Maksimov and Kolovsky, Equation (3)

Percentage Accurate: 73.1% → 99.4%

Time: 6.4s

Alternatives: 10

Speedup: 0.4×

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(j, k, u)
use fmin_fmax_functions
    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: 73.1% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(j, k, u)
use fmin_fmax_functions
    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: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(0.5 \cdot K\right)\\ t_1 := \cos \left(\frac{K}{2}\right)\\ t_2 := \left(\left(-2 \cdot J\_m\right) \cdot t\_1\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_1}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\left(0.5 \cdot U\_m\right) \cdot -2\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+296}:\\ \;\;\;\;\left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(-1 \cdot \mathsf{fma}\left(0.5, U\_m, \frac{J\_m \cdot J\_m}{U\_m}\right)\right) \cdot -2\\ \end{array} \end{array} \end{array} \]

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 (* 0.5 K)))
        (t_1 (cos (/ K 2.0)))
        (t_2
         (*
          (* (* -2.0 J_m) t_1)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_1)) 2.0))))))
   (*
    J_s
    (if (<= t_2 (- INFINITY))
      (* (* 0.5 U_m) -2.0)
      (if (<= t_2 2e+296)
        (*
         (* (* -2.0 J_m) t_0)
         (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_0)) 2.0))))
        (* (* -1.0 (fma 0.5 U_m (/ (* J_m J_m) U_m))) -2.0))))))

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((0.5 * K));
	double t_1 = cos((K / 2.0));
	double t_2 = ((-2.0 * J_m) * t_1) * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_1)), 2.0)));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = (0.5 * U_m) * -2.0;
	} else if (t_2 <= 2e+296) {
		tmp = ((-2.0 * J_m) * t_0) * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_0)), 2.0)));
	} else {
		tmp = (-1.0 * fma(0.5, U_m, ((J_m * J_m) / U_m))) * -2.0;
	}
	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(0.5 * K))
	t_1 = cos(Float64(K / 2.0))
	t_2 = Float64(Float64(Float64(-2.0 * J_m) * t_1) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * t_1)) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64(0.5 * U_m) * -2.0);
	elseif (t_2 <= 2e+296)
		tmp = Float64(Float64(Float64(-2.0 * J_m) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * t_0)) ^ 2.0))));
	else
		tmp = Float64(Float64(-1.0 * fma(0.5, U_m, Float64(Float64(J_m * J_m) / U_m))) * -2.0);
	end
	return Float64(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[(0.5 * K), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$1), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$2, (-Infinity)], N[(N[(0.5 * U$95$m), $MachinePrecision] * -2.0), $MachinePrecision], If[LessEqual[t$95$2, 2e+296], 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[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 * N[(0.5 * U$95$m + N[(N[(J$95$m * J$95$m), $MachinePrecision] / U$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]]]), $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 73.1%

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

   2. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 32.9%

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

   5. Taylor expanded in J around 0

      \[\leadsto \sqrt{\frac{1}{4} \cdot {U}^{2}} \cdot -2 \]
   6. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{1}{4} \cdot {U}^{2}} \cdot -2 \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{4} \cdot {U}^{2}} \cdot -2 \]

      3. pow2 N/A

         \[\leadsto \sqrt{\frac{1}{4} \cdot \left(U \cdot U\right)} \cdot -2 \]

      4. lift-*.f64 22.2

         \[\leadsto \sqrt{0.25 \cdot \left(U \cdot U\right)} \cdot -2 \]

   7. Applied rewrites 22.2%

      \[\leadsto \sqrt{0.25 \cdot \left(U \cdot U\right)} \cdot -2 \]

   8. Taylor expanded in U around 0

      \[\leadsto \left(\frac{1}{2} \cdot U\right) \cdot -2 \]
   9. Step-by-step derivation

      1. lower-*.f64 39.3

         \[\leadsto \left(0.5 \cdot U\right) \cdot -2 \]

   10. Applied rewrites 39.3%

       \[\leadsto \left(0.5 \cdot U\right) \cdot -2 \]

   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))))) < 1.99999999999999996e296

   1. Initial program 73.1%

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

   2. Taylor expanded in K around 0

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

      1. lower-*.f64 73.1

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

   4. Applied rewrites 73.1%

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

   5. Taylor expanded in K around 0

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

      1. lower-*.f64 73.1

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

   7. Applied rewrites 73.1%

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

   if 1.99999999999999996e296 < (*.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 73.1%

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

   2. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 32.9%

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

   5. Taylor expanded in J around 0

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

      1. sqrt-prod N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{{U}^{2}}, \frac{1}{2} \cdot \frac{{J}^{2}}{\sqrt{\frac{1}{4} \cdot {U}^{2}}}\right) \cdot -2 \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{{U}^{2}}, \frac{1}{2} \cdot \frac{{J}^{2}}{\sqrt{\frac{1}{4} \cdot {U}^{2}}}\right) \cdot -2 \]

      5. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{U \cdot U}, \frac{1}{2} \cdot \frac{{J}^{2}}{\sqrt{\frac{1}{4} \cdot {U}^{2}}}\right) \cdot -2 \]

      6. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{U \cdot U}, \frac{1}{2} \cdot \frac{{J}^{2}}{\sqrt{\frac{1}{4} \cdot {U}^{2}}}\right) \cdot -2 \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{U \cdot U}, \frac{1}{2} \cdot \frac{{J}^{2}}{\sqrt{\frac{1}{4} \cdot {U}^{2}}}\right) \cdot -2 \]

      8. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{U \cdot U}, \frac{1}{2} \cdot \frac{{J}^{2}}{\sqrt{\frac{1}{4} \cdot {U}^{2}}}\right) \cdot -2 \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{U \cdot U}, \frac{1}{2} \cdot \frac{J \cdot J}{\sqrt{\frac{1}{4} \cdot {U}^{2}}}\right) \cdot -2 \]

      10. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{U \cdot U}, \frac{1}{2} \cdot \frac{J \cdot J}{\sqrt{\frac{1}{4} \cdot {U}^{2}}}\right) \cdot -2 \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{U \cdot U}, \frac{1}{2} \cdot \frac{J \cdot J}{\sqrt{\frac{1}{4} \cdot {U}^{2}}}\right) \cdot -2 \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{U \cdot U}, \frac{1}{2} \cdot \frac{J \cdot J}{\sqrt{\frac{1}{4} \cdot {U}^{2}}}\right) \cdot -2 \]

      13. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{U \cdot U}, \frac{1}{2} \cdot \frac{J \cdot J}{\sqrt{\frac{1}{4} \cdot \left(U \cdot U\right)}}\right) \cdot -2 \]

      14. lift-*.f64 22.0

          \[\leadsto \mathsf{fma}\left(0.5, \sqrt{U \cdot U}, 0.5 \cdot \frac{J \cdot J}{\sqrt{0.25 \cdot \left(U \cdot U\right)}}\right) \cdot -2 \]

   7. Applied rewrites 22.0%

      \[\leadsto \mathsf{fma}\left(0.5, \sqrt{U \cdot U}, 0.5 \cdot \frac{J \cdot J}{\sqrt{0.25 \cdot \left(U \cdot U\right)}}\right) \cdot -2 \]

   8. Taylor expanded in U around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 12.3

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

   10. Applied rewrites 12.3%

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

   11. Taylor expanded in J around 0

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

       1. lower-fma.f64 N/A

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

       2. lower-/.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 13.5

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

   13. Applied rewrites 13.5%

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

Alternative 2: 97.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(0.5 \cdot K\right)\\ t_1 := \cos \left(\frac{K}{2}\right)\\ t_2 := \left(\left(-2 \cdot J\_m\right) \cdot t\_1\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_1}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\left(0.5 \cdot U\_m\right) \cdot -2\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+296}:\\ \;\;\;\;\left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \cosh \sinh^{-1} \left(\frac{U\_m}{\left(J\_m + J\_m\right) \cdot t\_0}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-1 \cdot \mathsf{fma}\left(0.5, U\_m, \frac{J\_m \cdot J\_m}{U\_m}\right)\right) \cdot -2\\ \end{array} \end{array} \end{array} \]

