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

Percentage Accurate: 68.8% → 93.5%

Time: 4.2s

Alternatives: 16

Speedup: 1.5×

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: 68.8% 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: 93.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{+94}:\\ \;\;\;\;\frac{\frac{y}{x}}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{y}{y + x}}{\left(\left(y + x\right) + 1\right) \cdot \left(y + x\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.2e+94)
   (/ (/ y x) (+ 1.0 x))
   (* x (/ (/ y (+ y x)) (* (+ (+ y x) 1.0) (+ y x))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.2e+94) {
		tmp = (y / x) / (1.0 + x);
	} else {
		tmp = x * ((y / (y + x)) / (((y + x) + 1.0) * (y + x)));
	}
	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 <= (-1.2d+94)) then
        tmp = (y / x) / (1.0d0 + x)
    else
        tmp = x * ((y / (y + x)) / (((y + x) + 1.0d0) * (y + x)))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -1.2e+94) {
		tmp = (y / x) / (1.0 + x);
	} else {
		tmp = x * ((y / (y + x)) / (((y + x) + 1.0) * (y + x)));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1.2e+94:
		tmp = (y / x) / (1.0 + x)
	else:
		tmp = x * ((y / (y + x)) / (((y + x) + 1.0) * (y + x)))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.2e+94)
		tmp = Float64(Float64(y / x) / Float64(1.0 + x));
	else
		tmp = Float64(x * Float64(Float64(y / Float64(y + x)) / Float64(Float64(Float64(y + x) + 1.0) * Float64(y + x))));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.2e+94)
		tmp = (y / x) / (1.0 + x);
	else
		tmp = x * ((y / (y + x)) / (((y + x) + 1.0) * (y + x)));
	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, -1.2e+94], N[(N[(y / x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.19999999999999991e94

   1. Initial program 53.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   4. 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 87.6

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

   5. Applied rewrites 87.6%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-/r* N/A

         \[\leadsto \frac{\frac{y}{x}}{\color{blue}{1 + x}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\frac{y}{x}}{\color{blue}{1 + x}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\frac{y}{x}}{\color{blue}{1} + x} \]

      8. lift-+.f64 91.1

         \[\leadsto \frac{\frac{y}{x}}{1 + \color{blue}{x}} \]

   7. Applied rewrites 91.1%

      \[\leadsto \frac{\frac{y}{x}}{\color{blue}{1 + x}} \]

   if -1.19999999999999991e94 < x

   1. Initial program 72.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. 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

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      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. pow2 N/A

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

      18. lower-pow.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 85.3

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

   4. Applied rewrites 85.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. pow2 N/A

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

      9. +-commutative N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lift-+.f64 N/A

