Maksimov and Kolovsky, Equation (3)

Percentage Accurate: 73.0% → 99.8%

Time: 5.9s

Alternatives: 10

Speedup: 0.4×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(-2 \cdot J\right) \cdot t\_0\\ t_2 := t\_1 \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\right) \cdot t\_0}\right)}^{2}}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+307}:\\ \;\;\;\;t\_1 \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\right) \cdot \cos \left(0.5 \cdot K\right)}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \]

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 5e+307)
       (* t_1 (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J) (cos (* 0.5 K)))) 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;
	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 <= 5e+307) {
		tmp = t_1 * sqrt((1.0 + pow((U_m / ((2.0 * J) * cos((0.5 * K)))), 2.0)));
	} else {
		tmp = 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((K / 2.0));
	double t_1 = (-2.0 * J) * t_0;
	double t_2 = t_1 * Math.sqrt((1.0 + Math.pow((U_m / ((2.0 * J) * t_0)), 2.0)));
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = -U_m;
	} else if (t_2 <= 5e+307) {
		tmp = t_1 * Math.sqrt((1.0 + Math.pow((U_m / ((2.0 * J) * Math.cos((0.5 * K)))), 2.0)));
	} else {
		tmp = U_m;
	}
	return tmp;
}

Python

U_m = math.fabs(U)
def code(J, K, U_m):
	t_0 = math.cos((K / 2.0))
	t_1 = (-2.0 * J) * t_0
	t_2 = t_1 * math.sqrt((1.0 + math.pow((U_m / ((2.0 * J) * t_0)), 2.0)))
	tmp = 0
	if t_2 <= -math.inf:
		tmp = -U_m
	elif t_2 <= 5e+307:
		tmp = t_1 * math.sqrt((1.0 + math.pow((U_m / ((2.0 * J) * math.cos((0.5 * K)))), 2.0)))
	else:
		tmp = 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(-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 <= 5e+307)
		tmp = Float64(t_1 * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J) * cos(Float64(0.5 * K)))) ^ 2.0))));
	else
		tmp = U_m;
	end
	return tmp
end

MATLAB

U_m = abs(U);
function tmp_2 = code(J, K, U_m)
	t_0 = cos((K / 2.0));
	t_1 = (-2.0 * J) * t_0;
	t_2 = t_1 * sqrt((1.0 + ((U_m / ((2.0 * J) * t_0)) ^ 2.0)));
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = -U_m;
	elseif (t_2 <= 5e+307)
		tmp = t_1 * sqrt((1.0 + ((U_m / ((2.0 * J) * cos((0.5 * K)))) ^ 2.0)));
	else
		tmp = 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[(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, 5e+307], N[(t$95$1 * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J), $MachinePrecision] * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], U$95$m]]]]]

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 6.2%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 48.4

         \[\leadsto -U \]

   5. Applied rewrites 48.4%

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

   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))))) < 5e307

   1. Initial program 99.8%

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

   3. Taylor expanded in K around 0

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

      1. lower-*.f64 99.8

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

   5. Applied rewrites 99.8%

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

   if 5e307 < (*.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 5.8%

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

   3. Taylor expanded in U around -inf

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

   5. Applied rewrites 39.3%

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

Alternative 2: 58.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \frac{J}{U\_m} \cdot \frac{J}{U\_m}\\ t_1 := \cos \left(\frac{K}{2}\right)\\ t_2 := \left(\left(-2 \cdot J\right) \cdot t\_1\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\right) \cdot t\_1}\right)}^{2}}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_2 \leq -7 \cdot 10^{+159}:\\ \;\;\;\;J \cdot \left(-0.25 \cdot \left(\frac{U\_m}{J} \cdot \frac{U\_m}{J}\right) - 2\right)\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{-159}:\\ \;\;\;\;\left(J \cdot -2\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U\_m \cdot U\_m}{J \cdot J}, 0.25, 1\right)}\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{-295}:\\ \;\;\;\;U\_m \cdot \left(-2 \cdot t\_0 - 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-U\_m\right) \cdot \left(t\_0 \cdot -2 - 1\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (* (/ J U_m) (/ J U_m)))
        (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 -7e+159)
       (* J (- (* -0.25 (* (/ U_m J) (/ U_m J))) 2.0))
       (if (<= t_2 -1e-159)
         (* (* J -2.0) (sqrt (fma (/ (* U_m U_m) (* J J)) 0.25 1.0)))
         (if (<= t_2 -5e-295)
           (* U_m (- (* -2.0 t_0) 1.0))
           (* (- U_m) (- (* t_0 -2.0) 1.0))))))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = (J / U_m) * (J / U_m);
	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 <= -7e+159) {
		tmp = J * ((-0.25 * ((U_m / J) * (U_m / J))) - 2.0);
	} else if (t_2 <= -1e-159) {
		tmp = (J * -2.0) * sqrt(fma(((U_m * U_m) / (J * J)), 0.25, 1.0));
	} else if (t_2 <= -5e-295) {
		tmp = U_m * ((-2.0 * t_0) - 1.0);
	} else {
		tmp = -U_m * ((t_0 * -2.0) - 1.0);
	}
	return tmp;
}

