11.2%

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

      8. *-commutative 11.2%

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

      9. *-commutative 11.2%

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

   8. Simplified 11.2%

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

      1. expm1-undefine 11.2%

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

      2. sub-neg 11.2%

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

   10. Applied egg-rr 87.8%

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

       1. associate-+l+ 87.8%

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

       2. metadata-eval 87.8%

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

       3. sub-neg 87.8%

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

       4. associate-*r* 87.8%

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

       5. *-commutative 87.8%

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

       6. fma-neg 87.8%

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

       7. associate-/l/ 87.8%

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

       8. *-commutative 87.8%

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

       9. metadata-eval 87.8%

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

   12. Simplified 87.8%

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

   13. Taylor expanded in K around 0 47.1%

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

       1. associate-*r* 47.1%

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

       2. metadata-eval 47.1%

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

       3. unpow2 47.1%

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

       4. unpow2 47.1%

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

       5. times-frac 55.3%

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

       6. swap-sqr 55.3%

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

       7. unpow2 55.3%

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

       8. metadata-eval 55.3%

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

       9. unpow2 55.3%

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

       10. hypot-undefine 71.2%

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

       11. associate-*r/ 71.2%

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

   15. Simplified 71.2%

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

   if 2.09999999999999998e170 < U < 3.6000000000000002e186

   1. Initial program 86.2%

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

   3. Simplified 99.6%

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

      1. expm1-log1p-u 53.8%

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

      2. expm1-undefine 53.4%

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

   6. Applied egg-rr 53.4%

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

      1. expm1-define 53.8%

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

      2. *-commutative 53.8%

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

      3. times-frac 53.8%

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

      4. *-commutative 53.8%

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

      5. associate-*r/ 53.8%

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

      6. associate-/r* 53.8%

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

      7. *-commutative 53.8%

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

      8. *-commutative 53.8%

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

      9. *-commutative 53.8%

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

   8. Simplified 53.8%

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

      1. expm1-undefine 53.4%

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

      2. sub-neg 53.4%

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

   10. Applied egg-rr 99.3%

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

       1. associate-+l+ 99.8%

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

       2. metadata-eval 99.8%

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

       3. sub-neg 99.8%

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

       4. associate-*r* 99.8%

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

       5. *-commutative 99.8%

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

       6. fma-neg 99.8%

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

       7. associate-/l/ 99.8%

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

       8. *-commutative 99.8%

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

       9. metadata-eval 99.8%

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

   12. Simplified 99.8%

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

   13. Taylor expanded in K around 0 77.5%

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

   if 3.6000000000000002e186 < U

   1. Initial program 39.7%

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

   3. Simplified 68.1%

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

   5. Taylor expanded in J around 0 36.1%

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

      1. neg-mul-1 36.1%

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

   7. Simplified 36.1%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 220000000000:\\ \;\;\;\;J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot 2}\right)\right)\\ \mathbf{elif}\;U \leq 2.1 \cdot 10^{+170}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \mathsf{hypot}\left(1, \frac{U \cdot 0.5}{J}\right)\\ \mathbf{elif}\;U \leq 3.6 \cdot 10^{+186}:\\ \;\;\;\;J \cdot \left(1 + \mathsf{fma}\left(-2, \cos \left(K \cdot 0.5\right) \cdot \mathsf{hypot}\left(1, 0.5 \cdot \frac{U}{J}\right), -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;U\_m \leq 1.95 \cdot 10^{+247}:\\ \;\;\;\;-2 \cdot \left(J \cdot \left(t\_0 \cdot \mathsf{hypot}\left(1, \frac{U\_m}{t\_0} \cdot \frac{0.5}{J}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= U_m 1.95e+247)
     (* -2.0 (* J (* t_0 (hypot 1.0 (* (/ U_m t_0) (/ 0.5 J))))))
     (- U_m))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (U_m <= 1.95e+247) {
		tmp = -2.0 * (J * (t_0 * hypot(1.0, ((U_m / t_0) * (0.5 / J)))));
	} else {
		tmp = -U_m;
	}
	return tmp;
}

Java

U_m = Math.abs(U);
public static double code(double J, double K, double U_m) {
	double t_0 = Math.cos((K / 2.0));
	double tmp;
	if (U_m <= 1.95e+247) {
		tmp = -2.0 * (J * (t_0 * Math.hypot(1.0, ((U_m / t_0) * (0.5 / J)))));
	} else {
		tmp = -U_m;
	}
	return tmp;
}

Python

U_m = math.fabs(U)
def code(J, K, U_m):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if U_m <= 1.95e+247:
		tmp = -2.0 * (J * (t_0 * math.hypot(1.0, ((U_m / t_0) * (0.5 / J)))))
	else:
		tmp = -U_m
	return tmp

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (U_m <= 1.95e+247)
		tmp = Float64(-2.0 * Float64(J * Float64(t_0 * hypot(1.0, Float64(Float64(U_m / t_0) * Float64(0.5 / J))))));
	else
		tmp = Float64(-U_m);
	end
	return tmp
end