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

      19. lift-+.f64 85.4

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

   6. Applied rewrites 85.4%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. lift-+.f64 92.9

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

   8. Applied rewrites 92.9%

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

Alternative 2: 92.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{+94}:\\ \;\;\;\;\frac{\frac{y}{x}}{1 + x}\\ \mathbf{elif}\;x \leq -6.2 \cdot 10^{-15}:\\ \;\;\;\;x \cdot \frac{y}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(\left(y + x\right) + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{y}{y + x}}{\left(y + 1\right) \cdot \left(y + x\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.2e+94)
   (/ (/ y x) (+ 1.0 x))
   (if (<= x -6.2e-15)
     (* x (/ y (* (+ y x) (* (+ y x) (+ (+ y x) 1.0)))))
     (* x (/ (/ y (+ y x)) (* (+ y 1.0) (+ y x)))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.2e+94) {
		tmp = (y / x) / (1.0 + x);
	} else if (x <= -6.2e-15) {
		tmp = x * (y / ((y + x) * ((y + x) * ((y + x) + 1.0))));
	} else {
		tmp = x * ((y / (y + x)) / ((y + 1.0) * (y + x)));
	}
	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 <= (-1.2d+94)) then
        tmp = (y / x) / (1.0d0 + x)
    else if (x <= (-6.2d-15)) then
        tmp = x * (y / ((y + x) * ((y + x) * ((y + x) + 1.0d0))))
    else
        tmp = x * ((y / (y + x)) / ((y + 1.0d0) * (y + x)))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -1.2e+94) {
		tmp = (y / x) / (1.0 + x);
	} else if (x <= -6.2e-15) {
		tmp = x * (y / ((y + x) * ((y + x) * ((y + x) + 1.0))));
	} else {
		tmp = x * ((y / (y + x)) / ((y + 1.0) * (y + x)));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1.2e+94:
		tmp = (y / x) / (1.0 + x)
	elif x <= -6.2e-15:
		tmp = x * (y / ((y + x) * ((y + x) * ((y + x) + 1.0))))
	else:
		tmp = x * ((y / (y + x)) / ((y + 1.0) * (y + x)))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.2e+94)
		tmp = Float64(Float64(y / x) / Float64(1.0 + x));
	elseif (x <= -6.2e-15)
		tmp = Float64(x * Float64(y / Float64(Float64(y + x) * Float64(Float64(y + x) * Float64(Float64(y + x) + 1.0)))));
	else
		tmp = Float64(x * Float64(Float64(y / Float64(y + x)) / Float64(Float64(y + 1.0) * Float64(y + x))));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.2e+94)
		tmp = (y / x) / (1.0 + x);
	elseif (x <= -6.2e-15)
		tmp = x * (y / ((y + x) * ((y + x) * ((y + x) + 1.0))));
	else
		tmp = x * ((y / (y + x)) / ((y + 1.0) * (y + x)));
	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, -1.2e+94], N[(N[(y / x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -6.2e-15], N[(x * N[(y / N[(N[(y + x), $MachinePrecision] * N[(N[(y + x), $MachinePrecision] * N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(N[(y + 1.0), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.19999999999999991e94

   1. Initial program 53.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   4. 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 87.6

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

   5. Applied rewrites 87.6%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-/r* N/A

         \[\leadsto \frac{\frac{y}{x}}{\color{blue}{1 + x}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\frac{y}{x}}{\color{blue}{1 + x}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\frac{y}{x}}{\color{blue}{1} + x} \]

      8. lift-+.f64 91.1

         \[\leadsto \frac{\frac{y}{x}}{1 + \color{blue}{x}} \]

   7. Applied rewrites 91.1%

      \[\leadsto \frac{\frac{y}{x}}{\color{blue}{1 + x}} \]

   if -1.19999999999999991e94 < x < -6.1999999999999998e-15

   1. Initial program 81.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. 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

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      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. pow2 N/A

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

      18. lower-pow.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 96.8

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

   4. Applied rewrites 96.8%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. pow2 N/A

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

      9. +-commutative N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lift-+.f64 N/A

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

      19. lift-+.f64 96.8

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

   6. Applied rewrites 96.8%

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

   if -6.1999999999999998e-15 < x

   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. Add Preprocessing
   3. 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

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      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. pow2 N/A

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

      18. lower-pow.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 83.2

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

   4. Applied rewrites 83.2%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. pow2 N/A

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

      9. +-commutative N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lift-+.f64 N/A

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

      19. lift-+.f64 83.2

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

   6. Applied rewrites 83.2%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. lift-+.f64 91.6

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

   8. Applied rewrites 91.6%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 86.1%

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

Alternative 3: 89.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{+94}:\\ \;\;\;\;\frac{\frac{y}{x}}{1 + x}\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-162}:\\ \;\;\;\;x \cdot \frac{y}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(\left(y + x\right) + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, 1, y \cdot y\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.2e+94)
   (/ (/ y x) (+ 1.0 x))
   (if (<= x -1e-162)
     (* x (/ y (* (+ y x) (* (+ y x) (+ (+ y x) 1.0)))))
     (/ x (fma y 1.0 (* y y))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.2e+94) {
		tmp = (y / x) / (1.0 + x);
	} else if (x <= -1e-162) {
		tmp = x * (y / ((y + x) * ((y + x) * ((y + x) + 1.0))));
	} else {
		tmp = x / fma(y, 1.0, (y * y));
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.2e+94)
		tmp = Float64(Float64(y / x) / Float64(1.0 + x));
	elseif (x <= -1e-162)
		tmp = Float64(x * Float64(y / Float64(Float64(y + x) * Float64(Float64(y + x) * Float64(Float64(y + x) + 1.0)))));
	else
		tmp = Float64(x / fma(y, 1.0, Float64(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[x, -1.2e+94], N[(N[(y / x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1e-162], N[(x * N[(y / N[(N[(y + x), $MachinePrecision] * N[(N[(y + x), $MachinePrecision] * N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * 1.0 + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.19999999999999991e94

