Maksimov and Kolovsky, Equation (3)

Percentage Accurate: 73.0% → 99.5%

Time: 6.8s

Alternatives: 13

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: 73.0% 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: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\right) \cdot t\_0}\right)}^{2}}\\ t_2 := \cos \left(0.5 \cdot K\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(-1 \cdot \left(U\_m \cdot \mathsf{fma}\left(0.5, \frac{J}{\left(U\_m \cdot U\_m\right) \cdot \frac{0.5}{\sqrt{J \cdot J}}}, -0.5\right)\right)\right) \cdot -2\\ \mathbf{elif}\;t\_1 \leq 10^{+304}:\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot t\_2\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\right) \cdot t\_2}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(-0.5 \cdot U\_m\right) \cdot -2\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* (* -2.0 J) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J) t_0)) 2.0)))))
        (t_2 (cos (* 0.5 K))))
   (if (<= t_1 (- INFINITY))
     (*
      (*
       -1.0
       (* U_m (fma 0.5 (/ J (* (* U_m U_m) (/ 0.5 (sqrt (* J J))))) -0.5)))
      -2.0)
     (if (<= t_1 1e+304)
       (*
        (* (* -2.0 J) t_2)
        (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J) t_2)) 2.0))))
       (* (* -0.5 U_m) -2.0)))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * J) * t_0) * sqrt((1.0 + pow((U_m / ((2.0 * J) * t_0)), 2.0)));
	double t_2 = cos((0.5 * K));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (-1.0 * (U_m * fma(0.5, (J / ((U_m * U_m) * (0.5 / sqrt((J * J))))), -0.5))) * -2.0;
	} else if (t_1 <= 1e+304) {
		tmp = ((-2.0 * J) * t_2) * sqrt((1.0 + pow((U_m / ((2.0 * J) * t_2)), 2.0)));
	} else {
		tmp = (-0.5 * U_m) * -2.0;
	}
	return tmp;
}

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * J) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J) * t_0)) ^ 2.0))))
	t_2 = cos(Float64(0.5 * K))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(-1.0 * Float64(U_m * fma(0.5, Float64(J / Float64(Float64(U_m * U_m) * Float64(0.5 / sqrt(Float64(J * J))))), -0.5))) * -2.0);
	elseif (t_1 <= 1e+304)
		tmp = Float64(Float64(Float64(-2.0 * J) * t_2) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J) * t_2)) ^ 2.0))));
	else
		tmp = Float64(Float64(-0.5 * U_m) * -2.0);
	end
	return tmp
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(-1.0 * N[(U$95$m * N[(0.5 * N[(J / N[(N[(U$95$m * U$95$m), $MachinePrecision] * N[(0.5 / N[Sqrt[N[(J * J), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], If[LessEqual[t$95$1, 1e+304], N[(N[(N[(-2.0 * J), $MachinePrecision] * t$95$2), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * U$95$m), $MachinePrecision] * -2.0), $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 73.0%

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

   2. 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 32.2%

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

   5. Taylor expanded in U around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-fma.f64 N/A

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

   7. Applied rewrites 7.6%

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

   8. Taylor expanded in J around -inf

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

   10. Applied rewrites 23.0%

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

   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))))) < 9.9999999999999994e303

   1. Initial program 73.0%

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

   2. 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 73.0

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

   4. Applied rewrites 73.0%

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

   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 73.0

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

   7. Applied rewrites 73.0%

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

   if 9.9999999999999994e303 < (*.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 73.0%

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

   2. 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 32.2%

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

   5. Taylor expanded in U around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-fma.f64 N/A

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

   7. Applied rewrites 7.6%

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

   8. Taylor expanded in J around 0

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

      1. lower-*.f64 27.5

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

   10. Applied rewrites 27.5%

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

Alternative 2: 97.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(-2 \cdot J\right) \cdot t\_0\\ t_2 := t\_1 \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\right) \cdot t\_0}\right)}^{2}}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\left(-1 \cdot \left(U\_m \cdot \mathsf{fma}\left(0.5, \frac{J}{\left(U\_m \cdot U\_m\right) \cdot \frac{0.5}{\sqrt{J \cdot J}}}, -0.5\right)\right)\right) \cdot -2\\ \mathbf{elif}\;t\_2 \leq 10^{+304}:\\ \;\;\;\;t\_1 \cdot \cosh \sinh^{-1} \left(\frac{U\_m}{\left(J + J\right) \cdot t\_0}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-0.5 \cdot U\_m\right) \cdot -2\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1 (* (* -2.0 J) t_0))
        (t_2 (* t_1 (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J) t_0)) 2.0))))))
   (if (<= t_2 (- INFINITY))
     (*
      (*
       -1.0
       (* U_m (fma 0.5 (/ J (* (* U_m U_m) (/ 0.5 (sqrt (* J J))))) -0.5)))
      -2.0)
     (if (<= t_2 1e+304)
       (* t_1 (cosh (asinh (/ U_m (* (+ J J) t_0)))))
       (* (* -0.5 U_m) -2.0)))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = (-2.0 * J) * t_0;
	double t_2 = t_1 * sqrt((1.0 + pow((U_m / ((2.0 * J) * t_0)), 2.0)));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = (-1.0 * (U_m * fma(0.5, (J / ((U_m * U_m) * (0.5 / sqrt((J * J))))), -0.5))) * -2.0;
	} else if (t_2 <= 1e+304) {
		tmp = t_1 * cosh(asinh((U_m / ((J + J) * t_0))));
	} else {
		tmp = (-0.5 * U_m) * -2.0;
	}
	return tmp;
}

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(-2.0 * J) * t_0)
	t_2 = Float64(t_1 * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J) * t_0)) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64(-1.0 * Float64(U_m * fma(0.5, Float64(J / Float64(Float64(U_m * U_m) * Float64(0.5 / sqrt(Float64(J * J))))), -0.5))) * -2.0);
	elseif (t_2 <= 1e+304)
		tmp = Float64(t_1 * cosh(asinh(Float64(U_m / Float64(Float64(J + J) * t_0)))));
	else
		tmp = Float64(Float64(-0.5 * U_m) * -2.0);
	end
	return tmp
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(-2.0 * J), $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), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[(-1.0 * N[(U$95$m * N[(0.5 * N[(J / N[(N[(U$95$m * U$95$m), $MachinePrecision] * N[(0.5 / N[Sqrt[N[(J * J), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], If[LessEqual[t$95$2, 1e+304], N[(t$95$1 * N[Cosh[N[ArcSinh[N[(U$95$m / N[(N[(J + J), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * U$95$m), $MachinePrecision] * -2.0), $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 73.0%

