Maksimov and Kolovsky, Equation (3)

Percentage Accurate: 73.9% → 99.4%

Time: 10.1s

Alternatives: 12

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: 73.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(-0.5 \cdot K\right)\\ t_1 := \cos \left(\frac{K}{2}\right)\\ t_2 := \left(\left(-2 \cdot J\right) \cdot t\_1\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\right) \cdot t\_1}\right)}^{2}}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+304}:\\ \;\;\;\;\left(\sqrt{{\left(\frac{\frac{\frac{U\_m}{J}}{2}}{t\_0}\right)}^{2} + 1} \cdot \cos \left(\frac{K}{-2}\right)\right) \cdot \left(J \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left({t\_0}^{2} \cdot \frac{J \cdot J}{U\_m \cdot U\_m}\right) \cdot \left(--2\right) - -1\right) \cdot U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (* -0.5 K)))
        (t_1 (cos (/ K 2.0)))
        (t_2
         (*
          (* (* -2.0 J) t_1)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J) t_1)) 2.0))))))
   (if (<= t_2 (- INFINITY))
     (- U_m)
     (if (<= t_2 5e+304)
       (*
        (* (sqrt (+ (pow (/ (/ (/ U_m J) 2.0) t_0) 2.0) 1.0)) (cos (/ K -2.0)))
        (* J -2.0))
       (*
        (- (* (* (pow t_0 2.0) (/ (* J J) (* U_m U_m))) (- -2.0)) -1.0)
        U_m)))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = cos((-0.5 * K));
	double t_1 = cos((K / 2.0));
	double t_2 = ((-2.0 * J) * t_1) * sqrt((1.0 + pow((U_m / ((2.0 * J) * t_1)), 2.0)));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_2 <= 5e+304) {
		tmp = (sqrt((pow((((U_m / J) / 2.0) / t_0), 2.0) + 1.0)) * cos((K / -2.0))) * (J * -2.0);
	} else {
		tmp = (((pow(t_0, 2.0) * ((J * J) / (U_m * U_m))) * -(-2.0)) - -1.0) * U_m;
	}
	return tmp;
}

Java

U_m = Math.abs(U);
public static double code(double J, double K, double U_m) {
	double t_0 = Math.cos((-0.5 * K));
	double t_1 = Math.cos((K / 2.0));
	double t_2 = ((-2.0 * J) * t_1) * Math.sqrt((1.0 + Math.pow((U_m / ((2.0 * J) * t_1)), 2.0)));
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = -U_m;
	} else if (t_2 <= 5e+304) {
		tmp = (Math.sqrt((Math.pow((((U_m / J) / 2.0) / t_0), 2.0) + 1.0)) * Math.cos((K / -2.0))) * (J * -2.0);
	} else {
		tmp = (((Math.pow(t_0, 2.0) * ((J * J) / (U_m * U_m))) * -(-2.0)) - -1.0) * U_m;
	}
	return tmp;
}

Python

U_m = math.fabs(U)
def code(J, K, U_m):
	t_0 = math.cos((-0.5 * K))
	t_1 = math.cos((K / 2.0))
	t_2 = ((-2.0 * J) * t_1) * math.sqrt((1.0 + math.pow((U_m / ((2.0 * J) * t_1)), 2.0)))
	tmp = 0
	if t_2 <= -math.inf:
		tmp = -U_m
	elif t_2 <= 5e+304:
		tmp = (math.sqrt((math.pow((((U_m / J) / 2.0) / t_0), 2.0) + 1.0)) * math.cos((K / -2.0))) * (J * -2.0)
	else:
		tmp = (((math.pow(t_0, 2.0) * ((J * J) / (U_m * U_m))) * -(-2.0)) - -1.0) * U_m
	return tmp

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(-0.5 * K))
	t_1 = cos(Float64(K / 2.0))
	t_2 = Float64(Float64(Float64(-2.0 * J) * t_1) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J) * t_1)) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_2 <= 5e+304)
		tmp = Float64(Float64(sqrt(Float64((Float64(Float64(Float64(U_m / J) / 2.0) / t_0) ^ 2.0) + 1.0)) * cos(Float64(K / -2.0))) * Float64(J * -2.0));
	else
		tmp = Float64(Float64(Float64(Float64((t_0 ^ 2.0) * Float64(Float64(J * J) / Float64(U_m * U_m))) * Float64(-(-2.0))) - -1.0) * U_m);
	end
	return tmp
end

MATLAB

U_m = abs(U);
function tmp_2 = code(J, K, U_m)
	t_0 = cos((-0.5 * K));
	t_1 = cos((K / 2.0));
	t_2 = ((-2.0 * J) * t_1) * sqrt((1.0 + ((U_m / ((2.0 * J) * t_1)) ^ 2.0)));
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = -U_m;
	elseif (t_2 <= 5e+304)
		tmp = (sqrt((((((U_m / J) / 2.0) / t_0) ^ 2.0) + 1.0)) * cos((K / -2.0))) * (J * -2.0);
	else
		tmp = ((((t_0 ^ 2.0) * ((J * J) / (U_m * U_m))) * -(-2.0)) - -1.0) * U_m;
	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[(-0.5 * K), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(-2.0 * J), $MachinePrecision] * t$95$1), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], (-U$95$m), If[LessEqual[t$95$2, 5e+304], N[(N[(N[Sqrt[N[(N[Power[N[(N[(N[(U$95$m / J), $MachinePrecision] / 2.0), $MachinePrecision] / t$95$0), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(K / -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(J * -2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[Power[t$95$0, 2.0], $MachinePrecision] * N[(N[(J * J), $MachinePrecision] / N[(U$95$m * U$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * (--2.0)), $MachinePrecision] - -1.0), $MachinePrecision] * U$95$m), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 5.5%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 47.4

