Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, A

Percentage Accurate: 68.7% → 89.2%

Time: 3.6s

Alternatives: 15

Speedup: 1.7×

Specification

Math

\[\begin{array}{l} \\ \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0))))

C

double code(double x, double y) {
	return (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
}

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(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0d0))
end function

Java

public static double code(double x, double y) {
	return (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
}

Python

def code(x, y):
	return (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0))

Julia

function code(x, y)
	return Float64(Float64(x * y) / Float64(Float64(Float64(x + y) * Float64(x + y)) * Float64(Float64(x + y) + 1.0)))
end

MATLAB

function tmp = code(x, y)
	tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
end

Wolfram

code[x_, y_] := N[(N[(x * y), $MachinePrecision] / N[(N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(N[(x + y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 68.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0))))

C

double code(double x, double y) {
	return (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
}

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(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0d0))
end function

Java

public static double code(double x, double y) {
	return (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
}

Python

def code(x, y):
	return (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0))

Julia

function code(x, y)
	return Float64(Float64(x * y) / Float64(Float64(Float64(x + y) * Float64(x + y)) * Float64(Float64(x + y) + 1.0)))
end

MATLAB

function tmp = code(x, y)
	tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
end

Wolfram

code[x_, y_] := N[(N[(x * y), $MachinePrecision] / N[(N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(N[(x + y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 89.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.95 \cdot 10^{-136}:\\ \;\;\;\;2 \cdot \frac{y}{\mathsf{fma}\left(2, x, 2 \cdot \left(x \cdot x\right)\right)}\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+97}:\\ \;\;\;\;x \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \mathsf{fma}\left(y, y - -2 \cdot x, x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(x \cdot \frac{x}{y}\right) \cdot -3 + x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 2.95e-136)
   (* 2.0 (/ y (fma 2.0 x (* 2.0 (* x x)))))
   (if (<= y 1.5e+97)
     (* x (/ y (* (+ (+ y x) 1.0) (fma y (- y (* -2.0 x)) (* x x)))))
     (/ (/ (+ (* (* x (/ x y)) -3.0) x) y) y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 2.95e-136) {
		tmp = 2.0 * (y / fma(2.0, x, (2.0 * (x * x))));
	} else if (y <= 1.5e+97) {
		tmp = x * (y / (((y + x) + 1.0) * fma(y, (y - (-2.0 * x)), (x * x))));
	} else {
		tmp = ((((x * (x / y)) * -3.0) + x) / y) / y;
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 2.95e-136)
		tmp = Float64(2.0 * Float64(y / fma(2.0, x, Float64(2.0 * Float64(x * x)))));
	elseif (y <= 1.5e+97)
		tmp = Float64(x * Float64(y / Float64(Float64(Float64(y + x) + 1.0) * fma(y, Float64(y - Float64(-2.0 * x)), Float64(x * x)))));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(x * Float64(x / y)) * -3.0) + x) / y) / y);
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 2.95e-136], N[(2.0 * N[(y / N[(2.0 * x + N[(2.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.5e+97], N[(x * N[(y / N[(N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(y * N[(y - N[(-2.0 * x), $MachinePrecision]), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision] * -3.0), $MachinePrecision] + x), $MachinePrecision] / y), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.95000000000000007e-136

   1. Initial program 64.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\left(x + y\right)} + 1\right)} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(\left(x + y\right) + 1\right)}} \]

      3. flip-+ N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\frac{x \cdot x - y \cdot y}{x - y}} + 1\right)} \]

      4. metadata-eval N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\frac{x \cdot x - y \cdot y}{x - y} + \color{blue}{\frac{2}{2}}\right)} \]

      5. frac-add N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\frac{\left(x \cdot x - y \cdot y\right) \cdot 2 + \left(x - y\right) \cdot 2}{\left(x - y\right) \cdot 2}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\frac{\left(x \cdot x - y \cdot y\right) \cdot 2 + \left(x - y\right) \cdot 2}{\left(x - y\right) \cdot 2}}} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \frac{\color{blue}{\mathsf{fma}\left(x \cdot x - y \cdot y, 2, \left(x - y\right) \cdot 2\right)}}{\left(x - y\right) \cdot 2}} \]

      8. unpow2 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \frac{\mathsf{fma}\left(\color{blue}{{x}^{2}} - y \cdot y, 2, \left(x - y\right) \cdot 2\right)}{\left(x - y\right) \cdot 2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \frac{\mathsf{fma}\left({x}^{2} - \color{blue}{{y}^{2}}, 2, \left(x - y\right) \cdot 2\right)}{\left(x - y\right) \cdot 2}} \]

      10. lower--.f64 N/A

          \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \frac{\mathsf{fma}\left(\color{blue}{{x}^{2} - {y}^{2}}, 2, \left(x - y\right) \cdot 2\right)}{\left(x - y\right) \cdot 2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \frac{\mathsf{fma}\left(\color{blue}{x \cdot x} - {y}^{2}, 2, \left(x - y\right) \cdot 2\right)}{\left(x - y\right) \cdot 2}} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \frac{\mathsf{fma}\left(\color{blue}{x \cdot x} - {y}^{2}, 2, \left(x - y\right) \cdot 2\right)}{\left(x - y\right) \cdot 2}} \]

      13. unpow2 N/A

          \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \frac{\mathsf{fma}\left(x \cdot x - \color{blue}{y \cdot y}, 2, \left(x - y\right) \cdot 2\right)}{\left(x - y\right) \cdot 2}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \frac{\mathsf{fma}\left(x \cdot x - \color{blue}{y \cdot y}, 2, \left(x - y\right) \cdot 2\right)}{\left(x - y\right) \cdot 2}} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \frac{\mathsf{fma}\left(x \cdot x - y \cdot y, 2, \color{blue}{\left(x - y\right) \cdot 2}\right)}{\left(x - y\right) \cdot 2}} \]

      16. lower--.f64 N/A

          \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \frac{\mathsf{fma}\left(x \cdot x - y \cdot y, 2, \color{blue}{\left(x - y\right)} \cdot 2\right)}{\left(x - y\right) \cdot 2}} \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \frac{\mathsf{fma}\left(x \cdot x - y \cdot y, 2, \left(x - y\right) \cdot 2\right)}{\color{blue}{\left(x - y\right) \cdot 2}}} \]

      18. lower--.f64 64.2

          \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \frac{\mathsf{fma}\left(x \cdot x - y \cdot y, 2, \left(x - y\right) \cdot 2\right)}{\color{blue}{\left(x - y\right)} \cdot 2}} \]

   3. Applied rewrites 64.2%

      \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\frac{\mathsf{fma}\left(x \cdot x - y \cdot y, 2, \left(x - y\right) \cdot 2\right)}{\left(x - y\right) \cdot 2}}} \]

   4. Taylor expanded in y around 0

      \[\leadsto \color{blue}{2 \cdot \frac{y}{2 \cdot x + 2 \cdot {x}^{2}}} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\frac{y}{2 \cdot x + 2 \cdot {x}^{2}}} \]

      2. lower-/.f64 N/A

         \[\leadsto 2 \cdot \frac{y}{\color{blue}{2 \cdot x + 2 \cdot {x}^{2}}} \]

      3. lower-fma.f64 N/A

         \[\leadsto 2 \cdot \frac{y}{\mathsf{fma}\left(2, \color{blue}{x}, 2 \cdot {x}^{2}\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{y}{\mathsf{fma}\left(2, x, 2 \cdot {x}^{2}\right)} \]

      5. pow2 N/A

         \[\leadsto 2 \cdot \frac{y}{\mathsf{fma}\left(2, x, 2 \cdot \left(x \cdot x\right)\right)} \]

      6. lift-*.f64 85.8

         \[\leadsto 2 \cdot \frac{y}{\mathsf{fma}\left(2, x, 2 \cdot \left(x \cdot x\right)\right)} \]

   6. Applied rewrites 85.8%

      \[\leadsto \color{blue}{2 \cdot \frac{y}{\mathsf{fma}\left(2, x, 2 \cdot \left(x \cdot x\right)\right)}} \]

   if 2.95000000000000007e-136 < y < 1.4999999999999999e97

   1. Initial program 84.9%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\color{blue}{\left(x + y\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\left(x + y\right)} + 1\right)} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(\left(x + y\right) + 1\right)}} \]

      9. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      11. lower-/.f64 N/A

          \[\leadsto x \cdot \color{blue}{\frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      12. *-commutative N/A

          \[\leadsto x \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      13. lower-*.f64 N/A

          \[\leadsto x \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      14. lift-+.f64 N/A

          \[\leadsto x \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right)} \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)} \]

      15. +-commutative N/A

          \[\leadsto x \cdot \frac{y}{\left(\color{blue}{\left(y + x\right)} + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)} \]

      16. lower-+.f64 N/A

          \[\leadsto x \cdot \frac{y}{\left(\color{blue}{\left(y + x\right)} + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)} \]

      17. lower-*.f64 N/A

          \[\leadsto x \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      18. +-commutative N/A

          \[\leadsto x \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \left(\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)\right)} \]

      19. lower-+.f64 N/A

          \[\leadsto x \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \left(\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)\right)} \]

      20. +-commutative N/A

          \[\leadsto x \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}\right)} \]

      21. lower-+.f64 93.3

          \[\leadsto x \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}\right)} \]

   3. Applied rewrites 93.3%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

   4. Taylor expanded in y around 0

      \[\leadsto x \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \color{blue}{\left(y \cdot \left(y + 2 \cdot x\right) + {x}^{2}\right)}} \]
   5. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto x \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \mathsf{fma}\left(y, \color{blue}{y + 2 \cdot x}, {x}^{2}\right)} \]

      2. fp-cancel-sign-sub-inv N/A

         \[\leadsto x \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \mathsf{fma}\left(y, y - \color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot x}, {x}^{2}\right)} \]

      3. metadata-eval N/A

         \[\leadsto x \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \mathsf{fma}\left(y, y - -2 \cdot x, {x}^{2}\right)} \]

      4. lower--.f64 N/A

         \[\leadsto x \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \mathsf{fma}\left(y, y - \color{blue}{-2 \cdot x}, {x}^{2}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto x \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \mathsf{fma}\left(y, y - -2 \cdot \color{blue}{x}, {x}^{2}\right)} \]

      6. pow2 N/A

         \[\leadsto x \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \mathsf{fma}\left(y, y - -2 \cdot x, x \cdot x\right)} \]

      7. lift-*.f64 93.2

         \[\leadsto x \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \mathsf{fma}\left(y, y - -2 \cdot x, x \cdot x\right)} \]

   6. Applied rewrites 93.2%

      \[\leadsto x \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \color{blue}{\mathsf{fma}\left(y, y - -2 \cdot x, x \cdot x\right)}} \]

   if 1.4999999999999999e97 < y

   1. Initial program 60.4%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{x + -1 \cdot \frac{x \cdot \left(1 + \left(x + 2 \cdot x\right)\right)}{y}}{\color{blue}{{y}^{2}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{-1 \cdot \frac{x \cdot \left(1 + \left(x + 2 \cdot x\right)\right)}{y} + x}{{\color{blue}{y}}^{2}} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{-1 \cdot \frac{x \cdot \left(1 + \left(x + 2 \cdot x\right)\right)}{y} + x}{{\color{blue}{y}}^{2}} \]

      4. mul-1-neg N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\frac{x \cdot \left(1 + \left(x + 2 \cdot x\right)\right)}{y}\right)\right) + x}{{y}^{2}} \]

      5. lower-neg.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. distribute-rgt1-in N/A

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

      11. metadata-eval N/A

          \[\leadsto \frac{\left(-\frac{\left(3 \cdot x + 1\right) \cdot x}{y}\right) + x}{{y}^{2}} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\left(-\frac{\mathsf{fma}\left(3, x, 1\right) \cdot x}{y}\right) + x}{{y}^{2}} \]

      13. unpow2 N/A

          \[\leadsto \frac{\left(-\frac{\mathsf{fma}\left(3, x, 1\right) \cdot x}{y}\right) + x}{y \cdot \color{blue}{y}} \]

      14. lower-*.f64 76.2

          \[\leadsto \frac{\left(-\frac{\mathsf{fma}\left(3, x, 1\right) \cdot x}{y}\right) + x}{y \cdot \color{blue}{y}} \]

   4. Applied rewrites 76.2%

      \[\leadsto \color{blue}{\frac{\left(-\frac{\mathsf{fma}\left(3, x, 1\right) \cdot x}{y}\right) + x}{y \cdot y}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{-3 \cdot \frac{{x}^{2}}{y} + x}{y \cdot y} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-3 \cdot \frac{{x}^{2}}{y} + x}{y \cdot y} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{-3 \cdot \frac{{x}^{2}}{y} + x}{y \cdot y} \]

      3. pow2 N/A

         \[\leadsto \frac{-3 \cdot \frac{x \cdot x}{y} + x}{y \cdot y} \]

      4. lift-*.f64 76.2

         \[\leadsto \frac{-3 \cdot \frac{x \cdot x}{y} + x}{y \cdot y} \]

   7. Applied rewrites 76.2%

      \[\leadsto \frac{-3 \cdot \frac{x \cdot x}{y} + x}{y \cdot y} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{-3 \cdot \frac{x \cdot x}{y} + x}{\color{blue}{y \cdot y}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{-3 \cdot \frac{x \cdot x}{y} + x}{y \cdot \color{blue}{y}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{-3 \cdot \frac{x \cdot x}{y} + x}{y}}{\color{blue}{y}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-3 \cdot \frac{x \cdot x}{y} + x}{y}}{\color{blue}{y}} \]