   1. Initial program 53.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   4. 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 87.6

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

   5. Applied rewrites 87.6%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-/r* N/A

         \[\leadsto \frac{\frac{y}{x}}{\color{blue}{1 + x}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\frac{y}{x}}{\color{blue}{1 + x}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\frac{y}{x}}{\color{blue}{1} + x} \]

      8. lift-+.f64 91.1

         \[\leadsto \frac{\frac{y}{x}}{1 + \color{blue}{x}} \]

   7. Applied rewrites 91.1%

      \[\leadsto \frac{\frac{y}{x}}{\color{blue}{1 + x}} \]

   if -1.19999999999999991e94 < x < -9.99999999999999954e-163

   1. Initial program 87.4%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. 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

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      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. pow2 N/A

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

      18. lower-pow.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 98.0

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

   4. Applied rewrites 98.0%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. pow2 N/A

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

      9. +-commutative N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lift-+.f64 N/A

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

      19. lift-+.f64 98.0

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

   6. Applied rewrites 98.0%

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

   if -9.99999999999999954e-163 < x

   1. Initial program 66.0%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   4. 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 53.1

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

   5. Applied rewrites 53.1%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. distribute-lft-in N/A

         \[\leadsto \frac{x}{y \cdot 1 + \color{blue}{y \cdot y}} \]

      5. pow2 N/A

         \[\leadsto \frac{x}{y \cdot 1 + {y}^{\color{blue}{2}}} \]

      6. lower-fma.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 53.1

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

   7. Applied rewrites 53.1%

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

Alternative 4: 85.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{+94}:\\ \;\;\;\;\frac{\frac{y}{x}}{1 + x}\\ \mathbf{elif}\;x \leq -7.5 \cdot 10^{-159}:\\ \;\;\;\;x \cdot \frac{y}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, 1, y \cdot y\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.1e+94)
   (/ (/ y x) (+ 1.0 x))
   (if (<= x -7.5e-159)
     (* x (/ y (* (+ y x) (* (+ y x) (+ x 1.0)))))
     (/ x (fma y 1.0 (* y y))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.1e+94) {
		tmp = (y / x) / (1.0 + x);
	} else if (x <= -7.5e-159) {
		tmp = x * (y / ((y + x) * ((y + x) * (x + 1.0))));
	} else {
		tmp = x / fma(y, 1.0, (y * y));
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.1e+94)
		tmp = Float64(Float64(y / x) / Float64(1.0 + x));
	elseif (x <= -7.5e-159)
		tmp = Float64(x * Float64(y / Float64(Float64(y + x) * Float64(Float64(y + x) * Float64(x + 1.0)))));
	else
		tmp = Float64(x / fma(y, 1.0, Float64(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[x, -1.1e+94], N[(N[(y / x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -7.5e-159], N[(x * N[(y / N[(N[(y + x), $MachinePrecision] * N[(N[(y + x), $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * 1.0 + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.10000000000000006e94

   1. Initial program 53.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   4. 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 87.6

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

   5. Applied rewrites 87.6%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-/r* N/A

         \[\leadsto \frac{\frac{y}{x}}{\color{blue}{1 + x}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\frac{y}{x}}{\color{blue}{1 + x}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\frac{y}{x}}{\color{blue}{1} + x} \]

      8. lift-+.f64 91.1

         \[\leadsto \frac{\frac{y}{x}}{1 + \color{blue}{x}} \]

   7. Applied rewrites 91.1%

      \[\leadsto \frac{\frac{y}{x}}{\color{blue}{1 + x}} \]

   if -1.10000000000000006e94 < x < -7.5e-159

   1. Initial program 87.4%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. 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

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      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. pow2 N/A

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

      18. lower-pow.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 98.0

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

   4. Applied rewrites 98.0%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. pow2 N/A

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

      9. +-commutative N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lift-+.f64 N/A

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

      19. lift-+.f64 98.0

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

   6. Applied rewrites 98.0%

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

   7. Taylor expanded in x around inf

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

   9. Applied rewrites 79.9%

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

   if -7.5e-159 < x

   1. Initial program 66.0%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   4. 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 53.1