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

   5. Applied rewrites 47.4%

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

   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))))) < 4.9999999999999997e304

   1. Initial program 99.7%

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

   3. Taylor expanded in K around inf

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

      1. cos-neg-rev N/A

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

      2. lower-cos.f64 N/A

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

      3. distribute-lft-neg-in N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 99.7

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

   5. Applied rewrites 99.7%

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

   7. Applied rewrites 97.8%

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

      1. lift-cosh.f64 N/A

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

      2. lift-asinh.f64 N/A

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

      3. cosh-asinh N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-+.f64 N/A

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

   9. Applied rewrites 99.7%

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

   if 4.9999999999999997e304 < (*.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 5.7%

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

   3. Taylor expanded in U around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 53.4%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \leq 5 \cdot 10^{+304}:\\ \;\;\;\;\left(\sqrt{{\left(\frac{\frac{\frac{U}{J}}{2}}{\cos \left(-0.5 \cdot K\right)}\right)}^{2} + 1} \cdot \cos \left(\frac{K}{-2}\right)\right) \cdot \left(J \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left({\cos \left(-0.5 \cdot K\right)}^{2} \cdot \frac{J \cdot J}{U \cdot U}\right) \cdot \left(--2\right) - -1\right) \cdot U\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 63.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{U\_m}{J}, 0.5, \sqrt{\mathsf{fma}\left(\frac{0.25}{J}, \frac{U\_m \cdot U\_m}{J}, 1\right)}\right)\\ t_1 := \cos \left(\frac{K}{2}\right)\\ t_2 := \left(\left(-2 \cdot J\right) \cdot t\_1\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\right) \cdot t\_1}\right)}^{2}}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{+166} \lor \neg \left(t\_2 \leq -5 \cdot 10^{-203}\right):\\ \;\;\;\;\cos \left(0.5 \cdot K\right) \cdot \left(J \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-J\right) \cdot \left(t\_0 + {t\_0}^{-1}\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (fma (/ U_m J) 0.5 (sqrt (fma (/ 0.25 J) (/ (* U_m U_m) J) 1.0))))
        (t_1 (cos (/ K 2.0)))
        (t_2
         (*
          (* (* -2.0 J) t_1)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J) t_1)) 2.0))))))
   (if (<= t_2 (- INFINITY))
     (- U_m)
     (if (or (<= t_2 -2e+166) (not (<= t_2 -5e-203)))
       (* (cos (* 0.5 K)) (* J -2.0))
       (* (- J) (+ t_0 (pow t_0 -1.0)))))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = fma((U_m / J), 0.5, sqrt(fma((0.25 / J), ((U_m * U_m) / J), 1.0)));
	double t_1 = cos((K / 2.0));
	double t_2 = ((-2.0 * J) * t_1) * sqrt((1.0 + pow((U_m / ((2.0 * J) * t_1)), 2.0)));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if ((t_2 <= -2e+166) || !(t_2 <= -5e-203)) {
		tmp = cos((0.5 * K)) * (J * -2.0);
	} else {
		tmp = -J * (t_0 + pow(t_0, -1.0));
	}
	return tmp;
}

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = fma(Float64(U_m / J), 0.5, sqrt(fma(Float64(0.25 / J), Float64(Float64(U_m * U_m) / J), 1.0)))
	t_1 = cos(Float64(K / 2.0))
	t_2 = Float64(Float64(Float64(-2.0 * J) * t_1) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J) * t_1)) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif ((t_2 <= -2e+166) || !(t_2 <= -5e-203))
		tmp = Float64(cos(Float64(0.5 * K)) * Float64(J * -2.0));
	else
		tmp = Float64(Float64(-J) * Float64(t_0 + (t_0 ^ -1.0)));
	end
	return tmp
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := Block[{t$95$0 = N[(N[(U$95$m / J), $MachinePrecision] * 0.5 + N[Sqrt[N[(N[(0.25 / J), $MachinePrecision] * N[(N[(U$95$m * U$95$m), $MachinePrecision] / J), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(-2.0 * J), $MachinePrecision] * t$95$1), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], (-U$95$m), If[Or[LessEqual[t$95$2, -2e+166], N[Not[LessEqual[t$95$2, -5e-203]], $MachinePrecision]], N[(N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] * N[(J * -2.0), $MachinePrecision]), $MachinePrecision], N[((-J) * N[(t$95$0 + N[Power[t$95$0, -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 5.5%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 47.4

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

   5. Applied rewrites 47.4%

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

   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))))) < -1.99999999999999988e166 or -5.0000000000000002e-203 < (*.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 81.8%

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

   3. Taylor expanded in K around inf

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

      1. cos-neg-rev N/A

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

      2. lower-cos.f64 N/A

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

      3. distribute-lft-neg-in N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 81.8

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

   5. Applied rewrites 81.8%

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

   7. Applied rewrites 89.7%

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

   8. Taylor expanded in J around inf

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

      1. cos-neg-rev N/A

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

      2. lower-cos.f64 N/A

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

      3. distribute-lft-neg-in N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 61.9

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

   10. Applied rewrites 61.9%

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

   if -1.99999999999999988e166 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -5.0000000000000002e-203