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

   2. 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 32.2%

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

   5. Taylor expanded in U around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-fma.f64 N/A

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

   7. Applied rewrites 7.6%

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

   8. Taylor expanded in J around -inf

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

   10. Applied rewrites 23.0%

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

   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))))) < 9.9999999999999994e303

   1. Initial program 73.0%

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

      1. 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 84.3%

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

   if 9.9999999999999994e303 < (*.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 73.0%

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

   2. 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 32.2%

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

   5. Taylor expanded in U around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-fma.f64 N/A

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

   7. Applied rewrites 7.6%

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

   8. Taylor expanded in J around 0

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

      1. lower-*.f64 27.5

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

   10. Applied rewrites 27.5%

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

Alternative 3: 91.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(-2 \cdot J\right) \cdot t\_0\\ t_2 := t\_1 \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\right) \cdot t\_0}\right)}^{2}}\\ t_3 := \frac{U\_m}{J + J}\\ t_4 := t\_1 \cdot \sqrt{1 + t\_3 \cdot t\_3}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\left(-1 \cdot \left(U\_m \cdot \mathsf{fma}\left(0.5, \frac{J}{\left(U\_m \cdot U\_m\right) \cdot \frac{0.5}{\sqrt{J \cdot J}}}, -0.5\right)\right)\right) \cdot -2\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{+156}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{-139}:\\ \;\;\;\;t\_1 \cdot \sqrt{\frac{\frac{U\_m \cdot U\_m}{0.5 + 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot K\right)\right)} \cdot 0.25}{J \cdot J} + 1}\\ \mathbf{elif}\;t\_2 \leq 10^{+304}:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;\left(-0.5 \cdot U\_m\right) \cdot -2\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1 (* (* -2.0 J) t_0))
        (t_2 (* t_1 (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J) t_0)) 2.0)))))
        (t_3 (/ U_m (+ J J)))
        (t_4 (* t_1 (sqrt (+ 1.0 (* t_3 t_3))))))
   (if (<= t_2 (- INFINITY))
     (*
      (*
       -1.0
       (* U_m (fma 0.5 (/ J (* (* U_m U_m) (/ 0.5 (sqrt (* J J))))) -0.5)))
      -2.0)
     (if (<= t_2 -2e+156)
       t_4
       (if (<= t_2 -2e-139)
         (*
          t_1
          (sqrt
           (+
            (/
             (* (/ (* U_m U_m) (+ 0.5 (* 0.5 (cos (* 2.0 (* 0.5 K)))))) 0.25)
             (* J J))
            1.0)))
         (if (<= t_2 1e+304) t_4 (* (* -0.5 U_m) -2.0)))))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = (-2.0 * J) * t_0;
	double t_2 = t_1 * sqrt((1.0 + pow((U_m / ((2.0 * J) * t_0)), 2.0)));
	double t_3 = U_m / (J + J);
	double t_4 = t_1 * sqrt((1.0 + (t_3 * t_3)));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = (-1.0 * (U_m * fma(0.5, (J / ((U_m * U_m) * (0.5 / sqrt((J * J))))), -0.5))) * -2.0;
	} else if (t_2 <= -2e+156) {
		tmp = t_4;
	} else if (t_2 <= -2e-139) {
		tmp = t_1 * sqrt((((((U_m * U_m) / (0.5 + (0.5 * cos((2.0 * (0.5 * K)))))) * 0.25) / (J * J)) + 1.0));
	} else if (t_2 <= 1e+304) {
		tmp = t_4;
	} else {
		tmp = (-0.5 * U_m) * -2.0;
	}
	return tmp;
}

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(-2.0 * J) * t_0)
	t_2 = Float64(t_1 * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J) * t_0)) ^ 2.0))))
	t_3 = Float64(U_m / Float64(J + J))
	t_4 = Float64(t_1 * sqrt(Float64(1.0 + Float64(t_3 * t_3))))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64(-1.0 * Float64(U_m * fma(0.5, Float64(J / Float64(Float64(U_m * U_m) * Float64(0.5 / sqrt(Float64(J * J))))), -0.5))) * -2.0);
	elseif (t_2 <= -2e+156)
		tmp = t_4;
	elseif (t_2 <= -2e-139)
		tmp = Float64(t_1 * sqrt(Float64(Float64(Float64(Float64(Float64(U_m * U_m) / Float64(0.5 + Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * K)))))) * 0.25) / Float64(J * J)) + 1.0)));
	elseif (t_2 <= 1e+304)
		tmp = t_4;
	else
		tmp = Float64(Float64(-0.5 * U_m) * -2.0);
	end
	return tmp
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(-2.0 * J), $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), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(U$95$m / N[(J + J), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$1 * N[Sqrt[N[(1.0 + N[(t$95$3 * t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[(-1.0 * N[(U$95$m * N[(0.5 * N[(J / N[(N[(U$95$m * U$95$m), $MachinePrecision] * N[(0.5 / N[Sqrt[N[(J * J), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], If[LessEqual[t$95$2, -2e+156], t$95$4, If[LessEqual[t$95$2, -2e-139], N[(t$95$1 * N[Sqrt[N[(N[(N[(N[(N[(U$95$m * U$95$m), $MachinePrecision] / N[(0.5 + N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * K), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision] / N[(J * J), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+304], t$95$4, N[(N[(-0.5 * U$95$m), $MachinePrecision] * -2.0), $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 73.0%

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

   2. 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 32.2%

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

   5. Taylor expanded in U around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-fma.f64 N/A

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

   7. Applied rewrites 7.6%

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

   8. Taylor expanded in J around -inf

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

   10. Applied rewrites 23.0%

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

   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))))) < -2e156 or -2.00000000000000006e-139 < (*.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))))) < 9.9999999999999994e303

   1. Initial program 73.0%

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

   2. Taylor expanded in K around 0

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

      1. count-2-rev N/A

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

      2. lower-+.f64 64.1

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

   4. Applied rewrites 64.1%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 64.1

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

   6. Applied rewrites 64.1%

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

   if -2e156 < (*.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))))) < -2.00000000000000006e-139