MATLAB

U_m = abs(U);
function tmp_2 = code(J, K, U_m)
	t_0 = cos((K / 2.0));
	tmp = 0.0;
	if (U_m <= 1.95e+247)
		tmp = -2.0 * (J * (t_0 * hypot(1.0, ((U_m / t_0) * (0.5 / J)))));
	else
		tmp = -U_m;
	end
	tmp_2 = tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[U$95$m, 1.95e+247], N[(-2.0 * N[(J * N[(t$95$0 * N[Sqrt[1.0 ^ 2 + N[(N[(U$95$m / t$95$0), $MachinePrecision] * N[(0.5 / J), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-U$95$m)]]

Derivation

1. Split input into 2 regimes

2. if U < 1.95000000000000001e247

   1. Initial program 74.3%

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

   3. Simplified 88.6%

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

   if 1.95000000000000001e247 < U

   1. Initial program 32.0%

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

   3. Simplified 54.8%

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

   5. Taylor expanded in J around 0 44.1%

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

      1. neg-mul-1 44.1%

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

   7. Simplified 44.1%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 1.95 \cdot 10^{+247}:\\ \;\;\;\;-2 \cdot \left(J \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\cos \left(\frac{K}{2}\right)} \cdot \frac{0.5}{J}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;U\_m \leq 6.6 \cdot 10^{+247}:\\ \;\;\;\;-2 \cdot \left(\left(J \cdot t\_0\right) \cdot \mathsf{hypot}\left(1, \frac{\frac{U\_m}{J \cdot 2}}{t\_0}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= U_m 6.6e+247)
     (* -2.0 (* (* J t_0) (hypot 1.0 (/ (/ U_m (* J 2.0)) t_0))))
     (- U_m))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (U_m <= 6.6e+247) {
		tmp = -2.0 * ((J * t_0) * hypot(1.0, ((U_m / (J * 2.0)) / t_0)));
	} else {
		tmp = -U_m;
	}
	return tmp;
}

Java

U_m = Math.abs(U);
public static double code(double J, double K, double U_m) {
	double t_0 = Math.cos((K / 2.0));
	double tmp;
	if (U_m <= 6.6e+247) {
		tmp = -2.0 * ((J * t_0) * Math.hypot(1.0, ((U_m / (J * 2.0)) / t_0)));
	} else {
		tmp = -U_m;
	}
	return tmp;
}

Python

U_m = math.fabs(U)
def code(J, K, U_m):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if U_m <= 6.6e+247:
		tmp = -2.0 * ((J * t_0) * math.hypot(1.0, ((U_m / (J * 2.0)) / t_0)))
	else:
		tmp = -U_m
	return tmp

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (U_m <= 6.6e+247)
		tmp = Float64(-2.0 * Float64(Float64(J * t_0) * hypot(1.0, Float64(Float64(U_m / Float64(J * 2.0)) / t_0))));
	else
		tmp = Float64(-U_m);
	end
	return tmp
end

MATLAB

U_m = abs(U);
function tmp_2 = code(J, K, U_m)
	t_0 = cos((K / 2.0));
	tmp = 0.0;
	if (U_m <= 6.6e+247)
		tmp = -2.0 * ((J * t_0) * hypot(1.0, ((U_m / (J * 2.0)) / t_0)));
	else
		tmp = -U_m;
	end
	tmp_2 = tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[U$95$m, 6.6e+247], N[(-2.0 * N[(N[(J * t$95$0), $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[(N[(U$95$m / N[(J * 2.0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-U$95$m)]]

Derivation

1. Split input into 2 regimes

2. if U < 6.60000000000000003e247

   1. Initial program 74.3%

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

   3. Simplified 88.7%

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

   if 6.60000000000000003e247 < U

   1. Initial program 32.0%

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

   3. Simplified 54.8%

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

   5. Taylor expanded in J around 0 44.1%

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

      1. neg-mul-1 44.1%

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

   7. Simplified 44.1%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 6.6 \cdot 10^{+247}:\\ \;\;\;\;-2 \cdot \left(\left(J \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{\frac{U}{J \cdot 2}}{\cos \left(\frac{K}{2}\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;U\_m \leq 5.8 \cdot 10^{+246}:\\ \;\;\;\;J \cdot \left(\left(-2 \cdot t\_0\right) \cdot \mathsf{hypot}\left(1, \frac{U\_m}{t\_0 \cdot \left(J \cdot 2\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= U_m 5.8e+246)
     (* J (* (* -2.0 t_0) (hypot 1.0 (/ U_m (* t_0 (* J 2.0))))))
     (- U_m))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (U_m <= 5.8e+246) {
		tmp = J * ((-2.0 * t_0) * hypot(1.0, (U_m / (t_0 * (J * 2.0)))));
	} else {
		tmp = -U_m;
	}
	return tmp;
}