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 (* 0.5 K)))
        (t_1 (cos (/ K 2.0)))
        (t_2
         (*
          (* (* -2.0 J_m) t_1)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_1)) 2.0))))))
   (*
    J_s
    (if (<= t_2 (- INFINITY))
      (* (* 0.5 U_m) -2.0)
      (if (<= t_2 2e+296)
        (* (* (* -2.0 J_m) t_0) (cosh (asinh (/ U_m (* (+ J_m J_m) t_0)))))
        (* (* -1.0 (fma 0.5 U_m (/ (* J_m J_m) U_m))) -2.0))))))

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((0.5 * K));
	double t_1 = cos((K / 2.0));
	double t_2 = ((-2.0 * J_m) * t_1) * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_1)), 2.0)));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = (0.5 * U_m) * -2.0;
	} else if (t_2 <= 2e+296) {
		tmp = ((-2.0 * J_m) * t_0) * cosh(asinh((U_m / ((J_m + J_m) * t_0))));
	} else {
		tmp = (-1.0 * fma(0.5, U_m, ((J_m * J_m) / U_m))) * -2.0;
	}
	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(0.5 * K))
	t_1 = cos(Float64(K / 2.0))
	t_2 = Float64(Float64(Float64(-2.0 * J_m) * t_1) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * t_1)) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64(0.5 * U_m) * -2.0);
	elseif (t_2 <= 2e+296)
		tmp = Float64(Float64(Float64(-2.0 * J_m) * t_0) * cosh(asinh(Float64(U_m / Float64(Float64(J_m + J_m) * t_0)))));
	else
		tmp = Float64(Float64(-1.0 * fma(0.5, U_m, Float64(Float64(J_m * J_m) / U_m))) * -2.0);
	end
	return Float64(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[(0.5 * K), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$1), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$2, (-Infinity)], N[(N[(0.5 * U$95$m), $MachinePrecision] * -2.0), $MachinePrecision], If[LessEqual[t$95$2, 2e+296], N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Cosh[N[ArcSinh[N[(U$95$m / N[(N[(J$95$m + J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 * N[(0.5 * U$95$m + N[(N[(J$95$m * J$95$m), $MachinePrecision] / U$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]]]), $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 73.1%

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

   2. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 32.9%

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

   5. Taylor expanded in J around 0

      \[\leadsto \sqrt{\frac{1}{4} \cdot {U}^{2}} \cdot -2 \]
   6. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{1}{4} \cdot {U}^{2}} \cdot -2 \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{4} \cdot {U}^{2}} \cdot -2 \]

      3. pow2 N/A

         \[\leadsto \sqrt{\frac{1}{4} \cdot \left(U \cdot U\right)} \cdot -2 \]

      4. lift-*.f64 22.2

         \[\leadsto \sqrt{0.25 \cdot \left(U \cdot U\right)} \cdot -2 \]

   7. Applied rewrites 22.2%

      \[\leadsto \sqrt{0.25 \cdot \left(U \cdot U\right)} \cdot -2 \]

   8. Taylor expanded in U around 0

      \[\leadsto \left(\frac{1}{2} \cdot U\right) \cdot -2 \]
   9. Step-by-step derivation

      1. lower-*.f64 39.3

         \[\leadsto \left(0.5 \cdot U\right) \cdot -2 \]

   10. Applied rewrites 39.3%

       \[\leadsto \left(0.5 \cdot U\right) \cdot -2 \]

   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))))) < 1.99999999999999996e296

   1. Initial program 73.1%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. +-commutative N/A

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

      10. unpow2 N/A

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

      11. cosh-asinh-rev N/A

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

      12. lower-cosh.f64 N/A

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

      13. lower-asinh.f64 N/A

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

   3. Applied rewrites 84.8%

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

   4. Taylor expanded in K around 0

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

      1. lift-*.f64 84.8

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

   6. Applied rewrites 84.8%

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

   7. Taylor expanded in K around 0

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

      1. lift-*.f64 84.8

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

   9. Applied rewrites 84.8%

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

   if 1.99999999999999996e296 < (*.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 73.1%

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

   2. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 32.9%

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

   5. Taylor expanded in J around 0

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

      1. sqrt-prod N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{{U}^{2}}, \frac{1}{2} \cdot \frac{{J}^{2}}{\sqrt{\frac{1}{4} \cdot {U}^{2}}}\right) \cdot -2 \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{{U}^{2}}, \frac{1}{2} \cdot \frac{{J}^{2}}{\sqrt{\frac{1}{4} \cdot {U}^{2}}}\right) \cdot -2 \]

      5. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{U \cdot U}, \frac{1}{2} \cdot \frac{{J}^{2}}{\sqrt{\frac{1}{4} \cdot {U}^{2}}}\right) \cdot -2 \]

      6. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{U \cdot U}, \frac{1}{2} \cdot \frac{{J}^{2}}{\sqrt{\frac{1}{4} \cdot {U}^{2}}}\right) \cdot -2 \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{U \cdot U}, \frac{1}{2} \cdot \frac{{J}^{2}}{\sqrt{\frac{1}{4} \cdot {U}^{2}}}\right) \cdot -2 \]

      8. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{U \cdot U}, \frac{1}{2} \cdot \frac{{J}^{2}}{\sqrt{\frac{1}{4} \cdot {U}^{2}}}\right) \cdot -2 \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{U \cdot U}, \frac{1}{2} \cdot \frac{J \cdot J}{\sqrt{\frac{1}{4} \cdot {U}^{2}}}\right) \cdot -2 \]

      10. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{U \cdot U}, \frac{1}{2} \cdot \frac{J \cdot J}{\sqrt{\frac{1}{4} \cdot {U}^{2}}}\right) \cdot -2 \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{U \cdot U}, \frac{1}{2} \cdot \frac{J \cdot J}{\sqrt{\frac{1}{4} \cdot {U}^{2}}}\right) \cdot -2 \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{U \cdot U}, \frac{1}{2} \cdot \frac{J \cdot J}{\sqrt{\frac{1}{4} \cdot {U}^{2}}}\right) \cdot -2 \]

      13. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{U \cdot U}, \frac{1}{2} \cdot \frac{J \cdot J}{\sqrt{\frac{1}{4} \cdot \left(U \cdot U\right)}}\right) \cdot -2 \]

      14. lift-*.f64 22.0

          \[\leadsto \mathsf{fma}\left(0.5, \sqrt{U \cdot U}, 0.5 \cdot \frac{J \cdot J}{\sqrt{0.25 \cdot \left(U \cdot U\right)}}\right) \cdot -2 \]

   7. Applied rewrites 22.0%

      \[\leadsto \mathsf{fma}\left(0.5, \sqrt{U \cdot U}, 0.5 \cdot \frac{J \cdot J}{\sqrt{0.25 \cdot \left(U \cdot U\right)}}\right) \cdot -2 \]

   8. Taylor expanded in U around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 12.3

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

   10. Applied rewrites 12.3%

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

   11. Taylor expanded in J around 0

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

       1. lower-fma.f64 N/A

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

       2. lower-/.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 13.5

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

   13. Applied rewrites 13.5%

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

Alternative 3: 90.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(-2 \cdot J\_m\right) \cdot t\_0\\ t_2 := t\_1 \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\left(0.5 \cdot U\_m\right) \cdot -2\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+296}:\\ \;\;\;\;t\_1 \cdot \sqrt{1 + {\left(\frac{U\_m}{J\_m} \cdot 0.5\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(-1 \cdot \mathsf{fma}\left(0.5, U\_m, \frac{J\_m \cdot J\_m}{U\_m}\right)\right) \cdot -2\\ \end{array} \end{array} \end{array} \]

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))
        (t_2 (* t_1 (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_0)) 2.0))))))
   (*
    J_s
    (if (<= t_2 (- INFINITY))
      (* (* 0.5 U_m) -2.0)
      (if (<= t_2 2e+296)
        (* t_1 (sqrt (+ 1.0 (pow (* (/ U_m J_m) 0.5) 2.0))))
        (* (* -1.0 (fma 0.5 U_m (/ (* J_m J_m) U_m))) -2.0))))))