   1. Initial program 73.0%

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

   2. Taylor expanded in J around 0

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

      1. div-add N/A

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

      2. pow-div N/A

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

      3. metadata-eval N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\frac{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}{{J}^{2}} + {J}^{0}} \]

      4. metadata-eval N/A

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

      5. lower-+.f64 N/A

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

   4. Applied rewrites 54.7%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\color{blue}{\frac{\frac{U \cdot U}{0.5 + 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot K\right)\right)} \cdot 0.25}{J \cdot J} + 1}} \]

   if 9.9999999999999994e303 < (*.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 73.0%

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

   2. 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 32.2%

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

   5. Taylor expanded in U around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-fma.f64 N/A

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

   7. Applied rewrites 7.6%

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

   8. Taylor expanded in J around 0

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

      1. lower-*.f64 27.5

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

   10. Applied rewrites 27.5%

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

Alternative 4: 91.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(-2 \cdot J\right) \cdot t\_0\\ t_2 := t\_1 \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\right) \cdot t\_0}\right)}^{2}}\\ t_3 := \frac{U\_m}{J + J}\\ t_4 := t\_1 \cdot \sqrt{1 + t\_3 \cdot t\_3}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\left(-1 \cdot \left(U\_m \cdot \mathsf{fma}\left(0.5, \frac{J}{\left(U\_m \cdot U\_m\right) \cdot \frac{0.5}{\sqrt{J \cdot J}}}, -0.5\right)\right)\right) \cdot -2\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{+156}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{-139}:\\ \;\;\;\;\left(\left(\cos \left(0.5 \cdot K\right) \cdot J\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U\_m \cdot U\_m}{\left(J \cdot J\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot K\right)\right)\right)}, 0.25, 1\right)}\right) \cdot -2\\ \mathbf{elif}\;t\_2 \leq 10^{+304}:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;\left(-0.5 \cdot U\_m\right) \cdot -2\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1 (* (* -2.0 J) t_0))
        (t_2 (* t_1 (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J) t_0)) 2.0)))))
        (t_3 (/ U_m (+ J J)))
        (t_4 (* t_1 (sqrt (+ 1.0 (* t_3 t_3))))))
   (if (<= t_2 (- INFINITY))
     (*
      (*
       -1.0
       (* U_m (fma 0.5 (/ J (* (* U_m U_m) (/ 0.5 (sqrt (* J J))))) -0.5)))
      -2.0)
     (if (<= t_2 -2e+156)
       t_4
       (if (<= t_2 -2e-139)
         (*
          (*
           (* (cos (* 0.5 K)) J)
           (sqrt
            (fma
             (/
              (* U_m U_m)
              (* (* J J) (+ 0.5 (* 0.5 (cos (* 2.0 (* 0.5 K)))))))
             0.25
             1.0)))
          -2.0)
         (if (<= t_2 1e+304) t_4 (* (* -0.5 U_m) -2.0)))))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = (-2.0 * J) * t_0;
	double t_2 = t_1 * sqrt((1.0 + pow((U_m / ((2.0 * J) * t_0)), 2.0)));
	double t_3 = U_m / (J + J);
	double t_4 = t_1 * sqrt((1.0 + (t_3 * t_3)));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = (-1.0 * (U_m * fma(0.5, (J / ((U_m * U_m) * (0.5 / sqrt((J * J))))), -0.5))) * -2.0;
	} else if (t_2 <= -2e+156) {
		tmp = t_4;
	} else if (t_2 <= -2e-139) {
		tmp = ((cos((0.5 * K)) * J) * sqrt(fma(((U_m * U_m) / ((J * J) * (0.5 + (0.5 * cos((2.0 * (0.5 * K))))))), 0.25, 1.0))) * -2.0;
	} else if (t_2 <= 1e+304) {
		tmp = t_4;
	} else {
		tmp = (-0.5 * U_m) * -2.0;
	}
	return tmp;
}

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(-2.0 * J) * t_0)
	t_2 = Float64(t_1 * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J) * t_0)) ^ 2.0))))
	t_3 = Float64(U_m / Float64(J + J))
	t_4 = Float64(t_1 * sqrt(Float64(1.0 + Float64(t_3 * t_3))))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64(-1.0 * Float64(U_m * fma(0.5, Float64(J / Float64(Float64(U_m * U_m) * Float64(0.5 / sqrt(Float64(J * J))))), -0.5))) * -2.0);
	elseif (t_2 <= -2e+156)
		tmp = t_4;
	elseif (t_2 <= -2e-139)
		tmp = Float64(Float64(Float64(cos(Float64(0.5 * K)) * J) * sqrt(fma(Float64(Float64(U_m * U_m) / Float64(Float64(J * J) * Float64(0.5 + Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * K))))))), 0.25, 1.0))) * -2.0);
	elseif (t_2 <= 1e+304)
		tmp = t_4;
	else
		tmp = Float64(Float64(-0.5 * U_m) * -2.0);
	end
	return tmp
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(-2.0 * J), $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), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(U$95$m / N[(J + J), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$1 * N[Sqrt[N[(1.0 + N[(t$95$3 * t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[(-1.0 * N[(U$95$m * N[(0.5 * N[(J / N[(N[(U$95$m * U$95$m), $MachinePrecision] * N[(0.5 / N[Sqrt[N[(J * J), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], If[LessEqual[t$95$2, -2e+156], t$95$4, If[LessEqual[t$95$2, -2e-139], N[(N[(N[(N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] * J), $MachinePrecision] * N[Sqrt[N[(N[(N[(U$95$m * U$95$m), $MachinePrecision] / N[(N[(J * J), $MachinePrecision] * N[(0.5 + N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * K), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.25 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], If[LessEqual[t$95$2, 1e+304], t$95$4, N[(N[(-0.5 * U$95$m), $MachinePrecision] * -2.0), $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 73.0%

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

   2. 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 32.2%

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

   5. Taylor expanded in U around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-fma.f64 N/A

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

   7. Applied rewrites 7.6%

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

   8. Taylor expanded in J around -inf

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

   10. Applied rewrites 23.0%

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

   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))))) < -2e156 or -2.00000000000000006e-139 < (*.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))))) < 9.9999999999999994e303