   1. Initial program 99.7%

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

   3. Taylor expanded in K around inf

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

      1. cos-neg-rev N/A

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

      2. lower-cos.f64 N/A

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

      3. distribute-lft-neg-in N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 99.7

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

   5. Applied rewrites 99.7%

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

   7. Applied rewrites 97.6%

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

   8. Taylor expanded in K around 0

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

      1. mul-1-neg N/A

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

      2. distribute-lft-neg-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. associate-+r+ N/A

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

      6. lower-+.f64 N/A

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

   10. Applied rewrites 45.4%

       \[\leadsto \color{blue}{\left(-J\right) \cdot \left(\mathsf{fma}\left(\frac{U}{J}, 0.5, \sqrt{\mathsf{fma}\left(\frac{0.25}{J}, \frac{U \cdot U}{J}, 1\right)}\right) + \frac{1}{\mathsf{fma}\left(\frac{U}{J}, 0.5, \sqrt{\mathsf{fma}\left(\frac{0.25}{J}, \frac{U \cdot U}{J}, 1\right)}\right)}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 56.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \leq -2 \cdot 10^{+166} \lor \neg \left(\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \leq -5 \cdot 10^{-203}\right):\\ \;\;\;\;\cos \left(0.5 \cdot K\right) \cdot \left(J \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-J\right) \cdot \left(\mathsf{fma}\left(\frac{U}{J}, 0.5, \sqrt{\mathsf{fma}\left(\frac{0.25}{J}, \frac{U \cdot U}{J}, 1\right)}\right) + {\left(\mathsf{fma}\left(\frac{U}{J}, 0.5, \sqrt{\mathsf{fma}\left(\frac{0.25}{J}, \frac{U \cdot U}{J}, 1\right)}\right)\right)}^{-1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(-2 \cdot J\right) \cdot t\_0\\ t_2 := t\_1 \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\right) \cdot t\_0}\right)}^{2}}\\ t_3 := \cos \left(0.5 \cdot K\right)\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_2 \leq -1.45 \cdot 10^{+168}:\\ \;\;\;\;t\_3 \cdot \left(J \cdot -2\right)\\ \mathbf{elif}\;t\_2 \leq 10^{+263}:\\ \;\;\;\;t\_1 \cdot \sqrt{\mathsf{fma}\left(\frac{0.25}{J}, \frac{U\_m \cdot U\_m}{J}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_3 \cdot U\_m}{t\_3}\\ \end{array} \end{array} \]

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 (cos (* 0.5 K))))
   (if (<= t_2 (- INFINITY))
     (- U_m)
     (if (<= t_2 -1.45e+168)
       (* t_3 (* J -2.0))
       (if (<= t_2 1e+263)
         (* t_1 (sqrt (fma (/ 0.25 J) (/ (* U_m U_m) J) 1.0)))
         (/ (* t_3 U_m) t_3))))))

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 = cos((0.5 * K));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_2 <= -1.45e+168) {
		tmp = t_3 * (J * -2.0);
	} else if (t_2 <= 1e+263) {
		tmp = t_1 * sqrt(fma((0.25 / J), ((U_m * U_m) / J), 1.0));
	} else {
		tmp = (t_3 * U_m) / t_3;
	}
	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 = cos(Float64(0.5 * K))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_2 <= -1.45e+168)
		tmp = Float64(t_3 * Float64(J * -2.0));
	elseif (t_2 <= 1e+263)
		tmp = Float64(t_1 * sqrt(fma(Float64(0.25 / J), Float64(Float64(U_m * U_m) / J), 1.0)));
	else
		tmp = Float64(Float64(t_3 * U_m) / t_3);
	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[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], (-U$95$m), If[LessEqual[t$95$2, -1.45e+168], N[(t$95$3 * N[(J * -2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+263], N[(t$95$1 * N[Sqrt[N[(N[(0.25 / J), $MachinePrecision] * N[(N[(U$95$m * U$95$m), $MachinePrecision] / J), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(t$95$3 * U$95$m), $MachinePrecision] / t$95$3), $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 5.5%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 47.4

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

   5. Applied rewrites 47.4%

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

   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))))) < -1.45e168

   1. Initial program 99.7%

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

   3. Taylor expanded in K around inf

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

      1. cos-neg-rev N/A

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

      2. lower-cos.f64 N/A

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

      3. distribute-lft-neg-in N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 99.7

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

   5. Applied rewrites 99.7%

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

   7. Applied rewrites 97.7%

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

   8. Taylor expanded in J around inf

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

      1. cos-neg-rev N/A

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

      2. lower-cos.f64 N/A

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

      3. distribute-lft-neg-in N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 74.8

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

   10. Applied rewrites 74.8%

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

   if -1.45e168 < (*.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))))) < 1.00000000000000002e263

   1. Initial program 99.7%

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

   3. Taylor expanded in K around 0

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

      1. lower-sqrt.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. unpow2 N/A

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

      5. times-frac N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 82.0

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

   5. Applied rewrites 82.0%

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

   if 1.00000000000000002e263 < (*.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 22.6%

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

   4. Applied rewrites 3.0%

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

   5. Taylor expanded in U around -inf

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

      1. cos-neg-rev N/A

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

      2. distribute-lft-neg-in N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. distribute-lft-neg-in N/A

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

      9. lower-cos.f64 N/A

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

      10. distribute-lft-neg-in N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval N/A