   9. Applied rewrites 90.2%

      \[\leadsto \frac{\frac{\left(x \cdot \frac{x}{y}\right) \cdot -3 + x}{y}}{\color{blue}{y}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 89.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.95 \cdot 10^{-136}:\\ \;\;\;\;2 \cdot \frac{y}{\mathsf{fma}\left(2, x, 2 \cdot \left(x \cdot x\right)\right)}\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+97}:\\ \;\;\;\;x \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(x \cdot \frac{x}{y}\right) \cdot -3 + x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 2.95e-136)
   (* 2.0 (/ y (fma 2.0 x (* 2.0 (* x x)))))
   (if (<= y 1.5e+97)
     (* x (/ y (* (+ (+ y x) 1.0) (* (+ y x) (+ y x)))))
     (/ (/ (+ (* (* x (/ x y)) -3.0) x) y) y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 2.95e-136) {
		tmp = 2.0 * (y / fma(2.0, x, (2.0 * (x * x))));
	} else if (y <= 1.5e+97) {
		tmp = x * (y / (((y + x) + 1.0) * ((y + x) * (y + x))));
	} else {
		tmp = ((((x * (x / y)) * -3.0) + x) / y) / y;
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 2.95e-136)
		tmp = Float64(2.0 * Float64(y / fma(2.0, x, Float64(2.0 * Float64(x * x)))));
	elseif (y <= 1.5e+97)
		tmp = Float64(x * Float64(y / Float64(Float64(Float64(y + x) + 1.0) * Float64(Float64(y + x) * Float64(y + x)))));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(x * Float64(x / y)) * -3.0) + x) / y) / y);
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 2.95e-136], N[(2.0 * N[(y / N[(2.0 * x + N[(2.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.5e+97], N[(x * N[(y / N[(N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision] * -3.0), $MachinePrecision] + x), $MachinePrecision] / y), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.95000000000000007e-136

   1. Initial program 64.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\left(x + y\right)} + 1\right)} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(\left(x + y\right) + 1\right)}} \]

      3. flip-+ N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\frac{x \cdot x - y \cdot y}{x - y}} + 1\right)} \]

      4. metadata-eval N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\frac{x \cdot x - y \cdot y}{x - y} + \color{blue}{\frac{2}{2}}\right)} \]

      5. frac-add N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\frac{\left(x \cdot x - y \cdot y\right) \cdot 2 + \left(x - y\right) \cdot 2}{\left(x - y\right) \cdot 2}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\frac{\left(x \cdot x - y \cdot y\right) \cdot 2 + \left(x - y\right) \cdot 2}{\left(x - y\right) \cdot 2}}} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \frac{\color{blue}{\mathsf{fma}\left(x \cdot x - y \cdot y, 2, \left(x - y\right) \cdot 2\right)}}{\left(x - y\right) \cdot 2}} \]

      8. unpow2 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \frac{\mathsf{fma}\left(\color{blue}{{x}^{2}} - y \cdot y, 2, \left(x - y\right) \cdot 2\right)}{\left(x - y\right) \cdot 2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \frac{\mathsf{fma}\left({x}^{2} - \color{blue}{{y}^{2}}, 2, \left(x - y\right) \cdot 2\right)}{\left(x - y\right) \cdot 2}} \]

      10. lower--.f64 N/A

          \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \frac{\mathsf{fma}\left(\color{blue}{{x}^{2} - {y}^{2}}, 2, \left(x - y\right) \cdot 2\right)}{\left(x - y\right) \cdot 2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \frac{\mathsf{fma}\left(\color{blue}{x \cdot x} - {y}^{2}, 2, \left(x - y\right) \cdot 2\right)}{\left(x - y\right) \cdot 2}} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \frac{\mathsf{fma}\left(\color{blue}{x \cdot x} - {y}^{2}, 2, \left(x - y\right) \cdot 2\right)}{\left(x - y\right) \cdot 2}} \]

      13. unpow2 N/A

          \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \frac{\mathsf{fma}\left(x \cdot x - \color{blue}{y \cdot y}, 2, \left(x - y\right) \cdot 2\right)}{\left(x - y\right) \cdot 2}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \frac{\mathsf{fma}\left(x \cdot x - \color{blue}{y \cdot y}, 2, \left(x - y\right) \cdot 2\right)}{\left(x - y\right) \cdot 2}} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \frac{\mathsf{fma}\left(x \cdot x - y \cdot y, 2, \color{blue}{\left(x - y\right) \cdot 2}\right)}{\left(x - y\right) \cdot 2}} \]

      16. lower--.f64 N/A

          \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \frac{\mathsf{fma}\left(x \cdot x - y \cdot y, 2, \color{blue}{\left(x - y\right)} \cdot 2\right)}{\left(x - y\right) \cdot 2}} \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \frac{\mathsf{fma}\left(x \cdot x - y \cdot y, 2, \left(x - y\right) \cdot 2\right)}{\color{blue}{\left(x - y\right) \cdot 2}}} \]

      18. lower--.f64 64.2

          \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \frac{\mathsf{fma}\left(x \cdot x - y \cdot y, 2, \left(x - y\right) \cdot 2\right)}{\color{blue}{\left(x - y\right)} \cdot 2}} \]

   3. Applied rewrites 64.2%

      \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\frac{\mathsf{fma}\left(x \cdot x - y \cdot y, 2, \left(x - y\right) \cdot 2\right)}{\left(x - y\right) \cdot 2}}} \]

   4. Taylor expanded in y around 0

      \[\leadsto \color{blue}{2 \cdot \frac{y}{2 \cdot x + 2 \cdot {x}^{2}}} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\frac{y}{2 \cdot x + 2 \cdot {x}^{2}}} \]

      2. lower-/.f64 N/A

         \[\leadsto 2 \cdot \frac{y}{\color{blue}{2 \cdot x + 2 \cdot {x}^{2}}} \]

      3. lower-fma.f64 N/A

         \[\leadsto 2 \cdot \frac{y}{\mathsf{fma}\left(2, \color{blue}{x}, 2 \cdot {x}^{2}\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{y}{\mathsf{fma}\left(2, x, 2 \cdot {x}^{2}\right)} \]

      5. pow2 N/A

         \[\leadsto 2 \cdot \frac{y}{\mathsf{fma}\left(2, x, 2 \cdot \left(x \cdot x\right)\right)} \]

      6. lift-*.f64 85.8

         \[\leadsto 2 \cdot \frac{y}{\mathsf{fma}\left(2, x, 2 \cdot \left(x \cdot x\right)\right)} \]

   6. Applied rewrites 85.8%

      \[\leadsto \color{blue}{2 \cdot \frac{y}{\mathsf{fma}\left(2, x, 2 \cdot \left(x \cdot x\right)\right)}} \]

   if 2.95000000000000007e-136 < y < 1.4999999999999999e97

   1. Initial program 84.9%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\color{blue}{\left(x + y\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\left(x + y\right)} + 1\right)} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(\left(x + y\right) + 1\right)}} \]

      9. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      11. lower-/.f64 N/A

          \[\leadsto x \cdot \color{blue}{\frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      12. *-commutative N/A

          \[\leadsto x \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      13. lower-*.f64 N/A

          \[\leadsto x \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      14. lift-+.f64 N/A

          \[\leadsto x \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right)} \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)} \]

      15. +-commutative N/A

          \[\leadsto x \cdot \frac{y}{\left(\color{blue}{\left(y + x\right)} + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)} \]

      16. lower-+.f64 N/A

          \[\leadsto x \cdot \frac{y}{\left(\color{blue}{\left(y + x\right)} + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)} \]

      17. lower-*.f64 N/A

          \[\leadsto x \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      18. +-commutative N/A

          \[\leadsto x \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \left(\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)\right)} \]

      19. lower-+.f64 N/A

          \[\leadsto x \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \left(\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)\right)} \]

      20. +-commutative N/A

          \[\leadsto x \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}\right)} \]

      21. lower-+.f64 93.3

          \[\leadsto x \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}\right)} \]

   3. Applied rewrites 93.3%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

   if 1.4999999999999999e97 < y

   1. Initial program 60.4%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{x + -1 \cdot \frac{x \cdot \left(1 + \left(x + 2 \cdot x\right)\right)}{y}}{\color{blue}{{y}^{2}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{-1 \cdot \frac{x \cdot \left(1 + \left(x + 2 \cdot x\right)\right)}{y} + x}{{\color{blue}{y}}^{2}} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{-1 \cdot \frac{x \cdot \left(1 + \left(x + 2 \cdot x\right)\right)}{y} + x}{{\color{blue}{y}}^{2}} \]

      4. mul-1-neg N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\frac{x \cdot \left(1 + \left(x + 2 \cdot x\right)\right)}{y}\right)\right) + x}{{y}^{2}} \]

      5. lower-neg.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. distribute-rgt1-in N/A

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

      11. metadata-eval N/A

          \[\leadsto \frac{\left(-\frac{\left(3 \cdot x + 1\right) \cdot x}{y}\right) + x}{{y}^{2}} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\left(-\frac{\mathsf{fma}\left(3, x, 1\right) \cdot x}{y}\right) + x}{{y}^{2}} \]

      13. unpow2 N/A

          \[\leadsto \frac{\left(-\frac{\mathsf{fma}\left(3, x, 1\right) \cdot x}{y}\right) + x}{y \cdot \color{blue}{y}} \]

      14. lower-*.f64 76.2

          \[\leadsto \frac{\left(-\frac{\mathsf{fma}\left(3, x, 1\right) \cdot x}{y}\right) + x}{y \cdot \color{blue}{y}} \]

   4. Applied rewrites 76.2%

      \[\leadsto \color{blue}{\frac{\left(-\frac{\mathsf{fma}\left(3, x, 1\right) \cdot x}{y}\right) + x}{y \cdot y}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{-3 \cdot \frac{{x}^{2}}{y} + x}{y \cdot y} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-3 \cdot \frac{{x}^{2}}{y} + x}{y \cdot y} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{-3 \cdot \frac{{x}^{2}}{y} + x}{y \cdot y} \]

      3. pow2 N/A

         \[\leadsto \frac{-3 \cdot \frac{x \cdot x}{y} + x}{y \cdot y} \]

      4. lift-*.f64 76.2

         \[\leadsto \frac{-3 \cdot \frac{x \cdot x}{y} + x}{y \cdot y} \]

   7. Applied rewrites 76.2%

      \[\leadsto \frac{-3 \cdot \frac{x \cdot x}{y} + x}{y \cdot y} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{-3 \cdot \frac{x \cdot x}{y} + x}{\color{blue}{y \cdot y}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{-3 \cdot \frac{x \cdot x}{y} + x}{y \cdot \color{blue}{y}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{-3 \cdot \frac{x \cdot x}{y} + x}{y}}{\color{blue}{y}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-3 \cdot \frac{x \cdot x}{y} + x}{y}}{\color{blue}{y}} \]

   9. Applied rewrites 90.2%

      \[\leadsto \frac{\frac{\left(x \cdot \frac{x}{y}\right) \cdot -3 + x}{y}}{\color{blue}{y}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 87.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3 \cdot 10^{-136}:\\ \;\;\;\;2 \cdot \frac{y}{\mathsf{fma}\left(2, x, 2 \cdot \left(x \cdot x\right)\right)}\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+139}:\\ \;\;\;\;x \cdot \frac{\frac{y}{1 + y}}{\left(y + x\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(x \cdot \frac{x}{y}\right) \cdot -3 + x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 3e-136)
   (* 2.0 (/ y (fma 2.0 x (* 2.0 (* x x)))))
   (if (<= y 1.15e+139)
     (* x (/ (/ y (+ 1.0 y)) (* (+ y x) (+ y x))))
     (/ (/ (+ (* (* x (/ x y)) -3.0) x) y) y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 3e-136) {
		tmp = 2.0 * (y / fma(2.0, x, (2.0 * (x * x))));
	} else if (y <= 1.15e+139) {
		tmp = x * ((y / (1.0 + y)) / ((y + x) * (y + x)));
	} else {
		tmp = ((((x * (x / y)) * -3.0) + x) / y) / y;
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 3e-136)
		tmp = Float64(2.0 * Float64(y / fma(2.0, x, Float64(2.0 * Float64(x * x)))));
	elseif (y <= 1.15e+139)
		tmp = Float64(x * Float64(Float64(y / Float64(1.0 + y)) / Float64(Float64(y + x) * Float64(y + x))));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(x * Float64(x / y)) * -3.0) + x) / y) / y);
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 3e-136], N[(2.0 * N[(y / N[(2.0 * x + N[(2.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.15e+139], N[(x * N[(N[(y / N[(1.0 + y), $MachinePrecision]), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision] * -3.0), $MachinePrecision] + x), $MachinePrecision] / y), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.9999999999999998e-136

   1. Initial program 64.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\left(x + y\right)} + 1\right)} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(\left(x + y\right) + 1\right)}} \]

      3. flip-+ N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\frac{x \cdot x - y \cdot y}{x - y}} + 1\right)} \]

      4. metadata-eval N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\frac{x \cdot x - y \cdot y}{x - y} + \color{blue}{\frac{2}{2}}\right)} \]

      5. frac-add N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\frac{\left(x \cdot x - y \cdot y\right) \cdot 2 + \left(x - y\right) \cdot 2}{\left(x - y\right) \cdot 2}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\frac{\left(x \cdot x - y \cdot y\right) \cdot 2 + \left(x - y\right) \cdot 2}{\left(x - y\right) \cdot 2}}} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \frac{\color{blue}{\mathsf{fma}\left(x \cdot x - y \cdot y, 2, \left(x - y\right) \cdot 2\right)}}{\left(x - y\right) \cdot 2}} \]

      8. unpow2 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \frac{\mathsf{fma}\left(\color{blue}{{x}^{2}} - y \cdot y, 2, \left(x - y\right) \cdot 2\right)}{\left(x - y\right) \cdot 2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \frac{\mathsf{fma}\left({x}^{2} - \color{blue}{{y}^{2}}, 2, \left(x - y\right) \cdot 2\right)}{\left(x - y\right) \cdot 2}} \]

      10. lower--.f64 N/A

          \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \frac{\mathsf{fma}\left(\color{blue}{{x}^{2} - {y}^{2}}, 2, \left(x - y\right) \cdot 2\right)}{\left(x - y\right) \cdot 2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \frac{\mathsf{fma}\left(\color{blue}{x \cdot x} - {y}^{2}, 2, \left(x - y\right) \cdot 2\right)}{\left(x - y\right) \cdot 2}} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \frac{\mathsf{fma}\left(\color{blue}{x \cdot x} - {y}^{2}, 2, \left(x - y\right) \cdot 2\right)}{\left(x - y\right) \cdot 2}} \]

      13. unpow2 N/A

          \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \frac{\mathsf{fma}\left(x \cdot x - \color{blue}{y \cdot y}, 2, \left(x - y\right) \cdot 2\right)}{\left(x - y\right) \cdot 2}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \frac{\mathsf{fma}\left(x \cdot x - \color{blue}{y \cdot y}, 2, \left(x - y\right) \cdot 2\right)}{\left(x - y\right) \cdot 2}} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \frac{\mathsf{fma}\left(x \cdot x - y \cdot y, 2, \color{blue}{\left(x - y\right) \cdot 2}\right)}{\left(x - y\right) \cdot 2}} \]

