Maksimov and Kolovsky, Equation (3)

Percentage Accurate: 73.4% → 99.6%

Time: 11.8s

Alternatives: 9

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: 73.4% 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: 99.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(t\_0 \cdot \left(-2 \cdot J\_m\right)\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_0 \cdot \left(J\_m \cdot 2\right)}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\frac{J\_m \cdot \left(-2 \cdot J\_m\right)}{U\_m} - U\_m\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+298}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* t_0 (* -2.0 J_m))
          (sqrt (+ 1.0 (pow (/ U_m (* t_0 (* J_m 2.0))) 2.0))))))
   (*
    J_s
    (if (<= t_1 (- INFINITY))
      (- (/ (* J_m (* -2.0 J_m)) U_m) U_m)
      (if (<= t_1 5e+298) t_1 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 t_0 = cos((K / 2.0));
	double t_1 = (t_0 * (-2.0 * J_m)) * sqrt((1.0 + pow((U_m / (t_0 * (J_m * 2.0))), 2.0)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = ((J_m * (-2.0 * J_m)) / U_m) - U_m;
	} else if (t_1 <= 5e+298) {
		tmp = t_1;
	} 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 t_0 = Math.cos((K / 2.0));
	double t_1 = (t_0 * (-2.0 * J_m)) * Math.sqrt((1.0 + Math.pow((U_m / (t_0 * (J_m * 2.0))), 2.0)));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = ((J_m * (-2.0 * J_m)) / U_m) - U_m;
	} else if (t_1 <= 5e+298) {
		tmp = t_1;
	} 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):
	t_0 = math.cos((K / 2.0))
	t_1 = (t_0 * (-2.0 * J_m)) * math.sqrt((1.0 + math.pow((U_m / (t_0 * (J_m * 2.0))), 2.0)))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = ((J_m * (-2.0 * J_m)) / U_m) - U_m
	elif t_1 <= 5e+298:
		tmp = t_1
	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)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(t_0 * Float64(-2.0 * J_m)) * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_0 * Float64(J_m * 2.0))) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(J_m * Float64(-2.0 * J_m)) / U_m) - U_m);
	elseif (t_1 <= 5e+298)
		tmp = t_1;
	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)
	t_0 = cos((K / 2.0));
	t_1 = (t_0 * (-2.0 * J_m)) * sqrt((1.0 + ((U_m / (t_0 * (J_m * 2.0))) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = ((J_m * (-2.0 * J_m)) / U_m) - U_m;
	elseif (t_1 <= 5e+298)
		tmp = t_1;
	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_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * N[(-2.0 * J$95$m), $MachinePrecision]), $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]}, N[(J$95$s * If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(J$95$m * N[(-2.0 * J$95$m), $MachinePrecision]), $MachinePrecision] / U$95$m), $MachinePrecision] - U$95$m), $MachinePrecision], If[LessEqual[t$95$1, 5e+298], t$95$1, U$95$m]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 5.7%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({J}^{2} \cdot \left(-2 \cdot \frac{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}{U}\right)\right), \color{blue}{U}\right) \]

   5. Simplified 45.5%

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

   6. Taylor expanded in K around 0

      \[\leadsto \color{blue}{-2 \cdot \frac{{J}^{2}}{U} - U} \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(-2 \cdot \frac{{J}^{2}}{U}\right), \color{blue}{U}\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{-2 \cdot {J}^{2}}{U}\right), U\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(-2 \cdot {J}^{2}\right), U\right), U\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left({J}^{2} \cdot -2\right), U\right), U\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(\left(J \cdot J\right) \cdot -2\right), U\right), U\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(J \cdot \left(J \cdot -2\right)\right), U\right), U\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(J \cdot \left(-2 \cdot J\right)\right), U\right), U\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(J, \left(-2 \cdot J\right)\right), U\right), U\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(J, \left(J \cdot -2\right)\right), U\right), U\right) \]

      10. *-lowering-*.f64 45.5%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(J, -2\right)\right), U\right), U\right) \]