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

      14. distribute-lft-neg-in N/A

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

      15. lower-cos.f64 N/A

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

      16. distribute-lft-neg-in N/A

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

      17. metadata-eval N/A

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

      18. lower-*.f64 51.8

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

   7. Applied rewrites 51.8%

      \[\leadsto \color{blue}{\frac{\cos \left(0.5 \cdot K\right) \cdot U}{\cos \left(0.5 \cdot K\right)}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 4: 63.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\right) \cdot t\_0}\right)}^{2}}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{+166} \lor \neg \left(t\_1 \leq -5 \cdot 10^{-129}\right):\\ \;\;\;\;\cos \left(0.5 \cdot K\right) \cdot \left(J \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{0.25}{J}, \frac{U\_m \cdot U\_m}{J}, 1\right)} \cdot \left(-2 \cdot J\right)\\ \end{array} \end{array} \]

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))
     (- U_m)
     (if (or (<= t_1 -2e+166) (not (<= t_1 -5e-129)))
       (* (cos (* 0.5 K)) (* J -2.0))
       (* (sqrt (fma (/ 0.25 J) (/ (* U_m U_m) J) 1.0)) (* -2.0 J))))))

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 = -U_m;
	} else if ((t_1 <= -2e+166) || !(t_1 <= -5e-129)) {
		tmp = cos((0.5 * K)) * (J * -2.0);
	} else {
		tmp = sqrt(fma((0.25 / J), ((U_m * U_m) / J), 1.0)) * (-2.0 * J);
	}
	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(-U_m);
	elseif ((t_1 <= -2e+166) || !(t_1 <= -5e-129))
		tmp = Float64(cos(Float64(0.5 * K)) * Float64(J * -2.0));
	else
		tmp = Float64(sqrt(fma(Float64(0.25 / J), Float64(Float64(U_m * U_m) / J), 1.0)) * Float64(-2.0 * J));
	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)], (-U$95$m), If[Or[LessEqual[t$95$1, -2e+166], N[Not[LessEqual[t$95$1, -5e-129]], $MachinePrecision]], N[(N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] * N[(J * -2.0), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[(0.25 / J), $MachinePrecision] * N[(N[(U$95$m * U$95$m), $MachinePrecision] / J), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(-2.0 * J), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 5.5%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 47.4

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

   5. Applied rewrites 47.4%

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

   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))))) < -1.99999999999999988e166 or -5.00000000000000027e-129 < (*.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 82.4%

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

   3. Taylor expanded in K around inf

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

      1. cos-neg-rev N/A

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

      2. lower-cos.f64 N/A

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

      3. distribute-lft-neg-in N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 82.4

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

   5. Applied rewrites 82.4%

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

   7. Applied rewrites 89.9%

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

   8. Taylor expanded in J around inf

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

      1. cos-neg-rev N/A

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

      2. lower-cos.f64 N/A

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

      3. distribute-lft-neg-in N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 61.1

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

   10. Applied rewrites 61.1%

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

   if -1.99999999999999988e166 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -5.00000000000000027e-129

   1. Initial program 99.7%

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

   3. Taylor expanded in K around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-*r/ N/A

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

      7. unpow2 N/A

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

      8. times-frac N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 61.3

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

   5. Applied rewrites 61.3%

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(\frac{0.25}{J}, \frac{U \cdot U}{J}, 1\right)} \cdot \left(-2 \cdot J\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 59.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \leq -2 \cdot 10^{+166} \lor \neg \left(\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \leq -5 \cdot 10^{-129}\right):\\ \;\;\;\;\cos \left(0.5 \cdot K\right) \cdot \left(J \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{0.25}{J}, \frac{U \cdot U}{J}, 1\right)} \cdot \left(-2 \cdot J\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(-0.5 \cdot K\right)\\ t_1 := \cos \left(\frac{K}{2}\right)\\ t_2 := \left(\left(-2 \cdot J\right) \cdot t\_1\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\right) \cdot t\_1}\right)}^{2}}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+304}:\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\right) \cdot t\_0}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left({t\_0}^{2} \cdot \frac{J \cdot J}{U\_m \cdot U\_m}\right) \cdot \left(--2\right) - -1\right) \cdot U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (* -0.5 K)))
        (t_1 (cos (/ K 2.0)))
        (t_2
         (*
          (* (* -2.0 J) t_1)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J) t_1)) 2.0))))))
   (if (<= t_2 (- INFINITY))
     (- U_m)
     (if (<= t_2 5e+304)
       (*
        (* (* -2.0 J) (cos (* 0.5 K)))
        (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J) t_0)) 2.0))))
       (*
        (- (* (* (pow t_0 2.0) (/ (* J J) (* U_m U_m))) (- -2.0)) -1.0)
        U_m)))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = cos((-0.5 * K));
	double t_1 = cos((K / 2.0));
	double t_2 = ((-2.0 * J) * t_1) * sqrt((1.0 + pow((U_m / ((2.0 * J) * t_1)), 2.0)));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_2 <= 5e+304) {
		tmp = ((-2.0 * J) * cos((0.5 * K))) * sqrt((1.0 + pow((U_m / ((2.0 * J) * t_0)), 2.0)));
	} else {
		tmp = (((pow(t_0, 2.0) * ((J * J) / (U_m * U_m))) * -(-2.0)) - -1.0) * U_m;
	}
	return tmp;
}