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

   5. Applied rewrites 53.1%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. distribute-lft-in N/A

         \[\leadsto \frac{x}{y \cdot 1 + \color{blue}{y \cdot y}} \]

      5. pow2 N/A

         \[\leadsto \frac{x}{y \cdot 1 + {y}^{\color{blue}{2}}} \]

      6. lower-fma.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 53.1

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

   7. Applied rewrites 53.1%

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

Alternative 5: 80.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{+94}:\\ \;\;\;\;\frac{\frac{y}{x}}{1 + x}\\ \mathbf{elif}\;x \leq -3.2 \cdot 10^{-86}:\\ \;\;\;\;x \cdot \frac{y}{\left(y + x\right) \cdot \left(\left(1 + x\right) \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, 1, y \cdot y\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.1e+94)
   (/ (/ y x) (+ 1.0 x))
   (if (<= x -3.2e-86)
     (* x (/ y (* (+ y x) (* (+ 1.0 x) x))))
     (/ x (fma y 1.0 (* y y))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.1e+94) {
		tmp = (y / x) / (1.0 + x);
	} else if (x <= -3.2e-86) {
		tmp = x * (y / ((y + x) * ((1.0 + x) * x)));
	} else {
		tmp = x / fma(y, 1.0, (y * y));
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.1e+94)
		tmp = Float64(Float64(y / x) / Float64(1.0 + x));
	elseif (x <= -3.2e-86)
		tmp = Float64(x * Float64(y / Float64(Float64(y + x) * Float64(Float64(1.0 + x) * x))));
	else
		tmp = Float64(x / fma(y, 1.0, Float64(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[x, -1.1e+94], N[(N[(y / x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -3.2e-86], N[(x * N[(y / N[(N[(y + x), $MachinePrecision] * N[(N[(1.0 + x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * 1.0 + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.10000000000000006e94

   1. Initial program 53.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   4. 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 87.6

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

   5. Applied rewrites 87.6%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-/r* N/A

         \[\leadsto \frac{\frac{y}{x}}{\color{blue}{1 + x}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\frac{y}{x}}{\color{blue}{1 + x}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\frac{y}{x}}{\color{blue}{1} + x} \]

      8. lift-+.f64 91.1

         \[\leadsto \frac{\frac{y}{x}}{1 + \color{blue}{x}} \]

   7. Applied rewrites 91.1%

      \[\leadsto \frac{\frac{y}{x}}{\color{blue}{1 + x}} \]

   if -1.10000000000000006e94 < x < -3.20000000000000006e-86

   1. Initial program 87.4%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. 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

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      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. pow2 N/A

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

      18. lower-pow.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 97.8

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

   4. Applied rewrites 97.8%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. pow2 N/A

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

      9. +-commutative N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lift-+.f64 N/A

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

      19. lift-+.f64 97.9

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

   6. Applied rewrites 97.9%

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

   7. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 63.0

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

   9. Applied rewrites 63.0%

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

   if -3.20000000000000006e-86 < x

   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. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   4. 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 53.6

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

   5. Applied rewrites 53.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. distribute-lft-in N/A

         \[\leadsto \frac{x}{y \cdot 1 + \color{blue}{y \cdot y}} \]

      5. pow2 N/A

         \[\leadsto \frac{x}{y \cdot 1 + {y}^{\color{blue}{2}}} \]

      6. lower-fma.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 53.6

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

   7. Applied rewrites 53.6%

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

Alternative 6: 72.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq -6 \cdot 10^{-114}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-x, y, y\right)}{x}\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-184}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.0)
   (/ y (* x x))
   (if (<= x -6e-114)
     (/ (fma (- x) y y) x)
     (if (<= x 2.8e-184) (/ x y) (/ x (* y y))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = y / (x * x);
	} else if (x <= -6e-114) {
		tmp = fma(-x, y, y) / x;
	} else if (x <= 2.8e-184) {
		tmp = x / y;
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(y / Float64(x * x));
	elseif (x <= -6e-114)
		tmp = Float64(fma(Float64(-x), y, y) / x);
	elseif (x <= 2.8e-184)
		tmp = Float64(x / y);
	else
		tmp = Float64(x / Float64(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[x, -1.0], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -6e-114], N[(N[((-x) * y + y), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 2.8e-184], N[(x / y), $MachinePrecision], N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1

