Maksimov and Kolovsky, Equation (3)

Percentage Accurate: 73.9% → 97.2%

Time: 20.9s

Alternatives: 7

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.9% 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: 97.2% accurate, 0.6× speedup

Math

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

FPCore

U_m = (fabs.f64 U)
J_m = (fabs.f64 J)
J_s = (copysign.f64 1 J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (*
    J_s
    (if (<=
         (*
          (* (* -2.0 J_m) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* t_0 (* J_m 2.0))) 2.0))))
         (- INFINITY))
      (- U_m)
      (* -2.0 (* t_0 (* J_m (hypot 1.0 (/ (/ (/ U_m 2.0) J_m) t_0)))))))))

C

U_m = fabs(U);
J_m = fabs(J);
J_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if ((((-2.0 * J_m) * t_0) * sqrt((1.0 + pow((U_m / (t_0 * (J_m * 2.0))), 2.0)))) <= -((double) INFINITY)) {
		tmp = -U_m;
	} else {
		tmp = -2.0 * (t_0 * (J_m * hypot(1.0, (((U_m / 2.0) / J_m) / t_0))));
	}
	return J_s * tmp;
}

Java

U_m = Math.abs(U);
J_m = Math.abs(J);
J_s = Math.copySign(1.0, J);
public static double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = Math.cos((K / 2.0));
	double tmp;
	if ((((-2.0 * J_m) * t_0) * Math.sqrt((1.0 + Math.pow((U_m / (t_0 * (J_m * 2.0))), 2.0)))) <= -Double.POSITIVE_INFINITY) {
		tmp = -U_m;
	} else {
		tmp = -2.0 * (t_0 * (J_m * Math.hypot(1.0, (((U_m / 2.0) / J_m) / t_0))));
	}
	return J_s * tmp;
}

Python

U_m = math.fabs(U)
J_m = math.fabs(J)
J_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if (((-2.0 * J_m) * t_0) * math.sqrt((1.0 + math.pow((U_m / (t_0 * (J_m * 2.0))), 2.0)))) <= -math.inf:
		tmp = -U_m
	else:
		tmp = -2.0 * (t_0 * (J_m * math.hypot(1.0, (((U_m / 2.0) / J_m) / t_0))))
	return J_s * tmp

Julia

U_m = abs(U)
J_m = abs(J)
J_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (Float64(Float64(Float64(-2.0 * J_m) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_0 * Float64(J_m * 2.0))) ^ 2.0)))) <= Float64(-Inf))
		tmp = Float64(-U_m);
	else
		tmp = Float64(-2.0 * Float64(t_0 * Float64(J_m * hypot(1.0, Float64(Float64(Float64(U_m / 2.0) / J_m) / t_0)))));
	end
	return Float64(J_s * tmp)
end

MATLAB

U_m = abs(U);
J_m = abs(J);
J_s = sign(J) * abs(1.0);
function tmp_2 = code(J_s, J_m, K, U_m)
	t_0 = cos((K / 2.0));
	tmp = 0.0;
	if ((((-2.0 * J_m) * t_0) * sqrt((1.0 + ((U_m / (t_0 * (J_m * 2.0))) ^ 2.0)))) <= -Inf)
		tmp = -U_m;
	else
		tmp = -2.0 * (t_0 * (J_m * hypot(1.0, (((U_m / 2.0) / J_m) / t_0))));
	end
	tmp_2 = J_s * tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J_m = N[Abs[J], $MachinePrecision]
J_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, N[(J$95$s * If[LessEqual[N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(t$95$0 * N[(J$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], (-Infinity)], (-U$95$m), N[(-2.0 * N[(t$95$0 * N[(J$95$m * N[Sqrt[1.0 ^ 2 + N[(N[(N[(U$95$m / 2.0), $MachinePrecision] / J$95$m), $MachinePrecision] / t$95$0), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 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 5.0%

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

      1. unpow2 5.0%

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

      2. hypot-1-def 63.9%

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

      3. associate-/r* 63.7%

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

      4. cos-neg 63.7%

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

      5. distribute-frac-neg 63.7%

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

      6. associate-/r* 63.9%

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

      7. hypot-1-def 5.0%

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

      8. unpow2 5.0%

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

   3. Simplified 63.7%

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

   5. Taylor expanded in J around 0 39.1%

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

      1. neg-mul-1 39.1%

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

   7. Simplified 39.1%

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

   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))))

