Maksimov and Kolovsky, Equation (3)

Percentage Accurate: 72.5% → 99.2%

Time: 11.2s

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: 72.5% 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: 99.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_0 \cdot \left(J\_m \cdot 2\right)}\right)}^{2}}\\ t_2 := J\_m \cdot \cos \left(K \cdot 0.5\right)\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+306}:\\ \;\;\;\;-2 \cdot \left(t\_2 \cdot \frac{t\_2}{U\_m}\right) - U\_m\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+294}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \end{array} \]

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
         (*
          (* (* -2.0 J_m) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* t_0 (* J_m 2.0))) 2.0)))))
        (t_2 (* J_m (cos (* K 0.5)))))
   (*
    J_s
    (if (<= t_1 -5e+306)
      (- (* -2.0 (* t_2 (/ t_2 U_m))) U_m)
      (if (<= t_1 2e+294) 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 = ((-2.0 * J_m) * t_0) * sqrt((1.0 + pow((U_m / (t_0 * (J_m * 2.0))), 2.0)));
	double t_2 = J_m * cos((K * 0.5));
	double tmp;
	if (t_1 <= -5e+306) {
		tmp = (-2.0 * (t_2 * (t_2 / U_m))) - U_m;
	} else if (t_1 <= 2e+294) {
		tmp = t_1;
	} 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) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    t_1 = (((-2.0d0) * j_m) * t_0) * sqrt((1.0d0 + ((u_m / (t_0 * (j_m * 2.0d0))) ** 2.0d0)))
    t_2 = j_m * cos((k * 0.5d0))
    if (t_1 <= (-5d+306)) then
        tmp = ((-2.0d0) * (t_2 * (t_2 / u_m))) - u_m
    else if (t_1 <= 2d+294) then
        tmp = t_1
    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 t_0 = Math.cos((K / 2.0));
	double t_1 = ((-2.0 * J_m) * t_0) * Math.sqrt((1.0 + Math.pow((U_m / (t_0 * (J_m * 2.0))), 2.0)));
	double t_2 = J_m * Math.cos((K * 0.5));
	double tmp;
	if (t_1 <= -5e+306) {
		tmp = (-2.0 * (t_2 * (t_2 / U_m))) - U_m;
	} else if (t_1 <= 2e+294) {
		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 = ((-2.0 * J_m) * t_0) * math.sqrt((1.0 + math.pow((U_m / (t_0 * (J_m * 2.0))), 2.0)))
	t_2 = J_m * math.cos((K * 0.5))
	tmp = 0
	if t_1 <= -5e+306:
		tmp = (-2.0 * (t_2 * (t_2 / U_m))) - U_m
	elif t_1 <= 2e+294:
		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(Float64(-2.0 * J_m) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_0 * Float64(J_m * 2.0))) ^ 2.0))))
	t_2 = Float64(J_m * cos(Float64(K * 0.5)))
	tmp = 0.0
	if (t_1 <= -5e+306)
		tmp = Float64(Float64(-2.0 * Float64(t_2 * Float64(t_2 / U_m))) - U_m);
	elseif (t_1 <= 2e+294)
		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 = ((-2.0 * J_m) * t_0) * sqrt((1.0 + ((U_m / (t_0 * (J_m * 2.0))) ^ 2.0)));
	t_2 = J_m * cos((K * 0.5));
	tmp = 0.0;
	if (t_1 <= -5e+306)
		tmp = (-2.0 * (t_2 * (t_2 / U_m))) - U_m;
	elseif (t_1 <= 2e+294)
		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[(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]}, Block[{t$95$2 = N[(J$95$m * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$1, -5e+306], N[(N[(-2.0 * N[(t$95$2 * N[(t$95$2 / U$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - U$95$m), $MachinePrecision], If[LessEqual[t$95$1, 2e+294], 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))))) < -4.99999999999999993e306