Java

U_m = Math.abs(U);
public static double code(double J, double K, double U_m) {
	double t_0 = Math.cos((-0.5 * K));
	double t_1 = Math.cos((K / 2.0));
	double t_2 = ((-2.0 * J) * t_1) * Math.sqrt((1.0 + Math.pow((U_m / ((2.0 * J) * t_1)), 2.0)));
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = -U_m;
	} else if (t_2 <= 5e+304) {
		tmp = ((-2.0 * J) * Math.cos((0.5 * K))) * Math.sqrt((1.0 + Math.pow((U_m / ((2.0 * J) * t_0)), 2.0)));
	} else {
		tmp = (((Math.pow(t_0, 2.0) * ((J * J) / (U_m * U_m))) * -(-2.0)) - -1.0) * U_m;
	}
	return tmp;
}

Python

U_m = math.fabs(U)
def code(J, K, U_m):
	t_0 = math.cos((-0.5 * K))
	t_1 = math.cos((K / 2.0))
	t_2 = ((-2.0 * J) * t_1) * math.sqrt((1.0 + math.pow((U_m / ((2.0 * J) * t_1)), 2.0)))
	tmp = 0
	if t_2 <= -math.inf:
		tmp = -U_m
	elif t_2 <= 5e+304:
		tmp = ((-2.0 * J) * math.cos((0.5 * K))) * math.sqrt((1.0 + math.pow((U_m / ((2.0 * J) * t_0)), 2.0)))
	else:
		tmp = (((math.pow(t_0, 2.0) * ((J * J) / (U_m * U_m))) * -(-2.0)) - -1.0) * U_m
	return tmp

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(-0.5 * K))
	t_1 = cos(Float64(K / 2.0))
	t_2 = Float64(Float64(Float64(-2.0 * J) * t_1) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J) * t_1)) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_2 <= 5e+304)
		tmp = Float64(Float64(Float64(-2.0 * J) * cos(Float64(0.5 * K))) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J) * t_0)) ^ 2.0))));
	else
		tmp = Float64(Float64(Float64(Float64((t_0 ^ 2.0) * Float64(Float64(J * J) / Float64(U_m * U_m))) * Float64(-(-2.0))) - -1.0) * U_m);
	end
	return tmp
end

MATLAB

U_m = abs(U);
function tmp_2 = code(J, K, U_m)
	t_0 = cos((-0.5 * K));
	t_1 = cos((K / 2.0));
	t_2 = ((-2.0 * J) * t_1) * sqrt((1.0 + ((U_m / ((2.0 * J) * t_1)) ^ 2.0)));
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = -U_m;
	elseif (t_2 <= 5e+304)
		tmp = ((-2.0 * J) * cos((0.5 * K))) * sqrt((1.0 + ((U_m / ((2.0 * J) * t_0)) ^ 2.0)));
	else
		tmp = ((((t_0 ^ 2.0) * ((J * J) / (U_m * U_m))) * -(-2.0)) - -1.0) * U_m;
	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[(-0.5 * K), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(-2.0 * J), $MachinePrecision] * t$95$1), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], (-U$95$m), If[LessEqual[t$95$2, 5e+304], N[(N[(N[(-2.0 * J), $MachinePrecision] * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $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], N[(N[(N[(N[(N[Power[t$95$0, 2.0], $MachinePrecision] * N[(N[(J * J), $MachinePrecision] / N[(U$95$m * U$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * (--2.0)), $MachinePrecision] - -1.0), $MachinePrecision] * U$95$m), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 5.5%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 47.4

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

   5. Applied rewrites 47.4%

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

   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))))) < 4.9999999999999997e304

   1. Initial program 99.7%

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

   3. Taylor expanded in K around inf

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

      1. cos-neg-rev N/A

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

      2. lower-cos.f64 N/A

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

      3. distribute-lft-neg-in N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in K around inf

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

      1. metadata-eval N/A

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

      2. distribute-lft-neg-in N/A

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

      3. lower-cos.f64 N/A

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

      4. distribute-lft-neg-in N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 99.7

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

   8. Applied rewrites 99.7%

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

   if 4.9999999999999997e304 < (*.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 5.7%

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

   3. Taylor expanded in U around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 53.4%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \leq 5 \cdot 10^{+304}:\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(-0.5 \cdot K\right)}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left({\cos \left(-0.5 \cdot K\right)}^{2} \cdot \frac{J \cdot J}{U \cdot U}\right) \cdot \left(--2\right) - -1\right) \cdot U\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(0.5 \cdot K\right)\\ t_1 := \cos \left(\frac{K}{2}\right)\\ t_2 := \left(-2 \cdot J\right) \cdot t\_1\\ t_3 := t\_2 \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\right) \cdot t\_1}\right)}^{2}}\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_3 \leq 10^{+263}:\\ \;\;\;\;t\_2 \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\right) \cdot \mathsf{fma}\left(-0.125, K \cdot K, 1\right)}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 \cdot U\_m}{t\_0}\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (* 0.5 K)))
        (t_1 (cos (/ K 2.0)))
        (t_2 (* (* -2.0 J) t_1))
        (t_3 (* t_2 (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J) t_1)) 2.0))))))
   (if (<= t_3 (- INFINITY))
     (- U_m)
     (if (<= t_3 1e+263)
       (*
        t_2
        (sqrt
         (+ 1.0 (pow (/ U_m (* (* 2.0 J) (fma -0.125 (* K K) 1.0))) 2.0))))
       (/ (* t_0 U_m) t_0)))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = cos((0.5 * K));
	double t_1 = cos((K / 2.0));
	double t_2 = (-2.0 * J) * t_1;
	double t_3 = t_2 * sqrt((1.0 + pow((U_m / ((2.0 * J) * t_1)), 2.0)));
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_3 <= 1e+263) {
		tmp = t_2 * sqrt((1.0 + pow((U_m / ((2.0 * J) * fma(-0.125, (K * K), 1.0))), 2.0)));
	} else {
		tmp = (t_0 * U_m) / t_0;
	}
	return tmp;
}