      16. lower--.f64 N/A

          \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \frac{\mathsf{fma}\left(x \cdot x - y \cdot y, 2, \color{blue}{\left(x - y\right)} \cdot 2\right)}{\left(x - y\right) \cdot 2}} \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \frac{\mathsf{fma}\left(x \cdot x - y \cdot y, 2, \left(x - y\right) \cdot 2\right)}{\color{blue}{\left(x - y\right) \cdot 2}}} \]

      18. lower--.f64 64.2

          \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \frac{\mathsf{fma}\left(x \cdot x - y \cdot y, 2, \left(x - y\right) \cdot 2\right)}{\color{blue}{\left(x - y\right)} \cdot 2}} \]

   3. Applied rewrites 64.2%

      \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\frac{\mathsf{fma}\left(x \cdot x - y \cdot y, 2, \left(x - y\right) \cdot 2\right)}{\left(x - y\right) \cdot 2}}} \]

   4. Taylor expanded in y around 0

      \[\leadsto \color{blue}{2 \cdot \frac{y}{2 \cdot x + 2 \cdot {x}^{2}}} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\frac{y}{2 \cdot x + 2 \cdot {x}^{2}}} \]

      2. lower-/.f64 N/A

         \[\leadsto 2 \cdot \frac{y}{\color{blue}{2 \cdot x + 2 \cdot {x}^{2}}} \]

      3. lower-fma.f64 N/A

         \[\leadsto 2 \cdot \frac{y}{\mathsf{fma}\left(2, \color{blue}{x}, 2 \cdot {x}^{2}\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{y}{\mathsf{fma}\left(2, x, 2 \cdot {x}^{2}\right)} \]

      5. pow2 N/A

         \[\leadsto 2 \cdot \frac{y}{\mathsf{fma}\left(2, x, 2 \cdot \left(x \cdot x\right)\right)} \]

      6. lift-*.f64 85.8

         \[\leadsto 2 \cdot \frac{y}{\mathsf{fma}\left(2, x, 2 \cdot \left(x \cdot x\right)\right)} \]

   6. Applied rewrites 85.8%

      \[\leadsto \color{blue}{2 \cdot \frac{y}{\mathsf{fma}\left(2, x, 2 \cdot \left(x \cdot x\right)\right)}} \]

   if 2.9999999999999998e-136 < y < 1.15e139

   1. Initial program 79.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\color{blue}{\left(x + y\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\left(x + y\right)} + 1\right)} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(\left(x + y\right) + 1\right)}} \]

      9. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      11. lower-/.f64 N/A

          \[\leadsto x \cdot \color{blue}{\frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      12. *-commutative N/A

          \[\leadsto x \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      13. lower-*.f64 N/A

          \[\leadsto x \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      14. lift-+.f64 N/A

          \[\leadsto x \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right)} \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)} \]

      15. +-commutative N/A

          \[\leadsto x \cdot \frac{y}{\left(\color{blue}{\left(y + x\right)} + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)} \]

      16. lower-+.f64 N/A

          \[\leadsto x \cdot \frac{y}{\left(\color{blue}{\left(y + x\right)} + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)} \]

      17. lower-*.f64 N/A

          \[\leadsto x \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      18. +-commutative N/A

          \[\leadsto x \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \left(\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)\right)} \]

      19. lower-+.f64 N/A

          \[\leadsto x \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \left(\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)\right)} \]

      20. +-commutative N/A

          \[\leadsto x \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}\right)} \]

      21. lower-+.f64 88.0

          \[\leadsto x \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}\right)} \]

   3. Applied rewrites 88.0%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

   4. Taylor expanded in x around 0

      \[\leadsto x \cdot \frac{y}{\color{blue}{\left(1 + y\right)} \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)} \]
   5. Step-by-step derivation

      1. lower-+.f64 80.6

         \[\leadsto x \cdot \frac{y}{\left(1 + \color{blue}{y}\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)} \]

   6. Applied rewrites 80.6%

      \[\leadsto x \cdot \frac{y}{\color{blue}{\left(1 + y\right)} \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto x \cdot \color{blue}{\frac{y}{\left(1 + y\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto x \cdot \frac{y}{\color{blue}{\left(1 + y\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto x \cdot \frac{y}{\left(1 + y\right) \cdot \color{blue}{\left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      4. lift-+.f64 N/A

         \[\leadsto x \cdot \frac{y}{\left(1 + y\right) \cdot \left(\color{blue}{\left(y + x\right)} \cdot \left(y + x\right)\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto x \cdot \frac{y}{\left(1 + y\right) \cdot \left(\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}\right)} \]

      6. associate-/r* N/A

         \[\leadsto x \cdot \color{blue}{\frac{\frac{y}{1 + y}}{\left(y + x\right) \cdot \left(y + x\right)}} \]

      7. lower-/.f64 N/A

         \[\leadsto x \cdot \color{blue}{\frac{\frac{y}{1 + y}}{\left(y + x\right) \cdot \left(y + x\right)}} \]

      8. lower-/.f64 N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{y}{1 + y}}}{\left(y + x\right) \cdot \left(y + x\right)} \]

      9. lift-+.f64 N/A

         \[\leadsto x \cdot \frac{\frac{y}{1 + y}}{\color{blue}{\left(y + x\right)} \cdot \left(y + x\right)} \]

      10. lift-+.f64 N/A

          \[\leadsto x \cdot \frac{\frac{y}{1 + y}}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}} \]

      11. lift-*.f64 85.6

          \[\leadsto x \cdot \frac{\frac{y}{1 + y}}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right)}} \]

   8. Applied rewrites 85.6%

      \[\leadsto x \cdot \color{blue}{\frac{\frac{y}{1 + y}}{\left(y + x\right) \cdot \left(y + x\right)}} \]

   if 1.15e139 < y

   1. Initial program 61.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{x + -1 \cdot \frac{x \cdot \left(1 + \left(x + 2 \cdot x\right)\right)}{y}}{\color{blue}{{y}^{2}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{-1 \cdot \frac{x \cdot \left(1 + \left(x + 2 \cdot x\right)\right)}{y} + x}{{\color{blue}{y}}^{2}} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{-1 \cdot \frac{x \cdot \left(1 + \left(x + 2 \cdot x\right)\right)}{y} + x}{{\color{blue}{y}}^{2}} \]

      4. mul-1-neg N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\frac{x \cdot \left(1 + \left(x + 2 \cdot x\right)\right)}{y}\right)\right) + x}{{y}^{2}} \]

      5. lower-neg.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. distribute-rgt1-in N/A

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

      11. metadata-eval N/A

          \[\leadsto \frac{\left(-\frac{\left(3 \cdot x + 1\right) \cdot x}{y}\right) + x}{{y}^{2}} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\left(-\frac{\mathsf{fma}\left(3, x, 1\right) \cdot x}{y}\right) + x}{{y}^{2}} \]

      13. unpow2 N/A

          \[\leadsto \frac{\left(-\frac{\mathsf{fma}\left(3, x, 1\right) \cdot x}{y}\right) + x}{y \cdot \color{blue}{y}} \]

      14. lower-*.f64 74.5

          \[\leadsto \frac{\left(-\frac{\mathsf{fma}\left(3, x, 1\right) \cdot x}{y}\right) + x}{y \cdot \color{blue}{y}} \]

   4. Applied rewrites 74.5%

      \[\leadsto \color{blue}{\frac{\left(-\frac{\mathsf{fma}\left(3, x, 1\right) \cdot x}{y}\right) + x}{y \cdot y}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{-3 \cdot \frac{{x}^{2}}{y} + x}{y \cdot y} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-3 \cdot \frac{{x}^{2}}{y} + x}{y \cdot y} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{-3 \cdot \frac{{x}^{2}}{y} + x}{y \cdot y} \]

      3. pow2 N/A

         \[\leadsto \frac{-3 \cdot \frac{x \cdot x}{y} + x}{y \cdot y} \]

      4. lift-*.f64 74.5

         \[\leadsto \frac{-3 \cdot \frac{x \cdot x}{y} + x}{y \cdot y} \]

   7. Applied rewrites 74.5%

      \[\leadsto \frac{-3 \cdot \frac{x \cdot x}{y} + x}{y \cdot y} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{-3 \cdot \frac{x \cdot x}{y} + x}{\color{blue}{y \cdot y}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{-3 \cdot \frac{x \cdot x}{y} + x}{y \cdot \color{blue}{y}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{-3 \cdot \frac{x \cdot x}{y} + x}{y}}{\color{blue}{y}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-3 \cdot \frac{x \cdot x}{y} + x}{y}}{\color{blue}{y}} \]

   9. Applied rewrites 91.7%

      \[\leadsto \frac{\frac{\left(x \cdot \frac{x}{y}\right) \cdot -3 + x}{y}}{\color{blue}{y}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 86.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3 \cdot 10^{-136}:\\ \;\;\;\;2 \cdot \frac{y}{\mathsf{fma}\left(2, x, 2 \cdot \left(x \cdot x\right)\right)}\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+97}:\\ \;\;\;\;x \cdot \frac{y}{\left(1 + y\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(x \cdot \frac{x}{y}\right) \cdot -3 + x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 3e-136)
   (* 2.0 (/ y (fma 2.0 x (* 2.0 (* x x)))))
   (if (<= y 1.5e+97)
     (* x (/ y (* (+ 1.0 y) (* (+ y x) (+ y x)))))
     (/ (/ (+ (* (* x (/ x y)) -3.0) x) y) y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 3e-136) {
		tmp = 2.0 * (y / fma(2.0, x, (2.0 * (x * x))));
	} else if (y <= 1.5e+97) {
		tmp = x * (y / ((1.0 + y) * ((y + x) * (y + x))));
	} else {
		tmp = ((((x * (x / y)) * -3.0) + x) / y) / y;
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 3e-136)
		tmp = Float64(2.0 * Float64(y / fma(2.0, x, Float64(2.0 * Float64(x * x)))));
	elseif (y <= 1.5e+97)
		tmp = Float64(x * Float64(y / Float64(Float64(1.0 + y) * Float64(Float64(y + x) * Float64(y + x)))));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(x * Float64(x / y)) * -3.0) + x) / y) / y);
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 3e-136], N[(2.0 * N[(y / N[(2.0 * x + N[(2.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.5e+97], N[(x * N[(y / N[(N[(1.0 + y), $MachinePrecision] * N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision] * -3.0), $MachinePrecision] + x), $MachinePrecision] / y), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.9999999999999998e-136

   1. Initial program 64.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\left(x + y\right)} + 1\right)} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(\left(x + y\right) + 1\right)}} \]

      3. flip-+ N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\frac{x \cdot x - y \cdot y}{x - y}} + 1\right)} \]

      4. metadata-eval N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\frac{x \cdot x - y \cdot y}{x - y} + \color{blue}{\frac{2}{2}}\right)} \]

      5. frac-add N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\frac{\left(x \cdot x - y \cdot y\right) \cdot 2 + \left(x - y\right) \cdot 2}{\left(x - y\right) \cdot 2}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\frac{\left(x \cdot x - y \cdot y\right) \cdot 2 + \left(x - y\right) \cdot 2}{\left(x - y\right) \cdot 2}}} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \frac{\color{blue}{\mathsf{fma}\left(x \cdot x - y \cdot y, 2, \left(x - y\right) \cdot 2\right)}}{\left(x - y\right) \cdot 2}} \]

      8. unpow2 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \frac{\mathsf{fma}\left(\color{blue}{{x}^{2}} - y \cdot y, 2, \left(x - y\right) \cdot 2\right)}{\left(x - y\right) \cdot 2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \frac{\mathsf{fma}\left({x}^{2} - \color{blue}{{y}^{2}}, 2, \left(x - y\right) \cdot 2\right)}{\left(x - y\right) \cdot 2}} \]

      10. lower--.f64 N/A

          \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \frac{\mathsf{fma}\left(\color{blue}{{x}^{2} - {y}^{2}}, 2, \left(x - y\right) \cdot 2\right)}{\left(x - y\right) \cdot 2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \frac{\mathsf{fma}\left(\color{blue}{x \cdot x} - {y}^{2}, 2, \left(x - y\right) \cdot 2\right)}{\left(x - y\right) \cdot 2}} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \frac{\mathsf{fma}\left(\color{blue}{x \cdot x} - {y}^{2}, 2, \left(x - y\right) \cdot 2\right)}{\left(x - y\right) \cdot 2}} \]

      13. unpow2 N/A

          \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \frac{\mathsf{fma}\left(x \cdot x - \color{blue}{y \cdot y}, 2, \left(x - y\right) \cdot 2\right)}{\left(x - y\right) \cdot 2}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \frac{\mathsf{fma}\left(x \cdot x - \color{blue}{y \cdot y}, 2, \left(x - y\right) \cdot 2\right)}{\left(x - y\right) \cdot 2}} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \frac{\mathsf{fma}\left(x \cdot x - y \cdot y, 2, \color{blue}{\left(x - y\right) \cdot 2}\right)}{\left(x - y\right) \cdot 2}} \]

      16. lower--.f64 N/A

          \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \frac{\mathsf{fma}\left(x \cdot x - y \cdot y, 2, \color{blue}{\left(x - y\right)} \cdot 2\right)}{\left(x - y\right) \cdot 2}} \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \frac{\mathsf{fma}\left(x \cdot x - y \cdot y, 2, \left(x - y\right) \cdot 2\right)}{\color{blue}{\left(x - y\right) \cdot 2}}} \]

      18. lower--.f64 64.2

          \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \frac{\mathsf{fma}\left(x \cdot x - y \cdot y, 2, \left(x - y\right) \cdot 2\right)}{\color{blue}{\left(x - y\right)} \cdot 2}} \]

   3. Applied rewrites 64.2%

      \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\frac{\mathsf{fma}\left(x \cdot x - y \cdot y, 2, \left(x - y\right) \cdot 2\right)}{\left(x - y\right) \cdot 2}}} \]

   4. Taylor expanded in y around 0

      \[\leadsto \color{blue}{2 \cdot \frac{y}{2 \cdot x + 2 \cdot {x}^{2}}} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\frac{y}{2 \cdot x + 2 \cdot {x}^{2}}} \]

      2. lower-/.f64 N/A

         \[\leadsto 2 \cdot \frac{y}{\color{blue}{2 \cdot x + 2 \cdot {x}^{2}}} \]

      3. lower-fma.f64 N/A

         \[\leadsto 2 \cdot \frac{y}{\mathsf{fma}\left(2, \color{blue}{x}, 2 \cdot {x}^{2}\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{y}{\mathsf{fma}\left(2, x, 2 \cdot {x}^{2}\right)} \]