   1. Initial program 6.0%

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

   3. Simplified 57.1%

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

   5. Taylor expanded in J around 0 54.5%

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

      1. fma-define 54.5%

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

      2. unpow2 54.5%

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

      3. *-commutative 54.5%

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

      4. unpow2 54.5%

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

      5. swap-sqr 54.5%

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

      6. unpow2 54.5%

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

      7. *-commutative 54.5%

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

      8. neg-mul-1 54.5%

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

   7. Simplified 54.5%

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

      1. fmm-undef 54.5%

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

      2. add-sqr-sqrt 54.5%

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

      3. pow2 54.5%

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

      4. unpow2 54.5%

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

      5. sqrt-prod 54.5%

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

      6. add-sqr-sqrt 54.5%

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

      7. *-commutative 54.5%

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

   9. Applied egg-rr 54.5%

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

       1. unpow2 54.5%

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

       2. *-un-lft-identity 54.5%

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

       3. times-frac 54.5%

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

   11. Applied egg-rr 54.5%

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

   if -4.99999999999999993e306 < (*.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))))) < 2.00000000000000013e294

   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 2.00000000000000013e294 < (*.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 21.6%

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

   3. Simplified 59.9%

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

   5. Taylor expanded in U around -inf 52.6%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -5 \cdot 10^{+306}:\\ \;\;\;\;-2 \cdot \left(\left(J \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \frac{J \cdot \cos \left(K \cdot 0.5\right)}{U}\right) - U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 2 \cdot 10^{+294}:\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := J\_m \cdot \cos \left(K \cdot 0.5\right)\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;J\_m \leq 1.2 \cdot 10^{-195}:\\ \;\;\;\;-2 \cdot \left(t\_1 \cdot \frac{t\_1}{U\_m}\right) - U\_m\\ \mathbf{else}:\\ \;\;\;\;\left(J\_m \cdot \left(-2 \cdot t\_0\right)\right) \cdot \mathsf{hypot}\left(1, \frac{\frac{U\_m}{J\_m \cdot 2}}{t\_0}\right)\\ \end{array} \end{array} \end{array} \]

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 (cos (/ K 2.0))) (t_1 (* J_m (cos (* K 0.5)))))
   (*
    J_s
    (if (<= J_m 1.2e-195)
      (- (* -2.0 (* t_1 (/ t_1 U_m))) U_m)
      (* (* J_m (* -2.0 t_0)) (hypot 1.0 (/ (/ U_m (* J_m 2.0)) 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 = cos((K / 2.0));
	double t_1 = J_m * cos((K * 0.5));
	double tmp;
	if (J_m <= 1.2e-195) {
		tmp = (-2.0 * (t_1 * (t_1 / U_m))) - U_m;
	} else {
		tmp = (J_m * (-2.0 * t_0)) * hypot(1.0, ((U_m / (J_m * 2.0)) / 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 t_1 = J_m * Math.cos((K * 0.5));
	double tmp;
	if (J_m <= 1.2e-195) {
		tmp = (-2.0 * (t_1 * (t_1 / U_m))) - U_m;
	} else {
		tmp = (J_m * (-2.0 * t_0)) * Math.hypot(1.0, ((U_m / (J_m * 2.0)) / 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 = math.cos((K / 2.0))
	t_1 = J_m * math.cos((K * 0.5))
	tmp = 0
	if J_m <= 1.2e-195:
		tmp = (-2.0 * (t_1 * (t_1 / U_m))) - U_m
	else:
		tmp = (J_m * (-2.0 * t_0)) * math.hypot(1.0, ((U_m / (J_m * 2.0)) / 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 = cos(Float64(K / 2.0))
	t_1 = Float64(J_m * cos(Float64(K * 0.5)))
	tmp = 0.0
	if (J_m <= 1.2e-195)
		tmp = Float64(Float64(-2.0 * Float64(t_1 * Float64(t_1 / U_m))) - U_m);
	else
		tmp = Float64(Float64(J_m * Float64(-2.0 * t_0)) * hypot(1.0, Float64(Float64(U_m / Float64(J_m * 2.0)) / 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 = cos((K / 2.0));
	t_1 = J_m * cos((K * 0.5));
	tmp = 0.0;
	if (J_m <= 1.2e-195)
		tmp = (-2.0 * (t_1 * (t_1 / U_m))) - U_m;
	else
		tmp = (J_m * (-2.0 * t_0)) * hypot(1.0, ((U_m / (J_m * 2.0)) / 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[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(J$95$m * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[J$95$m, 1.2e-195], N[(N[(-2.0 * N[(t$95$1 * N[(t$95$1 / U$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - U$95$m), $MachinePrecision], N[(N[(J$95$m * N[(-2.0 * t$95$0), $MachinePrecision]), $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[(N[(U$95$m / N[(J$95$m * 2.0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if J < 1.2e-195