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(0.5 * K))
	t_1 = cos(Float64(K / 2.0))
	t_2 = Float64(Float64(-2.0 * J) * t_1)
	t_3 = Float64(t_2 * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J) * t_1)) ^ 2.0))))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_3 <= 1e+263)
		tmp = Float64(t_2 * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J) * fma(-0.125, Float64(K * K), 1.0))) ^ 2.0))));
	else
		tmp = Float64(Float64(t_0 * U_m) / t_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[(0.5 * K), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(-2.0 * J), $MachinePrecision] * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], (-U$95$m), If[LessEqual[t$95$3, 1e+263], N[(t$95$2 * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J), $MachinePrecision] * N[(-0.125 * N[(K * K), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * U$95$m), $MachinePrecision] / t$95$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 5.5%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 47.4

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

   5. Applied rewrites 47.4%

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

   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))))) < 1.00000000000000002e263

   1. Initial program 99.7%

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

   3. Taylor expanded in K around 0

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

      1. +-commutative N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \color{blue}{\left(\frac{-1}{8} \cdot {K}^{2} + 1\right)}}\right)}^{2}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{8}, {K}^{2}, 1\right)}}\right)}^{2}} \]

      3. unpow2 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \mathsf{fma}\left(\frac{-1}{8}, \color{blue}{K \cdot K}, 1\right)}\right)}^{2}} \]

      4. lower-*.f64 85.8

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \mathsf{fma}\left(-0.125, \color{blue}{K \cdot K}, 1\right)}\right)}^{2}} \]

   5. Applied rewrites 85.8%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \color{blue}{\mathsf{fma}\left(-0.125, K \cdot K, 1\right)}}\right)}^{2}} \]

   if 1.00000000000000002e263 < (*.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 22.6%

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

   4. Applied rewrites 3.0%

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

   5. Taylor expanded in U around -inf

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

      1. cos-neg-rev N/A

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

      2. distribute-lft-neg-in N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. distribute-lft-neg-in N/A

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

      9. lower-cos.f64 N/A

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

      10. distribute-lft-neg-in N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval N/A

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

      14. distribute-lft-neg-in N/A

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

      15. lower-cos.f64 N/A

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

      16. distribute-lft-neg-in N/A

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

      17. metadata-eval N/A

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

      18. lower-*.f64 51.8

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

   7. Applied rewrites 51.8%

      \[\leadsto \color{blue}{\frac{\cos \left(0.5 \cdot K\right) \cdot U}{\cos \left(0.5 \cdot K\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 71.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(-2 \cdot J\right) \cdot t\_0\\ t_2 := t\_1 \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\right) \cdot t\_0}\right)}^{2}}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_2 \leq -1.45 \cdot 10^{+168}:\\ \;\;\;\;\cos \left(0.5 \cdot K\right) \cdot \left(J \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \sqrt{\mathsf{fma}\left(\frac{0.25}{J}, \frac{U\_m \cdot U\_m}{J}, 1\right)}\\ \end{array} \end{array} \]

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))
     (- U_m)
     (if (<= t_2 -1.45e+168)
       (* (cos (* 0.5 K)) (* J -2.0))
       (* t_1 (sqrt (fma (/ 0.25 J) (/ (* U_m U_m) J) 1.0)))))))

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 = -U_m;
	} else if (t_2 <= -1.45e+168) {
		tmp = cos((0.5 * K)) * (J * -2.0);
	} else {
		tmp = t_1 * sqrt(fma((0.25 / J), ((U_m * U_m) / J), 1.0));
	}
	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(-U_m);
	elseif (t_2 <= -1.45e+168)
		tmp = Float64(cos(Float64(0.5 * K)) * Float64(J * -2.0));
	else
		tmp = Float64(t_1 * sqrt(fma(Float64(0.25 / J), Float64(Float64(U_m * U_m) / J), 1.0)));
	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)], (-U$95$m), If[LessEqual[t$95$2, -1.45e+168], N[(N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] * N[(J * -2.0), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[Sqrt[N[(N[(0.25 / J), $MachinePrecision] * N[(N[(U$95$m * U$95$m), $MachinePrecision] / J), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 5.5%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 47.4

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

   5. Applied rewrites 47.4%

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

   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))))) < -1.45e168

   1. Initial program 99.7%

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

   3. Taylor expanded in K around inf

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

      1. cos-neg-rev N/A

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

      2. lower-cos.f64 N/A

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

      3. distribute-lft-neg-in N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 99.7

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

   5. Applied rewrites 99.7%

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

   7. Applied rewrites 97.7%

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

   8. Taylor expanded in J around inf

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

      1. cos-neg-rev N/A

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

      2. lower-cos.f64 N/A

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

      3. distribute-lft-neg-in N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 74.8

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

   10. Applied rewrites 74.8%

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

   if -1.45e168 < (*.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 83.8%

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

   3. Taylor expanded in K around 0

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

      1. lower-sqrt.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. unpow2 N/A

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

      5. times-frac N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 67.4

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

   5. Applied rewrites 67.4%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\sqrt{\mathsf{fma}\left(\frac{0.25}{J}, \frac{U \cdot U}{J}, 1\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 71.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\right) \cdot t\_0}\right)}^{2}}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -1.45 \cdot 10^{+168}:\\ \;\;\;\;\cos \left(0.5 \cdot K\right) \cdot \left(J \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(\frac{0.25}{J}, \frac{U\_m \cdot U\_m}{J}, 1\right)} \cdot \cos \left(\frac{K}{-2}\right)\right) \cdot \left(J \cdot -2\right)\\ \end{array} \end{array} \]