Java

U_m = Math.abs(U);
public static double code(double J, double K, double U_m) {
	double t_0 = Math.cos((K / 2.0));
	double tmp;
	if (U_m <= 5.8e+246) {
		tmp = J * ((-2.0 * t_0) * Math.hypot(1.0, (U_m / (t_0 * (J * 2.0)))));
	} else {
		tmp = -U_m;
	}
	return tmp;
}

Python

U_m = math.fabs(U)
def code(J, K, U_m):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if U_m <= 5.8e+246:
		tmp = J * ((-2.0 * t_0) * math.hypot(1.0, (U_m / (t_0 * (J * 2.0)))))
	else:
		tmp = -U_m
	return tmp

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (U_m <= 5.8e+246)
		tmp = Float64(J * Float64(Float64(-2.0 * t_0) * hypot(1.0, Float64(U_m / Float64(t_0 * Float64(J * 2.0))))));
	else
		tmp = Float64(-U_m);
	end
	return tmp
end

MATLAB

U_m = abs(U);
function tmp_2 = code(J, K, U_m)
	t_0 = cos((K / 2.0));
	tmp = 0.0;
	if (U_m <= 5.8e+246)
		tmp = J * ((-2.0 * t_0) * hypot(1.0, (U_m / (t_0 * (J * 2.0)))));
	else
		tmp = -U_m;
	end
	tmp_2 = tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[U$95$m, 5.8e+246], N[(J * N[(N[(-2.0 * t$95$0), $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], (-U$95$m)]]

Derivation

1. Split input into 2 regimes

2. if U < 5.80000000000000028e246

   1. Initial program 74.3%

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

   3. Simplified 88.7%

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

   if 5.80000000000000028e246 < U

   1. Initial program 32.0%

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

   3. Simplified 54.8%

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

   5. Taylor expanded in J around 0 44.1%

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

      1. neg-mul-1 44.1%

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

   7. Simplified 44.1%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 5.8 \cdot 10^{+246}:\\ \;\;\;\;J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 71.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U\_m}{J \cdot 2}\right)\right)\\ \mathbf{if}\;U\_m \leq 4.6 \cdot 10^{+14}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;U\_m \leq 1.3 \cdot 10^{+171}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \mathsf{hypot}\left(1, \frac{U\_m \cdot 0.5}{J}\right)\\ \mathbf{elif}\;U\_m \leq 3.8 \cdot 10^{+186}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;-U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (* J (* (* -2.0 (cos (/ K 2.0))) (hypot 1.0 (/ U_m (* J 2.0)))))))
   (if (<= U_m 4.6e+14)
     t_0
     (if (<= U_m 1.3e+171)
       (* (* -2.0 J) (hypot 1.0 (/ (* U_m 0.5) J)))
       (if (<= U_m 3.8e+186) t_0 (- U_m))))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = J * ((-2.0 * cos((K / 2.0))) * hypot(1.0, (U_m / (J * 2.0))));
	double tmp;
	if (U_m <= 4.6e+14) {
		tmp = t_0;
	} else if (U_m <= 1.3e+171) {
		tmp = (-2.0 * J) * hypot(1.0, ((U_m * 0.5) / J));
	} else if (U_m <= 3.8e+186) {
		tmp = t_0;
	} else {
		tmp = -U_m;
	}
	return tmp;
}

Java

U_m = Math.abs(U);
public static double code(double J, double K, double U_m) {
	double t_0 = J * ((-2.0 * Math.cos((K / 2.0))) * Math.hypot(1.0, (U_m / (J * 2.0))));
	double tmp;
	if (U_m <= 4.6e+14) {
		tmp = t_0;
	} else if (U_m <= 1.3e+171) {
		tmp = (-2.0 * J) * Math.hypot(1.0, ((U_m * 0.5) / J));
	} else if (U_m <= 3.8e+186) {
		tmp = t_0;
	} else {
		tmp = -U_m;
	}
	return tmp;
}

Python

U_m = math.fabs(U)
def code(J, K, U_m):
	t_0 = J * ((-2.0 * math.cos((K / 2.0))) * math.hypot(1.0, (U_m / (J * 2.0))))
	tmp = 0
	if U_m <= 4.6e+14:
		tmp = t_0
	elif U_m <= 1.3e+171:
		tmp = (-2.0 * J) * math.hypot(1.0, ((U_m * 0.5) / J))
	elif U_m <= 3.8e+186:
		tmp = t_0
	else:
		tmp = -U_m
	return tmp

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = Float64(J * Float64(Float64(-2.0 * cos(Float64(K / 2.0))) * hypot(1.0, Float64(U_m / Float64(J * 2.0)))))
	tmp = 0.0
	if (U_m <= 4.6e+14)
		tmp = t_0;
	elseif (U_m <= 1.3e+171)
		tmp = Float64(Float64(-2.0 * J) * hypot(1.0, Float64(Float64(U_m * 0.5) / J)));
	elseif (U_m <= 3.8e+186)
		tmp = t_0;
	else
		tmp = Float64(-U_m);
	end
	return tmp
end