   1. Initial program 67.0%

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

   3. Simplified 82.3%

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

   5. Taylor expanded in J around 0 26.5%

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

      1. fma-define 26.5%

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

      2. unpow2 26.5%

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

      3. *-commutative 26.5%

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

      4. unpow2 26.5%

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

      5. swap-sqr 26.5%

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

      6. unpow2 26.5%

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

      7. *-commutative 26.5%

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

      8. neg-mul-1 26.5%

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

   7. Simplified 26.5%

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

      1. fmm-undef 26.5%

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

      2. add-sqr-sqrt 26.5%

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

      3. pow2 26.5%

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

      4. unpow2 26.5%

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

      5. sqrt-prod 11.4%

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

      6. add-sqr-sqrt 26.5%

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

      7. *-commutative 26.5%

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

   9. Applied egg-rr 26.5%

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

       1. unpow2 26.5%

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

       2. *-un-lft-identity 26.5%

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

       3. times-frac 28.6%

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

   11. Applied egg-rr 28.6%

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

   if 1.2e-195 < J

   1. Initial program 81.4%

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

      1. unpow2 81.4%

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

      2. hypot-1-def 92.5%

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

      3. associate-/r* 92.5%

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

      4. cos-neg 92.5%

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

      5. distribute-frac-neg 92.5%

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

      6. associate-/r* 92.5%

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

      7. hypot-1-def 81.4%

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

      8. unpow2 81.4%

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

   3. Simplified 92.5%

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

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq 1.2 \cdot 10^{-195}:\\ \;\;\;\;-2 \cdot \left(\left(J \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \frac{J \cdot \cos \left(K \cdot 0.5\right)}{U}\right) - U\\ \mathbf{else}:\\ \;\;\;\;\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{\frac{U}{J \cdot 2}}{\cos \left(\frac{K}{2}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := J\_m \cdot \cos \left(K \cdot 0.5\right)\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;J\_m \leq 5.5 \cdot 10^{-195}:\\ \;\;\;\;-2 \cdot \left(t\_1 \cdot \frac{t\_1}{U\_m}\right) - U\_m\\ \mathbf{else}:\\ \;\;\;\;J\_m \cdot \left(\left(-2 \cdot t\_0\right) \cdot \mathsf{hypot}\left(1, \frac{\frac{U\_m}{2}}{J\_m \cdot t\_0}\right)\right)\\ \end{array} \end{array} \end{array} \]

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 (cos (/ K 2.0))) (t_1 (* J_m (cos (* K 0.5)))))
   (*
    J_s
    (if (<= J_m 5.5e-195)
      (- (* -2.0 (* t_1 (/ t_1 U_m))) U_m)
      (* J_m (* (* -2.0 t_0) (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 t_1 = J_m * cos((K * 0.5));
	double tmp;
	if (J_m <= 5.5e-195) {
		tmp = (-2.0 * (t_1 * (t_1 / U_m))) - U_m;
	} else {
		tmp = J_m * ((-2.0 * t_0) * 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 t_1 = J_m * Math.cos((K * 0.5));
	double tmp;
	if (J_m <= 5.5e-195) {
		tmp = (-2.0 * (t_1 * (t_1 / U_m))) - U_m;
	} else {
		tmp = J_m * ((-2.0 * t_0) * 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))
	t_1 = J_m * math.cos((K * 0.5))
	tmp = 0
	if J_m <= 5.5e-195:
		tmp = (-2.0 * (t_1 * (t_1 / U_m))) - U_m
	else:
		tmp = J_m * ((-2.0 * t_0) * 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))
	t_1 = Float64(J_m * cos(Float64(K * 0.5)))
	tmp = 0.0
	if (J_m <= 5.5e-195)
		tmp = Float64(Float64(-2.0 * Float64(t_1 * Float64(t_1 / U_m))) - U_m);
	else
		tmp = Float64(J_m * Float64(Float64(-2.0 * t_0) * hypot(1.0, Float64(Float64(U_m / 2.0) / Float64(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));
	t_1 = J_m * cos((K * 0.5));
	tmp = 0.0;
	if (J_m <= 5.5e-195)
		tmp = (-2.0 * (t_1 * (t_1 / U_m))) - U_m;
	else
		tmp = J_m * ((-2.0 * t_0) * 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]}, Block[{t$95$1 = N[(J$95$m * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[J$95$m, 5.5e-195], N[(N[(-2.0 * N[(t$95$1 * N[(t$95$1 / U$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - U$95$m), $MachinePrecision], N[(J$95$m * N[(N[(-2.0 * t$95$0), $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[(N[(U$95$m / 2.0), $MachinePrecision] / N[(J$95$m * t$95$0), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if J < 5.5000000000000003e-195