   8. Simplified 45.5%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 5.0000000000000003e298

   1. Initial program 99.8%

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

   if 5.0000000000000003e298 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 13.3%

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

   3. Taylor expanded in U around -inf

      \[\leadsto \color{blue}{U} \]
   4. Step-by-step derivation

   5. Simplified 44.0%

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

4. Final simplification 83.2%

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

Alternative 2: 80.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := 0.5 + 0.5 \cdot \cos K\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;J\_m \leq 1.2 \cdot 10^{-122}:\\ \;\;\;\;\frac{\left(J\_m \cdot \left(-2 \cdot J\_m\right)\right) \cdot t\_0}{U\_m} - U\_m\\ \mathbf{else}:\\ \;\;\;\;J\_m \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot \sqrt{1 + \frac{\frac{U\_m}{J\_m} \cdot \frac{U\_m}{J\_m \cdot 4}}{t\_0}}\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (+ 0.5 (* 0.5 (cos K)))))
   (*
    J_s
    (if (<= J_m 1.2e-122)
      (- (/ (* (* J_m (* -2.0 J_m)) t_0) U_m) U_m)
      (*
       J_m
       (*
        (cos (/ K 2.0))
        (*
         -2.0
         (sqrt (+ 1.0 (/ (* (/ U_m J_m) (/ U_m (* J_m 4.0))) 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 = 0.5 + (0.5 * cos(K));
	double tmp;
	if (J_m <= 1.2e-122) {
		tmp = (((J_m * (-2.0 * J_m)) * t_0) / U_m) - U_m;
	} else {
		tmp = J_m * (cos((K / 2.0)) * (-2.0 * sqrt((1.0 + (((U_m / J_m) * (U_m / (J_m * 4.0))) / t_0)))));
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 + (0.5d0 * cos(k))
    if (j_m <= 1.2d-122) then
        tmp = (((j_m * ((-2.0d0) * j_m)) * t_0) / u_m) - u_m
    else
        tmp = j_m * (cos((k / 2.0d0)) * ((-2.0d0) * sqrt((1.0d0 + (((u_m / j_m) * (u_m / (j_m * 4.0d0))) / t_0)))))
    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 t_0 = 0.5 + (0.5 * Math.cos(K));
	double tmp;
	if (J_m <= 1.2e-122) {
		tmp = (((J_m * (-2.0 * J_m)) * t_0) / U_m) - U_m;
	} else {
		tmp = J_m * (Math.cos((K / 2.0)) * (-2.0 * Math.sqrt((1.0 + (((U_m / J_m) * (U_m / (J_m * 4.0))) / 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 = 0.5 + (0.5 * math.cos(K))
	tmp = 0
	if J_m <= 1.2e-122:
		tmp = (((J_m * (-2.0 * J_m)) * t_0) / U_m) - U_m
	else:
		tmp = J_m * (math.cos((K / 2.0)) * (-2.0 * math.sqrt((1.0 + (((U_m / J_m) * (U_m / (J_m * 4.0))) / 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 = Float64(0.5 + Float64(0.5 * cos(K)))
	tmp = 0.0
	if (J_m <= 1.2e-122)
		tmp = Float64(Float64(Float64(Float64(J_m * Float64(-2.0 * J_m)) * t_0) / U_m) - U_m);
	else
		tmp = Float64(J_m * Float64(cos(Float64(K / 2.0)) * Float64(-2.0 * sqrt(Float64(1.0 + Float64(Float64(Float64(U_m / J_m) * Float64(U_m / Float64(J_m * 4.0))) / 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 = 0.5 + (0.5 * cos(K));
	tmp = 0.0;
	if (J_m <= 1.2e-122)
		tmp = (((J_m * (-2.0 * J_m)) * t_0) / U_m) - U_m;
	else
		tmp = J_m * (cos((K / 2.0)) * (-2.0 * sqrt((1.0 + (((U_m / J_m) * (U_m / (J_m * 4.0))) / 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[(0.5 + N[(0.5 * N[Cos[K], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[J$95$m, 1.2e-122], N[(N[(N[(N[(J$95$m * N[(-2.0 * J$95$m), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] / U$95$m), $MachinePrecision] - U$95$m), $MachinePrecision], N[(J$95$m * N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(-2.0 * N[Sqrt[N[(1.0 + N[(N[(N[(U$95$m / J$95$m), $MachinePrecision] * N[(U$95$m / N[(J$95$m * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if J < 1.19999999999999994e-122