   1. Initial program 73.0%

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

   2. Taylor expanded in K around 0

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

      1. count-2-rev N/A

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

      2. lower-+.f64 64.1

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

   4. Applied rewrites 64.1%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 64.1

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

   6. Applied rewrites 64.1%

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

   if -2e156 < (*.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))))) < -2.00000000000000006e-139

   1. Initial program 73.0%

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

   2. Taylor expanded in K around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 54.7%

      \[\leadsto \color{blue}{\left(\left(\cos \left(0.5 \cdot K\right) \cdot J\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U \cdot U}{\left(J \cdot J\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot K\right)\right)\right)}, 0.25, 1\right)}\right) \cdot -2} \]

   if 9.9999999999999994e303 < (*.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 73.0%

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

   2. 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 32.2%

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

   5. Taylor expanded in U around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-fma.f64 N/A

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

   7. Applied rewrites 7.6%

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

   8. Taylor expanded in J around 0

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

      1. lower-*.f64 27.5

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

   10. Applied rewrites 27.5%

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

Alternative 5: 90.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(-2 \cdot J\right) \cdot t\_0\\ t_2 := t\_1 \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\right) \cdot t\_0}\right)}^{2}}\\ t_3 := \frac{U\_m}{J + J}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\left(-1 \cdot \left(U\_m \cdot \mathsf{fma}\left(0.5, \frac{J}{\left(U\_m \cdot U\_m\right) \cdot \frac{0.5}{\sqrt{J \cdot J}}}, -0.5\right)\right)\right) \cdot -2\\ \mathbf{elif}\;t\_2 \leq 10^{+304}:\\ \;\;\;\;t\_1 \cdot \sqrt{1 + t\_3 \cdot t\_3}\\ \mathbf{else}:\\ \;\;\;\;\left(-0.5 \cdot U\_m\right) \cdot -2\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1 (* (* -2.0 J) t_0))
        (t_2 (* t_1 (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J) t_0)) 2.0)))))
        (t_3 (/ U_m (+ J J))))
   (if (<= t_2 (- INFINITY))
     (*
      (*
       -1.0
       (* U_m (fma 0.5 (/ J (* (* U_m U_m) (/ 0.5 (sqrt (* J J))))) -0.5)))
      -2.0)
     (if (<= t_2 1e+304)
       (* t_1 (sqrt (+ 1.0 (* t_3 t_3))))
       (* (* -0.5 U_m) -2.0)))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = (-2.0 * J) * t_0;
	double t_2 = t_1 * sqrt((1.0 + pow((U_m / ((2.0 * J) * t_0)), 2.0)));
	double t_3 = U_m / (J + J);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = (-1.0 * (U_m * fma(0.5, (J / ((U_m * U_m) * (0.5 / sqrt((J * J))))), -0.5))) * -2.0;
	} else if (t_2 <= 1e+304) {
		tmp = t_1 * sqrt((1.0 + (t_3 * t_3)));
	} else {
		tmp = (-0.5 * U_m) * -2.0;
	}
	return tmp;
}

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(-2.0 * J) * t_0)
	t_2 = Float64(t_1 * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J) * t_0)) ^ 2.0))))
	t_3 = Float64(U_m / Float64(J + J))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64(-1.0 * Float64(U_m * fma(0.5, Float64(J / Float64(Float64(U_m * U_m) * Float64(0.5 / sqrt(Float64(J * J))))), -0.5))) * -2.0);
	elseif (t_2 <= 1e+304)
		tmp = Float64(t_1 * sqrt(Float64(1.0 + Float64(t_3 * t_3))));
	else
		tmp = Float64(Float64(-0.5 * U_m) * -2.0);
	end
	return tmp
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(-2.0 * J), $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), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(U$95$m / N[(J + J), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[(-1.0 * N[(U$95$m * N[(0.5 * N[(J / N[(N[(U$95$m * U$95$m), $MachinePrecision] * N[(0.5 / N[Sqrt[N[(J * J), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], If[LessEqual[t$95$2, 1e+304], N[(t$95$1 * N[Sqrt[N[(1.0 + N[(t$95$3 * t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * U$95$m), $MachinePrecision] * -2.0), $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 73.0%

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

   2. 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 32.2%

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

   5. Taylor expanded in U around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-fma.f64 N/A

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

   7. Applied rewrites 7.6%

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

   8. Taylor expanded in J around -inf

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

   10. Applied rewrites 23.0%

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

   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))))) < 9.9999999999999994e303

   1. Initial program 73.0%

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

   2. Taylor expanded in K around 0

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

      1. count-2-rev N/A

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

      2. lower-+.f64 64.1

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

   4. Applied rewrites 64.1%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 64.1

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

   6. Applied rewrites 64.1%

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

   if 9.9999999999999994e303 < (*.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 73.0%

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

   2. 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 32.2%

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

   5. Taylor expanded in U around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-fma.f64 N/A

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

   7. Applied rewrites 7.6%

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

   8. Taylor expanded in J around 0

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

      1. lower-*.f64 27.5

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

   10. Applied rewrites 27.5%

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

Alternative 6: 74.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\right) \cdot t\_0}\right)}^{2}}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(-1 \cdot \left(U\_m \cdot \mathsf{fma}\left(0.5, \frac{J}{\left(U\_m \cdot U\_m\right) \cdot \frac{0.5}{\sqrt{J \cdot J}}}, -0.5\right)\right)\right) \cdot -2\\ \mathbf{elif}\;t\_1 \leq -4 \cdot 10^{-180}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(\frac{U\_m}{J} \cdot \frac{U\_m}{J}, 0.25, 1\right)} \cdot J\right) \cdot -2\\ \mathbf{elif}\;t\_1 \leq 10^{+304}:\\ \;\;\;\;\left(\cos \left(0.5 \cdot K\right) \cdot J\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\left(-0.5 \cdot U\_m\right) \cdot -2\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* (* -2.0 J) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J) t_0)) 2.0))))))
   (if (<= t_1 (- INFINITY))
     (*
      (*
       -1.0
       (* U_m (fma 0.5 (/ J (* (* U_m U_m) (/ 0.5 (sqrt (* J J))))) -0.5)))
      -2.0)
     (if (<= t_1 -4e-180)
       (* (* (sqrt (fma (* (/ U_m J) (/ U_m J)) 0.25 1.0)) J) -2.0)
       (if (<= t_1 1e+304)
         (* (* (cos (* 0.5 K)) J) -2.0)
         (* (* -0.5 U_m) -2.0))))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * J) * t_0) * sqrt((1.0 + pow((U_m / ((2.0 * J) * t_0)), 2.0)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (-1.0 * (U_m * fma(0.5, (J / ((U_m * U_m) * (0.5 / sqrt((J * J))))), -0.5))) * -2.0;
	} else if (t_1 <= -4e-180) {
		tmp = (sqrt(fma(((U_m / J) * (U_m / J)), 0.25, 1.0)) * J) * -2.0;
	} else if (t_1 <= 1e+304) {
		tmp = (cos((0.5 * K)) * J) * -2.0;
	} else {
		tmp = (-0.5 * U_m) * -2.0;
	}
	return tmp;
}