   1. Initial program 67.0%

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

   3. Simplified 82.3%

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

   5. Taylor expanded in J around 0 26.5%

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

      1. fma-define 26.5%

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

      2. unpow2 26.5%

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

      3. *-commutative 26.5%

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

      4. unpow2 26.5%

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

      5. swap-sqr 26.5%

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

      6. unpow2 26.5%

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

      7. *-commutative 26.5%

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

      8. neg-mul-1 26.5%

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

   7. Simplified 26.5%

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

      1. fmm-undef 26.5%

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

      2. add-sqr-sqrt 26.5%

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

      3. pow2 26.5%

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

      4. unpow2 26.5%

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

      5. sqrt-prod 11.4%

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

      6. add-sqr-sqrt 26.5%

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

      7. *-commutative 26.5%

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

   9. Applied egg-rr 26.5%

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

       1. unpow2 26.5%

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

       2. *-un-lft-identity 26.5%

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

       3. times-frac 28.6%

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

   11. Applied egg-rr 28.6%

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

   if 5.5000000000000003e-195 < J

   1. Initial program 81.4%

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

   3. Simplified 92.5%

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

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq 5.5 \cdot 10^{-195}:\\ \;\;\;\;-2 \cdot \left(\left(J \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \frac{J \cdot \cos \left(K \cdot 0.5\right)}{U}\right) - U\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{\frac{U}{2}}{J \cdot \cos \left(\frac{K}{2}\right)}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := J\_m \cdot \cos \left(K \cdot 0.5\right)\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;J\_m \leq 8.6 \cdot 10^{-195}:\\ \;\;\;\;-2 \cdot \left(t\_0 \cdot \frac{t\_0}{U\_m}\right) - U\_m\\ \mathbf{else}:\\ \;\;\;\;J\_m \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{\frac{U\_m}{2}}{J\_m}\right)\right)\\ \end{array} \end{array} \end{array} \]

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 (* J_m (cos (* K 0.5)))))
   (*
    J_s
    (if (<= J_m 8.6e-195)
      (- (* -2.0 (* t_0 (/ t_0 U_m))) U_m)
      (* J_m (* (* -2.0 (cos (/ K 2.0))) (hypot 1.0 (/ (/ 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 t_0 = J_m * cos((K * 0.5));
	double tmp;
	if (J_m <= 8.6e-195) {
		tmp = (-2.0 * (t_0 * (t_0 / U_m))) - U_m;
	} else {
		tmp = J_m * ((-2.0 * cos((K / 2.0))) * hypot(1.0, ((U_m / 2.0) / 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 t_0 = J_m * Math.cos((K * 0.5));
	double tmp;
	if (J_m <= 8.6e-195) {
		tmp = (-2.0 * (t_0 * (t_0 / U_m))) - U_m;
	} else {
		tmp = J_m * ((-2.0 * Math.cos((K / 2.0))) * Math.hypot(1.0, ((U_m / 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):
	t_0 = J_m * math.cos((K * 0.5))
	tmp = 0
	if J_m <= 8.6e-195:
		tmp = (-2.0 * (t_0 * (t_0 / U_m))) - U_m
	else:
		tmp = J_m * ((-2.0 * math.cos((K / 2.0))) * math.hypot(1.0, ((U_m / 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)
	t_0 = Float64(J_m * cos(Float64(K * 0.5)))
	tmp = 0.0
	if (J_m <= 8.6e-195)
		tmp = Float64(Float64(-2.0 * Float64(t_0 * Float64(t_0 / U_m))) - U_m);
	else
		tmp = Float64(J_m * Float64(Float64(-2.0 * cos(Float64(K / 2.0))) * hypot(1.0, Float64(Float64(U_m / 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)
	t_0 = J_m * cos((K * 0.5));
	tmp = 0.0;
	if (J_m <= 8.6e-195)
		tmp = (-2.0 * (t_0 * (t_0 / U_m))) - U_m;
	else
		tmp = J_m * ((-2.0 * cos((K / 2.0))) * hypot(1.0, ((U_m / 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_] := Block[{t$95$0 = N[(J$95$m * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[J$95$m, 8.6e-195], N[(N[(-2.0 * N[(t$95$0 * N[(t$95$0 / U$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - U$95$m), $MachinePrecision], N[(J$95$m * N[(N[(-2.0 * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[(N[(U$95$m / 2.0), $MachinePrecision] / J$95$m), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if J < 8.6000000000000007e-195