   1. Initial program 61.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]
   4. 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 75.3

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

   5. Applied rewrites 75.3%

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

   if -1 < x < -6.0000000000000003e-114

   1. Initial program 94.0%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   4. 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 49.5

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

   5. Applied rewrites 49.5%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{y}{x} \]
   7. Step-by-step derivation

   8. Applied rewrites 48.6%

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

   9. Taylor expanded in x around 0

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

       1. lower-/.f64 N/A

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

       2. +-commutative N/A

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

       3. associate-*r* N/A

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

       4. mul-1-neg N/A

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

       5. lower-fma.f64 N/A

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

       6. lower-neg.f64 49.5

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

   11. Applied rewrites 49.5%

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

   if -6.0000000000000003e-114 < x < 2.7999999999999998e-184

   1. Initial program 63.4%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   4. 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 85.6

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

   5. Applied rewrites 85.6%

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

   6. Taylor expanded in y around 0

      \[\leadsto \frac{x}{y} \]
   7. Step-by-step derivation

   8. Applied rewrites 76.4%

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

   if 2.7999999999999998e-184 < x

   1. Initial program 68.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   4. 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 31.0

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

   5. Applied rewrites 31.0%

      \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 7: 72.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq -6 \cdot 10^{-114}:\\ \;\;\;\;\mathsf{fma}\left(-1, y, \frac{y}{x}\right)\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-184}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.0)
   (/ y (* x x))
   (if (<= x -6e-114)
     (fma -1.0 y (/ y x))
     (if (<= x 2.8e-184) (/ x y) (/ x (* y y))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = y / (x * x);
	} else if (x <= -6e-114) {
		tmp = fma(-1.0, y, (y / x));
	} else if (x <= 2.8e-184) {
		tmp = x / y;
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(y / Float64(x * x));
	elseif (x <= -6e-114)
		tmp = fma(-1.0, y, Float64(y / x));
	elseif (x <= 2.8e-184)
		tmp = Float64(x / y);
	else
		tmp = Float64(x / Float64(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[x, -1.0], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -6e-114], N[(-1.0 * y + N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.8e-184], N[(x / y), $MachinePrecision], N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1

   1. Initial program 61.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]
   4. 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 75.3

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

   5. Applied rewrites 75.3%

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

   if -1 < x < -6.0000000000000003e-114

   1. Initial program 94.0%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   4. 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 49.5

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

   5. Applied rewrites 49.5%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{y}{x} \]
   7. Step-by-step derivation

   8. Applied rewrites 48.6%

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

   9. Taylor expanded in x around 0

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

       1. lower-/.f64 N/A

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

       2. +-commutative N/A

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

       3. associate-*r* N/A

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

       4. mul-1-neg N/A

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

       5. lower-fma.f64 N/A

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

       6. lower-neg.f64 49.5

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

   11. Applied rewrites 49.5%

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

   12. Taylor expanded in x around inf

       \[\leadsto -1 \cdot y + \frac{y}{\color{blue}{x}} \]
   13. Step-by-step derivation

       1. lower-fma.f64 N/A

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

       2. lower-/.f64 49.5

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

   14. Applied rewrites 49.5%

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

   if -6.0000000000000003e-114 < x < 2.7999999999999998e-184

   1. Initial program 63.4%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   4. 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 85.6

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

   5. Applied rewrites 85.6%

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

   6. Taylor expanded in y around 0

      \[\leadsto \frac{x}{y} \]
   7. Step-by-step derivation

   8. Applied rewrites 76.4%

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

   if 2.7999999999999998e-184 < x

   1. Initial program 68.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   4. 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 31.0