   1. Initial program 86.3%

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

   3. Simplified 93.2%

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

4. Final simplification 87.7%

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

Alternative 2: 78.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J_m = \left|J\right| \\ J_s = \mathsf{copysign}\left(1, J\right) \\ J\_s \cdot \begin{array}{l} \mathbf{if}\;J\_m \leq 7.6 \cdot 10^{-162}:\\ \;\;\;\;-U\_m\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J\_m \cdot \mathsf{hypot}\left(1, U\_m \cdot \frac{0.5}{J\_m}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J_m = (fabs.f64 J)
J_s = (copysign.f64 1 J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (*
  J_s
  (if (<= J_m 7.6e-162)
    (- U_m)
    (* -2.0 (* (cos (/ K 2.0)) (* J_m (hypot 1.0 (* U_m (/ 0.5 J_m)))))))))

C

U_m = fabs(U);
J_m = fabs(J);
J_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double tmp;
	if (J_m <= 7.6e-162) {
		tmp = -U_m;
	} else {
		tmp = -2.0 * (cos((K / 2.0)) * (J_m * hypot(1.0, (U_m * (0.5 / J_m)))));
	}
	return J_s * tmp;
}

Java

U_m = Math.abs(U);
J_m = Math.abs(J);
J_s = Math.copySign(1.0, J);
public static double code(double J_s, double J_m, double K, double U_m) {
	double tmp;
	if (J_m <= 7.6e-162) {
		tmp = -U_m;
	} else {
		tmp = -2.0 * (Math.cos((K / 2.0)) * (J_m * Math.hypot(1.0, (U_m * (0.5 / J_m)))));
	}
	return J_s * tmp;
}

Python

U_m = math.fabs(U)
J_m = math.fabs(J)
J_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	tmp = 0
	if J_m <= 7.6e-162:
		tmp = -U_m
	else:
		tmp = -2.0 * (math.cos((K / 2.0)) * (J_m * math.hypot(1.0, (U_m * (0.5 / J_m)))))
	return J_s * tmp

Julia

U_m = abs(U)
J_m = abs(J)
J_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	tmp = 0.0
	if (J_m <= 7.6e-162)
		tmp = Float64(-U_m);
	else
		tmp = Float64(-2.0 * Float64(cos(Float64(K / 2.0)) * Float64(J_m * hypot(1.0, Float64(U_m * Float64(0.5 / J_m))))));
	end
	return Float64(J_s * tmp)
end

MATLAB

U_m = abs(U);
J_m = abs(J);
J_s = sign(J) * abs(1.0);
function tmp_2 = code(J_s, J_m, K, U_m)
	tmp = 0.0;
	if (J_m <= 7.6e-162)
		tmp = -U_m;
	else
		tmp = -2.0 * (cos((K / 2.0)) * (J_m * hypot(1.0, (U_m * (0.5 / J_m)))));
	end
	tmp_2 = J_s * tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J_m = N[Abs[J], $MachinePrecision]
J_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := N[(J$95$s * If[LessEqual[J$95$m, 7.6e-162], (-U$95$m), N[(-2.0 * N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(J$95$m * N[Sqrt[1.0 ^ 2 + N[(U$95$m * N[(0.5 / J$95$m), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if J < 7.6000000000000001e-162