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = Float64(Float64(J / U_m) * Float64(J / U_m))
	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 <= -7e+159)
		tmp = Float64(J * Float64(Float64(-0.25 * Float64(Float64(U_m / J) * Float64(U_m / J))) - 2.0));
	elseif (t_2 <= -1e-159)
		tmp = Float64(Float64(J * -2.0) * sqrt(fma(Float64(Float64(U_m * U_m) / Float64(J * J)), 0.25, 1.0)));
	elseif (t_2 <= -5e-295)
		tmp = Float64(U_m * Float64(Float64(-2.0 * t_0) - 1.0));
	else
		tmp = Float64(Float64(-U_m) * Float64(Float64(t_0 * -2.0) - 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[(J / U$95$m), $MachinePrecision] * N[(J / U$95$m), $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, -7e+159], N[(J * N[(N[(-0.25 * N[(N[(U$95$m / J), $MachinePrecision] * N[(U$95$m / J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -1e-159], N[(N[(J * -2.0), $MachinePrecision] * N[Sqrt[N[(N[(N[(U$95$m * U$95$m), $MachinePrecision] / N[(J * J), $MachinePrecision]), $MachinePrecision] * 0.25 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -5e-295], N[(U$95$m * N[(N[(-2.0 * t$95$0), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision], N[((-U$95$m) * N[(N[(t$95$0 * -2.0), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 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 6.2%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 48.4

         \[\leadsto -U \]

   5. Applied rewrites 48.4%

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

   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))))) < -6.9999999999999999e159

   1. Initial program 99.8%

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

   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

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 48.8

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

   5. Applied rewrites 48.8%

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

   6. Taylor expanded in J around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 59.1

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

   8. Applied rewrites 59.1%

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

   if -6.9999999999999999e159 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -9.99999999999999989e-160

   1. Initial program 99.8%

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

   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

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 59.4

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

   5. Applied rewrites 59.4%

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

   if -9.99999999999999989e-160 < (*.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))))) < -5.00000000000000008e-295

   1. Initial program 99.4%

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

   3. Taylor expanded in K around 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

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 0.0

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

   5. Applied rewrites 0.0%

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

   6. Taylor expanded in J around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 45.4

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

   8. Applied rewrites 45.4%

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

   9. Taylor expanded in J around inf

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

   11. Applied rewrites 43.0%

       \[\leadsto J \cdot -2 \]

   12. Taylor expanded in U around inf

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

       1. lower-*.f64 N/A

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

       2. lower--.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. pow2 N/A

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

       5. pow2 N/A

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

       6. times-frac N/A

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

       7. lower-*.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. lower-/.f64 7.4

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

   14. Applied rewrites 7.4%

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

   if -5.00000000000000008e-295 < (*.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 76.0%

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

   3. Taylor expanded in 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

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 N/A

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

   5. Applied rewrites 19.9%

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

   6. Taylor expanded in K around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 19.9

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

   8. Applied rewrites 19.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 24.0

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

   10. Applied rewrites 24.0%

       \[\leadsto \color{blue}{\left(-U\right) \cdot \left(\left(\frac{J}{U} \cdot \frac{J}{U}\right) \cdot -2 - 1\right)} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 3: 72.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\right) \cdot t\_0}\right)}^{2}}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-284}:\\ \;\;\;\;\left(J \cdot -2\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U\_m}{J} \cdot \frac{U\_m}{J}, 0.25, 1\right)}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+236}:\\ \;\;\;\;\left(J \cdot -2\right) \cdot \cos \left(0.5 \cdot K\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-U\_m\right) \cdot \left(\left(\frac{J}{U\_m} \cdot \frac{J}{U\_m}\right) \cdot -2 - 1\right)\\ \end{array} \end{array} \]