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * J) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J) * t_0)) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(-1.0 * Float64(U_m * fma(0.5, Float64(J / Float64(Float64(U_m * U_m) * Float64(0.5 / sqrt(Float64(J * J))))), -0.5))) * -2.0);
	elseif (t_1 <= -4e-180)
		tmp = Float64(Float64(sqrt(fma(Float64(Float64(U_m / J) * Float64(U_m / J)), 0.25, 1.0)) * J) * -2.0);
	elseif (t_1 <= 1e+304)
		tmp = Float64(Float64(cos(Float64(0.5 * K)) * J) * -2.0);
	else
		tmp = Float64(Float64(-0.5 * U_m) * -2.0);
	end
	return tmp
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(-1.0 * N[(U$95$m * N[(0.5 * N[(J / N[(N[(U$95$m * U$95$m), $MachinePrecision] * N[(0.5 / N[Sqrt[N[(J * J), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], If[LessEqual[t$95$1, -4e-180], N[(N[(N[Sqrt[N[(N[(N[(U$95$m / J), $MachinePrecision] * N[(U$95$m / J), $MachinePrecision]), $MachinePrecision] * 0.25 + 1.0), $MachinePrecision]], $MachinePrecision] * J), $MachinePrecision] * -2.0), $MachinePrecision], If[LessEqual[t$95$1, 1e+304], N[(N[(N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] * J), $MachinePrecision] * -2.0), $MachinePrecision], N[(N[(-0.5 * U$95$m), $MachinePrecision] * -2.0), $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 73.0%

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

   2. 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 32.2%

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

   5. Taylor expanded in U around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-fma.f64 N/A

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

   7. Applied rewrites 7.6%

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

   8. Taylor expanded in J around -inf

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

   10. Applied rewrites 23.0%

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

   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))))) < -4.0000000000000001e-180

   1. Initial program 73.0%

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

   2. 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 32.2%

      \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(\frac{U \cdot U}{J \cdot J}, 0.25, 1\right)} \cdot J\right) \cdot -2} \]
   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 -4.0000000000000001e-180 < (*.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))))) < 9.9999999999999994e303

   1. Initial program 73.0%

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

   2. Taylor expanded in J around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 51.1

         \[\leadsto \left(\cos \left(0.5 \cdot K\right) \cdot J\right) \cdot -2 \]

   4. Applied rewrites 51.1%

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

   if 9.9999999999999994e303 < (*.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 73.0%

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

   2. 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 32.2%

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

   5. Taylor expanded in U around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-fma.f64 N/A

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

   7. Applied rewrites 7.6%

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

   8. Taylor expanded in J around 0

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

      1. lower-*.f64 27.5

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

   10. Applied rewrites 27.5%

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

Alternative 7: 62.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\right) \cdot t\_0}\right)}^{2}}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(-1 \cdot \left(U\_m \cdot \mathsf{fma}\left(0.5, \frac{J}{\left(U\_m \cdot U\_m\right) \cdot \frac{0.5}{\sqrt{J \cdot J}}}, -0.5\right)\right)\right) \cdot -2\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-262}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(\frac{U\_m}{J} \cdot \frac{U\_m}{J}, 0.25, 1\right)} \cdot J\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\left(-0.5 \cdot U\_m\right) \cdot -2\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* (* -2.0 J) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J) t_0)) 2.0))))))
   (if (<= t_1 (- INFINITY))
     (*
      (*
       -1.0
       (* U_m (fma 0.5 (/ J (* (* U_m U_m) (/ 0.5 (sqrt (* J J))))) -0.5)))
      -2.0)
     (if (<= t_1 -1e-262)
       (* (* (sqrt (fma (* (/ U_m J) (/ U_m J)) 0.25 1.0)) J) -2.0)
       (* (* -0.5 U_m) -2.0)))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * J) * t_0) * sqrt((1.0 + pow((U_m / ((2.0 * J) * t_0)), 2.0)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (-1.0 * (U_m * fma(0.5, (J / ((U_m * U_m) * (0.5 / sqrt((J * J))))), -0.5))) * -2.0;
	} else if (t_1 <= -1e-262) {
		tmp = (sqrt(fma(((U_m / J) * (U_m / J)), 0.25, 1.0)) * J) * -2.0;
	} else {
		tmp = (-0.5 * U_m) * -2.0;
	}
	return tmp;
}

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * J) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J) * t_0)) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(-1.0 * Float64(U_m * fma(0.5, Float64(J / Float64(Float64(U_m * U_m) * Float64(0.5 / sqrt(Float64(J * J))))), -0.5))) * -2.0);
	elseif (t_1 <= -1e-262)
		tmp = Float64(Float64(sqrt(fma(Float64(Float64(U_m / J) * Float64(U_m / J)), 0.25, 1.0)) * J) * -2.0);
	else
		tmp = Float64(Float64(-0.5 * U_m) * -2.0);
	end
	return tmp
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(-1.0 * N[(U$95$m * N[(0.5 * N[(J / N[(N[(U$95$m * U$95$m), $MachinePrecision] * N[(0.5 / N[Sqrt[N[(J * J), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], If[LessEqual[t$95$1, -1e-262], N[(N[(N[Sqrt[N[(N[(N[(U$95$m / J), $MachinePrecision] * N[(U$95$m / J), $MachinePrecision]), $MachinePrecision] * 0.25 + 1.0), $MachinePrecision]], $MachinePrecision] * J), $MachinePrecision] * -2.0), $MachinePrecision], N[(N[(-0.5 * U$95$m), $MachinePrecision] * -2.0), $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 73.0%