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

   5. Applied rewrites 31.0%

      \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 8: 71.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq -6 \cdot 10^{-114}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-184}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.0)
   (/ y (* x x))
   (if (<= x -6e-114) (/ y x) (if (<= x 2.8e-184) (/ x y) (/ x (* y y))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = y / (x * x);
	} else if (x <= -6e-114) {
		tmp = y / x;
	} else if (x <= 2.8e-184) {
		tmp = x / y;
	} 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 <= (-1.0d0)) then
        tmp = y / (x * x)
    else if (x <= (-6d-114)) then
        tmp = y / x
    else if (x <= 2.8d-184) then
        tmp = x / y
    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 <= -1.0) {
		tmp = y / (x * x);
	} else if (x <= -6e-114) {
		tmp = y / x;
	} else if (x <= 2.8e-184) {
		tmp = x / y;
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1.0:
		tmp = y / (x * x)
	elif x <= -6e-114:
		tmp = y / x
	elif x <= 2.8e-184:
		tmp = x / y
	else:
		tmp = x / (y * y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(y / Float64(x * x));
	elseif (x <= -6e-114)
		tmp = Float64(y / x);
	elseif (x <= 2.8e-184)
		tmp = Float64(x / y);
	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 <= -1.0)
		tmp = y / (x * x);
	elseif (x <= -6e-114)
		tmp = y / x;
	elseif (x <= 2.8e-184)
		tmp = x / y;
	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, -1.0], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -6e-114], N[(y / x), $MachinePrecision], If[LessEqual[x, 2.8e-184], N[(x / y), $MachinePrecision], N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1

   1. Initial program 61.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]
   4. 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 75.3

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

   5. Applied rewrites 75.3%

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

   if -1 < x < -6.0000000000000003e-114

   1. Initial program 94.0%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   4. 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 49.5

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

   5. Applied rewrites 49.5%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{y}{x} \]
   7. Step-by-step derivation

   8. Applied rewrites 48.6%

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

   if -6.0000000000000003e-114 < x < 2.7999999999999998e-184

   1. Initial program 63.4%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   4. 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 85.6

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

   5. Applied rewrites 85.6%

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

   6. Taylor expanded in y around 0

      \[\leadsto \frac{x}{y} \]
   7. Step-by-step derivation

   8. Applied rewrites 76.4%

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

   if 2.7999999999999998e-184 < x

   1. Initial program 68.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   4. 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 31.0

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

   5. Applied rewrites 31.0%

      \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 9: 77.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq -3.2 \cdot 10^{-86}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-x, y, y\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(1 + y\right) \cdot y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.0)
   (/ y (* x x))
   (if (<= x -3.2e-86) (/ (fma (- x) y y) x) (/ x (* (+ 1.0 y) y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = y / (x * x);
	} else if (x <= -3.2e-86) {
		tmp = fma(-x, y, 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 <= -1.0)
		tmp = Float64(y / Float64(x * x));
	elseif (x <= -3.2e-86)
		tmp = Float64(fma(Float64(-x), y, y) / 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[x, -1.0], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -3.2e-86], N[(N[((-x) * y + y), $MachinePrecision] / x), $MachinePrecision], N[(x / N[(N[(1.0 + y), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 61.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]
   4. 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 75.3

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

   5. Applied rewrites 75.3%

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

   if -1 < x < -3.20000000000000006e-86

   1. Initial program 96.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   4. 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 49.4

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

   5. Applied rewrites 49.4%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{y}{x} \]
   7. Step-by-step derivation

   8. Applied rewrites 48.2%

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

   9. Taylor expanded in x around 0

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

       1. lower-/.f64 N/A

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

       2. +-commutative N/A

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

       3. associate-*r* N/A

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

       4. mul-1-neg N/A

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

       5. lower-fma.f64 N/A

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

       6. lower-neg.f64 49.4

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

   11. Applied rewrites 49.4%

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

   if -3.20000000000000006e-86 < x

   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. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   4. 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 53.6

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

   5. Applied rewrites 53.6%

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

Alternative 10: 80.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-86}:\\ \;\;\;\;\frac{\frac{y}{x}}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, 1, y \cdot y\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -3.2e-86) (/ (/ y x) (+ 1.0 x)) (/ x (fma y 1.0 (* y y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -3.2e-86) {
		tmp = (y / x) / (1.0 + x);
	} else {
		tmp = x / fma(y, 1.0, (y * y));
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -3.2e-86)
		tmp = Float64(Float64(y / x) / Float64(1.0 + x));
	else
		tmp = Float64(x / fma(y, 1.0, Float64(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[x, -3.2e-86], N[(N[(y / x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * 1.0 + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.20000000000000006e-86

   1. Initial program 72.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   4. 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 67.2