   1. Initial program 62.7%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({J}^{2} \cdot \left(-2 \cdot \frac{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}{U}\right)\right), \color{blue}{U}\right) \]

   5. Simplified 34.0%

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

      1. associate-*l/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{{\cos \left(\frac{1}{2} \cdot K\right)}^{2} \cdot \left(-2 \cdot \left(J \cdot J\right)\right)}{U}\right), U\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left({\cos \left(\frac{1}{2} \cdot K\right)}^{2} \cdot \left(-2 \cdot \left(J \cdot J\right)\right)\right), U\right), U\right) \]

   7. Applied egg-rr 34.0%

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

   if 1.19999999999999994e-122 < J

   1. Initial program 88.0%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(-2 \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\right)\right), \color{blue}{J}\right) \]

   4. Applied egg-rr 87.7%

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

   6. Applied egg-rr 75.6%

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + \frac{\frac{U \cdot U}{\left(J \cdot J\right) \cdot 4}}{0.5 + 0.5 \cdot \cos K}} \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right)} \cdot J \]
   7. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{U \cdot U}{J \cdot \left(J \cdot 4\right)}\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(K\right)\right)\right)\right)\right)\right), -2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), J\right) \]

      2. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{U}{J} \cdot \frac{U}{J \cdot 4}\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(K\right)\right)\right)\right)\right)\right), -2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), J\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{U}{J}\right), \left(\frac{U}{J \cdot 4}\right)\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(K\right)\right)\right)\right)\right)\right), -2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), J\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(U, J\right), \left(\frac{U}{J \cdot 4}\right)\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(K\right)\right)\right)\right)\right)\right), -2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), J\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(U, J\right), \mathsf{/.f64}\left(U, \left(J \cdot 4\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(K\right)\right)\right)\right)\right)\right), -2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), J\right) \]

      6. *-lowering-*.f64 87.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(U, J\right), \mathsf{/.f64}\left(U, \mathsf{*.f64}\left(J, 4\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(K\right)\right)\right)\right)\right)\right), -2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), J\right) \]

   8. Applied egg-rr 87.8%

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

4. Final simplification 55.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq 1.2 \cdot 10^{-122}:\\ \;\;\;\;\frac{\left(J \cdot \left(-2 \cdot J\right)\right) \cdot \left(0.5 + 0.5 \cdot \cos K\right)}{U} - U\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot \sqrt{1 + \frac{\frac{U}{J} \cdot \frac{U}{J \cdot 4}}{0.5 + 0.5 \cdot \cos K}}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ J\_s \cdot \begin{array}{l} \mathbf{if}\;U\_m \leq 1.45 \cdot 10^{+76}:\\ \;\;\;\;\left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\_m\right)\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{J\_m \cdot 2}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;0 - U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (*
  J_s
  (if (<= U_m 1.45e+76)
    (*
     (* (cos (/ K 2.0)) (* -2.0 J_m))
     (sqrt (+ 1.0 (pow (/ U_m (* J_m 2.0)) 2.0))))
    (- 0.0 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 <= 1.45e+76) {
		tmp = (cos((K / 2.0)) * (-2.0 * J_m)) * sqrt((1.0 + pow((U_m / (J_m * 2.0)), 2.0)));
	} else {
		tmp = 0.0 - 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 <= 1.45d+76) then
        tmp = (cos((k / 2.0d0)) * ((-2.0d0) * j_m)) * sqrt((1.0d0 + ((u_m / (j_m * 2.0d0)) ** 2.0d0)))
    else
        tmp = 0.0d0 - 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 <= 1.45e+76) {
		tmp = (Math.cos((K / 2.0)) * (-2.0 * J_m)) * Math.sqrt((1.0 + Math.pow((U_m / (J_m * 2.0)), 2.0)));
	} else {
		tmp = 0.0 - 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 <= 1.45e+76:
		tmp = (math.cos((K / 2.0)) * (-2.0 * J_m)) * math.sqrt((1.0 + math.pow((U_m / (J_m * 2.0)), 2.0)))
	else:
		tmp = 0.0 - 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 <= 1.45e+76)
		tmp = Float64(Float64(cos(Float64(K / 2.0)) * Float64(-2.0 * J_m)) * sqrt(Float64(1.0 + (Float64(U_m / Float64(J_m * 2.0)) ^ 2.0))));
	else
		tmp = Float64(0.0 - 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 <= 1.45e+76)
		tmp = (cos((K / 2.0)) * (-2.0 * J_m)) * sqrt((1.0 + ((U_m / (J_m * 2.0)) ^ 2.0)));
	else
		tmp = 0.0 - 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, 1.45e+76], N[(N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(-2.0 * J$95$m), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(J$95$m * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.0 - U$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if U < 1.4500000000000001e76