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)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J) t_0)) 2.0))))))
   (if (<= t_1 (- INFINITY))
     (- U_m)
     (if (<= t_1 -2e-284)
       (* (* J -2.0) (sqrt (fma (* (/ U_m J) (/ U_m J)) 0.25 1.0)))
       (if (<= t_1 2e+236)
         (* (* J -2.0) (cos (* 0.5 K)))
         (* (- U_m) (- (* (* (/ J U_m) (/ J U_m)) -2.0) 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) * 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-284) {
		tmp = (J * -2.0) * sqrt(fma(((U_m / J) * (U_m / J)), 0.25, 1.0));
	} else if (t_1 <= 2e+236) {
		tmp = (J * -2.0) * cos((0.5 * K));
	} else {
		tmp = -U_m * ((((J / U_m) * (J / U_m)) * -2.0) - 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(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-284)
		tmp = Float64(Float64(J * -2.0) * sqrt(fma(Float64(Float64(U_m / J) * Float64(U_m / J)), 0.25, 1.0)));
	elseif (t_1 <= 2e+236)
		tmp = Float64(Float64(J * -2.0) * cos(Float64(0.5 * K)));
	else
		tmp = Float64(Float64(-U_m) * Float64(Float64(Float64(Float64(J / U_m) * Float64(J / U_m)) * -2.0) - 1.0));
	end
	return tmp
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := Block[{t$95$0 = N[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, -2e-284], N[(N[(J * -2.0), $MachinePrecision] * N[Sqrt[N[(N[(N[(U$95$m / J), $MachinePrecision] * N[(U$95$m / J), $MachinePrecision]), $MachinePrecision] * 0.25 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+236], N[(N[(J * -2.0), $MachinePrecision] * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[((-U$95$m) * N[(N[(N[(N[(J / U$95$m), $MachinePrecision] * N[(J / U$95$m), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 6.2%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 48.4

         \[\leadsto -U \]

   5. Applied rewrites 48.4%

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

   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))))) < -2.00000000000000007e-284

   1. Initial program 99.8%

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

   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

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 51.7

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

   5. Applied rewrites 51.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 68.0

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

   7. Applied rewrites 68.0%

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

   if -2.00000000000000007e-284 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 2.00000000000000011e236

   1. Initial program 99.9%

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

   3. Taylor expanded in J around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 63.6

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

   5. Applied rewrites 63.6%

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

   if 2.00000000000000011e236 < (*.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 38.3%

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

   3. Taylor expanded in U around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 N/A

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

   5. Applied rewrites 27.0%

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

   6. Taylor expanded in K around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 27.0

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

   8. Applied rewrites 27.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 27.3

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

   10. Applied rewrites 27.3%

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

Alternative 4: 91.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\right) \cdot t\_0}\right)}^{2}}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+307}:\\ \;\;\;\;\left(\left(J \cdot -2\right) \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U\_m}{J} \cdot \frac{U\_m}{J}, 0.25, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \]

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 5e+307)
       (*
        (* (* J -2.0) (cos (* 0.5 K)))
        (sqrt (fma (* (/ U_m J) (/ U_m J)) 0.25 1.0)))
       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 <= 5e+307) {
		tmp = ((J * -2.0) * cos((0.5 * K))) * sqrt(fma(((U_m / J) * (U_m / J)), 0.25, 1.0));
	} else {
		tmp = 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 <= 5e+307)
		tmp = Float64(Float64(Float64(J * -2.0) * cos(Float64(0.5 * K))) * sqrt(fma(Float64(Float64(U_m / J) * Float64(U_m / J)), 0.25, 1.0)));
	else
		tmp = 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, 5e+307], N[(N[(N[(J * -2.0), $MachinePrecision] * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(N[(U$95$m / J), $MachinePrecision] * N[(U$95$m / J), $MachinePrecision]), $MachinePrecision] * 0.25 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], U$95$m]]]]