   1. Initial program 67.0%

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

   3. Simplified 82.3%

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

   5. Taylor expanded in J around 0 26.5%

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

      1. fma-define 26.5%

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

      2. unpow2 26.5%

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

      3. *-commutative 26.5%

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

      4. unpow2 26.5%

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

      5. swap-sqr 26.5%

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

      6. unpow2 26.5%

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

      7. *-commutative 26.5%

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

      8. neg-mul-1 26.5%

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

   7. Simplified 26.5%

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

      1. fmm-undef 26.5%

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

      2. add-sqr-sqrt 26.5%

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

      3. pow2 26.5%

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

      4. unpow2 26.5%

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

      5. sqrt-prod 11.4%

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

      6. add-sqr-sqrt 26.5%

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

      7. *-commutative 26.5%

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

   9. Applied egg-rr 26.5%

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

       1. unpow2 26.5%

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

       2. *-un-lft-identity 26.5%

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

       3. times-frac 28.6%

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

   11. Applied egg-rr 28.6%

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

   if 8.6000000000000007e-195 < J

   1. Initial program 81.4%

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

   3. Simplified 92.5%

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

   5. Taylor expanded in K around 0 82.0%

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

4. Final simplification 55.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq 8.6 \cdot 10^{-195}:\\ \;\;\;\;-2 \cdot \left(\left(J \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \frac{J \cdot \cos \left(K \cdot 0.5\right)}{U}\right) - U\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{\frac{U}{2}}{J}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ J\_s \cdot \begin{array}{l} \mathbf{if}\;J\_m \leq 8.6 \cdot 10^{-195}:\\ \;\;\;\;-U\_m\\ \mathbf{else}:\\ \;\;\;\;J\_m \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{\frac{U\_m}{2}}{J\_m}\right)\right)\\ \end{array} \end{array} \]

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 (<= J_m 8.6e-195)
    (- U_m)
    (* J_m (* (* -2.0 (cos (/ K 2.0))) (hypot 1.0 (/ (/ 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 <= 8.6e-195) {
		tmp = -U_m;
	} else {
		tmp = J_m * ((-2.0 * cos((K / 2.0))) * hypot(1.0, ((U_m / 2.0) / 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 <= 8.6e-195) {
		tmp = -U_m;
	} else {
		tmp = J_m * ((-2.0 * Math.cos((K / 2.0))) * Math.hypot(1.0, ((U_m / 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 <= 8.6e-195:
		tmp = -U_m
	else:
		tmp = J_m * ((-2.0 * math.cos((K / 2.0))) * math.hypot(1.0, ((U_m / 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 <= 8.6e-195)
		tmp = Float64(-U_m);
	else
		tmp = Float64(J_m * Float64(Float64(-2.0 * cos(Float64(K / 2.0))) * hypot(1.0, Float64(Float64(U_m / 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 <= 8.6e-195)
		tmp = -U_m;
	else
		tmp = J_m * ((-2.0 * cos((K / 2.0))) * hypot(1.0, ((U_m / 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, 8.6e-195], (-U$95$m), N[(J$95$m * N[(N[(-2.0 * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[(N[(U$95$m / 2.0), $MachinePrecision] / J$95$m), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if J < 8.6000000000000007e-195