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

   2. 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 32.2%

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

   5. Taylor expanded in U around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-fma.f64 N/A

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

   7. Applied rewrites 7.6%

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

   8. Taylor expanded in J around -inf

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

   10. Applied rewrites 23.0%

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

   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))))) < -1.00000000000000001e-262

   1. Initial program 73.0%

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

   2. 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 32.2%

      \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(\frac{U \cdot U}{J \cdot J}, 0.25, 1\right)} \cdot J\right) \cdot -2} \]
   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 -1.00000000000000001e-262 < (*.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 73.0%

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

   2. 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 32.2%

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

   5. Taylor expanded in U around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-fma.f64 N/A

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

   7. Applied rewrites 7.6%

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

   8. Taylor expanded in J around 0

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

      1. lower-*.f64 27.5

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

   10. Applied rewrites 27.5%

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

Alternative 8: 59.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\right) \cdot t\_0}\right)}^{2}}\\ t_2 := \frac{0.5}{\left|J\right|}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(t\_2 \cdot J, -2, -\frac{J}{U\_m \cdot \left(t\_2 \cdot U\_m\right)}\right) \cdot U\_m\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-262}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(\frac{U\_m}{J} \cdot \frac{U\_m}{J}, 0.25, 1\right)} \cdot J\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\left(-0.5 \cdot U\_m\right) \cdot -2\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* (* -2.0 J) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J) t_0)) 2.0)))))
        (t_2 (/ 0.5 (fabs J))))
   (if (<= t_1 (- INFINITY))
     (* (fma (* t_2 J) -2.0 (- (/ J (* U_m (* t_2 U_m))))) U_m)
     (if (<= t_1 -1e-262)
       (* (* (sqrt (fma (* (/ U_m J) (/ U_m J)) 0.25 1.0)) J) -2.0)
       (* (* -0.5 U_m) -2.0)))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * J) * t_0) * sqrt((1.0 + pow((U_m / ((2.0 * J) * t_0)), 2.0)));
	double t_2 = 0.5 / fabs(J);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma((t_2 * J), -2.0, -(J / (U_m * (t_2 * U_m)))) * U_m;
	} else if (t_1 <= -1e-262) {
		tmp = (sqrt(fma(((U_m / J) * (U_m / J)), 0.25, 1.0)) * J) * -2.0;
	} else {
		tmp = (-0.5 * U_m) * -2.0;
	}
	return tmp;
}

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * J) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J) * t_0)) ^ 2.0))))
	t_2 = Float64(0.5 / abs(J))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(Float64(t_2 * J), -2.0, Float64(-Float64(J / Float64(U_m * Float64(t_2 * U_m))))) * U_m);
	elseif (t_1 <= -1e-262)
		tmp = Float64(Float64(sqrt(fma(Float64(Float64(U_m / J) * Float64(U_m / J)), 0.25, 1.0)) * J) * -2.0);
	else
		tmp = Float64(Float64(-0.5 * U_m) * -2.0);
	end
	return tmp
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.5 / N[Abs[J], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(t$95$2 * J), $MachinePrecision] * -2.0 + (-N[(J / N[(U$95$m * N[(t$95$2 * U$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])), $MachinePrecision] * U$95$m), $MachinePrecision], If[LessEqual[t$95$1, -1e-262], N[(N[(N[Sqrt[N[(N[(N[(U$95$m / J), $MachinePrecision] * N[(U$95$m / J), $MachinePrecision]), $MachinePrecision] * 0.25 + 1.0), $MachinePrecision]], $MachinePrecision] * J), $MachinePrecision] * -2.0), $MachinePrecision], N[(N[(-0.5 * U$95$m), $MachinePrecision] * -2.0), $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 73.0%

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

   2. 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 32.2%

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

   5. Taylor expanded in U around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-fma.f64 N/A

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

   7. Applied rewrites 7.6%

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

   8. Taylor expanded in J around 0

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

      1. lower-*.f64 27.5

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

   10. Applied rewrites 27.5%

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

   11. Taylor expanded in U around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

   13. Applied rewrites 40.8%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -1.00000000000000001e-262

   1. Initial program 73.0%

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

   2. 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 32.2%

      \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(\frac{U \cdot U}{J \cdot J}, 0.25, 1\right)} \cdot J\right) \cdot -2} \]
   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 -1.00000000000000001e-262 < (*.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 73.0%

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

   2. 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 32.2%

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

   5. Taylor expanded in U around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-fma.f64 N/A

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

   7. Applied rewrites 7.6%

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

   8. Taylor expanded in J around 0

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

      1. lower-*.f64 27.5

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

   10. Applied rewrites 27.5%

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

Alternative 9: 54.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\right) \cdot t\_0}\right)}^{2}}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\sqrt{0.25 \cdot \left(U\_m \cdot U\_m\right)} \cdot -2\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-262}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(\frac{U\_m}{J} \cdot \frac{U\_m}{J}, 0.25, 1\right)} \cdot J\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\left(-0.5 \cdot U\_m\right) \cdot -2\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* (* -2.0 J) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J) t_0)) 2.0))))))
   (if (<= t_1 (- INFINITY))
     (* (sqrt (* 0.25 (* U_m U_m))) -2.0)
     (if (<= t_1 -1e-262)
       (* (* (sqrt (fma (* (/ U_m J) (/ U_m J)) 0.25 1.0)) J) -2.0)
       (* (* -0.5 U_m) -2.0)))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * J) * t_0) * sqrt((1.0 + pow((U_m / ((2.0 * J) * t_0)), 2.0)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = sqrt((0.25 * (U_m * U_m))) * -2.0;
	} else if (t_1 <= -1e-262) {
		tmp = (sqrt(fma(((U_m / J) * (U_m / J)), 0.25, 1.0)) * J) * -2.0;
	} else {
		tmp = (-0.5 * U_m) * -2.0;
	}
	return tmp;
}