      5. pow2 N/A

         \[\leadsto 2 \cdot \frac{y}{\mathsf{fma}\left(2, x, 2 \cdot \left(x \cdot x\right)\right)} \]

      6. lift-*.f64 85.8

         \[\leadsto 2 \cdot \frac{y}{\mathsf{fma}\left(2, x, 2 \cdot \left(x \cdot x\right)\right)} \]

   6. Applied rewrites 85.8%

      \[\leadsto \color{blue}{2 \cdot \frac{y}{\mathsf{fma}\left(2, x, 2 \cdot \left(x \cdot x\right)\right)}} \]

   if 2.9999999999999998e-136 < y < 1.4999999999999999e97

   1. Initial program 84.9%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\color{blue}{\left(x + y\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\left(x + y\right)} + 1\right)} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(\left(x + y\right) + 1\right)}} \]

      9. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      11. lower-/.f64 N/A

          \[\leadsto x \cdot \color{blue}{\frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      12. *-commutative N/A

          \[\leadsto x \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      13. lower-*.f64 N/A

          \[\leadsto x \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      14. lift-+.f64 N/A

          \[\leadsto x \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right)} \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)} \]

      15. +-commutative N/A

          \[\leadsto x \cdot \frac{y}{\left(\color{blue}{\left(y + x\right)} + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)} \]

      16. lower-+.f64 N/A

          \[\leadsto x \cdot \frac{y}{\left(\color{blue}{\left(y + x\right)} + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)} \]

      17. lower-*.f64 N/A

          \[\leadsto x \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      18. +-commutative N/A

          \[\leadsto x \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \left(\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)\right)} \]

      19. lower-+.f64 N/A

          \[\leadsto x \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \left(\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)\right)} \]

      20. +-commutative N/A

          \[\leadsto x \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}\right)} \]

      21. lower-+.f64 93.3

          \[\leadsto x \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}\right)} \]

   3. Applied rewrites 93.3%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

   4. Taylor expanded in x around 0

      \[\leadsto x \cdot \frac{y}{\color{blue}{\left(1 + y\right)} \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)} \]
   5. Step-by-step derivation

      1. lower-+.f64 84.4

         \[\leadsto x \cdot \frac{y}{\left(1 + \color{blue}{y}\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)} \]

   6. Applied rewrites 84.4%

      \[\leadsto x \cdot \frac{y}{\color{blue}{\left(1 + y\right)} \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)} \]

   if 1.4999999999999999e97 < y

   1. Initial program 60.4%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{x + -1 \cdot \frac{x \cdot \left(1 + \left(x + 2 \cdot x\right)\right)}{y}}{\color{blue}{{y}^{2}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{-1 \cdot \frac{x \cdot \left(1 + \left(x + 2 \cdot x\right)\right)}{y} + x}{{\color{blue}{y}}^{2}} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{-1 \cdot \frac{x \cdot \left(1 + \left(x + 2 \cdot x\right)\right)}{y} + x}{{\color{blue}{y}}^{2}} \]

      4. mul-1-neg N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\frac{x \cdot \left(1 + \left(x + 2 \cdot x\right)\right)}{y}\right)\right) + x}{{y}^{2}} \]

      5. lower-neg.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. distribute-rgt1-in N/A

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

      11. metadata-eval N/A

          \[\leadsto \frac{\left(-\frac{\left(3 \cdot x + 1\right) \cdot x}{y}\right) + x}{{y}^{2}} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\left(-\frac{\mathsf{fma}\left(3, x, 1\right) \cdot x}{y}\right) + x}{{y}^{2}} \]

      13. unpow2 N/A

          \[\leadsto \frac{\left(-\frac{\mathsf{fma}\left(3, x, 1\right) \cdot x}{y}\right) + x}{y \cdot \color{blue}{y}} \]

      14. lower-*.f64 76.2

          \[\leadsto \frac{\left(-\frac{\mathsf{fma}\left(3, x, 1\right) \cdot x}{y}\right) + x}{y \cdot \color{blue}{y}} \]

   4. Applied rewrites 76.2%

      \[\leadsto \color{blue}{\frac{\left(-\frac{\mathsf{fma}\left(3, x, 1\right) \cdot x}{y}\right) + x}{y \cdot y}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{-3 \cdot \frac{{x}^{2}}{y} + x}{y \cdot y} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-3 \cdot \frac{{x}^{2}}{y} + x}{y \cdot y} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{-3 \cdot \frac{{x}^{2}}{y} + x}{y \cdot y} \]

      3. pow2 N/A

         \[\leadsto \frac{-3 \cdot \frac{x \cdot x}{y} + x}{y \cdot y} \]

      4. lift-*.f64 76.2

         \[\leadsto \frac{-3 \cdot \frac{x \cdot x}{y} + x}{y \cdot y} \]

   7. Applied rewrites 76.2%

      \[\leadsto \frac{-3 \cdot \frac{x \cdot x}{y} + x}{y \cdot y} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{-3 \cdot \frac{x \cdot x}{y} + x}{\color{blue}{y \cdot y}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{-3 \cdot \frac{x \cdot x}{y} + x}{y \cdot \color{blue}{y}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{-3 \cdot \frac{x \cdot x}{y} + x}{y}}{\color{blue}{y}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-3 \cdot \frac{x \cdot x}{y} + x}{y}}{\color{blue}{y}} \]

   9. Applied rewrites 90.2%

      \[\leadsto \frac{\frac{\left(x \cdot \frac{x}{y}\right) \cdot -3 + x}{y}}{\color{blue}{y}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 85.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3 \cdot 10^{-136}:\\ \;\;\;\;2 \cdot \frac{y}{\mathsf{fma}\left(2, x, 2 \cdot \left(x \cdot x\right)\right)}\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{-112}:\\ \;\;\;\;x \cdot \frac{y}{1 \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}\\ \mathbf{elif}\;y \leq 21000000:\\ \;\;\;\;\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(x \cdot \frac{x}{y}\right) \cdot -3 + x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 3e-136)
   (* 2.0 (/ y (fma 2.0 x (* 2.0 (* x x)))))
   (if (<= y 2.9e-112)
     (* x (/ y (* 1.0 (* (+ y x) (+ y x)))))
     (if (<= y 21000000.0)
       (/ (* x y) (* (* (+ x y) (+ x y)) (+ 1.0 x)))
       (/ (/ (+ (* (* x (/ x y)) -3.0) x) y) y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 3e-136) {
		tmp = 2.0 * (y / fma(2.0, x, (2.0 * (x * x))));
	} else if (y <= 2.9e-112) {
		tmp = x * (y / (1.0 * ((y + x) * (y + x))));
	} else if (y <= 21000000.0) {
		tmp = (x * y) / (((x + y) * (x + y)) * (1.0 + x));
	} else {
		tmp = ((((x * (x / y)) * -3.0) + x) / y) / y;
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 3e-136)
		tmp = Float64(2.0 * Float64(y / fma(2.0, x, Float64(2.0 * Float64(x * x)))));
	elseif (y <= 2.9e-112)
		tmp = Float64(x * Float64(y / Float64(1.0 * Float64(Float64(y + x) * Float64(y + x)))));
	elseif (y <= 21000000.0)
		tmp = Float64(Float64(x * y) / Float64(Float64(Float64(x + y) * Float64(x + y)) * Float64(1.0 + x)));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(x * Float64(x / y)) * -3.0) + x) / y) / y);
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 3e-136], N[(2.0 * N[(y / N[(2.0 * x + N[(2.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.9e-112], N[(x * N[(y / N[(1.0 * N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 21000000.0], N[(N[(x * y), $MachinePrecision] / N[(N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision] * -3.0), $MachinePrecision] + x), $MachinePrecision] / y), $MachinePrecision] / y), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < 2.9999999999999998e-136

   1. Initial program 64.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\left(x + y\right)} + 1\right)} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(\left(x + y\right) + 1\right)}} \]

      3. flip-+ N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\frac{x \cdot x - y \cdot y}{x - y}} + 1\right)} \]

      4. metadata-eval N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\frac{x \cdot x - y \cdot y}{x - y} + \color{blue}{\frac{2}{2}}\right)} \]

      5. frac-add N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\frac{\left(x \cdot x - y \cdot y\right) \cdot 2 + \left(x - y\right) \cdot 2}{\left(x - y\right) \cdot 2}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\frac{\left(x \cdot x - y \cdot y\right) \cdot 2 + \left(x - y\right) \cdot 2}{\left(x - y\right) \cdot 2}}} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \frac{\color{blue}{\mathsf{fma}\left(x \cdot x - y \cdot y, 2, \left(x - y\right) \cdot 2\right)}}{\left(x - y\right) \cdot 2}} \]

      8. unpow2 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \frac{\mathsf{fma}\left(\color{blue}{{x}^{2}} - y \cdot y, 2, \left(x - y\right) \cdot 2\right)}{\left(x - y\right) \cdot 2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \frac{\mathsf{fma}\left({x}^{2} - \color{blue}{{y}^{2}}, 2, \left(x - y\right) \cdot 2\right)}{\left(x - y\right) \cdot 2}} \]

      10. lower--.f64 N/A

          \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \frac{\mathsf{fma}\left(\color{blue}{{x}^{2} - {y}^{2}}, 2, \left(x - y\right) \cdot 2\right)}{\left(x - y\right) \cdot 2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \frac{\mathsf{fma}\left(\color{blue}{x \cdot x} - {y}^{2}, 2, \left(x - y\right) \cdot 2\right)}{\left(x - y\right) \cdot 2}} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \frac{\mathsf{fma}\left(\color{blue}{x \cdot x} - {y}^{2}, 2, \left(x - y\right) \cdot 2\right)}{\left(x - y\right) \cdot 2}} \]

      13. unpow2 N/A

          \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \frac{\mathsf{fma}\left(x \cdot x - \color{blue}{y \cdot y}, 2, \left(x - y\right) \cdot 2\right)}{\left(x - y\right) \cdot 2}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \frac{\mathsf{fma}\left(x \cdot x - \color{blue}{y \cdot y}, 2, \left(x - y\right) \cdot 2\right)}{\left(x - y\right) \cdot 2}} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \frac{\mathsf{fma}\left(x \cdot x - y \cdot y, 2, \color{blue}{\left(x - y\right) \cdot 2}\right)}{\left(x - y\right) \cdot 2}} \]

      16. lower--.f64 N/A

          \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \frac{\mathsf{fma}\left(x \cdot x - y \cdot y, 2, \color{blue}{\left(x - y\right)} \cdot 2\right)}{\left(x - y\right) \cdot 2}} \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \frac{\mathsf{fma}\left(x \cdot x - y \cdot y, 2, \left(x - y\right) \cdot 2\right)}{\color{blue}{\left(x - y\right) \cdot 2}}} \]

      18. lower--.f64 64.2

          \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \frac{\mathsf{fma}\left(x \cdot x - y \cdot y, 2, \left(x - y\right) \cdot 2\right)}{\color{blue}{\left(x - y\right)} \cdot 2}} \]

   3. Applied rewrites 64.2%

      \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\frac{\mathsf{fma}\left(x \cdot x - y \cdot y, 2, \left(x - y\right) \cdot 2\right)}{\left(x - y\right) \cdot 2}}} \]

   4. Taylor expanded in y around 0

      \[\leadsto \color{blue}{2 \cdot \frac{y}{2 \cdot x + 2 \cdot {x}^{2}}} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\frac{y}{2 \cdot x + 2 \cdot {x}^{2}}} \]

      2. lower-/.f64 N/A

         \[\leadsto 2 \cdot \frac{y}{\color{blue}{2 \cdot x + 2 \cdot {x}^{2}}} \]

      3. lower-fma.f64 N/A

         \[\leadsto 2 \cdot \frac{y}{\mathsf{fma}\left(2, \color{blue}{x}, 2 \cdot {x}^{2}\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{y}{\mathsf{fma}\left(2, x, 2 \cdot {x}^{2}\right)} \]

      5. pow2 N/A

         \[\leadsto 2 \cdot \frac{y}{\mathsf{fma}\left(2, x, 2 \cdot \left(x \cdot x\right)\right)} \]

      6. lift-*.f64 85.8

         \[\leadsto 2 \cdot \frac{y}{\mathsf{fma}\left(2, x, 2 \cdot \left(x \cdot x\right)\right)} \]

   6. Applied rewrites 85.8%

      \[\leadsto \color{blue}{2 \cdot \frac{y}{\mathsf{fma}\left(2, x, 2 \cdot \left(x \cdot x\right)\right)}} \]

   if 2.9999999999999998e-136 < y < 2.89999999999999992e-112

   1. Initial program 70.4%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\color{blue}{\left(x + y\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\left(x + y\right)} + 1\right)} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(\left(x + y\right) + 1\right)}} \]

      9. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      11. lower-/.f64 N/A

          \[\leadsto x \cdot \color{blue}{\frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      12. *-commutative N/A

          \[\leadsto x \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      13. lower-*.f64 N/A

          \[\leadsto x \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      14. lift-+.f64 N/A

          \[\leadsto x \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right)} \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)} \]

      15. +-commutative N/A

          \[\leadsto x \cdot \frac{y}{\left(\color{blue}{\left(y + x\right)} + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)} \]

      16. lower-+.f64 N/A

          \[\leadsto x \cdot \frac{y}{\left(\color{blue}{\left(y + x\right)} + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)} \]

      17. lower-*.f64 N/A

          \[\leadsto x \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      18. +-commutative N/A

          \[\leadsto x \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \left(\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)\right)} \]

      19. lower-+.f64 N/A

          \[\leadsto x \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \left(\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)\right)} \]

      20. +-commutative N/A

          \[\leadsto x \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}\right)} \]

      21. lower-+.f64 98.9

          \[\leadsto x \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}\right)} \]

   3. Applied rewrites 98.9%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

   4. Taylor expanded in x around 0

      \[\leadsto x \cdot \frac{y}{\color{blue}{\left(1 + y\right)} \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)} \]
   5. Step-by-step derivation

      1. lower-+.f64 89.8

         \[\leadsto x \cdot \frac{y}{\left(1 + \color{blue}{y}\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)} \]

   6. Applied rewrites 89.8%

      \[\leadsto x \cdot \frac{y}{\color{blue}{\left(1 + y\right)} \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)} \]