   1. Initial program 69.9%

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

      1. unpow2 69.9%

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

      2. hypot-1-def 86.4%

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

      3. associate-/r* 86.3%

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

      4. cos-neg 86.3%

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

      5. distribute-frac-neg 86.3%

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

      6. associate-/r* 86.4%

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

      7. hypot-1-def 69.9%

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

      8. unpow2 69.9%

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

   3. Simplified 86.3%

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

   5. Taylor expanded in J around 0 22.4%

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

      1. neg-mul-1 22.4%

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

   7. Simplified 22.4%

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

   if 7.6000000000000001e-162 < J

   1. Initial program 91.9%

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

   3. Simplified 96.8%

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

   5. Taylor expanded in K around 0 81.9%

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

      1. associate-*r/ 81.9%

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

      2. *-commutative 81.9%

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

      3. associate-*r/ 81.9%

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

   7. Simplified 81.9%

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

4. Final simplification 44.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq 7.6 \cdot 10^{-162}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \mathsf{hypot}\left(1, U \cdot \frac{0.5}{J}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 69.5% accurate, 3.5× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J_m = \left|J\right| \\ J_s = \mathsf{copysign}\left(1, J\right) \\ J\_s \cdot \begin{array}{l} \mathbf{if}\;U\_m \leq 1.1 \cdot 10^{-68}:\\ \;\;\;\;\left(-2 \cdot J\_m\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{elif}\;U\_m \leq 5.4 \cdot 10^{+105}:\\ \;\;\;\;-2 \cdot \left(J\_m \cdot \mathsf{hypot}\left(1, U\_m \cdot \frac{0.5}{J\_m}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J_m = (fabs.f64 J)
J_s = (copysign.f64 1 J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (*
  J_s
  (if (<= U_m 1.1e-68)
    (* (* -2.0 J_m) (cos (* K 0.5)))
    (if (<= U_m 5.4e+105)
      (* -2.0 (* J_m (hypot 1.0 (* U_m (/ 0.5 J_m)))))
      (- U_m)))))

C

U_m = fabs(U);
J_m = fabs(J);
J_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double tmp;
	if (U_m <= 1.1e-68) {
		tmp = (-2.0 * J_m) * cos((K * 0.5));
	} else if (U_m <= 5.4e+105) {
		tmp = -2.0 * (J_m * hypot(1.0, (U_m * (0.5 / J_m))));
	} else {
		tmp = -U_m;
	}
	return J_s * tmp;
}

Java

U_m = Math.abs(U);
J_m = Math.abs(J);
J_s = Math.copySign(1.0, J);
public static double code(double J_s, double J_m, double K, double U_m) {
	double tmp;
	if (U_m <= 1.1e-68) {
		tmp = (-2.0 * J_m) * Math.cos((K * 0.5));
	} else if (U_m <= 5.4e+105) {
		tmp = -2.0 * (J_m * Math.hypot(1.0, (U_m * (0.5 / J_m))));
	} else {
		tmp = -U_m;
	}
	return J_s * tmp;
}

Python

U_m = math.fabs(U)
J_m = math.fabs(J)
J_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	tmp = 0
	if U_m <= 1.1e-68:
		tmp = (-2.0 * J_m) * math.cos((K * 0.5))
	elif U_m <= 5.4e+105:
		tmp = -2.0 * (J_m * math.hypot(1.0, (U_m * (0.5 / J_m))))
	else:
		tmp = -U_m
	return J_s * tmp

Julia

U_m = abs(U)
J_m = abs(J)
J_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	tmp = 0.0
	if (U_m <= 1.1e-68)
		tmp = Float64(Float64(-2.0 * J_m) * cos(Float64(K * 0.5)));
	elseif (U_m <= 5.4e+105)
		tmp = Float64(-2.0 * Float64(J_m * hypot(1.0, Float64(U_m * Float64(0.5 / J_m)))));
	else
		tmp = Float64(-U_m);
	end
	return Float64(J_s * tmp)
end

MATLAB

U_m = abs(U);
J_m = abs(J);
J_s = sign(J) * abs(1.0);
function tmp_2 = code(J_s, J_m, K, U_m)
	tmp = 0.0;
	if (U_m <= 1.1e-68)
		tmp = (-2.0 * J_m) * cos((K * 0.5));
	elseif (U_m <= 5.4e+105)
		tmp = -2.0 * (J_m * hypot(1.0, (U_m * (0.5 / J_m))));
	else
		tmp = -U_m;
	end
	tmp_2 = J_s * tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J_m = N[Abs[J], $MachinePrecision]
J_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := N[(J$95$s * If[LessEqual[U$95$m, 1.1e-68], N[(N[(-2.0 * J$95$m), $MachinePrecision] * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[U$95$m, 5.4e+105], N[(-2.0 * N[(J$95$m * N[Sqrt[1.0 ^ 2 + N[(U$95$m * N[(0.5 / J$95$m), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-U$95$m)]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if U < 1.10000000000000001e-68