MATLAB

U_m = abs(U);
function tmp_2 = code(J, K, U_m)
	t_0 = J * ((-2.0 * cos((K / 2.0))) * hypot(1.0, (U_m / (J * 2.0))));
	tmp = 0.0;
	if (U_m <= 4.6e+14)
		tmp = t_0;
	elseif (U_m <= 1.3e+171)
		tmp = (-2.0 * J) * hypot(1.0, ((U_m * 0.5) / J));
	elseif (U_m <= 3.8e+186)
		tmp = t_0;
	else
		tmp = -U_m;
	end
	tmp_2 = tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := Block[{t$95$0 = N[(J * N[(N[(-2.0 * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[(U$95$m / N[(J * 2.0), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[U$95$m, 4.6e+14], t$95$0, If[LessEqual[U$95$m, 1.3e+171], N[(N[(-2.0 * J), $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[(N[(U$95$m * 0.5), $MachinePrecision] / J), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[U$95$m, 3.8e+186], t$95$0, (-U$95$m)]]]]

Derivation

1. Split input into 3 regimes

2. if U < 4.6e14 or 1.3e171 < U < 3.7999999999999998e186

   1. Initial program 77.9%

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

   3. Simplified 89.4%

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

   5. Taylor expanded in K around 0 75.2%

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

      1. *-commutative 75.2%

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

   7. Simplified 75.2%

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

   if 4.6e14 < U < 1.3e171

   1. Initial program 60.0%

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

   3. Simplified 87.8%

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

      1. expm1-log1p-u 11.2%

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

      2. expm1-undefine 11.2%

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

   6. Applied egg-rr 11.2%

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

      1. expm1-define 11.2%

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

      2. *-commutative 11.2%

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

      3. times-frac 11.2%

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

      4. *-commutative 11.2%

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

      5. associate-*r/ 11.2%

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

      6. associate-/r* 11.2%

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

      7. *-commutative 11.2%

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

      8. *-commutative 11.2%

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

      9. *-commutative 11.2%

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

   8. Simplified 11.2%

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

      1. expm1-undefine 11.2%

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

      2. sub-neg 11.2%

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

   10. Applied egg-rr 87.8%

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

       1. associate-+l+ 87.8%

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

       2. metadata-eval 87.8%

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

       3. sub-neg 87.8%

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

       4. associate-*r* 87.8%

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

       5. *-commutative 87.8%

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

       6. fma-neg 87.8%

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

       7. associate-/l/ 87.8%

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

       8. *-commutative 87.8%

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

       9. metadata-eval 87.8%

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

   12. Simplified 87.8%

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

   13. Taylor expanded in K around 0 47.1%

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

       1. associate-*r* 47.1%

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

       2. metadata-eval 47.1%

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

       3. unpow2 47.1%

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

       4. unpow2 47.1%

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

       5. times-frac 55.3%

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

       6. swap-sqr 55.3%

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

       7. unpow2 55.3%

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

       8. metadata-eval 55.3%

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

       9. unpow2 55.3%

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

       10. hypot-undefine 71.2%

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

       11. associate-*r/ 71.2%

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

   15. Simplified 71.2%

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

   if 3.7999999999999998e186 < U

   1. Initial program 39.7%

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

   3. Simplified 68.1%

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

   5. Taylor expanded in J around 0 36.1%

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

      1. neg-mul-1 36.1%

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

   7. Simplified 36.1%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 4.6 \cdot 10^{+14}:\\ \;\;\;\;J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot 2}\right)\right)\\ \mathbf{elif}\;U \leq 1.3 \cdot 10^{+171}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \mathsf{hypot}\left(1, \frac{U \cdot 0.5}{J}\right)\\ \mathbf{elif}\;U \leq 3.8 \cdot 10^{+186}:\\ \;\;\;\;J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot 2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 56.6% accurate, 3.4× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} \mathbf{if}\;U\_m \leq 0.0036 \lor \neg \left(U\_m \leq 6.9 \cdot 10^{+171}\right) \land U\_m \leq 3.6 \cdot 10^{+186}:\\ \;\;\;\;J \cdot \left(-2 \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (if (or (<= U_m 0.0036) (and (not (<= U_m 6.9e+171)) (<= U_m 3.6e+186)))
   (* J (* -2.0 (cos (* K 0.5))))
   (- U_m)))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double tmp;
	if ((U_m <= 0.0036) || (!(U_m <= 6.9e+171) && (U_m <= 3.6e+186))) {
		tmp = J * (-2.0 * cos((K * 0.5)));
	} else {
		tmp = -U_m;
	}
	return tmp;
}

Fortran

U_m = abs(U)
real(8) function code(j, k, u_m)
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u_m
    real(8) :: tmp
    if ((u_m <= 0.0036d0) .or. (.not. (u_m <= 6.9d+171)) .and. (u_m <= 3.6d+186)) then
        tmp = j * ((-2.0d0) * cos((k * 0.5d0)))
    else
        tmp = -u_m
    end if
    code = tmp
end function