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))
     (- U_m)
     (if (<= t_1 -1.45e+168)
       (* (cos (* 0.5 K)) (* J -2.0))
       (*
        (* (sqrt (fma (/ 0.25 J) (/ (* U_m U_m) J) 1.0)) (cos (/ K -2.0)))
        (* J -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 = -U_m;
	} else if (t_1 <= -1.45e+168) {
		tmp = cos((0.5 * K)) * (J * -2.0);
	} else {
		tmp = (sqrt(fma((0.25 / J), ((U_m * U_m) / J), 1.0)) * cos((K / -2.0))) * (J * -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(-U_m);
	elseif (t_1 <= -1.45e+168)
		tmp = Float64(cos(Float64(0.5 * K)) * Float64(J * -2.0));
	else
		tmp = Float64(Float64(sqrt(fma(Float64(0.25 / J), Float64(Float64(U_m * U_m) / J), 1.0)) * cos(Float64(K / -2.0))) * Float64(J * -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)], (-U$95$m), If[LessEqual[t$95$1, -1.45e+168], N[(N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] * N[(J * -2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(0.25 / J), $MachinePrecision] * N[(N[(U$95$m * U$95$m), $MachinePrecision] / J), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(K / -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(J * -2.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 5.5%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 47.4

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

   5. Applied rewrites 47.4%

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

   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))))) < -1.45e168

   1. Initial program 99.7%

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

   3. Taylor expanded in K around inf

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

      1. cos-neg-rev N/A

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

      2. lower-cos.f64 N/A

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

      3. distribute-lft-neg-in N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 99.7

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

   5. Applied rewrites 99.7%

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

   7. Applied rewrites 97.7%

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

   8. Taylor expanded in J around inf

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

      1. cos-neg-rev N/A

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

      2. lower-cos.f64 N/A

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

      3. distribute-lft-neg-in N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 74.8

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

   10. Applied rewrites 74.8%

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

   if -1.45e168 < (*.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 83.8%

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

   3. Taylor expanded in K around inf

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

      1. cos-neg-rev N/A

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

      2. lower-cos.f64 N/A

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

      3. distribute-lft-neg-in N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 83.8

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

   5. Applied rewrites 83.8%

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

   7. Applied rewrites 90.6%

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

      1. lift-cosh.f64 N/A

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

      2. lift-asinh.f64 N/A

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

      3. cosh-asinh N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-+.f64 N/A

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

   9. Applied rewrites 83.8%

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

   10. Taylor expanded in K around 0

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

       1. +-commutative N/A

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

       2. associate-*r/ N/A

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

       3. unpow2 N/A

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

       4. times-frac N/A

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

       5. lower-fma.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. lower-/.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 67.4

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

   12. Applied rewrites 67.4%

       \[\leadsto \left(\sqrt{\color{blue}{\mathsf{fma}\left(\frac{0.25}{J}, \frac{U \cdot U}{J}, 1\right)}} \cdot \cos \left(\frac{K}{-2}\right)\right) \cdot \left(J \cdot -2\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 32.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\right) \cdot t\_0}\right)}^{2}}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-196}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{0.25}{J}, \frac{U\_m \cdot U\_m}{J}, 1\right)} \cdot \left(-2 \cdot J\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0006944444444444445, K \cdot K, 0.020833333333333332\right) \cdot \left(K \cdot K\right) - 0.25, K \cdot K, 1\right)}{U\_m} \cdot J, J \cdot -2, -U\_m\right)\\ \end{array} \end{array} \]

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))
     (- U_m)
     (if (<= t_1 -1e-196)
       (* (sqrt (fma (/ 0.25 J) (/ (* U_m U_m) J) 1.0)) (* -2.0 J))
       (fma
        (*
         (/
          (fma
           (-
            (*
             (fma -0.0006944444444444445 (* K K) 0.020833333333333332)
             (* K K))
            0.25)
           (* K K)
           1.0)
          U_m)
         J)
        (* J -2.0)
        (- U_m))))))

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 = -U_m;
	} else if (t_1 <= -1e-196) {
		tmp = sqrt(fma((0.25 / J), ((U_m * U_m) / J), 1.0)) * (-2.0 * J);
	} else {
		tmp = fma(((fma(((fma(-0.0006944444444444445, (K * K), 0.020833333333333332) * (K * K)) - 0.25), (K * K), 1.0) / U_m) * J), (J * -2.0), -U_m);
	}
	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(-U_m);
	elseif (t_1 <= -1e-196)
		tmp = Float64(sqrt(fma(Float64(0.25 / J), Float64(Float64(U_m * U_m) / J), 1.0)) * Float64(-2.0 * J));
	else
		tmp = fma(Float64(Float64(fma(Float64(Float64(fma(-0.0006944444444444445, Float64(K * K), 0.020833333333333332) * Float64(K * K)) - 0.25), Float64(K * K), 1.0) / U_m) * J), Float64(J * -2.0), Float64(-U_m));
	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)], (-U$95$m), If[LessEqual[t$95$1, -1e-196], N[(N[Sqrt[N[(N[(0.25 / J), $MachinePrecision] * N[(N[(U$95$m * U$95$m), $MachinePrecision] / J), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(-2.0 * J), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(-0.0006944444444444445 * N[(K * K), $MachinePrecision] + 0.020833333333333332), $MachinePrecision] * N[(K * K), $MachinePrecision]), $MachinePrecision] - 0.25), $MachinePrecision] * N[(K * K), $MachinePrecision] + 1.0), $MachinePrecision] / U$95$m), $MachinePrecision] * J), $MachinePrecision] * N[(J * -2.0), $MachinePrecision] + (-U$95$m)), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 5.5%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 47.4