   1. Initial program 83.7%

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

      1. unpow2 83.7%

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

      2. hypot-1-def 93.5%

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

      3. associate-/r* 93.4%

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

      4. cos-neg 93.4%

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

      5. distribute-frac-neg 93.4%

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

      6. associate-/r* 93.5%

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

      7. hypot-1-def 83.7%

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

      8. unpow2 83.7%

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

   3. Simplified 93.4%

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

   5. Taylor expanded in J around inf 65.2%

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

      1. associate-*r* 65.2%

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

      2. *-commutative 65.2%

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

   7. Simplified 65.2%

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

   if 1.10000000000000001e-68 < U < 5.40000000000000033e105

   1. Initial program 86.2%

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

   3. Taylor expanded in K around 0 75.1%

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

      1. *-commutative 75.1%

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

      2. associate-*l/ 75.1%

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

      3. unpow2 75.1%

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

      4. metadata-eval 75.1%

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

      5. swap-sqr 75.1%

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

      6. unpow2 75.1%

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

      7. unpow2 75.1%

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

      8. associate-/r* 75.2%

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

      9. unpow2 75.2%

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

      10. associate-*r/ 75.2%

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

      11. associate-*r/ 75.2%

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

      12. associate-*l/ 75.1%

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

      13. associate-*r/ 75.1%

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

      14. unpow2 75.1%

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

   5. Simplified 75.1%

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

   6. Taylor expanded in K around 0 55.1%

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

      1. metadata-eval 55.1%

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

      2. associate-*r/ 55.1%

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

      3. metadata-eval 55.1%

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

      4. unpow2 55.1%

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

      5. swap-sqr 55.1%

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

      6. *-commutative 55.1%

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

      7. *-commutative 55.1%

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

      8. unpow2 55.1%

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

      9. times-frac 55.2%

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

      10. associate-*r/ 55.1%

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

      11. associate-*r/ 55.1%

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

      12. hypot-undefine 64.0%

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

   8. Simplified 64.0%

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

   if 5.40000000000000033e105 < U

   1. Initial program 40.0%

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

      1. unpow2 40.0%

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

      2. hypot-1-def 68.6%

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

      3. associate-/r* 68.4%

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

      4. cos-neg 68.4%

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

      5. distribute-frac-neg 68.4%

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

      6. associate-/r* 68.6%

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

      7. hypot-1-def 40.0%

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

      8. unpow2 40.0%

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

   3. Simplified 68.4%

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

   5. Taylor expanded in J around 0 36.9%

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

      1. neg-mul-1 36.9%

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

   7. Simplified 36.9%

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

4. Final simplification 61.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 1.1 \cdot 10^{-68}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{elif}\;U \leq 5.4 \cdot 10^{+105}:\\ \;\;\;\;-2 \cdot \left(J \cdot \mathsf{hypot}\left(1, U \cdot \frac{0.5}{J}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 66.6% accurate, 3.7× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J_m = \left|J\right| \\ J_s = \mathsf{copysign}\left(1, J\right) \\ J\_s \cdot \begin{array}{l} \mathbf{if}\;U\_m \leq 1.5 \cdot 10^{+39}:\\ \;\;\;\;\left(-2 \cdot J\_m\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;-U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J_m = (fabs.f64 J)
J_s = (copysign.f64 1 J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (* J_s (if (<= U_m 1.5e+39) (* (* -2.0 J_m) (cos (* K 0.5))) (- U_m))))

C

U_m = fabs(U);
J_m = fabs(J);
J_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double tmp;
	if (U_m <= 1.5e+39) {
		tmp = (-2.0 * J_m) * cos((K * 0.5));
	} else {
		tmp = -U_m;
	}
	return J_s * tmp;
}

Fortran

U_m = abs(U)
J_m = abs(J)
J_s = copysign(1.0d0, J)
real(8) function code(j_s, j_m, k, u_m)
    real(8), intent (in) :: j_s
    real(8), intent (in) :: j_m
    real(8), intent (in) :: k
    real(8), intent (in) :: u_m
    real(8) :: tmp
    if (u_m <= 1.5d+39) then
        tmp = ((-2.0d0) * j_m) * cos((k * 0.5d0))
    else
        tmp = -u_m
    end if
    code = j_s * tmp
end function

Java

U_m = Math.abs(U);
J_m = Math.abs(J);
J_s = Math.copySign(1.0, J);
public static double code(double J_s, double J_m, double K, double U_m) {
	double tmp;
	if (U_m <= 1.5e+39) {
		tmp = (-2.0 * J_m) * Math.cos((K * 0.5));
	} else {
		tmp = -U_m;
	}
	return J_s * tmp;
}