   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. Add Preprocessing

   3. Taylor expanded in K around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(U, \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, J\right), \color{blue}{1}\right)\right), 2\right)\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 67.4%

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

   if 1.4500000000000001e76 < 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. Add Preprocessing

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(U\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{U} \]

      3. --lowering--.f64 38.5%

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{U}\right) \]

   5. Simplified 38.5%

      \[\leadsto \color{blue}{0 - U} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(U\right) \]

      2. neg-lowering-neg.f64 38.5%

         \[\leadsto \mathsf{neg.f64}\left(U\right) \]

   7. Applied egg-rr 38.5%

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

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 1.45 \cdot 10^{+76}:\\ \;\;\;\;\left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{J \cdot 2}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;0 - U\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ J\_s \cdot \begin{array}{l} \mathbf{if}\;U\_m \leq 1.45 \cdot 10^{+76}:\\ \;\;\;\;J\_m \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + \frac{\frac{U\_m}{J\_m \cdot 2}}{2 \cdot \frac{J\_m}{U\_m}}}\right)\\ \mathbf{else}:\\ \;\;\;\;0 - U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (*
  J_s
  (if (<= U_m 1.45e+76)
    (*
     J_m
     (*
      (* -2.0 (cos (/ K 2.0)))
      (sqrt (+ 1.0 (/ (/ U_m (* J_m 2.0)) (* 2.0 (/ J_m U_m)))))))
    (- 0.0 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 <= 1.45e+76) {
		tmp = J_m * ((-2.0 * cos((K / 2.0))) * sqrt((1.0 + ((U_m / (J_m * 2.0)) / (2.0 * (J_m / U_m))))));
	} else {
		tmp = 0.0 - 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 <= 1.45d+76) then
        tmp = j_m * (((-2.0d0) * cos((k / 2.0d0))) * sqrt((1.0d0 + ((u_m / (j_m * 2.0d0)) / (2.0d0 * (j_m / u_m))))))
    else
        tmp = 0.0d0 - 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 <= 1.45e+76) {
		tmp = J_m * ((-2.0 * Math.cos((K / 2.0))) * Math.sqrt((1.0 + ((U_m / (J_m * 2.0)) / (2.0 * (J_m / U_m))))));
	} else {
		tmp = 0.0 - 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 <= 1.45e+76:
		tmp = J_m * ((-2.0 * math.cos((K / 2.0))) * math.sqrt((1.0 + ((U_m / (J_m * 2.0)) / (2.0 * (J_m / U_m))))))
	else:
		tmp = 0.0 - 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 <= 1.45e+76)
		tmp = Float64(J_m * Float64(Float64(-2.0 * cos(Float64(K / 2.0))) * sqrt(Float64(1.0 + Float64(Float64(U_m / Float64(J_m * 2.0)) / Float64(2.0 * Float64(J_m / U_m)))))));
	else
		tmp = Float64(0.0 - 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 <= 1.45e+76)
		tmp = J_m * ((-2.0 * cos((K / 2.0))) * sqrt((1.0 + ((U_m / (J_m * 2.0)) / (2.0 * (J_m / U_m))))));
	else
		tmp = 0.0 - 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, 1.45e+76], N[(J$95$m * N[(N[(-2.0 * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 + N[(N[(U$95$m / N[(J$95$m * 2.0), $MachinePrecision]), $MachinePrecision] / N[(2.0 * N[(J$95$m / U$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.0 - U$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if U < 1.4500000000000001e76