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

   5. Applied rewrites 67.2%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-/r* N/A

         \[\leadsto \frac{\frac{y}{x}}{\color{blue}{1 + x}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\frac{y}{x}}{\color{blue}{1 + x}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\frac{y}{x}}{\color{blue}{1} + x} \]

      8. lift-+.f64 68.8

         \[\leadsto \frac{\frac{y}{x}}{1 + \color{blue}{x}} \]

   7. Applied rewrites 68.8%

      \[\leadsto \frac{\frac{y}{x}}{\color{blue}{1 + x}} \]

   if -3.20000000000000006e-86 < x

   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. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   4. 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 53.6

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

   5. Applied rewrites 53.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. distribute-lft-in N/A

         \[\leadsto \frac{x}{y \cdot 1 + \color{blue}{y \cdot y}} \]

      5. pow2 N/A

         \[\leadsto \frac{x}{y \cdot 1 + {y}^{\color{blue}{2}}} \]

      6. lower-fma.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 53.6

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

   7. Applied rewrites 53.6%

      \[\leadsto \frac{x}{\mathsf{fma}\left(y, \color{blue}{1}, y \cdot y\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 63.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 4.3 \cdot 10^{-145}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 4.3e-145) (/ y x) (if (<= y 1.0) (/ x y) (/ x (* y y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 4.3e-145) {
		tmp = y / x;
	} else if (y <= 1.0) {
		tmp = x / y;
	} 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 (y <= 4.3d-145) then
        tmp = y / x
    else if (y <= 1.0d0) then
        tmp = x / y
    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 (y <= 4.3e-145) {
		tmp = y / x;
	} else if (y <= 1.0) {
		tmp = x / y;
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 4.3e-145:
		tmp = y / x
	elif y <= 1.0:
		tmp = x / y
	else:
		tmp = x / (y * y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 4.3e-145)
		tmp = Float64(y / x);
	elseif (y <= 1.0)
		tmp = Float64(x / y);
	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 (y <= 4.3e-145)
		tmp = y / x;
	elseif (y <= 1.0)
		tmp = x / y;
	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[y, 4.3e-145], N[(y / x), $MachinePrecision], If[LessEqual[y, 1.0], N[(x / y), $MachinePrecision], N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 4.2999999999999999e-145

   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. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   4. 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 58.8

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

   5. Applied rewrites 58.8%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{y}{x} \]
   7. Step-by-step derivation

   8. Applied rewrites 40.9%

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

   if 4.2999999999999999e-145 < y < 1

   1. Initial program 89.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   4. 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 37.9

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

   5. Applied rewrites 37.9%

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

   6. Taylor expanded in y around 0

      \[\leadsto \frac{x}{y} \]
   7. Step-by-step derivation

   8. Applied rewrites 37.6%

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

   if 1 < y

   1. Initial program 68.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   4. 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 72.4

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

   5. Applied rewrites 72.4%

      \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 78.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-86}:\\ \;\;\;\;\frac{y}{\left(1 + x\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, 1, y \cdot y\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -3.2e-86) (/ y (* (+ 1.0 x) x)) (/ x (fma y 1.0 (* y y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -3.2e-86) {
		tmp = y / ((1.0 + x) * x);
	} else {
		tmp = x / fma(y, 1.0, (y * y));
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -3.2e-86)
		tmp = Float64(y / Float64(Float64(1.0 + x) * x));
	else
		tmp = Float64(x / fma(y, 1.0, Float64(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[x, -3.2e-86], N[(y / N[(N[(1.0 + x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * 1.0 + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.20000000000000006e-86

   1. Initial program 72.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   4. 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 67.2

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

   5. Applied rewrites 67.2%

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

   if -3.20000000000000006e-86 < x

   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. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   4. 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 53.6

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

   5. Applied rewrites 53.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. distribute-lft-in N/A

         \[\leadsto \frac{x}{y \cdot 1 + \color{blue}{y \cdot y}} \]

      5. pow2 N/A

         \[\leadsto \frac{x}{y \cdot 1 + {y}^{\color{blue}{2}}} \]