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 = (-2.0 * J_m) * t_0;
	double t_2 = t_1 * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_0)), 2.0)));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = (0.5 * U_m) * -2.0;
	} else if (t_2 <= 2e+296) {
		tmp = t_1 * sqrt((1.0 + pow(((U_m / J_m) * 0.5), 2.0)));
	} else {
		tmp = (-1.0 * fma(0.5, U_m, ((J_m * J_m) / U_m))) * -2.0;
	}
	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(Float64(-2.0 * J_m) * t_0)
	t_2 = Float64(t_1 * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * t_0)) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64(0.5 * U_m) * -2.0);
	elseif (t_2 <= 2e+296)
		tmp = Float64(t_1 * sqrt(Float64(1.0 + (Float64(Float64(U_m / J_m) * 0.5) ^ 2.0))));
	else
		tmp = Float64(Float64(-1.0 * fma(0.5, U_m, Float64(Float64(J_m * J_m) / U_m))) * -2.0);
	end
	return Float64(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[(-2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$2, (-Infinity)], N[(N[(0.5 * U$95$m), $MachinePrecision] * -2.0), $MachinePrecision], If[LessEqual[t$95$2, 2e+296], N[(t$95$1 * N[Sqrt[N[(1.0 + N[Power[N[(N[(U$95$m / J$95$m), $MachinePrecision] * 0.5), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 * N[(0.5 * U$95$m + N[(N[(J$95$m * J$95$m), $MachinePrecision] / U$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]]]), $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 73.1%

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

   2. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 32.9%

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

   5. Taylor expanded in J around 0

      \[\leadsto \sqrt{\frac{1}{4} \cdot {U}^{2}} \cdot -2 \]
   6. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{1}{4} \cdot {U}^{2}} \cdot -2 \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{4} \cdot {U}^{2}} \cdot -2 \]

      3. pow2 N/A

         \[\leadsto \sqrt{\frac{1}{4} \cdot \left(U \cdot U\right)} \cdot -2 \]

      4. lift-*.f64 22.2

         \[\leadsto \sqrt{0.25 \cdot \left(U \cdot U\right)} \cdot -2 \]

   7. Applied rewrites 22.2%

      \[\leadsto \sqrt{0.25 \cdot \left(U \cdot U\right)} \cdot -2 \]

   8. Taylor expanded in U around 0

      \[\leadsto \left(\frac{1}{2} \cdot U\right) \cdot -2 \]
   9. Step-by-step derivation

      1. lower-*.f64 39.3

         \[\leadsto \left(0.5 \cdot U\right) \cdot -2 \]

   10. Applied rewrites 39.3%

       \[\leadsto \left(0.5 \cdot U\right) \cdot -2 \]

   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))))) < 1.99999999999999996e296

   1. Initial program 73.1%

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

   2. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 64.2

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

   4. Applied rewrites 64.2%

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

   if 1.99999999999999996e296 < (*.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 73.1%

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

   2. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 32.9%

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

   5. Taylor expanded in J around 0

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

      1. sqrt-prod N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{{U}^{2}}, \frac{1}{2} \cdot \frac{{J}^{2}}{\sqrt{\frac{1}{4} \cdot {U}^{2}}}\right) \cdot -2 \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{{U}^{2}}, \frac{1}{2} \cdot \frac{{J}^{2}}{\sqrt{\frac{1}{4} \cdot {U}^{2}}}\right) \cdot -2 \]

      5. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{U \cdot U}, \frac{1}{2} \cdot \frac{{J}^{2}}{\sqrt{\frac{1}{4} \cdot {U}^{2}}}\right) \cdot -2 \]

      6. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{U \cdot U}, \frac{1}{2} \cdot \frac{{J}^{2}}{\sqrt{\frac{1}{4} \cdot {U}^{2}}}\right) \cdot -2 \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{U \cdot U}, \frac{1}{2} \cdot \frac{{J}^{2}}{\sqrt{\frac{1}{4} \cdot {U}^{2}}}\right) \cdot -2 \]

      8. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{U \cdot U}, \frac{1}{2} \cdot \frac{{J}^{2}}{\sqrt{\frac{1}{4} \cdot {U}^{2}}}\right) \cdot -2 \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{U \cdot U}, \frac{1}{2} \cdot \frac{J \cdot J}{\sqrt{\frac{1}{4} \cdot {U}^{2}}}\right) \cdot -2 \]

      10. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{U \cdot U}, \frac{1}{2} \cdot \frac{J \cdot J}{\sqrt{\frac{1}{4} \cdot {U}^{2}}}\right) \cdot -2 \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{U \cdot U}, \frac{1}{2} \cdot \frac{J \cdot J}{\sqrt{\frac{1}{4} \cdot {U}^{2}}}\right) \cdot -2 \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{U \cdot U}, \frac{1}{2} \cdot \frac{J \cdot J}{\sqrt{\frac{1}{4} \cdot {U}^{2}}}\right) \cdot -2 \]

      13. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{U \cdot U}, \frac{1}{2} \cdot \frac{J \cdot J}{\sqrt{\frac{1}{4} \cdot \left(U \cdot U\right)}}\right) \cdot -2 \]

      14. lift-*.f64 22.0

          \[\leadsto \mathsf{fma}\left(0.5, \sqrt{U \cdot U}, 0.5 \cdot \frac{J \cdot J}{\sqrt{0.25 \cdot \left(U \cdot U\right)}}\right) \cdot -2 \]

   7. Applied rewrites 22.0%

      \[\leadsto \mathsf{fma}\left(0.5, \sqrt{U \cdot U}, 0.5 \cdot \frac{J \cdot J}{\sqrt{0.25 \cdot \left(U \cdot U\right)}}\right) \cdot -2 \]

   8. Taylor expanded in U around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 12.3

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

   10. Applied rewrites 12.3%

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

   11. Taylor expanded in J around 0

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

       1. lower-fma.f64 N/A

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

       2. lower-/.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 13.5

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

   13. Applied rewrites 13.5%

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

Alternative 4: 89.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(-2 \cdot J\_m\right) \cdot t\_0\\ t_2 := t\_1 \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\left(0.5 \cdot U\_m\right) \cdot -2\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+296}:\\ \;\;\;\;t\_1 \cdot \cosh \sinh^{-1} \left(0.5 \cdot \frac{U\_m}{J\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-1 \cdot \mathsf{fma}\left(0.5, U\_m, \frac{J\_m \cdot J\_m}{U\_m}\right)\right) \cdot -2\\ \end{array} \end{array} \end{array} \]

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))
        (t_2 (* t_1 (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_0)) 2.0))))))
   (*
    J_s
    (if (<= t_2 (- INFINITY))
      (* (* 0.5 U_m) -2.0)
      (if (<= t_2 2e+296)
        (* t_1 (cosh (asinh (* 0.5 (/ U_m J_m)))))
        (* (* -1.0 (fma 0.5 U_m (/ (* J_m J_m) U_m))) -2.0))))))

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 = (-2.0 * J_m) * t_0;
	double t_2 = t_1 * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_0)), 2.0)));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = (0.5 * U_m) * -2.0;
	} else if (t_2 <= 2e+296) {
		tmp = t_1 * cosh(asinh((0.5 * (U_m / J_m))));
	} else {
		tmp = (-1.0 * fma(0.5, U_m, ((J_m * J_m) / U_m))) * -2.0;
	}
	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(Float64(-2.0 * J_m) * t_0)
	t_2 = Float64(t_1 * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * t_0)) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64(0.5 * U_m) * -2.0);
	elseif (t_2 <= 2e+296)
		tmp = Float64(t_1 * cosh(asinh(Float64(0.5 * Float64(U_m / J_m)))));
	else
		tmp = Float64(Float64(-1.0 * fma(0.5, U_m, Float64(Float64(J_m * J_m) / U_m))) * -2.0);
	end
	return Float64(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[(-2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$2, (-Infinity)], N[(N[(0.5 * U$95$m), $MachinePrecision] * -2.0), $MachinePrecision], If[LessEqual[t$95$2, 2e+296], N[(t$95$1 * N[Cosh[N[ArcSinh[N[(0.5 * N[(U$95$m / J$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 * N[(0.5 * U$95$m + N[(N[(J$95$m * J$95$m), $MachinePrecision] / U$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]]]), $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 73.1%