   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. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(-2 \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\right)\right), \color{blue}{J}\right) \]

   4. Applied egg-rr 77.8%

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

   5. Taylor expanded in K around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(U, \mathsf{*.f64}\left(J, 2\right)\right), \color{blue}{\left(2 \cdot \frac{J}{U}\right)}\right)\right)\right)\right), J\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(U, \mathsf{*.f64}\left(J, 2\right)\right), \mathsf{*.f64}\left(2, \left(\frac{J}{U}\right)\right)\right)\right)\right)\right), J\right) \]

      2. /-lowering-/.f64 67.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(U, \mathsf{*.f64}\left(J, 2\right)\right), \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(J, U\right)\right)\right)\right)\right)\right), J\right) \]

   7. Simplified 67.4%

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

   if 1.4500000000000001e76 < 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. Add Preprocessing

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(U\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{U} \]

      3. --lowering--.f64 38.5%

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{U}\right) \]

   5. Simplified 38.5%

      \[\leadsto \color{blue}{0 - U} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(U\right) \]

      2. neg-lowering-neg.f64 38.5%

         \[\leadsto \mathsf{neg.f64}\left(U\right) \]

   7. Applied egg-rr 38.5%

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

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 1.45 \cdot 10^{+76}:\\ \;\;\;\;J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + \frac{\frac{U}{J \cdot 2}}{2 \cdot \frac{J}{U}}}\right)\\ \mathbf{else}:\\ \;\;\;\;0 - U\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ J\_s \cdot \begin{array}{l} \mathbf{if}\;U\_m \leq 1.45 \cdot 10^{+76}:\\ \;\;\;\;\left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\_m\right)\right) \cdot \sqrt{1 + \frac{\frac{\frac{U\_m}{J\_m}}{\frac{J\_m}{U\_m}}}{4}}\\ \mathbf{else}:\\ \;\;\;\;0 - U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (*
  J_s
  (if (<= U_m 1.45e+76)
    (*
     (* (cos (/ K 2.0)) (* -2.0 J_m))
     (sqrt (+ 1.0 (/ (/ (/ U_m J_m) (/ J_m U_m)) 4.0))))
    (- 0.0 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 <= 1.45e+76) {
		tmp = (cos((K / 2.0)) * (-2.0 * J_m)) * sqrt((1.0 + (((U_m / J_m) / (J_m / U_m)) / 4.0)));
	} else {
		tmp = 0.0 - 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 <= 1.45d+76) then
        tmp = (cos((k / 2.0d0)) * ((-2.0d0) * j_m)) * sqrt((1.0d0 + (((u_m / j_m) / (j_m / u_m)) / 4.0d0)))
    else
        tmp = 0.0d0 - 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 <= 1.45e+76) {
		tmp = (Math.cos((K / 2.0)) * (-2.0 * J_m)) * Math.sqrt((1.0 + (((U_m / J_m) / (J_m / U_m)) / 4.0)));
	} else {
		tmp = 0.0 - 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 <= 1.45e+76:
		tmp = (math.cos((K / 2.0)) * (-2.0 * J_m)) * math.sqrt((1.0 + (((U_m / J_m) / (J_m / U_m)) / 4.0)))
	else:
		tmp = 0.0 - 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 <= 1.45e+76)
		tmp = Float64(Float64(cos(Float64(K / 2.0)) * Float64(-2.0 * J_m)) * sqrt(Float64(1.0 + Float64(Float64(Float64(U_m / J_m) / Float64(J_m / U_m)) / 4.0))));
	else
		tmp = Float64(0.0 - 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 <= 1.45e+76)
		tmp = (cos((K / 2.0)) * (-2.0 * J_m)) * sqrt((1.0 + (((U_m / J_m) / (J_m / U_m)) / 4.0)));
	else
		tmp = 0.0 - 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, 1.45e+76], N[(N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(-2.0 * J$95$m), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 + N[(N[(N[(U$95$m / J$95$m), $MachinePrecision] / N[(J$95$m / U$95$m), $MachinePrecision]), $MachinePrecision] / 4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.0 - U$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if U < 1.4500000000000001e76