Python

U_m = math.fabs(U)
J_m = math.fabs(J)
J_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	tmp = 0
	if U_m <= 1.5e+39:
		tmp = (-2.0 * J_m) * math.cos((K * 0.5))
	else:
		tmp = -U_m
	return J_s * tmp

Julia

U_m = abs(U)
J_m = abs(J)
J_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	tmp = 0.0
	if (U_m <= 1.5e+39)
		tmp = Float64(Float64(-2.0 * J_m) * cos(Float64(K * 0.5)));
	else
		tmp = Float64(-U_m);
	end
	return Float64(J_s * tmp)
end

MATLAB

U_m = abs(U);
J_m = abs(J);
J_s = sign(J) * abs(1.0);
function tmp_2 = code(J_s, J_m, K, U_m)
	tmp = 0.0;
	if (U_m <= 1.5e+39)
		tmp = (-2.0 * J_m) * cos((K * 0.5));
	else
		tmp = -U_m;
	end
	tmp_2 = J_s * tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J_m = N[Abs[J], $MachinePrecision]
J_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := N[(J$95$s * If[LessEqual[U$95$m, 1.5e+39], N[(N[(-2.0 * J$95$m), $MachinePrecision] * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], (-U$95$m)]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if U < 1.5e39

   1. Initial program 84.6%

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

      1. unpow2 84.6%

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

      2. hypot-1-def 93.9%

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

      3. associate-/r* 93.8%

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

      4. cos-neg 93.8%

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

      5. distribute-frac-neg 93.8%

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

      6. associate-/r* 93.9%

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

      7. hypot-1-def 84.6%

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

      8. unpow2 84.6%

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

   3. Simplified 93.8%

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

   5. Taylor expanded in J around inf 64.7%

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

      1. associate-*r* 64.7%

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

      2. *-commutative 64.7%

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

   7. Simplified 64.7%

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

   if 1.5e39 < U

   1. Initial program 50.9%

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

      1. unpow2 50.9%

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

      2. hypot-1-def 75.4%

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

      3. associate-/r* 75.2%

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

      4. cos-neg 75.2%

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

      5. distribute-frac-neg 75.2%

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

      6. associate-/r* 75.4%

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

      7. hypot-1-def 50.9%

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

      8. unpow2 50.9%

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

   3. Simplified 75.2%

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

   5. Taylor expanded in J around 0 32.9%

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

      1. neg-mul-1 32.9%

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

   7. Simplified 32.9%

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

4. Final simplification 58.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 1.5 \cdot 10^{+39}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 38.9% accurate, 34.9× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J_m = \left|J\right| \\ J_s = \mathsf{copysign}\left(1, J\right) \\ J\_s \cdot \begin{array}{l} \mathbf{if}\;K \leq 9 \cdot 10^{+66} \lor \neg \left(K \leq 2.2 \cdot 10^{+298}\right):\\ \;\;\;\;-U\_m\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J_m = (fabs.f64 J)
J_s = (copysign.f64 1 J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (* J_s (if (or (<= K 9e+66) (not (<= K 2.2e+298))) (- U_m) U_m)))

C

U_m = fabs(U);
J_m = fabs(J);
J_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double tmp;
	if ((K <= 9e+66) || !(K <= 2.2e+298)) {
		tmp = -U_m;
	} else {
		tmp = U_m;
	}
	return J_s * tmp;
}

Fortran

U_m = abs(U)
J_m = abs(J)
J_s = copysign(1.0d0, J)
real(8) function code(j_s, j_m, k, u_m)
    real(8), intent (in) :: j_s
    real(8), intent (in) :: j_m
    real(8), intent (in) :: k
    real(8), intent (in) :: u_m
    real(8) :: tmp
    if ((k <= 9d+66) .or. (.not. (k <= 2.2d+298))) then
        tmp = -u_m
    else
        tmp = u_m
    end if
    code = j_s * tmp
end function

Java

U_m = Math.abs(U);
J_m = Math.abs(J);
J_s = Math.copySign(1.0, J);
public static double code(double J_s, double J_m, double K, double U_m) {
	double tmp;
	if ((K <= 9e+66) || !(K <= 2.2e+298)) {
		tmp = -U_m;
	} else {
		tmp = U_m;
	}
	return J_s * tmp;
}