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

   2. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 32.9%

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

   5. Taylor expanded in J around 0

      \[\leadsto \sqrt{\frac{1}{4} \cdot {U}^{2}} \cdot -2 \]
   6. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{1}{4} \cdot {U}^{2}} \cdot -2 \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{4} \cdot {U}^{2}} \cdot -2 \]

      3. pow2 N/A

         \[\leadsto \sqrt{\frac{1}{4} \cdot \left(U \cdot U\right)} \cdot -2 \]

      4. lift-*.f64 22.2

         \[\leadsto \sqrt{0.25 \cdot \left(U \cdot U\right)} \cdot -2 \]

   7. Applied rewrites 22.2%

      \[\leadsto \sqrt{0.25 \cdot \left(U \cdot U\right)} \cdot -2 \]

   8. Taylor expanded in U around 0

      \[\leadsto \left(\frac{1}{2} \cdot U\right) \cdot -2 \]
   9. Step-by-step derivation

      1. lower-*.f64 39.3

         \[\leadsto \left(0.5 \cdot U\right) \cdot -2 \]

   10. Applied rewrites 39.3%

       \[\leadsto \left(0.5 \cdot U\right) \cdot -2 \]

   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))))) < 1.99999999999999996e296

   1. Initial program 73.1%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. +-commutative N/A

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

      10. unpow2 N/A

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

      11. cosh-asinh-rev N/A

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

      12. lower-cosh.f64 N/A

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

      13. lower-asinh.f64 N/A

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

   3. Applied rewrites 84.8%

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

   4. Taylor expanded in K around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 71.4

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

   6. Applied rewrites 71.4%

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

   if 1.99999999999999996e296 < (*.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 73.1%

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

   2. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 32.9%

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

   5. Taylor expanded in J around 0

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

      1. sqrt-prod N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{{U}^{2}}, \frac{1}{2} \cdot \frac{{J}^{2}}{\sqrt{\frac{1}{4} \cdot {U}^{2}}}\right) \cdot -2 \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{{U}^{2}}, \frac{1}{2} \cdot \frac{{J}^{2}}{\sqrt{\frac{1}{4} \cdot {U}^{2}}}\right) \cdot -2 \]

      5. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{U \cdot U}, \frac{1}{2} \cdot \frac{{J}^{2}}{\sqrt{\frac{1}{4} \cdot {U}^{2}}}\right) \cdot -2 \]

      6. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{U \cdot U}, \frac{1}{2} \cdot \frac{{J}^{2}}{\sqrt{\frac{1}{4} \cdot {U}^{2}}}\right) \cdot -2 \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{U \cdot U}, \frac{1}{2} \cdot \frac{{J}^{2}}{\sqrt{\frac{1}{4} \cdot {U}^{2}}}\right) \cdot -2 \]

      8. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{U \cdot U}, \frac{1}{2} \cdot \frac{{J}^{2}}{\sqrt{\frac{1}{4} \cdot {U}^{2}}}\right) \cdot -2 \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{U \cdot U}, \frac{1}{2} \cdot \frac{J \cdot J}{\sqrt{\frac{1}{4} \cdot {U}^{2}}}\right) \cdot -2 \]

      10. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{U \cdot U}, \frac{1}{2} \cdot \frac{J \cdot J}{\sqrt{\frac{1}{4} \cdot {U}^{2}}}\right) \cdot -2 \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{U \cdot U}, \frac{1}{2} \cdot \frac{J \cdot J}{\sqrt{\frac{1}{4} \cdot {U}^{2}}}\right) \cdot -2 \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{U \cdot U}, \frac{1}{2} \cdot \frac{J \cdot J}{\sqrt{\frac{1}{4} \cdot {U}^{2}}}\right) \cdot -2 \]

      13. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{U \cdot U}, \frac{1}{2} \cdot \frac{J \cdot J}{\sqrt{\frac{1}{4} \cdot \left(U \cdot U\right)}}\right) \cdot -2 \]

      14. lift-*.f64 22.0

          \[\leadsto \mathsf{fma}\left(0.5, \sqrt{U \cdot U}, 0.5 \cdot \frac{J \cdot J}{\sqrt{0.25 \cdot \left(U \cdot U\right)}}\right) \cdot -2 \]

   7. Applied rewrites 22.0%

      \[\leadsto \mathsf{fma}\left(0.5, \sqrt{U \cdot U}, 0.5 \cdot \frac{J \cdot J}{\sqrt{0.25 \cdot \left(U \cdot U\right)}}\right) \cdot -2 \]

   8. Taylor expanded in U around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 12.3

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

   10. Applied rewrites 12.3%

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

   11. Taylor expanded in J around 0

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

       1. lower-fma.f64 N/A

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

       2. lower-/.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 13.5

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

   13. Applied rewrites 13.5%

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

Alternative 5: 83.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(0.5 \cdot U\_m\right) \cdot -2\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-196}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(\frac{U\_m}{J\_m} \cdot \frac{U\_m}{J\_m}, 0.25, 1\right)} \cdot J\_m\right) \cdot -2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+296}:\\ \;\;\;\;\left(-2 \cdot \left(J\_m \cdot \cos \left(0.5 \cdot K\right)\right)\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\left(-1 \cdot \mathsf{fma}\left(0.5, U\_m, \frac{J\_m \cdot J\_m}{U\_m}\right)\right) \cdot -2\\ \end{array} \end{array} \end{array} \]

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 (* (* 2.0 J_m) t_0)) 2.0))))))
   (*
    J_s
    (if (<= t_1 (- INFINITY))
      (* (* 0.5 U_m) -2.0)
      (if (<= t_1 -5e-196)
        (* (* (sqrt (fma (* (/ U_m J_m) (/ U_m J_m)) 0.25 1.0)) J_m) -2.0)
        (if (<= t_1 2e+296)
          (* (* -2.0 (* J_m (cos (* 0.5 K)))) 1.0)
          (* (* -1.0 (fma 0.5 U_m (/ (* J_m J_m) U_m))) -2.0)))))))

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 = ((-2.0 * J_m) * t_0) * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_0)), 2.0)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (0.5 * U_m) * -2.0;
	} else if (t_1 <= -5e-196) {
		tmp = (sqrt(fma(((U_m / J_m) * (U_m / J_m)), 0.25, 1.0)) * J_m) * -2.0;
	} else if (t_1 <= 2e+296) {
		tmp = (-2.0 * (J_m * cos((0.5 * K)))) * 1.0;
	} else {
		tmp = (-1.0 * fma(0.5, U_m, ((J_m * J_m) / U_m))) * -2.0;
	}
	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(Float64(Float64(-2.0 * J_m) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * t_0)) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(0.5 * U_m) * -2.0);
	elseif (t_1 <= -5e-196)
		tmp = Float64(Float64(sqrt(fma(Float64(Float64(U_m / J_m) * Float64(U_m / J_m)), 0.25, 1.0)) * J_m) * -2.0);
	elseif (t_1 <= 2e+296)
		tmp = Float64(Float64(-2.0 * Float64(J_m * cos(Float64(0.5 * K)))) * 1.0);
	else
		tmp = Float64(Float64(-1.0 * fma(0.5, U_m, Float64(Float64(J_m * J_m) / U_m))) * -2.0);
	end
	return Float64(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[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$1, (-Infinity)], N[(N[(0.5 * U$95$m), $MachinePrecision] * -2.0), $MachinePrecision], If[LessEqual[t$95$1, -5e-196], N[(N[(N[Sqrt[N[(N[(N[(U$95$m / J$95$m), $MachinePrecision] * N[(U$95$m / J$95$m), $MachinePrecision]), $MachinePrecision] * 0.25 + 1.0), $MachinePrecision]], $MachinePrecision] * J$95$m), $MachinePrecision] * -2.0), $MachinePrecision], If[LessEqual[t$95$1, 2e+296], N[(N[(-2.0 * N[(J$95$m * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[(-1.0 * N[(0.5 * U$95$m + N[(N[(J$95$m * J$95$m), $MachinePrecision] / U$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 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 73.1%