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

   5. Applied rewrites 47.4%

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

   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))))) < -1e-196

   1. Initial program 99.7%

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

   3. Taylor expanded in K around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-*r/ N/A

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

      7. unpow2 N/A

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

      8. times-frac N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 52.4

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

   5. Applied rewrites 52.4%

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

   if -1e-196 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 77.4%

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

   3. Taylor expanded in J around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. cos-neg-rev N/A

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

      13. lower-cos.f64 N/A

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

      14. distribute-lft-neg-in N/A

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

      15. metadata-eval N/A

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

      16. lower-*.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-neg.f64 21.2

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

   5. Applied rewrites 21.2%

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

   7. Applied rewrites 23.2%

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

   8. Taylor expanded in K around 0

      \[\leadsto \mathsf{fma}\left(\frac{1 + {K}^{2} \cdot \left({K}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {K}^{2}\right) - \frac{1}{4}\right)}{U} \cdot J, J \cdot -2, -U\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 13.3%

       \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0006944444444444445, K \cdot K, 0.020833333333333332\right) \cdot \left(K \cdot K\right) - 0.25, K \cdot K, 1\right)}{U} \cdot J, J \cdot -2, -U\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 33.7% accurate, 0.5× speedup

Math

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

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))
     (- U_m)
     (if (<= t_1 -1e-196)
       (* (sqrt (fma (/ 0.25 J) (/ (* U_m U_m) J) 1.0)) (* -2.0 J))
       (- (+ (/ (* (* -2.0 J) J) U_m) U_m))))))

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 = -U_m;
	} else if (t_1 <= -1e-196) {
		tmp = sqrt(fma((0.25 / J), ((U_m * U_m) / J), 1.0)) * (-2.0 * J);
	} else {
		tmp = -((((-2.0 * J) * J) / U_m) + U_m);
	}
	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(-U_m);
	elseif (t_1 <= -1e-196)
		tmp = Float64(sqrt(fma(Float64(0.25 / J), Float64(Float64(U_m * U_m) / J), 1.0)) * Float64(-2.0 * J));
	else
		tmp = Float64(-Float64(Float64(Float64(Float64(-2.0 * J) * J) / U_m) + U_m));
	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)], (-U$95$m), If[LessEqual[t$95$1, -1e-196], N[(N[Sqrt[N[(N[(0.25 / J), $MachinePrecision] * N[(N[(U$95$m * U$95$m), $MachinePrecision] / J), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(-2.0 * J), $MachinePrecision]), $MachinePrecision], (-N[(N[(N[(N[(-2.0 * J), $MachinePrecision] * J), $MachinePrecision] / U$95$m), $MachinePrecision] + U$95$m), $MachinePrecision])]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 5.5%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 47.4

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

   5. Applied rewrites 47.4%

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

   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))))) < -1e-196

   1. Initial program 99.7%

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

   3. Taylor expanded in K around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-*r/ N/A

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

      7. unpow2 N/A

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

      8. times-frac N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 52.4

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

   5. Applied rewrites 52.4%

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

   if -1e-196 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 77.4%

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

   3. Taylor expanded in J around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. cos-neg-rev N/A

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

      13. lower-cos.f64 N/A

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

      14. distribute-lft-neg-in N/A

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

      15. metadata-eval N/A

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

      16. lower-*.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-neg.f64 21.2

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

   5. Applied rewrites 21.2%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 21.2%

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

   10. Applied rewrites 21.2%

       \[\leadsto \frac{\left(-2 \cdot J\right) \cdot J}{-U} - U \]
3. Recombined 3 regimes into one program.

4. Final simplification 35.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \leq -1 \cdot 10^{-196}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{0.25}{J}, \frac{U \cdot U}{J}, 1\right)} \cdot \left(-2 \cdot J\right)\\ \mathbf{else}:\\ \;\;\;\;-\left(\frac{\left(-2 \cdot J\right) \cdot J}{U} + U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 26.1% accurate, 14.9× speedup

Math

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

FPCore

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

C

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

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 = ((((-2.0d0) / u_m) * j) * j) - u_m
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.4%

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

3. Taylor expanded in J around 0

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-pow.f64 N/A

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

   12. cos-neg-rev N/A

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

   13. lower-cos.f64 N/A

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

   14. distribute-lft-neg-in N/A

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

   15. metadata-eval N/A

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

   16. lower-*.f64 N/A

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

   17. mul-1-neg N/A

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

   18. lower-neg.f64 21.3

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

5. Applied rewrites 21.3%

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

6. Taylor expanded in K around 0

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

8. Applied rewrites 21.2%

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

10. Applied rewrites 21.2%

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

12. Applied rewrites 22.5%

    \[\leadsto \left(\frac{-2}{U} \cdot J\right) \cdot J - U \]
13. Add Preprocessing

Alternative 12: 25.9% accurate, 124.3× speedup

Math

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

FPCore

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

C

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

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 = -u_m
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.4%

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

3. Taylor expanded in J around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 22.5

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

5. Applied rewrites 22.5%

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

Reproduce

herbie shell --seed 2024346 
(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)))))