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * J) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J) * t_0)) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(sqrt(Float64(0.25 * Float64(U_m * U_m))) * -2.0);
	elseif (t_1 <= -1e-262)
		tmp = Float64(Float64(sqrt(fma(Float64(Float64(U_m / J) * Float64(U_m / J)), 0.25, 1.0)) * J) * -2.0);
	else
		tmp = Float64(Float64(-0.5 * U_m) * -2.0);
	end
	return tmp
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[Sqrt[N[(0.25 * N[(U$95$m * U$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -2.0), $MachinePrecision], If[LessEqual[t$95$1, -1e-262], N[(N[(N[Sqrt[N[(N[(N[(U$95$m / J), $MachinePrecision] * N[(U$95$m / J), $MachinePrecision]), $MachinePrecision] * 0.25 + 1.0), $MachinePrecision]], $MachinePrecision] * J), $MachinePrecision] * -2.0), $MachinePrecision], N[(N[(-0.5 * U$95$m), $MachinePrecision] * -2.0), $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 73.0%

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

   2. 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 32.2%

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

   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 15.1

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

   7. Applied rewrites 15.1%

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

   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))))) < -1.00000000000000001e-262

   1. Initial program 73.0%

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

   2. 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 32.2%

      \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(\frac{U \cdot U}{J \cdot J}, 0.25, 1\right)} \cdot J\right) \cdot -2} \]
   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 -1.00000000000000001e-262 < (*.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 73.0%

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

   2. 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 32.2%

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

   5. Taylor expanded in U around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-fma.f64 N/A

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

   7. Applied rewrites 7.6%

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

   8. Taylor expanded in J around 0

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

      1. lower-*.f64 27.5

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

   10. Applied rewrites 27.5%

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

Alternative 10: 49.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\right) \cdot t\_0}\right)}^{2}}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\sqrt{0.25 \cdot \left(U\_m \cdot U\_m\right)} \cdot -2\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-139}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(\frac{U\_m \cdot U\_m}{J \cdot J}, 0.25, 1\right)} \cdot J\right) \cdot -2\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-262}:\\ \;\;\;\;\mathsf{fma}\left(-2, J, -0.25 \cdot \frac{U\_m \cdot U\_m}{J}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-0.5 \cdot U\_m\right) \cdot -2\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* (* -2.0 J) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J) t_0)) 2.0))))))
   (if (<= t_1 (- INFINITY))
     (* (sqrt (* 0.25 (* U_m U_m))) -2.0)
     (if (<= t_1 -2e-139)
       (* (* (sqrt (fma (/ (* U_m U_m) (* J J)) 0.25 1.0)) J) -2.0)
       (if (<= t_1 -1e-262)
         (fma -2.0 J (* -0.25 (/ (* U_m U_m) J)))
         (* (* -0.5 U_m) -2.0))))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * J) * t_0) * sqrt((1.0 + pow((U_m / ((2.0 * J) * t_0)), 2.0)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = sqrt((0.25 * (U_m * U_m))) * -2.0;
	} else if (t_1 <= -2e-139) {
		tmp = (sqrt(fma(((U_m * U_m) / (J * J)), 0.25, 1.0)) * J) * -2.0;
	} else if (t_1 <= -1e-262) {
		tmp = fma(-2.0, J, (-0.25 * ((U_m * U_m) / J)));
	} else {
		tmp = (-0.5 * U_m) * -2.0;
	}
	return tmp;
}

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * J) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J) * t_0)) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(sqrt(Float64(0.25 * Float64(U_m * U_m))) * -2.0);
	elseif (t_1 <= -2e-139)
		tmp = Float64(Float64(sqrt(fma(Float64(Float64(U_m * U_m) / Float64(J * J)), 0.25, 1.0)) * J) * -2.0);
	elseif (t_1 <= -1e-262)
		tmp = fma(-2.0, J, Float64(-0.25 * Float64(Float64(U_m * U_m) / J)));
	else
		tmp = Float64(Float64(-0.5 * U_m) * -2.0);
	end
	return tmp
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[Sqrt[N[(0.25 * N[(U$95$m * U$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -2.0), $MachinePrecision], If[LessEqual[t$95$1, -2e-139], N[(N[(N[Sqrt[N[(N[(N[(U$95$m * U$95$m), $MachinePrecision] / N[(J * J), $MachinePrecision]), $MachinePrecision] * 0.25 + 1.0), $MachinePrecision]], $MachinePrecision] * J), $MachinePrecision] * -2.0), $MachinePrecision], If[LessEqual[t$95$1, -1e-262], N[(-2.0 * J + N[(-0.25 * N[(N[(U$95$m * U$95$m), $MachinePrecision] / J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * U$95$m), $MachinePrecision] * -2.0), $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 73.0%

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

   2. 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 32.2%

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

   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 15.1

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

   7. Applied rewrites 15.1%

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

   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))))) < -2.00000000000000006e-139

   1. Initial program 73.0%

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

   2. 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 32.2%

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

   if -2.00000000000000006e-139 < (*.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))))) < -1.00000000000000001e-262

   1. Initial program 73.0%

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

   2. 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 32.2%

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

   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 27.6

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

   7. Applied rewrites 27.6%

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

   if -1.00000000000000001e-262 < (*.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 73.0%

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

   2. 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 32.2%

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

   5. Taylor expanded in U around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-fma.f64 N/A

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

   7. Applied rewrites 7.6%

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

   8. Taylor expanded in J around 0

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

      1. lower-*.f64 27.5

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

   10. Applied rewrites 27.5%

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

Alternative 11: 47.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\right) \cdot t\_0}\right)}^{2}}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\sqrt{0.25 \cdot \left(U\_m \cdot U\_m\right)} \cdot -2\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-262}:\\ \;\;\;\;J \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\left(-0.5 \cdot U\_m\right) \cdot -2\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* (* -2.0 J) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J) t_0)) 2.0))))))
   (if (<= t_1 (- INFINITY))
     (* (sqrt (* 0.25 (* U_m U_m))) -2.0)
     (if (<= t_1 -1e-262) (* J -2.0) (* (* -0.5 U_m) -2.0)))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * J) * t_0) * sqrt((1.0 + pow((U_m / ((2.0 * J) * t_0)), 2.0)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = sqrt((0.25 * (U_m * U_m))) * -2.0;
	} else if (t_1 <= -1e-262) {
		tmp = J * -2.0;
	} else {
		tmp = (-0.5 * U_m) * -2.0;
	}
	return tmp;
}