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

   2. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 32.9%

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

   5. Taylor expanded in J around 0

      \[\leadsto \sqrt{\frac{1}{4} \cdot {U}^{2}} \cdot -2 \]
   6. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{1}{4} \cdot {U}^{2}} \cdot -2 \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{4} \cdot {U}^{2}} \cdot -2 \]

      3. pow2 N/A

         \[\leadsto \sqrt{\frac{1}{4} \cdot \left(U \cdot U\right)} \cdot -2 \]

      4. lift-*.f64 22.2

         \[\leadsto \sqrt{0.25 \cdot \left(U \cdot U\right)} \cdot -2 \]

   7. Applied rewrites 22.2%

      \[\leadsto \sqrt{0.25 \cdot \left(U \cdot U\right)} \cdot -2 \]

   8. Taylor expanded in U around 0

      \[\leadsto \left(\frac{1}{2} \cdot U\right) \cdot -2 \]
   9. Step-by-step derivation

      1. lower-*.f64 39.3

         \[\leadsto \left(0.5 \cdot U\right) \cdot -2 \]

   10. Applied rewrites 39.3%

       \[\leadsto \left(0.5 \cdot U\right) \cdot -2 \]

   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))))) < -5.0000000000000005e-196

   1. Initial program 73.1%

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

   2. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 32.9%

      \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(\frac{U \cdot U}{J \cdot J}, 0.25, 1\right)} \cdot J\right) \cdot -2} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\mathsf{fma}\left(\frac{U \cdot U}{J \cdot J}, \frac{1}{4}, 1\right)} \cdot J\right) \cdot -2 \]

      2. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\mathsf{fma}\left(\frac{U \cdot U}{J \cdot J}, \frac{1}{4}, 1\right)} \cdot J\right) \cdot -2 \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\mathsf{fma}\left(\frac{U \cdot U}{J \cdot J}, \frac{1}{4}, 1\right)} \cdot J\right) \cdot -2 \]

      4. times-frac N/A

         \[\leadsto \left(\sqrt{\mathsf{fma}\left(\frac{U}{J} \cdot \frac{U}{J}, \frac{1}{4}, 1\right)} \cdot J\right) \cdot -2 \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\mathsf{fma}\left(\frac{U}{J} \cdot \frac{U}{J}, \frac{1}{4}, 1\right)} \cdot J\right) \cdot -2 \]

      6. lower-/.f64 N/A

         \[\leadsto \left(\sqrt{\mathsf{fma}\left(\frac{U}{J} \cdot \frac{U}{J}, \frac{1}{4}, 1\right)} \cdot J\right) \cdot -2 \]

      7. lower-/.f64 44.9

         \[\leadsto \left(\sqrt{\mathsf{fma}\left(\frac{U}{J} \cdot \frac{U}{J}, 0.25, 1\right)} \cdot J\right) \cdot -2 \]

   6. Applied rewrites 44.9%

      \[\leadsto \left(\sqrt{\mathsf{fma}\left(\frac{U}{J} \cdot \frac{U}{J}, 0.25, 1\right)} \cdot J\right) \cdot -2 \]

   if -5.0000000000000005e-196 < (*.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))))) < 1.99999999999999996e296

   1. Initial program 73.1%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. +-commutative N/A

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

      10. unpow2 N/A

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

      11. cosh-asinh-rev N/A

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

      12. lower-cosh.f64 N/A

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

      13. lower-asinh.f64 N/A

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

   3. Applied rewrites 84.8%

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

   4. Taylor expanded in J around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 84.8

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

   6. Applied rewrites 84.8%

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

   7. Taylor expanded in J around inf

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

   9. Applied rewrites 51.9%

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

   if 1.99999999999999996e296 < (*.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 73.1%

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

   2. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 32.9%

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

   5. Taylor expanded in J around 0

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

      1. sqrt-prod N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{{U}^{2}}, \frac{1}{2} \cdot \frac{{J}^{2}}{\sqrt{\frac{1}{4} \cdot {U}^{2}}}\right) \cdot -2 \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{{U}^{2}}, \frac{1}{2} \cdot \frac{{J}^{2}}{\sqrt{\frac{1}{4} \cdot {U}^{2}}}\right) \cdot -2 \]

      5. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{U \cdot U}, \frac{1}{2} \cdot \frac{{J}^{2}}{\sqrt{\frac{1}{4} \cdot {U}^{2}}}\right) \cdot -2 \]

      6. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{U \cdot U}, \frac{1}{2} \cdot \frac{{J}^{2}}{\sqrt{\frac{1}{4} \cdot {U}^{2}}}\right) \cdot -2 \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{U \cdot U}, \frac{1}{2} \cdot \frac{{J}^{2}}{\sqrt{\frac{1}{4} \cdot {U}^{2}}}\right) \cdot -2 \]

      8. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{U \cdot U}, \frac{1}{2} \cdot \frac{{J}^{2}}{\sqrt{\frac{1}{4} \cdot {U}^{2}}}\right) \cdot -2 \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{U \cdot U}, \frac{1}{2} \cdot \frac{J \cdot J}{\sqrt{\frac{1}{4} \cdot {U}^{2}}}\right) \cdot -2 \]

      10. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{U \cdot U}, \frac{1}{2} \cdot \frac{J \cdot J}{\sqrt{\frac{1}{4} \cdot {U}^{2}}}\right) \cdot -2 \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{U \cdot U}, \frac{1}{2} \cdot \frac{J \cdot J}{\sqrt{\frac{1}{4} \cdot {U}^{2}}}\right) \cdot -2 \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{U \cdot U}, \frac{1}{2} \cdot \frac{J \cdot J}{\sqrt{\frac{1}{4} \cdot {U}^{2}}}\right) \cdot -2 \]

      13. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{U \cdot U}, \frac{1}{2} \cdot \frac{J \cdot J}{\sqrt{\frac{1}{4} \cdot \left(U \cdot U\right)}}\right) \cdot -2 \]

      14. lift-*.f64 22.0

          \[\leadsto \mathsf{fma}\left(0.5, \sqrt{U \cdot U}, 0.5 \cdot \frac{J \cdot J}{\sqrt{0.25 \cdot \left(U \cdot U\right)}}\right) \cdot -2 \]

   7. Applied rewrites 22.0%

      \[\leadsto \mathsf{fma}\left(0.5, \sqrt{U \cdot U}, 0.5 \cdot \frac{J \cdot J}{\sqrt{0.25 \cdot \left(U \cdot U\right)}}\right) \cdot -2 \]

   8. Taylor expanded in U around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 12.3

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

   10. Applied rewrites 12.3%

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

   11. Taylor expanded in J around 0

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

       1. lower-fma.f64 N/A

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

       2. lower-/.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 13.5

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

   13. Applied rewrites 13.5%

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

Alternative 6: 77.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(0.5 \cdot U\_m\right) \cdot -2\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-292}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(\frac{U\_m}{J\_m} \cdot \frac{U\_m}{J\_m}, 0.25, 1\right)} \cdot J\_m\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\left(-0.5 \cdot U\_m\right) \cdot -2\\ \end{array} \end{array} \end{array} \]

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 (* (* 2.0 J_m) t_0)) 2.0))))))
   (*
    J_s
    (if (<= t_1 (- INFINITY))
      (* (* 0.5 U_m) -2.0)
      (if (<= t_1 -5e-292)
        (* (* (sqrt (fma (* (/ U_m J_m) (/ U_m J_m)) 0.25 1.0)) J_m) -2.0)
        (* (* -0.5 U_m) -2.0))))))