   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. Add Preprocessing

   3. Taylor expanded in K around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(U, \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, J\right), \color{blue}{1}\right)\right), 2\right)\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 67.4%

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

      1. *-rgt-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left({\left(\frac{U}{2 \cdot J}\right)}^{2}\right)\right)\right)\right) \]

      2. associate-/l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left({\left(\frac{\frac{U}{J}}{2}\right)}^{2}\right)\right)\right)\right) \]

      3. div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left({\left(\frac{U}{J} \cdot \frac{1}{2}\right)}^{2}\right)\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left({\left(\frac{U}{J} \cdot \frac{1}{2}\right)}^{2}\right)\right)\right)\right) \]

      5. unpow-prod-down N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left({\left(\frac{U}{J}\right)}^{2} \cdot {\frac{1}{2}}^{2}\right)\right)\right)\right) \]

      6. pow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\left(\frac{U}{J} \cdot \frac{U}{J}\right) \cdot {\frac{1}{2}}^{2}\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\left(\frac{U}{J} \cdot \frac{U}{J}\right) \cdot \frac{1}{4}\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\left(\frac{U}{J} \cdot \frac{U}{J}\right) \cdot \frac{1}{4}\right)\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\left(\frac{U}{J} \cdot \frac{U}{J}\right) \cdot \frac{1}{2 + 2}\right)\right)\right)\right) \]

      10. div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{U}{J} \cdot \frac{U}{J}}{2 + 2}\right)\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{U}{J} \cdot \frac{U}{J}\right), \left(2 + 2\right)\right)\right)\right)\right) \]

      12. clear-num N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{U}{J} \cdot \frac{1}{\frac{J}{U}}\right), \left(2 + 2\right)\right)\right)\right)\right) \]

      13. un-div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{\frac{U}{J}}{\frac{J}{U}}\right), \left(2 + 2\right)\right)\right)\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{U}{J}\right), \left(\frac{J}{U}\right)\right), \left(2 + 2\right)\right)\right)\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(U, J\right), \left(\frac{J}{U}\right)\right), \left(2 + 2\right)\right)\right)\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(U, J\right), \mathsf{/.f64}\left(J, U\right)\right), \left(2 + 2\right)\right)\right)\right)\right) \]

      17. metadata-eval 67.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, J\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(U, J\right), \mathsf{/.f64}\left(J, U\right)\right), 4\right)\right)\right)\right) \]

   7. Applied egg-rr 67.4%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + \color{blue}{\frac{\frac{\frac{U}{J}}{\frac{J}{U}}}{4}}} \]

   if 1.4500000000000001e76 < 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. Add Preprocessing

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(U\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{U} \]

      3. --lowering--.f64 38.5%

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{U}\right) \]

   5. Simplified 38.5%

      \[\leadsto \color{blue}{0 - U} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(U\right) \]

      2. neg-lowering-neg.f64 38.5%

         \[\leadsto \mathsf{neg.f64}\left(U\right) \]