   7. Taylor expanded in y around 0

      \[\leadsto x \cdot \frac{y}{1 \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 89.8%

      \[\leadsto x \cdot \frac{y}{1 \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)} \]

   if 2.89999999999999992e-112 < y < 2.1e7

   1. Initial program 86.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. Taylor expanded in y around 0

      \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(1 + x\right)}} \]
   3. Step-by-step derivation

      1. lower-+.f64 79.0

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + \color{blue}{x}\right)} \]

   4. Applied rewrites 79.0%

      \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(1 + x\right)}} \]

   if 2.1e7 < y

   1. Initial program 67.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{x + -1 \cdot \frac{x \cdot \left(1 + \left(x + 2 \cdot x\right)\right)}{y}}{\color{blue}{{y}^{2}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{-1 \cdot \frac{x \cdot \left(1 + \left(x + 2 \cdot x\right)\right)}{y} + x}{{\color{blue}{y}}^{2}} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{-1 \cdot \frac{x \cdot \left(1 + \left(x + 2 \cdot x\right)\right)}{y} + x}{{\color{blue}{y}}^{2}} \]

      4. mul-1-neg N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\frac{x \cdot \left(1 + \left(x + 2 \cdot x\right)\right)}{y}\right)\right) + x}{{y}^{2}} \]

      5. lower-neg.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. distribute-rgt1-in N/A

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

      11. metadata-eval N/A

          \[\leadsto \frac{\left(-\frac{\left(3 \cdot x + 1\right) \cdot x}{y}\right) + x}{{y}^{2}} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\left(-\frac{\mathsf{fma}\left(3, x, 1\right) \cdot x}{y}\right) + x}{{y}^{2}} \]

      13. unpow2 N/A

          \[\leadsto \frac{\left(-\frac{\mathsf{fma}\left(3, x, 1\right) \cdot x}{y}\right) + x}{y \cdot \color{blue}{y}} \]

      14. lower-*.f64 75.1

          \[\leadsto \frac{\left(-\frac{\mathsf{fma}\left(3, x, 1\right) \cdot x}{y}\right) + x}{y \cdot \color{blue}{y}} \]

   4. Applied rewrites 75.1%

      \[\leadsto \color{blue}{\frac{\left(-\frac{\mathsf{fma}\left(3, x, 1\right) \cdot x}{y}\right) + x}{y \cdot y}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{-3 \cdot \frac{{x}^{2}}{y} + x}{y \cdot y} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-3 \cdot \frac{{x}^{2}}{y} + x}{y \cdot y} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{-3 \cdot \frac{{x}^{2}}{y} + x}{y \cdot y} \]

      3. pow2 N/A

         \[\leadsto \frac{-3 \cdot \frac{x \cdot x}{y} + x}{y \cdot y} \]

      4. lift-*.f64 74.8

         \[\leadsto \frac{-3 \cdot \frac{x \cdot x}{y} + x}{y \cdot y} \]

   7. Applied rewrites 74.8%

      \[\leadsto \frac{-3 \cdot \frac{x \cdot x}{y} + x}{y \cdot y} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{-3 \cdot \frac{x \cdot x}{y} + x}{\color{blue}{y \cdot y}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{-3 \cdot \frac{x \cdot x}{y} + x}{y \cdot \color{blue}{y}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{-3 \cdot \frac{x \cdot x}{y} + x}{y}}{\color{blue}{y}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-3 \cdot \frac{x \cdot x}{y} + x}{y}}{\color{blue}{y}} \]

   9. Applied rewrites 85.2%

      \[\leadsto \frac{\frac{\left(x \cdot \frac{x}{y}\right) \cdot -3 + x}{y}}{\color{blue}{y}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 6: 84.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3 \cdot 10^{-136}:\\ \;\;\;\;2 \cdot \frac{y}{\mathsf{fma}\left(2, x, 2 \cdot \left(x \cdot x\right)\right)}\\ \mathbf{elif}\;y \leq 10500000:\\ \;\;\;\;x \cdot \frac{y}{1 \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(x \cdot \frac{x}{y}\right) \cdot -3 + x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 3e-136)
   (* 2.0 (/ y (fma 2.0 x (* 2.0 (* x x)))))
   (if (<= y 10500000.0)
     (* x (/ y (* 1.0 (* (+ y x) (+ y x)))))
     (/ (/ (+ (* (* x (/ x y)) -3.0) x) y) y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 3e-136) {
		tmp = 2.0 * (y / fma(2.0, x, (2.0 * (x * x))));
	} else if (y <= 10500000.0) {
		tmp = x * (y / (1.0 * ((y + x) * (y + x))));
	} else {
		tmp = ((((x * (x / y)) * -3.0) + x) / y) / y;
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 3e-136)
		tmp = Float64(2.0 * Float64(y / fma(2.0, x, Float64(2.0 * Float64(x * x)))));
	elseif (y <= 10500000.0)
		tmp = Float64(x * Float64(y / Float64(1.0 * Float64(Float64(y + x) * Float64(y + x)))));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(x * Float64(x / y)) * -3.0) + x) / y) / y);
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 3e-136], N[(2.0 * N[(y / N[(2.0 * x + N[(2.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 10500000.0], N[(x * N[(y / N[(1.0 * N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision] * -3.0), $MachinePrecision] + x), $MachinePrecision] / y), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.9999999999999998e-136

   1. Initial program 64.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\left(x + y\right)} + 1\right)} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(\left(x + y\right) + 1\right)}} \]

      3. flip-+ N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\frac{x \cdot x - y \cdot y}{x - y}} + 1\right)} \]

      4. metadata-eval N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\frac{x \cdot x - y \cdot y}{x - y} + \color{blue}{\frac{2}{2}}\right)} \]

      5. frac-add N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\frac{\left(x \cdot x - y \cdot y\right) \cdot 2 + \left(x - y\right) \cdot 2}{\left(x - y\right) \cdot 2}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\frac{\left(x \cdot x - y \cdot y\right) \cdot 2 + \left(x - y\right) \cdot 2}{\left(x - y\right) \cdot 2}}} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \frac{\color{blue}{\mathsf{fma}\left(x \cdot x - y \cdot y, 2, \left(x - y\right) \cdot 2\right)}}{\left(x - y\right) \cdot 2}} \]

      8. unpow2 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \frac{\mathsf{fma}\left(\color{blue}{{x}^{2}} - y \cdot y, 2, \left(x - y\right) \cdot 2\right)}{\left(x - y\right) \cdot 2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \frac{\mathsf{fma}\left({x}^{2} - \color{blue}{{y}^{2}}, 2, \left(x - y\right) \cdot 2\right)}{\left(x - y\right) \cdot 2}} \]

      10. lower--.f64 N/A

          \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \frac{\mathsf{fma}\left(\color{blue}{{x}^{2} - {y}^{2}}, 2, \left(x - y\right) \cdot 2\right)}{\left(x - y\right) \cdot 2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \frac{\mathsf{fma}\left(\color{blue}{x \cdot x} - {y}^{2}, 2, \left(x - y\right) \cdot 2\right)}{\left(x - y\right) \cdot 2}} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \frac{\mathsf{fma}\left(\color{blue}{x \cdot x} - {y}^{2}, 2, \left(x - y\right) \cdot 2\right)}{\left(x - y\right) \cdot 2}} \]

      13. unpow2 N/A

          \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \frac{\mathsf{fma}\left(x \cdot x - \color{blue}{y \cdot y}, 2, \left(x - y\right) \cdot 2\right)}{\left(x - y\right) \cdot 2}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \frac{\mathsf{fma}\left(x \cdot x - \color{blue}{y \cdot y}, 2, \left(x - y\right) \cdot 2\right)}{\left(x - y\right) \cdot 2}} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \frac{\mathsf{fma}\left(x \cdot x - y \cdot y, 2, \color{blue}{\left(x - y\right) \cdot 2}\right)}{\left(x - y\right) \cdot 2}} \]

      16. lower--.f64 N/A

          \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \frac{\mathsf{fma}\left(x \cdot x - y \cdot y, 2, \color{blue}{\left(x - y\right)} \cdot 2\right)}{\left(x - y\right) \cdot 2}} \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \frac{\mathsf{fma}\left(x \cdot x - y \cdot y, 2, \left(x - y\right) \cdot 2\right)}{\color{blue}{\left(x - y\right) \cdot 2}}} \]

      18. lower--.f64 64.2

          \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \frac{\mathsf{fma}\left(x \cdot x - y \cdot y, 2, \left(x - y\right) \cdot 2\right)}{\color{blue}{\left(x - y\right)} \cdot 2}} \]

   3. Applied rewrites 64.2%

      \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\frac{\mathsf{fma}\left(x \cdot x - y \cdot y, 2, \left(x - y\right) \cdot 2\right)}{\left(x - y\right) \cdot 2}}} \]

   4. Taylor expanded in y around 0

      \[\leadsto \color{blue}{2 \cdot \frac{y}{2 \cdot x + 2 \cdot {x}^{2}}} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\frac{y}{2 \cdot x + 2 \cdot {x}^{2}}} \]

      2. lower-/.f64 N/A

         \[\leadsto 2 \cdot \frac{y}{\color{blue}{2 \cdot x + 2 \cdot {x}^{2}}} \]

      3. lower-fma.f64 N/A

         \[\leadsto 2 \cdot \frac{y}{\mathsf{fma}\left(2, \color{blue}{x}, 2 \cdot {x}^{2}\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{y}{\mathsf{fma}\left(2, x, 2 \cdot {x}^{2}\right)} \]

      5. pow2 N/A

         \[\leadsto 2 \cdot \frac{y}{\mathsf{fma}\left(2, x, 2 \cdot \left(x \cdot x\right)\right)} \]

      6. lift-*.f64 85.8

         \[\leadsto 2 \cdot \frac{y}{\mathsf{fma}\left(2, x, 2 \cdot \left(x \cdot x\right)\right)} \]

   6. Applied rewrites 85.8%

      \[\leadsto \color{blue}{2 \cdot \frac{y}{\mathsf{fma}\left(2, x, 2 \cdot \left(x \cdot x\right)\right)}} \]

   if 2.9999999999999998e-136 < y < 1.05e7

   1. Initial program 84.2%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\color{blue}{\left(x + y\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\left(x + y\right)} + 1\right)} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(\left(x + y\right) + 1\right)}} \]

      9. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      11. lower-/.f64 N/A

          \[\leadsto x \cdot \color{blue}{\frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      12. *-commutative N/A

          \[\leadsto x \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      13. lower-*.f64 N/A

          \[\leadsto x \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      14. lift-+.f64 N/A

          \[\leadsto x \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right)} \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)} \]

      15. +-commutative N/A

          \[\leadsto x \cdot \frac{y}{\left(\color{blue}{\left(y + x\right)} + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)} \]

      16. lower-+.f64 N/A

          \[\leadsto x \cdot \frac{y}{\left(\color{blue}{\left(y + x\right)} + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)} \]

      17. lower-*.f64 N/A

          \[\leadsto x \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      18. +-commutative N/A

          \[\leadsto x \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \left(\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)\right)} \]

      19. lower-+.f64 N/A

          \[\leadsto x \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \left(\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)\right)} \]

      20. +-commutative N/A

          \[\leadsto x \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}\right)} \]

      21. lower-+.f64 94.8

          \[\leadsto x \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}\right)} \]

   3. Applied rewrites 94.8%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

   4. Taylor expanded in x around 0

      \[\leadsto x \cdot \frac{y}{\color{blue}{\left(1 + y\right)} \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)} \]
   5. Step-by-step derivation

      1. lower-+.f64 83.6

         \[\leadsto x \cdot \frac{y}{\left(1 + \color{blue}{y}\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)} \]

   6. Applied rewrites 83.6%

      \[\leadsto x \cdot \frac{y}{\color{blue}{\left(1 + y\right)} \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)} \]

   7. Taylor expanded in y around 0

      \[\leadsto x \cdot \frac{y}{1 \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 77.8%

      \[\leadsto x \cdot \frac{y}{1 \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)} \]

   if 1.05e7 < y

   1. Initial program 67.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{x + -1 \cdot \frac{x \cdot \left(1 + \left(x + 2 \cdot x\right)\right)}{y}}{\color{blue}{{y}^{2}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{-1 \cdot \frac{x \cdot \left(1 + \left(x + 2 \cdot x\right)\right)}{y} + x}{{\color{blue}{y}}^{2}} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{-1 \cdot \frac{x \cdot \left(1 + \left(x + 2 \cdot x\right)\right)}{y} + x}{{\color{blue}{y}}^{2}} \]

      4. mul-1-neg N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\frac{x \cdot \left(1 + \left(x + 2 \cdot x\right)\right)}{y}\right)\right) + x}{{y}^{2}} \]

      5. lower-neg.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. distribute-rgt1-in N/A

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

      11. metadata-eval N/A

          \[\leadsto \frac{\left(-\frac{\left(3 \cdot x + 1\right) \cdot x}{y}\right) + x}{{y}^{2}} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\left(-\frac{\mathsf{fma}\left(3, x, 1\right) \cdot x}{y}\right) + x}{{y}^{2}} \]

      13. unpow2 N/A

          \[\leadsto \frac{\left(-\frac{\mathsf{fma}\left(3, x, 1\right) \cdot x}{y}\right) + x}{y \cdot \color{blue}{y}} \]

      14. lower-*.f64 75.1

          \[\leadsto \frac{\left(-\frac{\mathsf{fma}\left(3, x, 1\right) \cdot x}{y}\right) + x}{y \cdot \color{blue}{y}} \]

   4. Applied rewrites 75.1%

      \[\leadsto \color{blue}{\frac{\left(-\frac{\mathsf{fma}\left(3, x, 1\right) \cdot x}{y}\right) + x}{y \cdot y}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{-3 \cdot \frac{{x}^{2}}{y} + x}{y \cdot y} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-3 \cdot \frac{{x}^{2}}{y} + x}{y \cdot y} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{-3 \cdot \frac{{x}^{2}}{y} + x}{y \cdot y} \]

      3. pow2 N/A

         \[\leadsto \frac{-3 \cdot \frac{x \cdot x}{y} + x}{y \cdot y} \]

      4. lift-*.f64 74.8

         \[\leadsto \frac{-3 \cdot \frac{x \cdot x}{y} + x}{y \cdot y} \]

   7. Applied rewrites 74.8%

      \[\leadsto \frac{-3 \cdot \frac{x \cdot x}{y} + x}{y \cdot y} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{-3 \cdot \frac{x \cdot x}{y} + x}{\color{blue}{y \cdot y}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{-3 \cdot \frac{x \cdot x}{y} + x}{y \cdot \color{blue}{y}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{-3 \cdot \frac{x \cdot x}{y} + x}{y}}{\color{blue}{y}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-3 \cdot \frac{x \cdot x}{y} + x}{y}}{\color{blue}{y}} \]