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 = ((-2.0 * J_m) * t_0) * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_0)), 2.0)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (0.5 * U_m) * -2.0;
	} else if (t_1 <= -5e-292) {
		tmp = (sqrt(fma(((U_m / J_m) * (U_m / J_m)), 0.25, 1.0)) * J_m) * -2.0;
	} else {
		tmp = (-0.5 * U_m) * -2.0;
	}
	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(Float64(Float64(-2.0 * J_m) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * t_0)) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(0.5 * U_m) * -2.0);
	elseif (t_1 <= -5e-292)
		tmp = Float64(Float64(sqrt(fma(Float64(Float64(U_m / J_m) * Float64(U_m / J_m)), 0.25, 1.0)) * J_m) * -2.0);
	else
		tmp = Float64(Float64(-0.5 * U_m) * -2.0);
	end
	return Float64(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[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$1, (-Infinity)], N[(N[(0.5 * U$95$m), $MachinePrecision] * -2.0), $MachinePrecision], If[LessEqual[t$95$1, -5e-292], N[(N[(N[Sqrt[N[(N[(N[(U$95$m / J$95$m), $MachinePrecision] * N[(U$95$m / J$95$m), $MachinePrecision]), $MachinePrecision] * 0.25 + 1.0), $MachinePrecision]], $MachinePrecision] * J$95$m), $MachinePrecision] * -2.0), $MachinePrecision], N[(N[(-0.5 * U$95$m), $MachinePrecision] * -2.0), $MachinePrecision]]]), $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 73.1%

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

   2. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 32.9%

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

   5. Taylor expanded in J around 0

      \[\leadsto \sqrt{\frac{1}{4} \cdot {U}^{2}} \cdot -2 \]
   6. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{1}{4} \cdot {U}^{2}} \cdot -2 \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{4} \cdot {U}^{2}} \cdot -2 \]

      3. pow2 N/A

         \[\leadsto \sqrt{\frac{1}{4} \cdot \left(U \cdot U\right)} \cdot -2 \]

      4. lift-*.f64 22.2

         \[\leadsto \sqrt{0.25 \cdot \left(U \cdot U\right)} \cdot -2 \]

   7. Applied rewrites 22.2%

      \[\leadsto \sqrt{0.25 \cdot \left(U \cdot U\right)} \cdot -2 \]

   8. Taylor expanded in U around 0

      \[\leadsto \left(\frac{1}{2} \cdot U\right) \cdot -2 \]
   9. Step-by-step derivation

      1. lower-*.f64 39.3

         \[\leadsto \left(0.5 \cdot U\right) \cdot -2 \]

   10. Applied rewrites 39.3%

       \[\leadsto \left(0.5 \cdot U\right) \cdot -2 \]

   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))))) < -4.99999999999999981e-292

   1. Initial program 73.1%

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

   2. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 32.9%

      \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(\frac{U \cdot U}{J \cdot J}, 0.25, 1\right)} \cdot J\right) \cdot -2} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\mathsf{fma}\left(\frac{U \cdot U}{J \cdot J}, \frac{1}{4}, 1\right)} \cdot J\right) \cdot -2 \]

      2. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\mathsf{fma}\left(\frac{U \cdot U}{J \cdot J}, \frac{1}{4}, 1\right)} \cdot J\right) \cdot -2 \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\mathsf{fma}\left(\frac{U \cdot U}{J \cdot J}, \frac{1}{4}, 1\right)} \cdot J\right) \cdot -2 \]

      4. times-frac N/A

         \[\leadsto \left(\sqrt{\mathsf{fma}\left(\frac{U}{J} \cdot \frac{U}{J}, \frac{1}{4}, 1\right)} \cdot J\right) \cdot -2 \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\mathsf{fma}\left(\frac{U}{J} \cdot \frac{U}{J}, \frac{1}{4}, 1\right)} \cdot J\right) \cdot -2 \]

      6. lower-/.f64 N/A

         \[\leadsto \left(\sqrt{\mathsf{fma}\left(\frac{U}{J} \cdot \frac{U}{J}, \frac{1}{4}, 1\right)} \cdot J\right) \cdot -2 \]

      7. lower-/.f64 44.9

         \[\leadsto \left(\sqrt{\mathsf{fma}\left(\frac{U}{J} \cdot \frac{U}{J}, 0.25, 1\right)} \cdot J\right) \cdot -2 \]

   6. Applied rewrites 44.9%

      \[\leadsto \left(\sqrt{\mathsf{fma}\left(\frac{U}{J} \cdot \frac{U}{J}, 0.25, 1\right)} \cdot J\right) \cdot -2 \]

   if -4.99999999999999981e-292 < (*.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 73.1%

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

   2. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 32.9%

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

   5. Taylor expanded in J around 0

      \[\leadsto \sqrt{\frac{1}{4} \cdot {U}^{2}} \cdot -2 \]
   6. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{1}{4} \cdot {U}^{2}} \cdot -2 \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{4} \cdot {U}^{2}} \cdot -2 \]

      3. pow2 N/A

         \[\leadsto \sqrt{\frac{1}{4} \cdot \left(U \cdot U\right)} \cdot -2 \]

      4. lift-*.f64 22.2

         \[\leadsto \sqrt{0.25 \cdot \left(U \cdot U\right)} \cdot -2 \]

   7. Applied rewrites 22.2%

      \[\leadsto \sqrt{0.25 \cdot \left(U \cdot U\right)} \cdot -2 \]

   8. Taylor expanded in U around -inf

      \[\leadsto \left(\frac{-1}{2} \cdot U\right) \cdot -2 \]
   9. Step-by-step derivation

      1. lower-*.f64 14.2

         \[\leadsto \left(-0.5 \cdot U\right) \cdot -2 \]

   10. Applied rewrites 14.2%

       \[\leadsto \left(-0.5 \cdot U\right) \cdot -2 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 66.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \left(0.5 \cdot U\_m\right) \cdot -2\\ t_1 := \cos \left(\frac{K}{2}\right)\\ t_2 := \left(\left(-2 \cdot J\_m\right) \cdot t\_1\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_1}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+266}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{-158}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(\frac{U\_m \cdot U\_m}{J\_m \cdot J\_m}, 0.25, 1\right)} \cdot J\_m\right) \cdot -2\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{-292}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(-0.5 \cdot U\_m\right) \cdot -2\\ \end{array} \end{array} \end{array} \]

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 U_m) -2.0))
        (t_1 (cos (/ K 2.0)))
        (t_2
         (*
          (* (* -2.0 J_m) t_1)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_1)) 2.0))))))
   (*
    J_s
    (if (<= t_2 -2e+266)
      t_0
      (if (<= t_2 -5e-158)
        (* (* (sqrt (fma (/ (* U_m U_m) (* J_m J_m)) 0.25 1.0)) J_m) -2.0)
        (if (<= t_2 -5e-292) t_0 (* (* -0.5 U_m) -2.0)))))))

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 * U_m) * -2.0;
	double t_1 = cos((K / 2.0));
	double t_2 = ((-2.0 * J_m) * t_1) * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_1)), 2.0)));
	double tmp;
	if (t_2 <= -2e+266) {
		tmp = t_0;
	} else if (t_2 <= -5e-158) {
		tmp = (sqrt(fma(((U_m * U_m) / (J_m * J_m)), 0.25, 1.0)) * J_m) * -2.0;
	} else if (t_2 <= -5e-292) {
		tmp = t_0;
	} else {
		tmp = (-0.5 * U_m) * -2.0;
	}
	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(Float64(0.5 * U_m) * -2.0)
	t_1 = cos(Float64(K / 2.0))
	t_2 = Float64(Float64(Float64(-2.0 * J_m) * t_1) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * t_1)) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= -2e+266)
		tmp = t_0;
	elseif (t_2 <= -5e-158)
		tmp = Float64(Float64(sqrt(fma(Float64(Float64(U_m * U_m) / Float64(J_m * J_m)), 0.25, 1.0)) * J_m) * -2.0);
	elseif (t_2 <= -5e-292)
		tmp = t_0;
	else
		tmp = Float64(Float64(-0.5 * U_m) * -2.0);
	end
	return Float64(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[(N[(0.5 * U$95$m), $MachinePrecision] * -2.0), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$1), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$2, -2e+266], t$95$0, If[LessEqual[t$95$2, -5e-158], N[(N[(N[Sqrt[N[(N[(N[(U$95$m * U$95$m), $MachinePrecision] / N[(J$95$m * J$95$m), $MachinePrecision]), $MachinePrecision] * 0.25 + 1.0), $MachinePrecision]], $MachinePrecision] * J$95$m), $MachinePrecision] * -2.0), $MachinePrecision], If[LessEqual[t$95$2, -5e-292], t$95$0, N[(N[(-0.5 * U$95$m), $MachinePrecision] * -2.0), $MachinePrecision]]]]), $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))))) < -2.0000000000000001e266 or -4.99999999999999972e-158 < (*.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))))) < -4.99999999999999981e-292