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 6.2%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 48.4

         \[\leadsto -U \]

   5. Applied rewrites 48.4%

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

   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))))) < 5e307

   1. Initial program 99.8%

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

   3. Taylor expanded in K around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

   5. Applied rewrites 74.8%

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

   6. Taylor expanded in K around 0

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

      1. pow2 N/A

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

      2. pow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 88.0

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

   8. Applied rewrites 88.0%

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

   if 5e307 < (*.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 5.8%

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

   3. Taylor expanded in U around -inf

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

   5. Applied rewrites 39.3%

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

Alternative 5: 62.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\right) \cdot t\_0}\right)}^{2}}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-295}:\\ \;\;\;\;\left(J \cdot -2\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U\_m}{J} \cdot \frac{U\_m}{J}, 0.25, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-U\_m\right) \cdot \left(\left(\frac{J}{U\_m} \cdot \frac{J}{U\_m}\right) \cdot -2 - 1\right)\\ \end{array} \end{array} \]

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)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J) t_0)) 2.0))))))
   (if (<= t_1 (- INFINITY))
     (- U_m)
     (if (<= t_1 -5e-295)
       (* (* J -2.0) (sqrt (fma (* (/ U_m J) (/ U_m J)) 0.25 1.0)))
       (* (- U_m) (- (* (* (/ J U_m) (/ J U_m)) -2.0) 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) * 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 <= -5e-295) {
		tmp = (J * -2.0) * sqrt(fma(((U_m / J) * (U_m / J)), 0.25, 1.0));
	} else {
		tmp = -U_m * ((((J / U_m) * (J / U_m)) * -2.0) - 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(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 <= -5e-295)
		tmp = Float64(Float64(J * -2.0) * sqrt(fma(Float64(Float64(U_m / J) * Float64(U_m / J)), 0.25, 1.0)));
	else
		tmp = Float64(Float64(-U_m) * Float64(Float64(Float64(Float64(J / U_m) * Float64(J / U_m)) * -2.0) - 1.0));
	end
	return tmp
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := Block[{t$95$0 = N[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, -5e-295], N[(N[(J * -2.0), $MachinePrecision] * N[Sqrt[N[(N[(N[(U$95$m / J), $MachinePrecision] * N[(U$95$m / J), $MachinePrecision]), $MachinePrecision] * 0.25 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[((-U$95$m) * N[(N[(N[(N[(J / U$95$m), $MachinePrecision] * N[(J / U$95$m), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision] - 1.0), $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 6.2%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 48.4

         \[\leadsto -U \]

   5. Applied rewrites 48.4%

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

   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))))) < -5.00000000000000008e-295

   1. Initial program 99.8%

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

   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

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 51.7

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

   5. Applied rewrites 51.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 68.0

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

   7. Applied rewrites 68.0%

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

   if -5.00000000000000008e-295 < (*.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 76.0%

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

   3. Taylor expanded in 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

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 N/A

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

   5. Applied rewrites 19.9%

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

   6. Taylor expanded in K around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 19.9

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

   8. Applied rewrites 19.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 24.0

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

   10. Applied rewrites 24.0%

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

Alternative 6: 54.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\right) \cdot t\_0}\right)}^{2}}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-295}:\\ \;\;\;\;J \cdot \left(-0.25 \cdot \left(\frac{U\_m}{J} \cdot \frac{U\_m}{J}\right) - 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-U\_m\right) \cdot \left(\left(\frac{J}{U\_m} \cdot \frac{J}{U\_m}\right) \cdot -2 - 1\right)\\ \end{array} \end{array} \]

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)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J) t_0)) 2.0))))))
   (if (<= t_1 (- INFINITY))
     (- U_m)
     (if (<= t_1 -5e-295)
       (* J (- (* -0.25 (* (/ U_m J) (/ U_m J))) 2.0))
       (* (- U_m) (- (* (* (/ J U_m) (/ J U_m)) -2.0) 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) * 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 <= -5e-295) {
		tmp = J * ((-0.25 * ((U_m / J) * (U_m / J))) - 2.0);
	} else {
		tmp = -U_m * ((((J / U_m) * (J / U_m)) * -2.0) - 1.0);
	}
	return tmp;
}