   9. Applied rewrites 85.1%

      \[\leadsto \frac{\frac{\left(x \cdot \frac{x}{y}\right) \cdot -3 + x}{y}}{\color{blue}{y}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 84.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{\frac{y}{-x} + y}{x}}{x}\\ \mathbf{elif}\;x \leq -75:\\ \;\;\;\;\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot x}\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-129}:\\ \;\;\;\;x \cdot \frac{y}{1 \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(1 + y\right) \cdot y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -5.5e+102)
   (/ (/ (+ (/ y (- x)) y) x) x)
   (if (<= x -75.0)
     (/ (* x y) (* (* (+ x y) (+ x y)) x))
     (if (<= x -2e-129)
       (* x (/ y (* 1.0 (* (+ y x) (+ y x)))))
       (/ x (* (+ 1.0 y) y))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -5.5e+102) {
		tmp = (((y / -x) + y) / x) / x;
	} else if (x <= -75.0) {
		tmp = (x * y) / (((x + y) * (x + y)) * x);
	} else if (x <= -2e-129) {
		tmp = x * (y / (1.0 * ((y + x) * (y + x))));
	} else {
		tmp = x / ((1.0 + y) * y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
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(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-5.5d+102)) then
        tmp = (((y / -x) + y) / x) / x
    else if (x <= (-75.0d0)) then
        tmp = (x * y) / (((x + y) * (x + y)) * x)
    else if (x <= (-2d-129)) then
        tmp = x * (y / (1.0d0 * ((y + x) * (y + x))))
    else
        tmp = x / ((1.0d0 + y) * y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -5.5e+102) {
		tmp = (((y / -x) + y) / x) / x;
	} else if (x <= -75.0) {
		tmp = (x * y) / (((x + y) * (x + y)) * x);
	} else if (x <= -2e-129) {
		tmp = x * (y / (1.0 * ((y + x) * (y + x))));
	} else {
		tmp = x / ((1.0 + y) * y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -5.5e+102:
		tmp = (((y / -x) + y) / x) / x
	elif x <= -75.0:
		tmp = (x * y) / (((x + y) * (x + y)) * x)
	elif x <= -2e-129:
		tmp = x * (y / (1.0 * ((y + x) * (y + x))))
	else:
		tmp = x / ((1.0 + y) * y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -5.5e+102)
		tmp = Float64(Float64(Float64(Float64(y / Float64(-x)) + y) / x) / x);
	elseif (x <= -75.0)
		tmp = Float64(Float64(x * y) / Float64(Float64(Float64(x + y) * Float64(x + y)) * x));
	elseif (x <= -2e-129)
		tmp = Float64(x * Float64(y / Float64(1.0 * Float64(Float64(y + x) * Float64(y + x)))));
	else
		tmp = Float64(x / Float64(Float64(1.0 + y) * y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -5.5e+102)
		tmp = (((y / -x) + y) / x) / x;
	elseif (x <= -75.0)
		tmp = (x * y) / (((x + y) * (x + y)) * x);
	elseif (x <= -2e-129)
		tmp = x * (y / (1.0 * ((y + x) * (y + x))));
	else
		tmp = x / ((1.0 + y) * y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -5.5e+102], N[(N[(N[(N[(y / (-x)), $MachinePrecision] + y), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, -75.0], N[(N[(x * y), $MachinePrecision] / N[(N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2e-129], N[(x * N[(y / N[(1.0 * N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[(1.0 + y), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -5.49999999999999981e102

   1. Initial program 58.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{y + -1 \cdot \frac{y \cdot \left(1 + \left(y + 2 \cdot y\right)\right)}{x}}{\color{blue}{{x}^{2}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{-1 \cdot \frac{y \cdot \left(1 + \left(y + 2 \cdot y\right)\right)}{x} + y}{{\color{blue}{x}}^{2}} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{-1 \cdot \frac{y \cdot \left(1 + \left(y + 2 \cdot y\right)\right)}{x} + y}{{\color{blue}{x}}^{2}} \]

      4. mul-1-neg N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\frac{y \cdot \left(1 + \left(y + 2 \cdot y\right)\right)}{x}\right)\right) + y}{{x}^{2}} \]

      5. lower-neg.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. distribute-rgt1-in N/A

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

      11. metadata-eval N/A

          \[\leadsto \frac{\left(-\frac{\left(3 \cdot y + 1\right) \cdot y}{x}\right) + y}{{x}^{2}} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\left(-\frac{\mathsf{fma}\left(3, y, 1\right) \cdot y}{x}\right) + y}{{x}^{2}} \]

      13. unpow2 N/A

          \[\leadsto \frac{\left(-\frac{\mathsf{fma}\left(3, y, 1\right) \cdot y}{x}\right) + y}{x \cdot \color{blue}{x}} \]

      14. lower-*.f64 75.2

          \[\leadsto \frac{\left(-\frac{\mathsf{fma}\left(3, y, 1\right) \cdot y}{x}\right) + y}{x \cdot \color{blue}{x}} \]

   4. Applied rewrites 75.2%

      \[\leadsto \color{blue}{\frac{\left(-\frac{\mathsf{fma}\left(3, y, 1\right) \cdot y}{x}\right) + y}{x \cdot x}} \]

   5. Taylor expanded in y around 0

      \[\leadsto \frac{\left(-\frac{y}{x}\right) + y}{x \cdot x} \]
   6. Step-by-step derivation

   7. Applied rewrites 83.4%

      \[\leadsto \frac{\left(-\frac{y}{x}\right) + y}{x \cdot x} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\left(-\frac{y}{x}\right) + y}{x \cdot \color{blue}{x}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\left(-\frac{y}{x}\right) + y}{\color{blue}{x \cdot x}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{\left(-\frac{y}{x}\right) + y}{x}}{\color{blue}{x}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\left(-\frac{y}{x}\right) + y}{x}}{\color{blue}{x}} \]

      5. lower-/.f64 90.5

         \[\leadsto \frac{\frac{\left(-\frac{y}{x}\right) + y}{x}}{x} \]

      6. lift-neg.f64 N/A

         \[\leadsto \frac{\frac{\left(\mathsf{neg}\left(\frac{y}{x}\right)\right) + y}{x}}{x} \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{\frac{\left(\mathsf{neg}\left(\frac{y}{x}\right)\right) + y}{x}}{x} \]

      8. distribute-neg-frac2 N/A

         \[\leadsto \frac{\frac{\frac{y}{\mathsf{neg}\left(x\right)} + y}{x}}{x} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\frac{y}{\mathsf{neg}\left(x\right)} + y}{x}}{x} \]

      10. lower-neg.f64 90.5

          \[\leadsto \frac{\frac{\frac{y}{-x} + y}{x}}{x} \]

   9. Applied rewrites 90.5%

      \[\leadsto \frac{\frac{\frac{y}{-x} + y}{x}}{\color{blue}{x}} \]

   if -5.49999999999999981e102 < x < -75

   1. Initial program 86.2%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. Taylor expanded in x around inf

      \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{x}} \]
   3. Step-by-step derivation

   4. Applied rewrites 76.9%

      \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{x}} \]

   if -75 < x < -1.9999999999999999e-129

   1. Initial program 83.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\color{blue}{\left(x + y\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\left(x + y\right)} + 1\right)} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(\left(x + y\right) + 1\right)}} \]

      9. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      11. lower-/.f64 N/A

          \[\leadsto x \cdot \color{blue}{\frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      12. *-commutative N/A

          \[\leadsto x \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      13. lower-*.f64 N/A

          \[\leadsto x \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      14. lift-+.f64 N/A

          \[\leadsto x \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right)} \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)} \]

      15. +-commutative N/A

          \[\leadsto x \cdot \frac{y}{\left(\color{blue}{\left(y + x\right)} + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)} \]

      16. lower-+.f64 N/A

          \[\leadsto x \cdot \frac{y}{\left(\color{blue}{\left(y + x\right)} + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)} \]

      17. lower-*.f64 N/A

          \[\leadsto x \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      18. +-commutative N/A

          \[\leadsto x \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \left(\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)\right)} \]

      19. lower-+.f64 N/A

          \[\leadsto x \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \left(\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)\right)} \]

      20. +-commutative N/A

          \[\leadsto x \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}\right)} \]

      21. lower-+.f64 96.7

          \[\leadsto x \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}\right)} \]

   3. Applied rewrites 96.7%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

   4. Taylor expanded in x around 0

      \[\leadsto x \cdot \frac{y}{\color{blue}{\left(1 + y\right)} \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)} \]
   5. Step-by-step derivation

      1. lower-+.f64 92.5

         \[\leadsto x \cdot \frac{y}{\left(1 + \color{blue}{y}\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)} \]

   6. Applied rewrites 92.5%

      \[\leadsto x \cdot \frac{y}{\color{blue}{\left(1 + y\right)} \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)} \]

   7. Taylor expanded in y around 0

      \[\leadsto x \cdot \frac{y}{1 \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 78.3%

      \[\leadsto x \cdot \frac{y}{1 \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)} \]

   if -1.9999999999999999e-129 < x

   1. Initial program 65.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot \left(1 + y\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{x}{\left(1 + y\right) \cdot \color{blue}{y}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{x}{\left(1 + y\right) \cdot \color{blue}{y}} \]

      4. lower-+.f64 85.8

         \[\leadsto \frac{x}{\left(1 + y\right) \cdot y} \]

   4. Applied rewrites 85.8%

      \[\leadsto \color{blue}{\frac{x}{\left(1 + y\right) \cdot y}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 8: 84.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -75:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-129}:\\ \;\;\;\;x \cdot \frac{y}{1 \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(1 + y\right) \cdot y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -75.0)
   (* (/ x (* (+ x y) (+ x y))) (/ y x))
   (if (<= x -2e-129)
     (* x (/ y (* 1.0 (* (+ y x) (+ y x)))))
     (/ x (* (+ 1.0 y) y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -75.0) {
		tmp = (x / ((x + y) * (x + y))) * (y / x);
	} else if (x <= -2e-129) {
		tmp = x * (y / (1.0 * ((y + x) * (y + x))));
	} else {
		tmp = x / ((1.0 + y) * y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
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(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-75.0d0)) then
        tmp = (x / ((x + y) * (x + y))) * (y / x)
    else if (x <= (-2d-129)) then
        tmp = x * (y / (1.0d0 * ((y + x) * (y + x))))
    else
        tmp = x / ((1.0d0 + y) * y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -75.0) {
		tmp = (x / ((x + y) * (x + y))) * (y / x);
	} else if (x <= -2e-129) {
		tmp = x * (y / (1.0 * ((y + x) * (y + x))));
	} else {
		tmp = x / ((1.0 + y) * y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -75.0:
		tmp = (x / ((x + y) * (x + y))) * (y / x)
	elif x <= -2e-129:
		tmp = x * (y / (1.0 * ((y + x) * (y + x))))
	else:
		tmp = x / ((1.0 + y) * y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -75.0)
		tmp = Float64(Float64(x / Float64(Float64(x + y) * Float64(x + y))) * Float64(y / x));
	elseif (x <= -2e-129)
		tmp = Float64(x * Float64(y / Float64(1.0 * Float64(Float64(y + x) * Float64(y + x)))));
	else
		tmp = Float64(x / Float64(Float64(1.0 + y) * y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -75.0)
		tmp = (x / ((x + y) * (x + y))) * (y / x);
	elseif (x <= -2e-129)
		tmp = x * (y / (1.0 * ((y + x) * (y + x))));
	else
		tmp = x / ((1.0 + y) * y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -75.0], N[(N[(x / N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2e-129], N[(x * N[(y / N[(1.0 * N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[(1.0 + y), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -75

   1. Initial program 66.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\left(x + y\right)} + 1\right)} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(\left(x + y\right) + 1\right)}} \]

      3. flip-+ N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\frac{x \cdot x - y \cdot y}{x - y}} + 1\right)} \]

      4. metadata-eval N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\frac{x \cdot x - y \cdot y}{x - y} + \color{blue}{\frac{2}{2}}\right)} \]

      5. frac-add N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\frac{\left(x \cdot x - y \cdot y\right) \cdot 2 + \left(x - y\right) \cdot 2}{\left(x - y\right) \cdot 2}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\frac{\left(x \cdot x - y \cdot y\right) \cdot 2 + \left(x - y\right) \cdot 2}{\left(x - y\right) \cdot 2}}} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \frac{\color{blue}{\mathsf{fma}\left(x \cdot x - y \cdot y, 2, \left(x - y\right) \cdot 2\right)}}{\left(x - y\right) \cdot 2}} \]

      8. unpow2 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \frac{\mathsf{fma}\left(\color{blue}{{x}^{2}} - y \cdot y, 2, \left(x - y\right) \cdot 2\right)}{\left(x - y\right) \cdot 2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \frac{\mathsf{fma}\left({x}^{2} - \color{blue}{{y}^{2}}, 2, \left(x - y\right) \cdot 2\right)}{\left(x - y\right) \cdot 2}} \]

      10. lower--.f64 N/A

          \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \frac{\mathsf{fma}\left(\color{blue}{{x}^{2} - {y}^{2}}, 2, \left(x - y\right) \cdot 2\right)}{\left(x - y\right) \cdot 2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \frac{\mathsf{fma}\left(\color{blue}{x \cdot x} - {y}^{2}, 2, \left(x - y\right) \cdot 2\right)}{\left(x - y\right) \cdot 2}} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \frac{\mathsf{fma}\left(\color{blue}{x \cdot x} - {y}^{2}, 2, \left(x - y\right) \cdot 2\right)}{\left(x - y\right) \cdot 2}} \]

      13. unpow2 N/A

          \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \frac{\mathsf{fma}\left(x \cdot x - \color{blue}{y \cdot y}, 2, \left(x - y\right) \cdot 2\right)}{\left(x - y\right) \cdot 2}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \frac{\mathsf{fma}\left(x \cdot x - \color{blue}{y \cdot y}, 2, \left(x - y\right) \cdot 2\right)}{\left(x - y\right) \cdot 2}} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \frac{\mathsf{fma}\left(x \cdot x - y \cdot y, 2, \color{blue}{\left(x - y\right) \cdot 2}\right)}{\left(x - y\right) \cdot 2}} \]