   7. Applied egg-rr 38.5%

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

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 1.45 \cdot 10^{+76}:\\ \;\;\;\;\left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right) \cdot \sqrt{1 + \frac{\frac{\frac{U}{J}}{\frac{J}{U}}}{4}}\\ \mathbf{else}:\\ \;\;\;\;0 - U\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 63.5% accurate, 3.7× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ J\_s \cdot \begin{array}{l} \mathbf{if}\;U\_m \leq 5.5 \cdot 10^{-87}:\\ \;\;\;\;\left(-2 \cdot J\_m\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0 - U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (* J_s (if (<= U_m 5.5e-87) (* (* -2.0 J_m) (cos (* K 0.5))) (- 0.0 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 <= 5.5e-87) {
		tmp = (-2.0 * J_m) * cos((K * 0.5));
	} else {
		tmp = 0.0 - 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 <= 5.5d-87) then
        tmp = ((-2.0d0) * j_m) * cos((k * 0.5d0))
    else
        tmp = 0.0d0 - 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 <= 5.5e-87) {
		tmp = (-2.0 * J_m) * Math.cos((K * 0.5));
	} else {
		tmp = 0.0 - 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 <= 5.5e-87:
		tmp = (-2.0 * J_m) * math.cos((K * 0.5))
	else:
		tmp = 0.0 - 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 <= 5.5e-87)
		tmp = Float64(Float64(-2.0 * J_m) * cos(Float64(K * 0.5)));
	else
		tmp = Float64(0.0 - 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 <= 5.5e-87)
		tmp = (-2.0 * J_m) * cos((K * 0.5));
	else
		tmp = 0.0 - 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, 5.5e-87], N[(N[(-2.0 * J$95$m), $MachinePrecision] * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.0 - U$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if U < 5.5000000000000004e-87

   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. Add Preprocessing

   3. Taylor expanded in J around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\cos \left(\frac{1}{2} \cdot K\right), \color{blue}{\left(-2 \cdot J\right)}\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot K\right)\right), \left(\color{blue}{-2} \cdot J\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, K\right)\right), \left(-2 \cdot J\right)\right) \]

      6. *-lowering-*.f64 57.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, K\right)\right), \mathsf{*.f64}\left(-2, \color{blue}{J}\right)\right) \]

   5. Simplified 57.8%

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

   if 5.5000000000000004e-87 < U

   1. Initial program 60.2%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(U\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{U} \]

      3. --lowering--.f64 32.4%

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{U}\right) \]

   5. Simplified 32.4%

      \[\leadsto \color{blue}{0 - U} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(U\right) \]

      2. neg-lowering-neg.f64 32.4%

         \[\leadsto \mathsf{neg.f64}\left(U\right) \]

   7. Applied egg-rr 32.4%

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

4. Final simplification 50.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 5.5 \cdot 10^{-87}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0 - U\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 39.4% accurate, 32.2× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ J\_s \cdot \begin{array}{l} \mathbf{if}\;K \leq 3.8 \cdot 10^{+231}:\\ \;\;\;\;0 - U\_m\\ \mathbf{elif}\;K \leq 7.5 \cdot 10^{+258}:\\ \;\;\;\;U\_m\\ \mathbf{else}:\\ \;\;\;\;0 - U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (* J_s (if (<= K 3.8e+231) (- 0.0 U_m) (if (<= K 7.5e+258) U_m (- 0.0 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 <= 3.8e+231) {
		tmp = 0.0 - U_m;
	} else if (K <= 7.5e+258) {
		tmp = U_m;
	} else {
		tmp = 0.0 - 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 <= 3.8d+231) then
        tmp = 0.0d0 - u_m
    else if (k <= 7.5d+258) then
        tmp = u_m
    else
        tmp = 0.0d0 - 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 <= 3.8e+231) {
		tmp = 0.0 - U_m;
	} else if (K <= 7.5e+258) {
		tmp = U_m;
	} else {
		tmp = 0.0 - 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 <= 3.8e+231:
		tmp = 0.0 - U_m
	elif K <= 7.5e+258:
		tmp = U_m
	else:
		tmp = 0.0 - 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 <= 3.8e+231)
		tmp = Float64(0.0 - U_m);
	elseif (K <= 7.5e+258)
		tmp = U_m;
	else
		tmp = Float64(0.0 - 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 <= 3.8e+231)
		tmp = 0.0 - U_m;
	elseif (K <= 7.5e+258)
		tmp = U_m;
	else
		tmp = 0.0 - 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[K, 3.8e+231], N[(0.0 - U$95$m), $MachinePrecision], If[LessEqual[K, 7.5e+258], U$95$m, N[(0.0 - U$95$m), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if K < 3.8000000000000001e231 or 7.50000000000000032e258 < K