Python

U_m = math.fabs(U)
J_m = math.fabs(J)
J_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	tmp = 0
	if (K <= 9e+66) or not (K <= 2.2e+298):
		tmp = -U_m
	else:
		tmp = U_m
	return J_s * tmp

Julia

U_m = abs(U)
J_m = abs(J)
J_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	tmp = 0.0
	if ((K <= 9e+66) || !(K <= 2.2e+298))
		tmp = Float64(-U_m);
	else
		tmp = U_m;
	end
	return Float64(J_s * tmp)
end

MATLAB

U_m = abs(U);
J_m = abs(J);
J_s = sign(J) * abs(1.0);
function tmp_2 = code(J_s, J_m, K, U_m)
	tmp = 0.0;
	if ((K <= 9e+66) || ~((K <= 2.2e+298)))
		tmp = -U_m;
	else
		tmp = U_m;
	end
	tmp_2 = J_s * tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J_m = N[Abs[J], $MachinePrecision]
J_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := N[(J$95$s * If[Or[LessEqual[K, 9e+66], N[Not[LessEqual[K, 2.2e+298]], $MachinePrecision]], (-U$95$m), U$95$m]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if K < 8.9999999999999997e66 or 2.2000000000000001e298 < K

   1. Initial program 76.6%

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

      1. unpow2 76.6%

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

      2. hypot-1-def 88.7%

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

      3. associate-/r* 88.7%

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

      4. cos-neg 88.7%

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

      5. distribute-frac-neg 88.7%

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

      6. associate-/r* 88.7%

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

      7. hypot-1-def 76.6%

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

      8. unpow2 76.6%

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

   3. Simplified 88.7%

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

   5. Taylor expanded in J around 0 20.6%

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

      1. neg-mul-1 20.6%

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

   7. Simplified 20.6%

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

   if 8.9999999999999997e66 < K < 2.2000000000000001e298

   1. Initial program 84.7%

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

      1. unpow2 84.7%

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

      2. hypot-1-def 97.6%

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

      3. associate-/r* 97.4%

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

      4. cos-neg 97.4%

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

      5. distribute-frac-neg 97.4%

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

      6. associate-/r* 97.6%

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

      7. hypot-1-def 84.7%

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

      8. unpow2 84.7%

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

   3. Simplified 97.4%

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

   5. Taylor expanded in U around -inf 22.3%

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

4. Final simplification 20.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;K \leq 9 \cdot 10^{+66} \lor \neg \left(K \leq 2.2 \cdot 10^{+298}\right):\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 50.4% accurate, 52.4× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J_m = \left|J\right| \\ J_s = \mathsf{copysign}\left(1, J\right) \\ J\_s \cdot \begin{array}{l} \mathbf{if}\;J\_m \leq 1.65 \cdot 10^{+17}:\\ \;\;\;\;-U\_m\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot J\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J_m = (fabs.f64 J)
J_s = (copysign.f64 1 J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (* J_s (if (<= J_m 1.65e+17) (- U_m) (* -2.0 J_m))))

C

U_m = fabs(U);
J_m = fabs(J);
J_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double tmp;
	if (J_m <= 1.65e+17) {
		tmp = -U_m;
	} else {
		tmp = -2.0 * J_m;
	}
	return J_s * tmp;
}

Fortran

U_m = abs(U)
J_m = abs(J)
J_s = copysign(1.0d0, J)
real(8) function code(j_s, j_m, k, u_m)
    real(8), intent (in) :: j_s
    real(8), intent (in) :: j_m
    real(8), intent (in) :: k
    real(8), intent (in) :: u_m
    real(8) :: tmp
    if (j_m <= 1.65d+17) then
        tmp = -u_m
    else
        tmp = (-2.0d0) * j_m
    end if
    code = j_s * tmp
end function

Java

U_m = Math.abs(U);
J_m = Math.abs(J);
J_s = Math.copySign(1.0, J);
public static double code(double J_s, double J_m, double K, double U_m) {
	double tmp;
	if (J_m <= 1.65e+17) {
		tmp = -U_m;
	} else {
		tmp = -2.0 * J_m;
	}
	return J_s * tmp;
}

Python

U_m = math.fabs(U)
J_m = math.fabs(J)
J_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	tmp = 0
	if J_m <= 1.65e+17:
		tmp = -U_m
	else:
		tmp = -2.0 * J_m
	return J_s * tmp