      16. lower--.f64 N/A

          \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \frac{\mathsf{fma}\left(x \cdot x - y \cdot y, 2, \color{blue}{\left(x - y\right)} \cdot 2\right)}{\left(x - y\right) \cdot 2}} \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \frac{\mathsf{fma}\left(x \cdot x - y \cdot y, 2, \left(x - y\right) \cdot 2\right)}{\color{blue}{\left(x - y\right) \cdot 2}}} \]

      18. lower--.f64 66.6

          \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \frac{\mathsf{fma}\left(x \cdot x - y \cdot y, 2, \left(x - y\right) \cdot 2\right)}{\color{blue}{\left(x - y\right)} \cdot 2}} \]

   3. Applied rewrites 66.6%

      \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\frac{\mathsf{fma}\left(x \cdot x - y \cdot y, 2, \left(x - y\right) \cdot 2\right)}{\left(x - y\right) \cdot 2}}} \]

   4. Taylor expanded in x around inf

      \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{x}} \]
   5. Step-by-step derivation

   6. Applied rewrites 64.0%

      \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{x}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot x} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot x}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot x}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot x} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\color{blue}{\left(x + y\right)} \cdot \left(x + y\right)\right) \cdot x} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}\right) \cdot x} \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x}} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x}} \]

   8. Applied rewrites 84.6%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x}} \]

   if -75 < x < -1.9999999999999999e-129

   1. Initial program 83.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\color{blue}{\left(x + y\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\left(x + y\right)} + 1\right)} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(\left(x + y\right) + 1\right)}} \]

      9. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      11. lower-/.f64 N/A

          \[\leadsto x \cdot \color{blue}{\frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      12. *-commutative N/A

          \[\leadsto x \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      13. lower-*.f64 N/A

          \[\leadsto x \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      14. lift-+.f64 N/A

          \[\leadsto x \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right)} \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)} \]

      15. +-commutative N/A

          \[\leadsto x \cdot \frac{y}{\left(\color{blue}{\left(y + x\right)} + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)} \]

      16. lower-+.f64 N/A

          \[\leadsto x \cdot \frac{y}{\left(\color{blue}{\left(y + x\right)} + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)} \]

      17. lower-*.f64 N/A

          \[\leadsto x \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      18. +-commutative N/A

          \[\leadsto x \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \left(\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)\right)} \]

      19. lower-+.f64 N/A

          \[\leadsto x \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \left(\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)\right)} \]

      20. +-commutative N/A

          \[\leadsto x \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}\right)} \]

      21. lower-+.f64 96.7

          \[\leadsto x \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}\right)} \]

   3. Applied rewrites 96.7%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

   4. Taylor expanded in x around 0

      \[\leadsto x \cdot \frac{y}{\color{blue}{\left(1 + y\right)} \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)} \]
   5. Step-by-step derivation

      1. lower-+.f64 92.5

         \[\leadsto x \cdot \frac{y}{\left(1 + \color{blue}{y}\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)} \]

   6. Applied rewrites 92.5%

      \[\leadsto x \cdot \frac{y}{\color{blue}{\left(1 + y\right)} \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)} \]

   7. Taylor expanded in y around 0

      \[\leadsto x \cdot \frac{y}{1 \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 78.3%

      \[\leadsto x \cdot \frac{y}{1 \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)} \]

   if -1.9999999999999999e-129 < x

   1. Initial program 65.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot \left(1 + y\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{x}{\left(1 + y\right) \cdot \color{blue}{y}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{x}{\left(1 + y\right) \cdot \color{blue}{y}} \]

      4. lower-+.f64 85.8

         \[\leadsto \frac{x}{\left(1 + y\right) \cdot y} \]

   4. Applied rewrites 85.8%

      \[\leadsto \color{blue}{\frac{x}{\left(1 + y\right) \cdot y}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 84.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -10600000:\\ \;\;\;\;\frac{\frac{\frac{y}{-x} + y}{x}}{x}\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-129}:\\ \;\;\;\;x \cdot \frac{y}{1 \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(1 + y\right) \cdot y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -10600000.0)
   (/ (/ (+ (/ y (- x)) y) x) x)
   (if (<= x -2e-129)
     (* x (/ y (* 1.0 (* (+ y x) (+ y x)))))
     (/ x (* (+ 1.0 y) y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -10600000.0) {
		tmp = (((y / -x) + y) / x) / x;
	} else if (x <= -2e-129) {
		tmp = x * (y / (1.0 * ((y + x) * (y + x))));
	} else {
		tmp = x / ((1.0 + y) * y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
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(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-10600000.0d0)) then
        tmp = (((y / -x) + y) / x) / x
    else if (x <= (-2d-129)) then
        tmp = x * (y / (1.0d0 * ((y + x) * (y + x))))
    else
        tmp = x / ((1.0d0 + y) * y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -10600000.0) {
		tmp = (((y / -x) + y) / x) / x;
	} else if (x <= -2e-129) {
		tmp = x * (y / (1.0 * ((y + x) * (y + x))));
	} else {
		tmp = x / ((1.0 + y) * y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -10600000.0:
		tmp = (((y / -x) + y) / x) / x
	elif x <= -2e-129:
		tmp = x * (y / (1.0 * ((y + x) * (y + x))))
	else:
		tmp = x / ((1.0 + y) * y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -10600000.0)
		tmp = Float64(Float64(Float64(Float64(y / Float64(-x)) + y) / x) / x);
	elseif (x <= -2e-129)
		tmp = Float64(x * Float64(y / Float64(1.0 * Float64(Float64(y + x) * Float64(y + x)))));
	else
		tmp = Float64(x / Float64(Float64(1.0 + y) * y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -10600000.0)
		tmp = (((y / -x) + y) / x) / x;
	elseif (x <= -2e-129)
		tmp = x * (y / (1.0 * ((y + x) * (y + x))));
	else
		tmp = x / ((1.0 + y) * y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -10600000.0], N[(N[(N[(N[(y / (-x)), $MachinePrecision] + y), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, -2e-129], N[(x * N[(y / N[(1.0 * N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[(1.0 + y), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.06e7

   1. Initial program 66.4%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{y + -1 \cdot \frac{y \cdot \left(1 + \left(y + 2 \cdot y\right)\right)}{x}}{\color{blue}{{x}^{2}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{-1 \cdot \frac{y \cdot \left(1 + \left(y + 2 \cdot y\right)\right)}{x} + y}{{\color{blue}{x}}^{2}} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{-1 \cdot \frac{y \cdot \left(1 + \left(y + 2 \cdot y\right)\right)}{x} + y}{{\color{blue}{x}}^{2}} \]

      4. mul-1-neg N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\frac{y \cdot \left(1 + \left(y + 2 \cdot y\right)\right)}{x}\right)\right) + y}{{x}^{2}} \]

      5. lower-neg.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. distribute-rgt1-in N/A

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

      11. metadata-eval N/A

          \[\leadsto \frac{\left(-\frac{\left(3 \cdot y + 1\right) \cdot y}{x}\right) + y}{{x}^{2}} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\left(-\frac{\mathsf{fma}\left(3, y, 1\right) \cdot y}{x}\right) + y}{{x}^{2}} \]

      13. unpow2 N/A

          \[\leadsto \frac{\left(-\frac{\mathsf{fma}\left(3, y, 1\right) \cdot y}{x}\right) + y}{x \cdot \color{blue}{x}} \]

      14. lower-*.f64 74.9

          \[\leadsto \frac{\left(-\frac{\mathsf{fma}\left(3, y, 1\right) \cdot y}{x}\right) + y}{x \cdot \color{blue}{x}} \]

   4. Applied rewrites 74.9%

      \[\leadsto \color{blue}{\frac{\left(-\frac{\mathsf{fma}\left(3, y, 1\right) \cdot y}{x}\right) + y}{x \cdot x}} \]

   5. Taylor expanded in y around 0

      \[\leadsto \frac{\left(-\frac{y}{x}\right) + y}{x \cdot x} \]
   6. Step-by-step derivation

   7. Applied rewrites 80.6%

      \[\leadsto \frac{\left(-\frac{y}{x}\right) + y}{x \cdot x} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\left(-\frac{y}{x}\right) + y}{x \cdot \color{blue}{x}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\left(-\frac{y}{x}\right) + y}{\color{blue}{x \cdot x}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{\left(-\frac{y}{x}\right) + y}{x}}{\color{blue}{x}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\left(-\frac{y}{x}\right) + y}{x}}{\color{blue}{x}} \]

      5. lower-/.f64 85.7

         \[\leadsto \frac{\frac{\left(-\frac{y}{x}\right) + y}{x}}{x} \]

      6. lift-neg.f64 N/A

         \[\leadsto \frac{\frac{\left(\mathsf{neg}\left(\frac{y}{x}\right)\right) + y}{x}}{x} \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{\frac{\left(\mathsf{neg}\left(\frac{y}{x}\right)\right) + y}{x}}{x} \]

      8. distribute-neg-frac2 N/A

         \[\leadsto \frac{\frac{\frac{y}{\mathsf{neg}\left(x\right)} + y}{x}}{x} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\frac{y}{\mathsf{neg}\left(x\right)} + y}{x}}{x} \]

      10. lower-neg.f64 85.7

          \[\leadsto \frac{\frac{\frac{y}{-x} + y}{x}}{x} \]

   9. Applied rewrites 85.7%

      \[\leadsto \frac{\frac{\frac{y}{-x} + y}{x}}{\color{blue}{x}} \]

   if -1.06e7 < x < -1.9999999999999999e-129

   1. Initial program 84.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\color{blue}{\left(x + y\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\left(x + y\right)} + 1\right)} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(\left(x + y\right) + 1\right)}} \]

      9. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      11. lower-/.f64 N/A

          \[\leadsto x \cdot \color{blue}{\frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      12. *-commutative N/A

          \[\leadsto x \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      13. lower-*.f64 N/A

          \[\leadsto x \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      14. lift-+.f64 N/A

          \[\leadsto x \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right)} \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)} \]

      15. +-commutative N/A

          \[\leadsto x \cdot \frac{y}{\left(\color{blue}{\left(y + x\right)} + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)} \]

      16. lower-+.f64 N/A

          \[\leadsto x \cdot \frac{y}{\left(\color{blue}{\left(y + x\right)} + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)} \]

      17. lower-*.f64 N/A

          \[\leadsto x \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      18. +-commutative N/A

          \[\leadsto x \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \left(\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)\right)} \]

      19. lower-+.f64 N/A

          \[\leadsto x \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \left(\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)\right)} \]

      20. +-commutative N/A

          \[\leadsto x \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}\right)} \]

      21. lower-+.f64 96.5

          \[\leadsto x \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}\right)} \]

   3. Applied rewrites 96.5%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

   4. Taylor expanded in x around 0

      \[\leadsto x \cdot \frac{y}{\color{blue}{\left(1 + y\right)} \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)} \]
   5. Step-by-step derivation

      1. lower-+.f64 90.1

         \[\leadsto x \cdot \frac{y}{\left(1 + \color{blue}{y}\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)} \]

   6. Applied rewrites 90.1%

      \[\leadsto x \cdot \frac{y}{\color{blue}{\left(1 + y\right)} \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)} \]

   7. Taylor expanded in y around 0

      \[\leadsto x \cdot \frac{y}{1 \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 76.0%

      \[\leadsto x \cdot \frac{y}{1 \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)} \]

   if -1.9999999999999999e-129 < x

   1. Initial program 65.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot \left(1 + y\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{x}{\left(1 + y\right) \cdot \color{blue}{y}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{x}{\left(1 + y\right) \cdot \color{blue}{y}} \]

      4. lower-+.f64 85.8

         \[\leadsto \frac{x}{\left(1 + y\right) \cdot y} \]

   4. Applied rewrites 85.8%

      \[\leadsto \color{blue}{\frac{x}{\left(1 + y\right) \cdot y}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 78.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3.6 \cdot 10^{-72}:\\ \;\;\;\;2 \cdot \frac{y}{\mathsf{fma}\left(2, x, 2 \cdot \left(x \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(1 + y\right) \cdot y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 3.6e-72)
   (* 2.0 (/ y (fma 2.0 x (* 2.0 (* x x)))))
   (/ x (* (+ 1.0 y) y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 3.6e-72) {
		tmp = 2.0 * (y / fma(2.0, x, (2.0 * (x * x))));
	} else {
		tmp = x / ((1.0 + y) * y);
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 3.6e-72)
		tmp = Float64(2.0 * Float64(y / fma(2.0, x, Float64(2.0 * Float64(x * x)))));
	else
		tmp = Float64(x / Float64(Float64(1.0 + y) * y));
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 3.6e-72], N[(2.0 * N[(y / N[(2.0 * x + N[(2.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[(1.0 + y), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.6e-72

   1. Initial program 66.2%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\left(x + y\right)} + 1\right)} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(\left(x + y\right) + 1\right)}} \]

      3. flip-+ N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\frac{x \cdot x - y \cdot y}{x - y}} + 1\right)} \]

      4. metadata-eval N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\frac{x \cdot x - y \cdot y}{x - y} + \color{blue}{\frac{2}{2}}\right)} \]

      5. frac-add N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\frac{\left(x \cdot x - y \cdot y\right) \cdot 2 + \left(x - y\right) \cdot 2}{\left(x - y\right) \cdot 2}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\frac{\left(x \cdot x - y \cdot y\right) \cdot 2 + \left(x - y\right) \cdot 2}{\left(x - y\right) \cdot 2}}} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \frac{\color{blue}{\mathsf{fma}\left(x \cdot x - y \cdot y, 2, \left(x - y\right) \cdot 2\right)}}{\left(x - y\right) \cdot 2}} \]

      8. unpow2 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \frac{\mathsf{fma}\left(\color{blue}{{x}^{2}} - y \cdot y, 2, \left(x - y\right) \cdot 2\right)}{\left(x - y\right) \cdot 2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \frac{\mathsf{fma}\left({x}^{2} - \color{blue}{{y}^{2}}, 2, \left(x - y\right) \cdot 2\right)}{\left(x - y\right) \cdot 2}} \]

      10. lower--.f64 N/A

          \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \frac{\mathsf{fma}\left(\color{blue}{{x}^{2} - {y}^{2}}, 2, \left(x - y\right) \cdot 2\right)}{\left(x - y\right) \cdot 2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \frac{\mathsf{fma}\left(\color{blue}{x \cdot x} - {y}^{2}, 2, \left(x - y\right) \cdot 2\right)}{\left(x - y\right) \cdot 2}} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \frac{\mathsf{fma}\left(\color{blue}{x \cdot x} - {y}^{2}, 2, \left(x - y\right) \cdot 2\right)}{\left(x - y\right) \cdot 2}} \]