Java

U_m = Math.abs(U);
public static double code(double J, double K, double U_m) {
	double tmp;
	if ((U_m <= 0.0036) || (!(U_m <= 6.9e+171) && (U_m <= 3.6e+186))) {
		tmp = J * (-2.0 * Math.cos((K * 0.5)));
	} else {
		tmp = -U_m;
	}
	return tmp;
}

Python

U_m = math.fabs(U)
def code(J, K, U_m):
	tmp = 0
	if (U_m <= 0.0036) or (not (U_m <= 6.9e+171) and (U_m <= 3.6e+186)):
		tmp = J * (-2.0 * math.cos((K * 0.5)))
	else:
		tmp = -U_m
	return tmp

Julia

U_m = abs(U)
function code(J, K, U_m)
	tmp = 0.0
	if ((U_m <= 0.0036) || (!(U_m <= 6.9e+171) && (U_m <= 3.6e+186)))
		tmp = Float64(J * Float64(-2.0 * cos(Float64(K * 0.5))));
	else
		tmp = Float64(-U_m);
	end
	return tmp
end

MATLAB

U_m = abs(U);
function tmp_2 = code(J, K, U_m)
	tmp = 0.0;
	if ((U_m <= 0.0036) || (~((U_m <= 6.9e+171)) && (U_m <= 3.6e+186)))
		tmp = J * (-2.0 * cos((K * 0.5)));
	else
		tmp = -U_m;
	end
	tmp_2 = tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := If[Or[LessEqual[U$95$m, 0.0036], And[N[Not[LessEqual[U$95$m, 6.9e+171]], $MachinePrecision], LessEqual[U$95$m, 3.6e+186]]], N[(J * N[(-2.0 * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-U$95$m)]

Derivation

1. Split input into 2 regimes

2. if U < 0.0035999999999999999 or 6.90000000000000009e171 < U < 3.6000000000000002e186

   1. Initial program 77.8%

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

   3. Simplified 89.1%

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

   5. Taylor expanded in J around inf 61.7%

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

      1. *-commutative 61.7%

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

      2. *-commutative 61.7%

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

      3. metadata-eval 61.7%

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

      4. distribute-rgt-neg-in 61.7%

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

      5. associate-*r* 61.7%

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

      6. distribute-rgt-neg-in 61.7%

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

      7. *-commutative 61.7%

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

      8. distribute-lft-neg-in 61.7%

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

      9. metadata-eval 61.7%

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

      10. *-commutative 61.7%

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

   7. Simplified 61.7%

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

   if 0.0035999999999999999 < U < 6.90000000000000009e171 or 3.6000000000000002e186 < U

   1. Initial program 52.2%

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

   3. Simplified 79.5%

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

   5. Taylor expanded in J around 0 25.5%

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

      1. neg-mul-1 25.5%

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

   7. Simplified 25.5%

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

4. Final simplification 53.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 0.0036 \lor \neg \left(U \leq 6.9 \cdot 10^{+171}\right) \land U \leq 3.6 \cdot 10^{+186}:\\ \;\;\;\;J \cdot \left(-2 \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 56.6% accurate, 3.4× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := J \cdot \left(-2 \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{if}\;U\_m \leq 0.0031:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;U\_m \leq 6.6 \cdot 10^{+171}:\\ \;\;\;\;\frac{-2 \cdot {J}^{2}}{U\_m} - U\_m\\ \mathbf{elif}\;U\_m \leq 3.6 \cdot 10^{+186}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;-U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (* J (* -2.0 (cos (* K 0.5))))))
   (if (<= U_m 0.0031)
     t_0
     (if (<= U_m 6.6e+171)
       (- (/ (* -2.0 (pow J 2.0)) U_m) U_m)
       (if (<= U_m 3.6e+186) t_0 (- U_m))))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = J * (-2.0 * cos((K * 0.5)));
	double tmp;
	if (U_m <= 0.0031) {
		tmp = t_0;
	} else if (U_m <= 6.6e+171) {
		tmp = ((-2.0 * pow(J, 2.0)) / U_m) - U_m;
	} else if (U_m <= 3.6e+186) {
		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 = j * ((-2.0d0) * cos((k * 0.5d0)))
    if (u_m <= 0.0031d0) then
        tmp = t_0
    else if (u_m <= 6.6d+171) then
        tmp = (((-2.0d0) * (j ** 2.0d0)) / u_m) - u_m
    else if (u_m <= 3.6d+186) 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 = J * (-2.0 * Math.cos((K * 0.5)));
	double tmp;
	if (U_m <= 0.0031) {
		tmp = t_0;
	} else if (U_m <= 6.6e+171) {
		tmp = ((-2.0 * Math.pow(J, 2.0)) / U_m) - U_m;
	} else if (U_m <= 3.6e+186) {
		tmp = t_0;
	} else {
		tmp = -U_m;
	}
	return tmp;
}