   1. Initial program 73.1%

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

   2. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 32.9%

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

   5. Taylor expanded in J around 0

      \[\leadsto \sqrt{\frac{1}{4} \cdot {U}^{2}} \cdot -2 \]
   6. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{1}{4} \cdot {U}^{2}} \cdot -2 \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{4} \cdot {U}^{2}} \cdot -2 \]

      3. pow2 N/A

         \[\leadsto \sqrt{\frac{1}{4} \cdot \left(U \cdot U\right)} \cdot -2 \]

      4. lift-*.f64 22.2

         \[\leadsto \sqrt{0.25 \cdot \left(U \cdot U\right)} \cdot -2 \]

   7. Applied rewrites 22.2%

      \[\leadsto \sqrt{0.25 \cdot \left(U \cdot U\right)} \cdot -2 \]

   8. Taylor expanded in U around 0

      \[\leadsto \left(\frac{1}{2} \cdot U\right) \cdot -2 \]
   9. Step-by-step derivation

      1. lower-*.f64 39.3

         \[\leadsto \left(0.5 \cdot U\right) \cdot -2 \]

   10. Applied rewrites 39.3%

       \[\leadsto \left(0.5 \cdot U\right) \cdot -2 \]

   if -2.0000000000000001e266 < (*.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))))) < -4.99999999999999972e-158

   1. Initial program 73.1%

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

   2. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 32.9%

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

   if -4.99999999999999981e-292 < (*.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 73.1%

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

   2. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 32.9%

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

   5. Taylor expanded in J around 0

      \[\leadsto \sqrt{\frac{1}{4} \cdot {U}^{2}} \cdot -2 \]
   6. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{1}{4} \cdot {U}^{2}} \cdot -2 \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{4} \cdot {U}^{2}} \cdot -2 \]

      3. pow2 N/A

         \[\leadsto \sqrt{\frac{1}{4} \cdot \left(U \cdot U\right)} \cdot -2 \]

      4. lift-*.f64 22.2

         \[\leadsto \sqrt{0.25 \cdot \left(U \cdot U\right)} \cdot -2 \]

   7. Applied rewrites 22.2%

      \[\leadsto \sqrt{0.25 \cdot \left(U \cdot U\right)} \cdot -2 \]

   8. Taylor expanded in U around -inf

      \[\leadsto \left(\frac{-1}{2} \cdot U\right) \cdot -2 \]
   9. Step-by-step derivation

      1. lower-*.f64 14.2

         \[\leadsto \left(-0.5 \cdot U\right) \cdot -2 \]

   10. Applied rewrites 14.2%

       \[\leadsto \left(-0.5 \cdot U\right) \cdot -2 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 60.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(0.5 \cdot U\_m\right) \cdot -2\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-292}:\\ \;\;\;\;\mathsf{fma}\left(-2, J\_m, -0.25 \cdot \frac{U\_m \cdot U\_m}{J\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-0.5 \cdot U\_m\right) \cdot -2\\ \end{array} \end{array} \end{array} \]

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 (* (* 2.0 J_m) t_0)) 2.0))))))
   (*
    J_s
    (if (<= t_1 (- INFINITY))
      (* (* 0.5 U_m) -2.0)
      (if (<= t_1 -5e-292)
        (fma -2.0 J_m (* -0.25 (/ (* U_m U_m) J_m)))
        (* (* -0.5 U_m) -2.0))))))

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 = ((-2.0 * J_m) * t_0) * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_0)), 2.0)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (0.5 * U_m) * -2.0;
	} else if (t_1 <= -5e-292) {
		tmp = fma(-2.0, J_m, (-0.25 * ((U_m * U_m) / J_m)));
	} else {
		tmp = (-0.5 * U_m) * -2.0;
	}
	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(Float64(Float64(-2.0 * J_m) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * t_0)) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(0.5 * U_m) * -2.0);
	elseif (t_1 <= -5e-292)
		tmp = fma(-2.0, J_m, Float64(-0.25 * Float64(Float64(U_m * U_m) / J_m)));
	else
		tmp = Float64(Float64(-0.5 * U_m) * -2.0);
	end
	return Float64(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[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$1, (-Infinity)], N[(N[(0.5 * U$95$m), $MachinePrecision] * -2.0), $MachinePrecision], If[LessEqual[t$95$1, -5e-292], N[(-2.0 * J$95$m + N[(-0.25 * N[(N[(U$95$m * U$95$m), $MachinePrecision] / J$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * U$95$m), $MachinePrecision] * -2.0), $MachinePrecision]]]), $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 73.1%

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

   2. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 32.9%

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

   5. Taylor expanded in J around 0

      \[\leadsto \sqrt{\frac{1}{4} \cdot {U}^{2}} \cdot -2 \]
   6. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{1}{4} \cdot {U}^{2}} \cdot -2 \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{4} \cdot {U}^{2}} \cdot -2 \]

      3. pow2 N/A

         \[\leadsto \sqrt{\frac{1}{4} \cdot \left(U \cdot U\right)} \cdot -2 \]

      4. lift-*.f64 22.2

         \[\leadsto \sqrt{0.25 \cdot \left(U \cdot U\right)} \cdot -2 \]

   7. Applied rewrites 22.2%

      \[\leadsto \sqrt{0.25 \cdot \left(U \cdot U\right)} \cdot -2 \]

   8. Taylor expanded in U around 0

      \[\leadsto \left(\frac{1}{2} \cdot U\right) \cdot -2 \]
   9. Step-by-step derivation

      1. lower-*.f64 39.3

         \[\leadsto \left(0.5 \cdot U\right) \cdot -2 \]

   10. Applied rewrites 39.3%

       \[\leadsto \left(0.5 \cdot U\right) \cdot -2 \]

   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))))) < -4.99999999999999981e-292

   1. Initial program 73.1%

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

   2. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 32.9%

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

   5. Taylor expanded in U around 0

      \[\leadsto -2 \cdot J + \color{blue}{\frac{-1}{4} \cdot \frac{{U}^{2}}{J}} \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(-2, J, \frac{-1}{4} \cdot \frac{{U}^{2}}{J}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-2, J, \frac{-1}{4} \cdot \frac{{U}^{2}}{J}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(-2, J, \frac{-1}{4} \cdot \frac{{U}^{2}}{J}\right) \]

      4. pow2 N/A

         \[\leadsto \mathsf{fma}\left(-2, J, \frac{-1}{4} \cdot \frac{U \cdot U}{J}\right) \]

      5. lift-*.f64 28.2

         \[\leadsto \mathsf{fma}\left(-2, J, -0.25 \cdot \frac{U \cdot U}{J}\right) \]

   7. Applied rewrites 28.2%

      \[\leadsto \mathsf{fma}\left(-2, \color{blue}{J}, -0.25 \cdot \frac{U \cdot U}{J}\right) \]

   if -4.99999999999999981e-292 < (*.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 73.1%

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

   2. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 32.9%

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

   5. Taylor expanded in J around 0

      \[\leadsto \sqrt{\frac{1}{4} \cdot {U}^{2}} \cdot -2 \]
   6. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{1}{4} \cdot {U}^{2}} \cdot -2 \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{4} \cdot {U}^{2}} \cdot -2 \]

      3. pow2 N/A

         \[\leadsto \sqrt{\frac{1}{4} \cdot \left(U \cdot U\right)} \cdot -2 \]

      4. lift-*.f64 22.2

         \[\leadsto \sqrt{0.25 \cdot \left(U \cdot U\right)} \cdot -2 \]

   7. Applied rewrites 22.2%

      \[\leadsto \sqrt{0.25 \cdot \left(U \cdot U\right)} \cdot -2 \]

   8. Taylor expanded in U around -inf

      \[\leadsto \left(\frac{-1}{2} \cdot U\right) \cdot -2 \]
   9. Step-by-step derivation

      1. lower-*.f64 14.2

         \[\leadsto \left(-0.5 \cdot U\right) \cdot -2 \]