      13. unpow2 N/A

          \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \frac{\mathsf{fma}\left(x \cdot x - \color{blue}{y \cdot y}, 2, \left(x - y\right) \cdot 2\right)}{\left(x - y\right) \cdot 2}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \frac{\mathsf{fma}\left(x \cdot x - \color{blue}{y \cdot y}, 2, \left(x - y\right) \cdot 2\right)}{\left(x - y\right) \cdot 2}} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \frac{\mathsf{fma}\left(x \cdot x - y \cdot y, 2, \color{blue}{\left(x - y\right) \cdot 2}\right)}{\left(x - y\right) \cdot 2}} \]

      16. lower--.f64 N/A

          \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \frac{\mathsf{fma}\left(x \cdot x - y \cdot y, 2, \color{blue}{\left(x - y\right)} \cdot 2\right)}{\left(x - y\right) \cdot 2}} \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \frac{\mathsf{fma}\left(x \cdot x - y \cdot y, 2, \left(x - y\right) \cdot 2\right)}{\color{blue}{\left(x - y\right) \cdot 2}}} \]

      18. lower--.f64 66.1

          \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \frac{\mathsf{fma}\left(x \cdot x - y \cdot y, 2, \left(x - y\right) \cdot 2\right)}{\color{blue}{\left(x - y\right)} \cdot 2}} \]

   3. Applied rewrites 66.1%

      \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\frac{\mathsf{fma}\left(x \cdot x - y \cdot y, 2, \left(x - y\right) \cdot 2\right)}{\left(x - y\right) \cdot 2}}} \]

   4. Taylor expanded in y around 0

      \[\leadsto \color{blue}{2 \cdot \frac{y}{2 \cdot x + 2 \cdot {x}^{2}}} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\frac{y}{2 \cdot x + 2 \cdot {x}^{2}}} \]

      2. lower-/.f64 N/A

         \[\leadsto 2 \cdot \frac{y}{\color{blue}{2 \cdot x + 2 \cdot {x}^{2}}} \]

      3. lower-fma.f64 N/A

         \[\leadsto 2 \cdot \frac{y}{\mathsf{fma}\left(2, \color{blue}{x}, 2 \cdot {x}^{2}\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{y}{\mathsf{fma}\left(2, x, 2 \cdot {x}^{2}\right)} \]

      5. pow2 N/A

         \[\leadsto 2 \cdot \frac{y}{\mathsf{fma}\left(2, x, 2 \cdot \left(x \cdot x\right)\right)} \]

      6. lift-*.f64 80.5

         \[\leadsto 2 \cdot \frac{y}{\mathsf{fma}\left(2, x, 2 \cdot \left(x \cdot x\right)\right)} \]

   6. Applied rewrites 80.5%

      \[\leadsto \color{blue}{2 \cdot \frac{y}{\mathsf{fma}\left(2, x, 2 \cdot \left(x \cdot x\right)\right)}} \]

   if 3.6e-72 < y

   1. Initial program 71.0%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot \left(1 + y\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{x}{\left(1 + y\right) \cdot \color{blue}{y}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{x}{\left(1 + y\right) \cdot \color{blue}{y}} \]

      4. lower-+.f64 76.8

         \[\leadsto \frac{x}{\left(1 + y\right) \cdot y} \]

   4. Applied rewrites 76.8%

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

Alternative 11: 78.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3.6 \cdot 10^{-72}:\\ \;\;\;\;\frac{y}{\left(1 + x\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(1 + y\right) \cdot y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 3.6e-72) (/ y (* (+ 1.0 x) x)) (/ x (* (+ 1.0 y) y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 3.6e-72) {
		tmp = y / ((1.0 + x) * x);
	} else {
		tmp = x / ((1.0 + y) * y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
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(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 3.6d-72) then
        tmp = y / ((1.0d0 + x) * x)
    else
        tmp = x / ((1.0d0 + y) * y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 3.6e-72) {
		tmp = y / ((1.0 + x) * x);
	} else {
		tmp = x / ((1.0 + y) * y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 3.6e-72:
		tmp = y / ((1.0 + x) * x)
	else:
		tmp = x / ((1.0 + y) * y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 3.6e-72)
		tmp = Float64(y / Float64(Float64(1.0 + x) * x));
	else
		tmp = Float64(x / Float64(Float64(1.0 + y) * y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.6e-72)
		tmp = y / ((1.0 + x) * x);
	else
		tmp = x / ((1.0 + y) * y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 3.6e-72], N[(y / N[(N[(1.0 + x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[(1.0 + y), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.6e-72

   1. Initial program 66.2%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{x \cdot \left(1 + x\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{y}{\left(1 + x\right) \cdot \color{blue}{x}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{y}{\left(1 + x\right) \cdot \color{blue}{x}} \]

      4. lower-+.f64 80.5

         \[\leadsto \frac{y}{\left(1 + x\right) \cdot x} \]

   4. Applied rewrites 80.5%

      \[\leadsto \color{blue}{\frac{y}{\left(1 + x\right) \cdot x}} \]

   if 3.6e-72 < y

   1. Initial program 71.0%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot \left(1 + y\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{x}{\left(1 + y\right) \cdot \color{blue}{y}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{x}{\left(1 + y\right) \cdot \color{blue}{y}} \]

      4. lower-+.f64 76.8

         \[\leadsto \frac{x}{\left(1 + y\right) \cdot y} \]

   4. Applied rewrites 76.8%

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

Alternative 12: 77.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -5100000000:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(1 + y\right) \cdot y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -5100000000.0) (/ y (* x x)) (/ x (* (+ 1.0 y) y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -5100000000.0) {
		tmp = y / (x * x);
	} else {
		tmp = x / ((1.0 + y) * y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
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(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-5100000000.0d0)) then
        tmp = y / (x * x)
    else
        tmp = x / ((1.0d0 + y) * y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -5100000000.0) {
		tmp = y / (x * x);
	} else {
		tmp = x / ((1.0 + y) * y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -5100000000.0:
		tmp = y / (x * x)
	else:
		tmp = x / ((1.0 + y) * y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -5100000000.0)
		tmp = Float64(y / Float64(x * x));
	else
		tmp = Float64(x / Float64(Float64(1.0 + y) * y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -5100000000.0)
		tmp = y / (x * x);
	else
		tmp = x / ((1.0 + y) * y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -5100000000.0], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[(1.0 + y), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.1e9

   1. Initial program 66.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{{x}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{y}{x \cdot \color{blue}{x}} \]

      3. lower-*.f64 80.6

         \[\leadsto \frac{y}{x \cdot \color{blue}{x}} \]

   4. Applied rewrites 80.6%

      \[\leadsto \color{blue}{\frac{y}{x \cdot x}} \]

   if -5.1e9 < x

   1. Initial program 70.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot \left(1 + y\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{x}{\left(1 + y\right) \cdot \color{blue}{y}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{x}{\left(1 + y\right) \cdot \color{blue}{y}} \]

      4. lower-+.f64 74.4

         \[\leadsto \frac{x}{\left(1 + y\right) \cdot y} \]

   4. Applied rewrites 74.4%

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

Alternative 13: 70.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{x}{y \cdot y}\\ \mathbf{if}\;x \leq -5100000000:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq -3.7 \cdot 10^{-180}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{-163}:\\ \;\;\;\;x \cdot \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (* y y))))
   (if (<= x -5100000000.0)
     (/ y (* x x))
     (if (<= x -3.7e-180) t_0 (if (<= x 1.75e-163) (* x (/ 1.0 y)) t_0)))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = x / (y * y);
	double tmp;
	if (x <= -5100000000.0) {
		tmp = y / (x * x);
	} else if (x <= -3.7e-180) {
		tmp = t_0;
	} else if (x <= 1.75e-163) {
		tmp = x * (1.0 / y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
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(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / (y * y)
    if (x <= (-5100000000.0d0)) then
        tmp = y / (x * x)
    else if (x <= (-3.7d-180)) then
        tmp = t_0
    else if (x <= 1.75d-163) then
        tmp = x * (1.0d0 / y)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = x / (y * y);
	double tmp;
	if (x <= -5100000000.0) {
		tmp = y / (x * x);
	} else if (x <= -3.7e-180) {
		tmp = t_0;
	} else if (x <= 1.75e-163) {
		tmp = x * (1.0 / y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = x / (y * y)
	tmp = 0
	if x <= -5100000000.0:
		tmp = y / (x * x)
	elif x <= -3.7e-180:
		tmp = t_0
	elif x <= 1.75e-163:
		tmp = x * (1.0 / y)
	else:
		tmp = t_0
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(x / Float64(y * y))
	tmp = 0.0
	if (x <= -5100000000.0)
		tmp = Float64(y / Float64(x * x));
	elseif (x <= -3.7e-180)
		tmp = t_0;
	elseif (x <= 1.75e-163)
		tmp = Float64(x * Float64(1.0 / y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = x / (y * y);
	tmp = 0.0;
	if (x <= -5100000000.0)
		tmp = y / (x * x);
	elseif (x <= -3.7e-180)
		tmp = t_0;
	elseif (x <= 1.75e-163)
		tmp = x * (1.0 / y);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5100000000.0], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -3.7e-180], t$95$0, If[LessEqual[x, 1.75e-163], N[(x * N[(1.0 / y), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.1e9

   1. Initial program 66.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{{x}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{y}{x \cdot \color{blue}{x}} \]

      3. lower-*.f64 80.6

         \[\leadsto \frac{y}{x \cdot \color{blue}{x}} \]

   4. Applied rewrites 80.6%

      \[\leadsto \color{blue}{\frac{y}{x \cdot x}} \]

   if -5.1e9 < x < -3.70000000000000016e-180 or 1.75000000000000014e-163 < x

   1. Initial program 74.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{{y}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{x}{y \cdot \color{blue}{y}} \]

      3. lower-*.f64 53.8

         \[\leadsto \frac{x}{y \cdot \color{blue}{y}} \]

   4. Applied rewrites 53.8%

      \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \]

   if -3.70000000000000016e-180 < x < 1.75000000000000014e-163

   1. Initial program 64.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\color{blue}{\left(x + y\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\left(x + y\right)} + 1\right)} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(\left(x + y\right) + 1\right)}} \]

      9. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      11. lower-/.f64 N/A

          \[\leadsto x \cdot \color{blue}{\frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      12. *-commutative N/A

          \[\leadsto x \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      13. lower-*.f64 N/A

          \[\leadsto x \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      14. lift-+.f64 N/A

          \[\leadsto x \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right)} \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)} \]

      15. +-commutative N/A

          \[\leadsto x \cdot \frac{y}{\left(\color{blue}{\left(y + x\right)} + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)} \]

      16. lower-+.f64 N/A

          \[\leadsto x \cdot \frac{y}{\left(\color{blue}{\left(y + x\right)} + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)} \]

      17. lower-*.f64 N/A

          \[\leadsto x \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      18. +-commutative N/A

          \[\leadsto x \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \left(\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)\right)} \]

      19. lower-+.f64 N/A

          \[\leadsto x \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \left(\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)\right)} \]

      20. +-commutative N/A

          \[\leadsto x \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}\right)} \]

      21. lower-+.f64 78.2

          \[\leadsto x \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}\right)} \]

   3. Applied rewrites 78.2%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

   4. Taylor expanded in x around 0

      \[\leadsto x \cdot \color{blue}{\frac{1}{y \cdot \left(1 + y\right)}} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto x \cdot \frac{1}{\color{blue}{y \cdot \left(1 + y\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto x \cdot \frac{1}{y \cdot \color{blue}{\left(1 + y\right)}} \]

      3. lower-+.f64 89.5

         \[\leadsto x \cdot \frac{1}{y \cdot \left(1 + \color{blue}{y}\right)} \]

   6. Applied rewrites 89.5%

      \[\leadsto x \cdot \color{blue}{\frac{1}{y \cdot \left(1 + y\right)}} \]

   7. Taylor expanded in y around 0

      \[\leadsto x \cdot \frac{1}{y} \]
   8. Step-by-step derivation

   9. Applied rewrites 78.5%

      \[\leadsto x \cdot \frac{1}{y} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 65.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -5100000000:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -5100000000.0) (/ y (* x x)) (/ x (* y y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -5100000000.0) {
		tmp = y / (x * x);
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
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(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-5100000000.0d0)) then
        tmp = y / (x * x)
    else
        tmp = x / (y * y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -5100000000.0) {
		tmp = y / (x * x);
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -5100000000.0:
		tmp = y / (x * x)
	else:
		tmp = x / (y * y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -5100000000.0)
		tmp = Float64(y / Float64(x * x));
	else
		tmp = Float64(x / Float64(y * y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -5100000000.0)
		tmp = y / (x * x);
	else
		tmp = x / (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -5100000000.0], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.1e9

   1. Initial program 66.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{{x}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{y}{x \cdot \color{blue}{x}} \]

      3. lower-*.f64 80.6

         \[\leadsto \frac{y}{x \cdot \color{blue}{x}} \]

   4. Applied rewrites 80.6%

      \[\leadsto \color{blue}{\frac{y}{x \cdot x}} \]

   if -5.1e9 < x

   1. Initial program 70.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{{y}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{x}{y \cdot \color{blue}{y}} \]

      3. lower-*.f64 53.7

         \[\leadsto \frac{x}{y \cdot \color{blue}{y}} \]

   4. Applied rewrites 53.7%

      \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 37.3% accurate, 3.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \frac{x}{y \cdot y} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (/ x (* y y)))

C

assert(x < y);
double code(double x, double y) {
	return x / (y * y);
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
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(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x / (y * y)
end function

Java

assert x < y;
public static double code(double x, double y) {
	return x / (y * y);
}

Python

[x, y] = sort([x, y])
def code(x, y):
	return x / (y * y)

Julia

x, y = sort([x, y])
function code(x, y)
	return Float64(x / Float64(y * y))
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = x / (y * y);
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 68.7%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

2. Taylor expanded in y around inf

   \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{x}{\color{blue}{{y}^{2}}} \]

   2. unpow2 N/A

      \[\leadsto \frac{x}{y \cdot \color{blue}{y}} \]

   3. lower-*.f64 37.3

      \[\leadsto \frac{x}{y \cdot \color{blue}{y}} \]

4. Applied rewrites 37.3%

   \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025120 
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, A"
  :precision binary64
  (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0))))