Python

U_m = math.fabs(U)
def code(J, K, U_m):
	t_0 = J * (-2.0 * math.cos((K * 0.5)))
	tmp = 0
	if U_m <= 0.0031:
		tmp = t_0
	elif U_m <= 6.6e+171:
		tmp = ((-2.0 * math.pow(J, 2.0)) / U_m) - U_m
	elif U_m <= 3.6e+186:
		tmp = t_0
	else:
		tmp = -U_m
	return tmp

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = Float64(J * Float64(-2.0 * cos(Float64(K * 0.5))))
	tmp = 0.0
	if (U_m <= 0.0031)
		tmp = t_0;
	elseif (U_m <= 6.6e+171)
		tmp = Float64(Float64(Float64(-2.0 * (J ^ 2.0)) / U_m) - U_m);
	elseif (U_m <= 3.6e+186)
		tmp = t_0;
	else
		tmp = Float64(-U_m);
	end
	return tmp
end

MATLAB

U_m = abs(U);
function tmp_2 = code(J, K, U_m)
	t_0 = J * (-2.0 * cos((K * 0.5)));
	tmp = 0.0;
	if (U_m <= 0.0031)
		tmp = t_0;
	elseif (U_m <= 6.6e+171)
		tmp = ((-2.0 * (J ^ 2.0)) / U_m) - U_m;
	elseif (U_m <= 3.6e+186)
		tmp = t_0;
	else
		tmp = -U_m;
	end
	tmp_2 = tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := Block[{t$95$0 = N[(J * N[(-2.0 * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[U$95$m, 0.0031], t$95$0, If[LessEqual[U$95$m, 6.6e+171], N[(N[(N[(-2.0 * N[Power[J, 2.0], $MachinePrecision]), $MachinePrecision] / U$95$m), $MachinePrecision] - U$95$m), $MachinePrecision], If[LessEqual[U$95$m, 3.6e+186], t$95$0, (-U$95$m)]]]]

Derivation

1. Split input into 3 regimes

2. if U < 0.00309999999999999989 or 6.59999999999999982e171 < U < 3.6000000000000002e186

   1. Initial program 77.8%

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

   3. Simplified 89.1%

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

   5. Taylor expanded in J around inf 61.7%

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

      1. *-commutative 61.7%

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

      2. *-commutative 61.7%

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

      3. metadata-eval 61.7%

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

      4. distribute-rgt-neg-in 61.7%

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

      5. associate-*r* 61.7%

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

      6. distribute-rgt-neg-in 61.7%

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

      7. *-commutative 61.7%

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

      8. distribute-lft-neg-in 61.7%

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

      9. metadata-eval 61.7%

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

      10. *-commutative 61.7%

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

   7. Simplified 61.7%

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

   if 0.00309999999999999989 < U < 6.59999999999999982e171

   1. Initial program 63.5%

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

   3. Simplified 89.8%

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

      1. expm1-log1p-u 12.7%

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

      2. expm1-undefine 12.7%

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

   6. Applied egg-rr 12.7%

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

      1. expm1-define 12.7%

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

      2. *-commutative 12.7%

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

      3. times-frac 12.7%

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

      4. *-commutative 12.7%

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

      5. associate-*r/ 12.7%

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

      6. associate-/r* 12.7%

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

      7. *-commutative 12.7%

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

      8. *-commutative 12.7%

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

      9. *-commutative 12.7%

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

   8. Simplified 12.7%

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

   9. Taylor expanded in J around 0 17.2%

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

       1. neg-mul-1 17.2%

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

       2. unsub-neg 17.2%

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

       3. associate-*r/ 17.2%

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

       4. *-commutative 17.2%

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

       5. unpow2 17.2%

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

       6. unpow2 17.2%

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

       7. swap-sqr 17.2%

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

       8. unpow2 17.2%

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

   11. Simplified 17.2%

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

   12. Taylor expanded in K around 0 16.5%

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

   if 3.6000000000000002e186 < U

   1. Initial program 39.7%

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

   3. Simplified 68.1%

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

   5. Taylor expanded in J around 0 36.1%

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

      1. neg-mul-1 36.1%

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

   7. Simplified 36.1%

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

4. Final simplification 53.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 0.0031:\\ \;\;\;\;J \cdot \left(-2 \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{elif}\;U \leq 6.6 \cdot 10^{+171}:\\ \;\;\;\;\frac{-2 \cdot {J}^{2}}{U} - U\\ \mathbf{elif}\;U \leq 3.6 \cdot 10^{+186}:\\ \;\;\;\;J \cdot \left(-2 \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 30.1% accurate, 19.0× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} \mathbf{if}\;U\_m \leq 10^{-99} \lor \neg \left(U\_m \leq 3.6 \cdot 10^{-53}\right) \land U\_m \leq 3.8 \cdot 10^{-38}:\\ \;\;\;\;J \cdot \left(\frac{U\_m}{J} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;-U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (if (or (<= U_m 1e-99) (and (not (<= U_m 3.6e-53)) (<= U_m 3.8e-38)))
   (* J (+ (/ U_m J) -1.0))
   (- U_m)))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double tmp;
	if ((U_m <= 1e-99) || (!(U_m <= 3.6e-53) && (U_m <= 3.8e-38))) {
		tmp = J * ((U_m / J) + -1.0);
	} 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) :: tmp
    if ((u_m <= 1d-99) .or. (.not. (u_m <= 3.6d-53)) .and. (u_m <= 3.8d-38)) then
        tmp = j * ((u_m / j) + (-1.0d0))
    else
        tmp = -u_m
    end if
    code = tmp
end function