Java

U_m = Math.abs(U);
public static double code(double J, double K, double U_m) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = ((-2.0 * J) * t_0) * Math.sqrt((1.0 + Math.pow((U_m / ((2.0 * J) * t_0)), 2.0)));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt((0.25 * (U_m * U_m))) * -2.0;
	} else if (t_1 <= -1e-262) {
		tmp = J * -2.0;
	} else {
		tmp = (-0.5 * U_m) * -2.0;
	}
	return tmp;
}

Python

U_m = math.fabs(U)
def code(J, K, U_m):
	t_0 = math.cos((K / 2.0))
	t_1 = ((-2.0 * J) * t_0) * math.sqrt((1.0 + math.pow((U_m / ((2.0 * J) * t_0)), 2.0)))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = math.sqrt((0.25 * (U_m * U_m))) * -2.0
	elif t_1 <= -1e-262:
		tmp = J * -2.0
	else:
		tmp = (-0.5 * U_m) * -2.0
	return tmp

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * J) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J) * t_0)) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(sqrt(Float64(0.25 * Float64(U_m * U_m))) * -2.0);
	elseif (t_1 <= -1e-262)
		tmp = Float64(J * -2.0);
	else
		tmp = Float64(Float64(-0.5 * U_m) * -2.0);
	end
	return tmp
end

MATLAB

U_m = abs(U);
function tmp_2 = code(J, K, U_m)
	t_0 = cos((K / 2.0));
	t_1 = ((-2.0 * J) * t_0) * sqrt((1.0 + ((U_m / ((2.0 * J) * t_0)) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = sqrt((0.25 * (U_m * U_m))) * -2.0;
	elseif (t_1 <= -1e-262)
		tmp = J * -2.0;
	else
		tmp = (-0.5 * U_m) * -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[Sqrt[N[(0.25 * N[(U$95$m * U$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -2.0), $MachinePrecision], If[LessEqual[t$95$1, -1e-262], N[(J * -2.0), $MachinePrecision], N[(N[(-0.5 * U$95$m), $MachinePrecision] * -2.0), $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 73.0%

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

   2. 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 32.2%

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

   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 15.1

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

   7. Applied rewrites 15.1%

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

   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))))) < -1.00000000000000001e-262

   1. Initial program 73.0%

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

   2. 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 32.2%

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

   5. Taylor expanded in J around inf

      \[\leadsto J \cdot -2 \]
   6. Step-by-step derivation

   7. Applied rewrites 28.6%

      \[\leadsto J \cdot -2 \]

   if -1.00000000000000001e-262 < (*.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 73.0%

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

   2. 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 32.2%

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

   5. Taylor expanded in U around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-fma.f64 N/A

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

   7. Applied rewrites 7.6%

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

   8. Taylor expanded in J around 0

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

      1. lower-*.f64 27.5

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

   10. Applied rewrites 27.5%

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

Alternative 12: 34.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(-0.5 \cdot U\_m\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;J \cdot -2\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -5e-310) (* (* -0.5 U_m) -2.0) (* J -2.0)))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double tmp;
	if (cos((K / 2.0)) <= -5e-310) {
		tmp = (-0.5 * U_m) * -2.0;
	} else {
		tmp = J * -2.0;
	}
	return tmp;
}

Fortran

U_m =     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, k, u_m)
use fmin_fmax_functions
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u_m
    real(8) :: tmp
    if (cos((k / 2.0d0)) <= (-5d-310)) then
        tmp = ((-0.5d0) * u_m) * (-2.0d0)
    else
        tmp = j * (-2.0d0)
    end if
    code = tmp
end function

Java

U_m = Math.abs(U);
public static double code(double J, double K, double U_m) {
	double tmp;
	if (Math.cos((K / 2.0)) <= -5e-310) {
		tmp = (-0.5 * U_m) * -2.0;
	} else {
		tmp = J * -2.0;
	}
	return tmp;
}

Python

U_m = math.fabs(U)
def code(J, K, U_m):
	tmp = 0
	if math.cos((K / 2.0)) <= -5e-310:
		tmp = (-0.5 * U_m) * -2.0
	else:
		tmp = J * -2.0
	return tmp

Julia

U_m = abs(U)
function code(J, K, U_m)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -5e-310)
		tmp = Float64(Float64(-0.5 * U_m) * -2.0);
	else
		tmp = Float64(J * -2.0);
	end
	return tmp
end

MATLAB

U_m = abs(U);
function tmp_2 = code(J, K, U_m)
	tmp = 0.0;
	if (cos((K / 2.0)) <= -5e-310)
		tmp = (-0.5 * U_m) * -2.0;
	else
		tmp = J * -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -5e-310], N[(N[(-0.5 * U$95$m), $MachinePrecision] * -2.0), $MachinePrecision], N[(J * -2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -4.999999999999985e-310

   1. Initial program 73.0%

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

   2. 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 32.2%

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

   5. Taylor expanded in U around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-fma.f64 N/A

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

   7. Applied rewrites 7.6%

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

   8. Taylor expanded in J around 0

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

      1. lower-*.f64 27.5

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

   10. Applied rewrites 27.5%

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

   if -4.999999999999985e-310 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 73.0%

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

   2. 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 32.2%

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

   5. Taylor expanded in J around inf

      \[\leadsto J \cdot -2 \]
   6. Step-by-step derivation

   7. Applied rewrites 28.6%

      \[\leadsto J \cdot -2 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 28.6% accurate, 27.8× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J \cdot -2 \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m) :precision binary64 (* J -2.0))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	return J * -2.0;
}

Fortran

U_m =     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, k, u_m)
use fmin_fmax_functions
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u_m
    code = j * (-2.0d0)
end function

Java

U_m = Math.abs(U);
public static double code(double J, double K, double U_m) {
	return J * -2.0;
}

Python

U_m = math.fabs(U)
def code(J, K, U_m):
	return J * -2.0

Julia

U_m = abs(U)
function code(J, K, U_m)
	return Float64(J * -2.0)
end

MATLAB

U_m = abs(U);
function tmp = code(J, K, U_m)
	tmp = J * -2.0;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := N[(J * -2.0), $MachinePrecision]

Derivation

1. Initial program 73.0%

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

2. 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 32.2%

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

5. Taylor expanded in J around inf

   \[\leadsto J \cdot -2 \]
6. Step-by-step derivation

7. Applied rewrites 28.6%

   \[\leadsto J \cdot -2 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025127 
(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)))))