Java

U_m = Math.abs(U);
public static double code(double J, double K, double U_m) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = ((-2.0 * J) * t_0) * Math.sqrt((1.0 + Math.pow((U_m / ((2.0 * J) * t_0)), 2.0)));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = -U_m;
	} else if (t_1 <= -5e-295) {
		tmp = J * ((-0.25 * ((U_m / J) * (U_m / J))) - 2.0);
	} else {
		tmp = -U_m * ((((J / U_m) * (J / U_m)) * -2.0) - 1.0);
	}
	return tmp;
}

Python

U_m = math.fabs(U)
def code(J, K, U_m):
	t_0 = math.cos((K / 2.0))
	t_1 = ((-2.0 * J) * t_0) * math.sqrt((1.0 + math.pow((U_m / ((2.0 * J) * t_0)), 2.0)))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = -U_m
	elif t_1 <= -5e-295:
		tmp = J * ((-0.25 * ((U_m / J) * (U_m / J))) - 2.0)
	else:
		tmp = -U_m * ((((J / U_m) * (J / U_m)) * -2.0) - 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(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 <= -5e-295)
		tmp = Float64(J * Float64(Float64(-0.25 * Float64(Float64(U_m / J) * Float64(U_m / J))) - 2.0));
	else
		tmp = Float64(Float64(-U_m) * Float64(Float64(Float64(Float64(J / U_m) * Float64(J / U_m)) * -2.0) - 1.0));
	end
	return tmp
end

MATLAB

U_m = abs(U);
function tmp_2 = code(J, K, U_m)
	t_0 = cos((K / 2.0));
	t_1 = ((-2.0 * J) * t_0) * sqrt((1.0 + ((U_m / ((2.0 * J) * t_0)) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = -U_m;
	elseif (t_1 <= -5e-295)
		tmp = J * ((-0.25 * ((U_m / J) * (U_m / J))) - 2.0);
	else
		tmp = -U_m * ((((J / U_m) * (J / U_m)) * -2.0) - 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], (-U$95$m), If[LessEqual[t$95$1, -5e-295], N[(J * N[(N[(-0.25 * N[(N[(U$95$m / J), $MachinePrecision] * N[(U$95$m / J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision], N[((-U$95$m) * N[(N[(N[(N[(J / U$95$m), $MachinePrecision] * N[(J / U$95$m), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision] - 1.0), $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 6.2%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 48.4

         \[\leadsto -U \]

   5. Applied rewrites 48.4%

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

   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))))) < -5.00000000000000008e-295

   1. Initial program 99.8%

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

   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

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 51.7

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

   5. Applied rewrites 51.7%

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

   6. Taylor expanded in J around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 52.3

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

   8. Applied rewrites 52.3%

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

   if -5.00000000000000008e-295 < (*.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 76.0%

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

   3. Taylor expanded in 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

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 N/A

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

   5. Applied rewrites 19.9%

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

   6. Taylor expanded in K around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 19.9

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

   8. Applied rewrites 19.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 24.0

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

   10. Applied rewrites 24.0%

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

Alternative 7: 54.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\right) \cdot t\_0}\right)}^{2}}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-295}:\\ \;\;\;\;J \cdot \left(-0.25 \cdot \left(\frac{U\_m}{J} \cdot \frac{U\_m}{J}\right) - 2\right)\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \]

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 -5e-295)
       (* J (- (* -0.25 (* (/ U_m J) (/ U_m 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 <= -5e-295) {
		tmp = J * ((-0.25 * ((U_m / J) * (U_m / J))) - 2.0);
	} else {
		tmp = 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((K / 2.0));
	double t_1 = ((-2.0 * J) * t_0) * Math.sqrt((1.0 + Math.pow((U_m / ((2.0 * J) * t_0)), 2.0)));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = -U_m;
	} else if (t_1 <= -5e-295) {
		tmp = J * ((-0.25 * ((U_m / J) * (U_m / J))) - 2.0);
	} else {
		tmp = U_m;
	}
	return tmp;
}