Java

U_m = Math.abs(U);
public static double code(double J, double K, double U_m) {
	double tmp;
	if ((U_m <= 1e-99) || (!(U_m <= 3.6e-53) && (U_m <= 3.8e-38))) {
		tmp = J * ((U_m / J) + -1.0);
	} else {
		tmp = -U_m;
	}
	return tmp;
}

Python

U_m = math.fabs(U)
def code(J, K, U_m):
	tmp = 0
	if (U_m <= 1e-99) or (not (U_m <= 3.6e-53) and (U_m <= 3.8e-38)):
		tmp = J * ((U_m / J) + -1.0)
	else:
		tmp = -U_m
	return tmp

Julia

U_m = abs(U)
function code(J, K, U_m)
	tmp = 0.0
	if ((U_m <= 1e-99) || (!(U_m <= 3.6e-53) && (U_m <= 3.8e-38)))
		tmp = Float64(J * Float64(Float64(U_m / J) + -1.0));
	else
		tmp = Float64(-U_m);
	end
	return tmp
end

MATLAB

U_m = abs(U);
function tmp_2 = code(J, K, U_m)
	tmp = 0.0;
	if ((U_m <= 1e-99) || (~((U_m <= 3.6e-53)) && (U_m <= 3.8e-38)))
		tmp = J * ((U_m / J) + -1.0);
	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, 1e-99], And[N[Not[LessEqual[U$95$m, 3.6e-53]], $MachinePrecision], LessEqual[U$95$m, 3.8e-38]]], N[(J * N[(N[(U$95$m / J), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], (-U$95$m)]

Derivation

1. Split input into 2 regimes

2. if U < 1e-99 or 3.5999999999999999e-53 < U < 3.8e-38

   1. Initial program 74.2%

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

   3. Simplified 86.8%

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

      1. expm1-log1p-u 24.5%

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

      2. expm1-undefine 24.3%

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

   6. Applied egg-rr 24.3%

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

      1. expm1-define 24.5%

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

      2. *-commutative 24.5%

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

      3. times-frac 24.5%

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

      4. *-commutative 24.5%

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

      5. associate-*r/ 24.5%

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

      6. associate-/r* 24.5%

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

      7. *-commutative 24.5%

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

      8. *-commutative 24.5%

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

      9. *-commutative 24.5%

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

   8. Simplified 24.5%

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

      1. expm1-undefine 24.3%

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

   10. Applied egg-rr 86.7%

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

   11. Taylor expanded in U around -inf 26.7%

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

   if 1e-99 < U < 3.5999999999999999e-53 or 3.8e-38 < U

   1. Initial program 68.9%

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

   3. Simplified 87.5%

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

   5. Taylor expanded in J around 0 19.7%

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

      1. neg-mul-1 19.7%

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

   7. Simplified 19.7%

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

4. Final simplification 24.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 10^{-99} \lor \neg \left(U \leq 3.6 \cdot 10^{-53}\right) \land U \leq 3.8 \cdot 10^{-38}:\\ \;\;\;\;J \cdot \left(\frac{U}{J} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 26.4% accurate, 24.6× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} \mathbf{if}\;K \leq 4.5 \cdot 10^{+74} \lor \neg \left(K \leq 6.4 \cdot 10^{+199}\right) \land K \leq 7 \cdot 10^{+224}:\\ \;\;\;\;-U\_m\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (if (or (<= K 4.5e+74) (and (not (<= K 6.4e+199)) (<= K 7e+224)))
   (- U_m)
   U_m))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double tmp;
	if ((K <= 4.5e+74) || (!(K <= 6.4e+199) && (K <= 7e+224))) {
		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 <= 4.5d+74) .or. (.not. (k <= 6.4d+199)) .and. (k <= 7d+224)) 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 <= 4.5e+74) || (!(K <= 6.4e+199) && (K <= 7e+224))) {
		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 <= 4.5e+74) or (not (K <= 6.4e+199) and (K <= 7e+224)):
		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 <= 4.5e+74) || (!(K <= 6.4e+199) && (K <= 7e+224)))
		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 <= 4.5e+74) || (~((K <= 6.4e+199)) && (K <= 7e+224)))
		tmp = -U_m;
	else
		tmp = U_m;
	end
	tmp_2 = tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := If[Or[LessEqual[K, 4.5e+74], And[N[Not[LessEqual[K, 6.4e+199]], $MachinePrecision], LessEqual[K, 7e+224]]], (-U$95$m), U$95$m]