   1. Initial program 67.0%

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

   3. Simplified 82.3%

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

   5. Taylor expanded in J around 0 28.5%

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

      1. neg-mul-1 28.5%

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

   7. Simplified 28.5%

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

   if 8.6000000000000007e-195 < J

   1. Initial program 81.4%

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

   3. Simplified 92.5%

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

   5. Taylor expanded in K around 0 82.0%

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

Alternative 6: 66.7% accurate, 3.5× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ J\_s \cdot \begin{array}{l} \mathbf{if}\;U\_m \leq 1.22 \cdot 10^{+66}:\\ \;\;\;\;\left(-2 \cdot J\_m\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{elif}\;U\_m \leq 7 \cdot 10^{+163}:\\ \;\;\;\;\left(-2 \cdot J\_m\right) \cdot \mathsf{hypot}\left(1, 0.5 \cdot \frac{U\_m}{J\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;-U\_m\\ \end{array} \end{array} \]

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 1.22e+66)
    (* (* -2.0 J_m) (cos (* K 0.5)))
    (if (<= U_m 7e+163)
      (* (* -2.0 J_m) (hypot 1.0 (* 0.5 (/ U_m 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 <= 1.22e+66) {
		tmp = (-2.0 * J_m) * cos((K * 0.5));
	} else if (U_m <= 7e+163) {
		tmp = (-2.0 * J_m) * hypot(1.0, (0.5 * (U_m / 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 <= 1.22e+66) {
		tmp = (-2.0 * J_m) * Math.cos((K * 0.5));
	} else if (U_m <= 7e+163) {
		tmp = (-2.0 * J_m) * Math.hypot(1.0, (0.5 * (U_m / 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 <= 1.22e+66:
		tmp = (-2.0 * J_m) * math.cos((K * 0.5))
	elif U_m <= 7e+163:
		tmp = (-2.0 * J_m) * math.hypot(1.0, (0.5 * (U_m / 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 <= 1.22e+66)
		tmp = Float64(Float64(-2.0 * J_m) * cos(Float64(K * 0.5)));
	elseif (U_m <= 7e+163)
		tmp = Float64(Float64(-2.0 * J_m) * hypot(1.0, Float64(0.5 * Float64(U_m / 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 <= 1.22e+66)
		tmp = (-2.0 * J_m) * cos((K * 0.5));
	elseif (U_m <= 7e+163)
		tmp = (-2.0 * J_m) * hypot(1.0, (0.5 * (U_m / 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, 1.22e+66], N[(N[(-2.0 * J$95$m), $MachinePrecision] * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[U$95$m, 7e+163], N[(N[(-2.0 * J$95$m), $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[(0.5 * N[(U$95$m / J$95$m), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], (-U$95$m)]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if U < 1.21999999999999993e66