      6. lower-fma.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 53.6

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

   7. Applied rewrites 53.6%

      \[\leadsto \frac{x}{\mathsf{fma}\left(y, \color{blue}{1}, y \cdot y\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 78.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-86}:\\ \;\;\;\;\frac{y}{\left(1 + x\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(1 + y\right) \cdot y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -3.2e-86) (/ y (* (+ 1.0 x) x)) (/ x (* (+ 1.0 y) y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -3.2e-86) {
		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 (x <= (-3.2d-86)) 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 (x <= -3.2e-86) {
		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 x <= -3.2e-86:
		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 (x <= -3.2e-86)
		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 (x <= -3.2e-86)
		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[x, -3.2e-86], 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 x < -3.20000000000000006e-86

   1. Initial program 72.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   4. 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 67.2

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

   5. Applied rewrites 67.2%

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

   if -3.20000000000000006e-86 < x

   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. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   4. 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 53.6

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

   5. Applied rewrites 53.6%

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

Alternative 14: 43.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{-114}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (if (<= x -6e-114) (/ y x) (/ x y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -6e-114) {
		tmp = y / x;
	} else {
		tmp = x / 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 <= (-6d-114)) then
        tmp = y / x
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -6e-114) {
		tmp = y / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -6e-114:
		tmp = y / x
	else:
		tmp = x / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -6e-114)
		tmp = Float64(y / x);
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -6e-114)
		tmp = y / x;
	else
		tmp = x / 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, -6e-114], N[(y / x), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6.0000000000000003e-114

   1. Initial program 72.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   4. 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 66.6

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

   5. Applied rewrites 66.6%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{y}{x} \]
   7. Step-by-step derivation

   8. Applied rewrites 36.6%

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

   if -6.0000000000000003e-114 < x

   1. Initial program 66.9%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   4. 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 53.7

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

   5. Applied rewrites 53.7%

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

   6. Taylor expanded in y around 0

      \[\leadsto \frac{x}{y} \]
   7. Step-by-step derivation

   8. Applied rewrites 34.8%

      \[\leadsto \frac{x}{y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 26.0% accurate, 3.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \frac{x}{y} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (/ x y))

C

assert(x < y);
double code(double x, double y) {
	return x / 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
end function

Java

assert x < y;
public static double code(double x, double y) {
	return x / y;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	return x / y

Julia

x, y = sort([x, y])
function code(x, y)
	return Float64(x / y)
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = x / y;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[(x / y), $MachinePrecision]

Derivation

1. Initial program 69.3%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
4. 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 45.4

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

5. Applied rewrites 45.4%

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

6. Taylor expanded in y around 0

   \[\leadsto \frac{x}{y} \]
7. Step-by-step derivation

8. Applied rewrites 22.4%

   \[\leadsto \frac{x}{y} \]
9. Add Preprocessing

Alternative 16: 3.8% accurate, 13.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ -y \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (- y))

C

assert(x < y);
double code(double x, double y) {
	return -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 = -y
end function

Java

assert x < y;
public static double code(double x, double y) {
	return -y;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	return -y

Julia

x, y = sort([x, y])
function code(x, y)
	return Float64(-y)
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = -y;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := (-y)

Derivation

1. Initial program 69.3%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
4. 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 52.1

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

5. Applied rewrites 52.1%

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

6. Taylor expanded in x around 0

   \[\leadsto \frac{y}{x} \]
7. Step-by-step derivation

8. Applied rewrites 30.0%

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

9. Taylor expanded in x around 0

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

    1. lower-/.f64 N/A

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

    2. +-commutative N/A

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

    3. associate-*r* N/A

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

    4. mul-1-neg N/A

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

    5. lower-fma.f64 N/A

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

    6. lower-neg.f64 16.0

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

11. Applied rewrites 16.0%

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

12. Taylor expanded in x around inf

    \[\leadsto -1 \cdot y \]
13. Step-by-step derivation

    1. mul-1-neg N/A

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

    2. lower-neg.f64 3.8

       \[\leadsto -y \]

14. Applied rewrites 3.8%

    \[\leadsto -y \]
15. Add Preprocessing

Developer Target 1: 99.8% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

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 + 1.0d0) + x)) / (y + x)) / (1.0d0 / (y / (y + x)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2025051 
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, A"
  :precision binary64

  :alt
  (! :herbie-platform default (/ (/ (/ x (+ (+ y 1) x)) (+ y x)) (/ 1 (/ y (+ y x)))))

  (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0))))