Derivation

1. Split input into 2 regimes

2. if K < 4.5e74 or 6.40000000000000011e199 < K < 7e224

   1. Initial program 72.0%

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

   3. Simplified 86.9%

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

   5. Taylor expanded in J around 0 16.7%

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

      1. neg-mul-1 16.7%

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

   7. Simplified 16.7%

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

   if 4.5e74 < K < 6.40000000000000011e199 or 7e224 < K

   1. Initial program 73.7%

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

   3. Simplified 87.7%

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

      1. expm1-log1p-u 47.0%

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

      2. expm1-undefine 46.8%

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

   6. Applied egg-rr 46.8%

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

      1. expm1-define 47.0%

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

      2. *-commutative 47.0%

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

      3. times-frac 47.0%

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

      4. *-commutative 47.0%

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

      5. associate-*r/ 47.0%

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

      6. associate-/r* 47.0%

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

      7. *-commutative 47.0%

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

      8. *-commutative 47.0%

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

      9. *-commutative 47.0%

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

   8. Simplified 47.0%

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

   9. Taylor expanded in U around -inf 29.5%

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

4. Final simplification 19.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;K \leq 4.5 \cdot 10^{+74} \lor \neg \left(K \leq 6.4 \cdot 10^{+199}\right) \land K \leq 7 \cdot 10^{+224}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 31.1% accurate, 35.0× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} \mathbf{if}\;U\_m \leq 2.25 \cdot 10^{+17}:\\ \;\;\;\;J \cdot \left(-1 - \frac{U\_m}{J}\right)\\ \mathbf{else}:\\ \;\;\;\;-U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (if (<= U_m 2.25e+17) (* J (- -1.0 (/ U_m J))) (- U_m)))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double tmp;
	if (U_m <= 2.25e+17) {
		tmp = J * (-1.0 - (U_m / 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 <= 2.25d+17) then
        tmp = j * ((-1.0d0) - (u_m / 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 <= 2.25e+17) {
		tmp = J * (-1.0 - (U_m / J));
	} else {
		tmp = -U_m;
	}
	return tmp;
}

Python

U_m = math.fabs(U)
def code(J, K, U_m):
	tmp = 0
	if U_m <= 2.25e+17:
		tmp = J * (-1.0 - (U_m / J))
	else:
		tmp = -U_m
	return tmp

Julia

U_m = abs(U)
function code(J, K, U_m)
	tmp = 0.0
	if (U_m <= 2.25e+17)
		tmp = Float64(J * Float64(-1.0 - Float64(U_m / 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 <= 2.25e+17)
		tmp = J * (-1.0 - (U_m / 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, 2.25e+17], N[(J * N[(-1.0 - N[(U$95$m / J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-U$95$m)]

Derivation

1. Split input into 2 regimes

2. if U < 2.25e17

   1. Initial program 77.6%

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

   3. Simplified 89.0%

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

      1. expm1-log1p-u 24.8%

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

      2. expm1-undefine 24.7%

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

   6. Applied egg-rr 24.7%

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

      1. expm1-define 24.8%

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

      2. *-commutative 24.8%

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

      3. times-frac 24.8%

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

      4. *-commutative 24.8%

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

      5. associate-*r/ 24.8%

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

      6. associate-/r* 24.8%

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

      7. *-commutative 24.8%

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

      8. *-commutative 24.8%

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

      9. *-commutative 24.8%

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

   8. Simplified 24.8%

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

      1. expm1-undefine 24.7%

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

   10. Applied egg-rr 88.9%

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

   11. Taylor expanded in U around inf 18.7%

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

       1. associate-*r/ 18.7%

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

       2. mul-1-neg 18.7%

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

   13. Simplified 18.7%

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

   if 2.25e17 < U

   1. Initial program 53.9%

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

   3. Simplified 80.3%

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

   5. Taylor expanded in J around 0 24.5%

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

      1. neg-mul-1 24.5%

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

   7. Simplified 24.5%

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

4. Final simplification 20.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 2.25 \cdot 10^{+17}:\\ \;\;\;\;J \cdot \left(-1 - \frac{U}{J}\right)\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 26.9% accurate, 420.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.3%

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

3. Simplified 87.1%

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

   1. expm1-log1p-u 23.3%

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

   2. expm1-undefine 23.2%

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

6. Applied egg-rr 23.2%

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

   1. expm1-define 23.3%

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

   2. *-commutative 23.3%

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

   3. times-frac 23.3%

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

   4. *-commutative 23.3%

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

   5. associate-*r/ 23.3%

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

   6. associate-/r* 23.3%

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

   7. *-commutative 23.3%

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

   8. *-commutative 23.3%

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

   9. *-commutative 23.3%

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

8. Simplified 23.3%

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

9. Taylor expanded in U around -inf 31.9%

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

10. Final simplification 31.9%

    \[\leadsto U \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024043 
(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)))))