   1. Initial program 73.3%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(U\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{U} \]

      3. --lowering--.f64 27.3%

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{U}\right) \]

   5. Simplified 27.3%

      \[\leadsto \color{blue}{0 - U} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(U\right) \]

      2. neg-lowering-neg.f64 27.3%

         \[\leadsto \mathsf{neg.f64}\left(U\right) \]

   7. Applied egg-rr 27.3%

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

   if 3.8000000000000001e231 < K < 7.50000000000000032e258

   1. Initial program 59.2%

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

   3. Taylor expanded in U around -inf

      \[\leadsto \color{blue}{U} \]
   4. Step-by-step derivation

   5. Simplified 45.2%

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

4. Final simplification 27.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;K \leq 3.8 \cdot 10^{+231}:\\ \;\;\;\;0 - U\\ \mathbf{elif}\;K \leq 7.5 \cdot 10^{+258}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;0 - U\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 50.2% accurate, 52.4× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ J\_s \cdot \begin{array}{l} \mathbf{if}\;U\_m \leq 3.5 \cdot 10^{-90}:\\ \;\;\;\;-2 \cdot J\_m\\ \mathbf{else}:\\ \;\;\;\;0 - U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (* J_s (if (<= U_m 3.5e-90) (* -2.0 J_m) (- 0.0 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 <= 3.5e-90) {
		tmp = -2.0 * J_m;
	} else {
		tmp = 0.0 - 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 <= 3.5d-90) then
        tmp = (-2.0d0) * j_m
    else
        tmp = 0.0d0 - 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 <= 3.5e-90) {
		tmp = -2.0 * J_m;
	} else {
		tmp = 0.0 - 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 <= 3.5e-90:
		tmp = -2.0 * J_m
	else:
		tmp = 0.0 - 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 <= 3.5e-90)
		tmp = Float64(-2.0 * J_m);
	else
		tmp = Float64(0.0 - 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 <= 3.5e-90)
		tmp = -2.0 * J_m;
	else
		tmp = 0.0 - 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, 3.5e-90], N[(-2.0 * J$95$m), $MachinePrecision], N[(0.0 - U$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if U < 3.4999999999999999e-90

   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. Add Preprocessing

   3. Taylor expanded in J around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\cos \left(\frac{1}{2} \cdot K\right), \color{blue}{\left(-2 \cdot J\right)}\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot K\right)\right), \left(\color{blue}{-2} \cdot J\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, K\right)\right), \left(-2 \cdot J\right)\right) \]

      6. *-lowering-*.f64 57.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, K\right)\right), \mathsf{*.f64}\left(-2, \color{blue}{J}\right)\right) \]

   5. Simplified 57.8%

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

   6. Taylor expanded in K around 0

      \[\leadsto \color{blue}{-2 \cdot J} \]
   7. Step-by-step derivation

      1. *-commutative N/A

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

      2. *-lowering-*.f64 33.2%

         \[\leadsto \mathsf{*.f64}\left(J, \color{blue}{-2}\right) \]

   8. Simplified 33.2%

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

   if 3.4999999999999999e-90 < U

   1. Initial program 60.2%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(U\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{U} \]

      3. --lowering--.f64 32.4%

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{U}\right) \]

   5. Simplified 32.4%

      \[\leadsto \color{blue}{0 - U} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(U\right) \]

      2. neg-lowering-neg.f64 32.4%

         \[\leadsto \mathsf{neg.f64}\left(U\right) \]

   7. Applied egg-rr 32.4%

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

4. Final simplification 33.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 3.5 \cdot 10^{-90}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{else}:\\ \;\;\;\;0 - U\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 14.1% 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 #s(literal 1 binary64) 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 72.9%

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

3. Taylor expanded in U around -inf

   \[\leadsto \color{blue}{U} \]
4. Step-by-step derivation

5. Simplified 29.2%

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

Reproduce

herbie shell --seed 2024185 
(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)))))