   10. Applied rewrites 14.2%

       \[\leadsto \left(-0.5 \cdot U\right) \cdot -2 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 52.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2}} \leq -5 \cdot 10^{-292}:\\ \;\;\;\;\left(0.5 \cdot U\_m\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\left(-0.5 \cdot U\_m\right) \cdot -2\\ \end{array} \end{array} \end{array} \]

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))))
   (*
    J_s
    (if (<=
         (*
          (* (* -2.0 J_m) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_0)) 2.0))))
         -5e-292)
      (* (* 0.5 U_m) -2.0)
      (* (* -0.5 U_m) -2.0)))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if ((((-2.0 * J_m) * t_0) * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_0)), 2.0)))) <= -5e-292) {
		tmp = (0.5 * U_m) * -2.0;
	} else {
		tmp = (-0.5 * U_m) * -2.0;
	}
	return J_s * tmp;
}

Fortran

U_m =     private
J\_m =     private
J\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(j_s, j_m, k, u_m)
use fmin_fmax_functions
    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 = cos((k / 2.0d0))
    if (((((-2.0d0) * j_m) * t_0) * sqrt((1.0d0 + ((u_m / ((2.0d0 * j_m) * t_0)) ** 2.0d0)))) <= (-5d-292)) then
        tmp = (0.5d0 * u_m) * (-2.0d0)
    else
        tmp = ((-0.5d0) * u_m) * (-2.0d0)
    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 tmp;
	if ((((-2.0 * J_m) * t_0) * Math.sqrt((1.0 + Math.pow((U_m / ((2.0 * J_m) * t_0)), 2.0)))) <= -5e-292) {
		tmp = (0.5 * U_m) * -2.0;
	} else {
		tmp = (-0.5 * U_m) * -2.0;
	}
	return J_s * tmp;
}

Python

U_m = math.fabs(U)
J\_m = math.fabs(J)
J\_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if (((-2.0 * J_m) * t_0) * math.sqrt((1.0 + math.pow((U_m / ((2.0 * J_m) * t_0)), 2.0)))) <= -5e-292:
		tmp = (0.5 * U_m) * -2.0
	else:
		tmp = (-0.5 * U_m) * -2.0
	return J_s * tmp

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (Float64(Float64(Float64(-2.0 * J_m) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * t_0)) ^ 2.0)))) <= -5e-292)
		tmp = Float64(Float64(0.5 * U_m) * -2.0);
	else
		tmp = Float64(Float64(-0.5 * U_m) * -2.0);
	end
	return Float64(J_s * tmp)
end

MATLAB

U_m = abs(U);
J\_m = abs(J);
J\_s = sign(J) * abs(1.0);
function tmp_2 = code(J_s, J_m, K, U_m)
	t_0 = cos((K / 2.0));
	tmp = 0.0;
	if ((((-2.0 * J_m) * t_0) * sqrt((1.0 + ((U_m / ((2.0 * J_m) * t_0)) ^ 2.0)))) <= -5e-292)
		tmp = (0.5 * U_m) * -2.0;
	else
		tmp = (-0.5 * U_m) * -2.0;
	end
	tmp_2 = J_s * tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, N[(J$95$s * If[LessEqual[N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -5e-292], N[(N[(0.5 * U$95$m), $MachinePrecision] * -2.0), $MachinePrecision], N[(N[(-0.5 * U$95$m), $MachinePrecision] * -2.0), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 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))))) < -4.99999999999999981e-292

   1. Initial program 73.1%

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

   2. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 32.9%

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

   5. Taylor expanded in J around 0

      \[\leadsto \sqrt{\frac{1}{4} \cdot {U}^{2}} \cdot -2 \]
   6. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{1}{4} \cdot {U}^{2}} \cdot -2 \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{4} \cdot {U}^{2}} \cdot -2 \]

      3. pow2 N/A

         \[\leadsto \sqrt{\frac{1}{4} \cdot \left(U \cdot U\right)} \cdot -2 \]

      4. lift-*.f64 22.2

         \[\leadsto \sqrt{0.25 \cdot \left(U \cdot U\right)} \cdot -2 \]

   7. Applied rewrites 22.2%

      \[\leadsto \sqrt{0.25 \cdot \left(U \cdot U\right)} \cdot -2 \]

   8. Taylor expanded in U around 0

      \[\leadsto \left(\frac{1}{2} \cdot U\right) \cdot -2 \]
   9. Step-by-step derivation

      1. lower-*.f64 39.3

         \[\leadsto \left(0.5 \cdot U\right) \cdot -2 \]

   10. Applied rewrites 39.3%

       \[\leadsto \left(0.5 \cdot U\right) \cdot -2 \]

   if -4.99999999999999981e-292 < (*.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 73.1%

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

   2. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 32.9%

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

   5. Taylor expanded in J around 0

      \[\leadsto \sqrt{\frac{1}{4} \cdot {U}^{2}} \cdot -2 \]
   6. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{1}{4} \cdot {U}^{2}} \cdot -2 \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{4} \cdot {U}^{2}} \cdot -2 \]

      3. pow2 N/A

         \[\leadsto \sqrt{\frac{1}{4} \cdot \left(U \cdot U\right)} \cdot -2 \]

      4. lift-*.f64 22.2

         \[\leadsto \sqrt{0.25 \cdot \left(U \cdot U\right)} \cdot -2 \]

   7. Applied rewrites 22.2%

      \[\leadsto \sqrt{0.25 \cdot \left(U \cdot U\right)} \cdot -2 \]

   8. Taylor expanded in U around -inf

      \[\leadsto \left(\frac{-1}{2} \cdot U\right) \cdot -2 \]
   9. Step-by-step derivation

      1. lower-*.f64 14.2

         \[\leadsto \left(-0.5 \cdot U\right) \cdot -2 \]

   10. Applied rewrites 14.2%

       \[\leadsto \left(-0.5 \cdot U\right) \cdot -2 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 14.2% accurate, 15.9× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ J\_s \cdot \left(\left(-0.5 \cdot U\_m\right) \cdot -2\right) \end{array} \]

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 (* (* -0.5 U_m) -2.0)))

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) {
	return J_s * ((-0.5 * U_m) * -2.0);
}

Fortran

U_m =     private
J\_m =     private
J\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(j_s, j_m, k, u_m)
use fmin_fmax_functions
    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 * (((-0.5d0) * u_m) * (-2.0d0))
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 * ((-0.5 * U_m) * -2.0);
}

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 * ((-0.5 * U_m) * -2.0)

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 * Float64(Float64(-0.5 * U_m) * -2.0))
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 * ((-0.5 * U_m) * -2.0);
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 * N[(N[(-0.5 * U$95$m), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 73.1%

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

2. Taylor expanded in K around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

4. Applied rewrites 32.9%

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

5. Taylor expanded in J around 0

   \[\leadsto \sqrt{\frac{1}{4} \cdot {U}^{2}} \cdot -2 \]
6. Step-by-step derivation

   1. lower-sqrt.f64 N/A

      \[\leadsto \sqrt{\frac{1}{4} \cdot {U}^{2}} \cdot -2 \]

   2. lower-*.f64 N/A

      \[\leadsto \sqrt{\frac{1}{4} \cdot {U}^{2}} \cdot -2 \]

   3. pow2 N/A

      \[\leadsto \sqrt{\frac{1}{4} \cdot \left(U \cdot U\right)} \cdot -2 \]

   4. lift-*.f64 22.2

      \[\leadsto \sqrt{0.25 \cdot \left(U \cdot U\right)} \cdot -2 \]

7. Applied rewrites 22.2%

   \[\leadsto \sqrt{0.25 \cdot \left(U \cdot U\right)} \cdot -2 \]

8. Taylor expanded in U around -inf

   \[\leadsto \left(\frac{-1}{2} \cdot U\right) \cdot -2 \]
9. Step-by-step derivation

   1. lower-*.f64 14.2

      \[\leadsto \left(-0.5 \cdot U\right) \cdot -2 \]

10. Applied rewrites 14.2%

    \[\leadsto \left(-0.5 \cdot U\right) \cdot -2 \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025142 
(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)))))