Python

U_m = math.fabs(U)
def code(J, K, U_m):
	t_0 = math.cos((K / 2.0))
	t_1 = ((-2.0 * J) * t_0) * math.sqrt((1.0 + math.pow((U_m / ((2.0 * J) * t_0)), 2.0)))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = -U_m
	elif t_1 <= -5e-295:
		tmp = J * ((-0.25 * ((U_m / J) * (U_m / J))) - 2.0)
	else:
		tmp = 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 <= -5e-295)
		tmp = Float64(J * Float64(Float64(-0.25 * Float64(Float64(U_m / J) * Float64(U_m / J))) - 2.0));
	else
		tmp = U_m;
	end
	return tmp
end

MATLAB

U_m = abs(U);
function tmp_2 = code(J, K, U_m)
	t_0 = cos((K / 2.0));
	t_1 = ((-2.0 * J) * t_0) * sqrt((1.0 + ((U_m / ((2.0 * J) * t_0)) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = -U_m;
	elseif (t_1 <= -5e-295)
		tmp = J * ((-0.25 * ((U_m / J) * (U_m / J))) - 2.0);
	else
		tmp = 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[(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, -5e-295], N[(J * N[(N[(-0.25 * N[(N[(U$95$m / J), $MachinePrecision] * N[(U$95$m / J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision], U$95$m]]]]

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 6.2%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 48.4

         \[\leadsto -U \]

   5. Applied rewrites 48.4%

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

   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))))) < -5.00000000000000008e-295

   1. Initial program 99.8%

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

   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

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 51.7

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

   5. Applied rewrites 51.7%

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

   6. Taylor expanded in J around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 52.3

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

   8. Applied rewrites 52.3%

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

   if -5.00000000000000008e-295 < (*.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 76.0%

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

   3. Taylor expanded in U around -inf

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

   5. Applied rewrites 23.6%

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

Alternative 8: 54.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\right) \cdot t\_0}\right)}^{2}}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-295}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \]

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 -5e-295) (* -2.0 J) 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 <= -5e-295) {
		tmp = -2.0 * J;
	} else {
		tmp = 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((K / 2.0));
	double t_1 = ((-2.0 * J) * t_0) * Math.sqrt((1.0 + Math.pow((U_m / ((2.0 * J) * t_0)), 2.0)));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = -U_m;
	} else if (t_1 <= -5e-295) {
		tmp = -2.0 * J;
	} else {
		tmp = U_m;
	}
	return tmp;
}

Python

U_m = math.fabs(U)
def code(J, K, U_m):
	t_0 = math.cos((K / 2.0))
	t_1 = ((-2.0 * J) * t_0) * math.sqrt((1.0 + math.pow((U_m / ((2.0 * J) * t_0)), 2.0)))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = -U_m
	elif t_1 <= -5e-295:
		tmp = -2.0 * J
	else:
		tmp = 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 <= -5e-295)
		tmp = Float64(-2.0 * J);
	else
		tmp = U_m;
	end
	return tmp
end

MATLAB

U_m = abs(U);
function tmp_2 = code(J, K, U_m)
	t_0 = cos((K / 2.0));
	t_1 = ((-2.0 * J) * t_0) * sqrt((1.0 + ((U_m / ((2.0 * J) * t_0)) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = -U_m;
	elseif (t_1 <= -5e-295)
		tmp = -2.0 * J;
	else
		tmp = 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[(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, -5e-295], N[(-2.0 * J), $MachinePrecision], U$95$m]]]]

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 6.2%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 48.4

         \[\leadsto -U \]

   5. Applied rewrites 48.4%

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

   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))))) < -5.00000000000000008e-295

   1. Initial program 99.8%

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

   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

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 51.7

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

   5. Applied rewrites 51.7%

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

   6. Taylor expanded in J around inf

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

      1. lift-*.f64 52.1

         \[\leadsto -2 \cdot J \]

   8. Applied rewrites 52.1%

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

   if -5.00000000000000008e-295 < (*.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 76.0%

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

   3. Taylor expanded in U around -inf

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

   5. Applied rewrites 23.6%

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

Alternative 9: 26.3% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Fortran

U_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 73.2%

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

   3. Taylor expanded in U around -inf

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

   5. Applied rewrites 22.7%

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

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

   1. Initial program 77.9%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 26.6

         \[\leadsto -U \]

   5. Applied rewrites 26.6%

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

Alternative 10: 26.7% accurate, 373.0× 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 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 76.7%

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

3. Taylor expanded in U around -inf

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

5. Applied rewrites 22.6%

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

Reproduce

herbie shell --seed 2025057 
(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)))))