   1. Initial program 81.5%

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

   3. Simplified 92.1%

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

   5. Taylor expanded in J around inf 63.0%

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

      1. *-commutative 63.0%

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

      2. *-commutative 63.0%

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

      3. *-commutative 63.0%

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

      4. associate-*r* 63.0%

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

      5. *-commutative 63.0%

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

      6. *-commutative 63.0%

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

   7. Simplified 63.0%

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

   if 1.21999999999999993e66 < U < 7.0000000000000005e163

   1. Initial program 71.8%

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

      1. unpow2 71.8%

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

      2. hypot-1-def 87.7%

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

      3. associate-/r* 87.7%

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

      4. cos-neg 87.7%

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

      5. distribute-frac-neg 87.7%

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

      6. associate-/r* 87.7%

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

      7. hypot-1-def 71.8%

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

      8. unpow2 71.8%

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

   3. Simplified 87.7%

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

      1. log1p-expm1-u 87.7%

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

      2. div-inv 87.7%

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

      3. metadata-eval 87.7%

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

   6. Applied egg-rr 87.7%

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

   7. Taylor expanded in K around 0 40.8%

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

      1. associate-*r* 40.8%

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

      2. metadata-eval 40.8%

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

      3. metadata-eval 40.8%

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

      4. unpow2 40.8%

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

      5. unpow2 40.8%

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

      6. times-frac 57.0%

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

      7. swap-sqr 57.0%

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

      8. associate-*r/ 57.0%

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

      9. *-commutative 57.0%

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

      10. associate-*r/ 57.0%

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

      11. associate-*r/ 57.0%

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

      12. *-commutative 57.0%

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

      13. associate-*r/ 57.0%

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

      14. hypot-undefine 73.0%

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

      15. associate-*r/ 73.0%

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

      16. *-commutative 73.0%

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

      17. associate-*r/ 73.0%

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

   9. Simplified 73.0%

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

   if 7.0000000000000005e163 < U

   1. Initial program 37.2%

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

   3. Simplified 62.4%

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

   5. Taylor expanded in J around 0 38.2%

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

      1. neg-mul-1 38.2%

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

   7. Simplified 38.2%

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

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 1.22 \cdot 10^{+66}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{elif}\;U \leq 7 \cdot 10^{+163}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \mathsf{hypot}\left(1, 0.5 \cdot \frac{U}{J}\right)\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 66.1% accurate, 3.7× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ J\_s \cdot \begin{array}{l} \mathbf{if}\;U\_m \leq 7.8 \cdot 10^{+80}:\\ \;\;\;\;\left(-2 \cdot J\_m\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;-U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (* J_s (if (<= U_m 7.8e+80) (* (* -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 <= 7.8e+80) {
		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 <= 7.8d+80) 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 <= 7.8e+80) {
		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 <= 7.8e+80:
		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 <= 7.8e+80)
		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 <= 7.8e+80)
		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, 7.8e+80], 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 < 7.79999999999999998e80

   1. Initial program 81.5%

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

   3. Simplified 92.1%

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

   5. Taylor expanded in J around inf 63.1%

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

      1. *-commutative 63.1%

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

      2. *-commutative 63.1%

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

      3. *-commutative 63.1%

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

      4. associate-*r* 63.1%

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

      5. *-commutative 63.1%

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

      6. *-commutative 63.1%

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

   7. Simplified 63.1%

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

   if 7.79999999999999998e80 < U

   1. Initial program 50.0%

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

   3. Simplified 71.9%

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

   5. Taylor expanded in J around 0 39.4%

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

      1. neg-mul-1 39.4%

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

   7. Simplified 39.4%

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

4. Final simplification 57.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 7.8 \cdot 10^{+80}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 40.2% accurate, 34.9× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ J\_s \cdot \begin{array}{l} \mathbf{if}\;K \leq 3.7 \cdot 10^{+53} \lor \neg \left(K \leq 8.5 \cdot 10^{+118}\right):\\ \;\;\;\;-U\_m\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (* J_s (if (or (<= K 3.7e+53) (not (<= K 8.5e+118))) (- 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 <= 3.7e+53) || !(K <= 8.5e+118)) {
		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 <= 3.7d+53) .or. (.not. (k <= 8.5d+118))) 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 <= 3.7e+53) || !(K <= 8.5e+118)) {
		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 <= 3.7e+53) or not (K <= 8.5e+118):
		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 <= 3.7e+53) || !(K <= 8.5e+118))
		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 <= 3.7e+53) || ~((K <= 8.5e+118)))
		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, 3.7e+53], N[Not[LessEqual[K, 8.5e+118]], $MachinePrecision]], (-U$95$m), U$95$m]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if K < 3.7e53 or 8.50000000000000033e118 < K

   1. Initial program 73.8%

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

   3. Simplified 87.1%

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

   5. Taylor expanded in J around 0 22.8%

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

      1. neg-mul-1 22.8%

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

   7. Simplified 22.8%

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

   if 3.7e53 < K < 8.50000000000000033e118

   1. Initial program 79.6%

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

   3. Simplified 92.8%

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

   5. Taylor expanded in U around -inf 16.5%

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

4. Final simplification 22.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;K \leq 3.7 \cdot 10^{+53} \lor \neg \left(K \leq 8.5 \cdot 10^{+118}\right):\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
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 74.2%

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

3. Simplified 87.4%

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

5. Taylor expanded in U around -inf 28.5%

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

Reproduce

herbie shell --seed 2024180 
(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)))))