Julia

U_m = abs(U)
J_m = abs(J)
J_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	tmp = 0.0
	if (J_m <= 1.65e+17)
		tmp = Float64(-U_m);
	else
		tmp = Float64(-2.0 * J_m);
	end
	return Float64(J_s * tmp)
end

MATLAB

U_m = abs(U);
J_m = abs(J);
J_s = sign(J) * abs(1.0);
function tmp_2 = code(J_s, J_m, K, U_m)
	tmp = 0.0;
	if (J_m <= 1.65e+17)
		tmp = -U_m;
	else
		tmp = -2.0 * J_m;
	end
	tmp_2 = J_s * tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J_m = N[Abs[J], $MachinePrecision]
J_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := N[(J$95$s * If[LessEqual[J$95$m, 1.65e+17], (-U$95$m), N[(-2.0 * J$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if J < 1.65e17

   1. Initial program 70.7%

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

      1. unpow2 70.7%

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

      2. hypot-1-def 87.1%

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

      3. associate-/r* 87.0%

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

      4. cos-neg 87.0%

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

      5. distribute-frac-neg 87.0%

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

      6. associate-/r* 87.1%

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

      7. hypot-1-def 70.7%

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

      8. unpow2 70.7%

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

   3. Simplified 87.0%

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

   5. Taylor expanded in J around 0 24.6%

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

      1. neg-mul-1 24.6%

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

   7. Simplified 24.6%

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

   if 1.65e17 < J

   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. Step-by-step derivation

      1. unpow2 99.8%

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

      2. hypot-1-def 99.8%

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

      3. associate-/r* 99.7%

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

      4. cos-neg 99.7%

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

      5. distribute-frac-neg 99.7%

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

      6. associate-/r* 99.8%

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

      7. hypot-1-def 99.8%

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

      8. unpow2 99.8%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in J around inf 79.0%

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

      1. associate-*r* 79.0%

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

      2. *-commutative 79.0%

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

   7. Simplified 79.0%

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

   8. Taylor expanded in K around 0 44.8%

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

4. Final simplification 29.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq 1.65 \cdot 10^{+17}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot J\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 14.2% accurate, 420.0× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J_m = \left|J\right| \\ J_s = \mathsf{copysign}\left(1, J\right) \\ J\_s \cdot U\_m \end{array} \]

FPCore

U_m = (fabs.f64 U)
J_m = (fabs.f64 J)
J_s = (copysign.f64 1 J)
(FPCore (J_s J_m K U_m) :precision binary64 (* J_s U_m))

C

U_m = fabs(U);
J_m = fabs(J);
J_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	return J_s * U_m;
}

Fortran

U_m = abs(U)
J_m = abs(J)
J_s = copysign(1.0d0, J)
real(8) function code(j_s, j_m, k, u_m)
    real(8), intent (in) :: j_s
    real(8), intent (in) :: j_m
    real(8), intent (in) :: k
    real(8), intent (in) :: u_m
    code = j_s * u_m
end function

Java

U_m = Math.abs(U);
J_m = Math.abs(J);
J_s = Math.copySign(1.0, J);
public static double code(double J_s, double J_m, double K, double U_m) {
	return J_s * U_m;
}

Python

U_m = math.fabs(U)
J_m = math.fabs(J)
J_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	return J_s * U_m

Julia

U_m = abs(U)
J_m = abs(J)
J_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	return Float64(J_s * U_m)
end

MATLAB

U_m = abs(U);
J_m = abs(J);
J_s = sign(J) * abs(1.0);
function tmp = code(J_s, J_m, K, U_m)
	tmp = J_s * U_m;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J_m = N[Abs[J], $MachinePrecision]
J_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := N[(J$95$s * U$95$m), $MachinePrecision]

Derivation

1. Initial program 78.0%

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

   1. unpow2 78.0%

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

   2. hypot-1-def 90.3%

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

   3. associate-/r* 90.2%

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

   4. cos-neg 90.2%

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

   5. distribute-frac-neg 90.2%

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

   6. associate-/r* 90.3%

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

   7. hypot-1-def 78.0%

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

   8. unpow2 78.0%

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

3. Simplified 90.2%

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

5. Taylor expanded in U around -inf 28.3%

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

6. Final simplification 28.3%

   \[\leadsto U \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024034 
(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)))))