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

Percentage Accurate: 69.3% → 99.8%

Time: 6.8s

Alternatives: 14

Speedup: 1.6×

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: 69.3% 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: 99.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \frac{y \cdot \frac{\frac{x}{x + y}}{\left(x + y\right) - -1}}{y + x} \end{array} \]

FPCore

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

C

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

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

Java

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

Python

[x, y] = sort([x, y])
def code(x, y):
	return (y * ((x / (x + y)) / ((x + y) - -1.0))) / (y + x)

Julia

x, y = sort([x, y])
function code(x, y)
	return Float64(Float64(y * Float64(Float64(x / Float64(x + y)) / Float64(Float64(x + y) - -1.0))) / Float64(y + x))
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = (y * ((x / (x + y)) / ((x + y) - -1.0))) / (y + x);
end

Wolfram

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

Derivation

1. Initial program 70.1%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. times-frac N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-+.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 94.2

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

   15. lift-+.f64 N/A

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

   16. metadata-eval N/A

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

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

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

   18. metadata-eval N/A

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

   19. metadata-eval N/A

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

   20. metadata-eval N/A

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

   21. lower--.f64 N/A

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

   22. lift-+.f64 N/A

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

   23. +-commutative N/A

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

   24. lower-+.f64 N/A

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

   25. metadata-eval 94.2

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

   26. lift-+.f64 N/A

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

4. Applied rewrites 94.2%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/r* N/A

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

   6. lift-+.f64 N/A

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

   7. +-commutative N/A

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

   8. associate-*l/ N/A

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

   9. lower-/.f64 N/A

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

6. Applied rewrites 99.8%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. associate-/l* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 99.8

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-+.f64 99.8

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 99.8

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

8. Applied rewrites 99.8%

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

Alternative 2: 87.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 6.5 \cdot 10^{-162}:\\ \;\;\;\;\frac{\frac{y}{\left(y + x\right) - -1} \cdot 1}{y + x}\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{-148}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \mathbf{elif}\;y \leq 3 \cdot 10^{+106}:\\ \;\;\;\;\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x, -3 \cdot \frac{x}{y}, x\right)}{y}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 6.5e-162)
   (/ (* (/ y (- (+ y x) -1.0)) 1.0) (+ y x))
   (if (<= y 2.1e-148)
     (/ x (fma y y y))
     (if (<= y 3e+106)
       (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0)))
       (/ (/ (fma x (* -3.0 (/ x y)) x) y) y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 6.5e-162) {
		tmp = ((y / ((y + x) - -1.0)) * 1.0) / (y + x);
	} else if (y <= 2.1e-148) {
		tmp = x / fma(y, y, y);
	} else if (y <= 3e+106) {
		tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
	} else {
		tmp = (fma(x, (-3.0 * (x / y)), x) / y) / y;
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 6.5e-162)
		tmp = Float64(Float64(Float64(y / Float64(Float64(y + x) - -1.0)) * 1.0) / Float64(y + x));
	elseif (y <= 2.1e-148)
		tmp = Float64(x / fma(y, y, y));
	elseif (y <= 3e+106)
		tmp = Float64(Float64(x * y) / Float64(Float64(Float64(x + y) * Float64(x + y)) * Float64(Float64(x + y) + 1.0)));
	else
		tmp = Float64(Float64(fma(x, Float64(-3.0 * Float64(x / y)), x) / y) / y);
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 6.5e-162], N[(N[(N[(y / N[(N[(y + x), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.1e-148], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3e+106], 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], N[(N[(N[(x * N[(-3.0 * N[(x / y), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] / y), $MachinePrecision] / y), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < 6.49999999999999989e-162

   1. Initial program 69.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 96.1

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

      15. lift-+.f64 N/A

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

      16. metadata-eval N/A

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

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

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

      21. lower--.f64 N/A

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

      22. lift-+.f64 N/A

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

      23. +-commutative N/A

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

      24. lower-+.f64 N/A

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

      25. metadata-eval 96.1

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

      26. lift-+.f64 N/A

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

   4. Applied rewrites 96.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*l/ N/A

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

      9. lower-/.f64 N/A

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in x around inf

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

   9. Applied rewrites 62.7%

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

   if 6.49999999999999989e-162 < y < 2.1e-148

   1. Initial program 28.1%

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

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 75.4

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

   5. Applied rewrites 75.4%

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

   if 2.1e-148 < y < 3.0000000000000001e106

   1. Initial program 89.1%

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

   if 3.0000000000000001e106 < y

   1. Initial program 52.7%

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

   3. Taylor expanded in y around inf

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

      3. lower-/.f64 N/A

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

   5. Applied rewrites 84.5%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 84.7%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 84.5%

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

   12. Taylor expanded in x around inf

       \[\leadsto \frac{\frac{\mathsf{fma}\left(x, -3 \cdot \frac{x}{y}, x\right)}{y}}{y} \]
   13. Step-by-step derivation

   14. Applied rewrites 84.5%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.5 \cdot 10^{-162}:\\ \;\;\;\;\frac{\frac{y}{\left(y + x\right) - -1} \cdot 1}{y + x}\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{-148}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \mathbf{elif}\;y \leq 3 \cdot 10^{+106}:\\ \;\;\;\;\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x, -3 \cdot \frac{x}{y}, x\right)}{y}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.32 \cdot 10^{+131}:\\ \;\;\;\;\frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y} \cdot \frac{y - \mathsf{fma}\left(3, x, 1\right)}{y}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1.32e+131)
   (* (/ x (+ y x)) (/ y (* (- (+ y x) -1.0) (+ y x))))
   (/ (* (/ x y) (/ (- y (fma 3.0 x 1.0)) y)) y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.32e+131) {
		tmp = (x / (y + x)) * (y / (((y + x) - -1.0) * (y + x)));
	} else {
		tmp = ((x / y) * ((y - fma(3.0, x, 1.0)) / y)) / y;
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 1.32e+131)
		tmp = Float64(Float64(x / Float64(y + x)) * Float64(y / Float64(Float64(Float64(y + x) - -1.0) * Float64(y + x))));
	else
		tmp = Float64(Float64(Float64(x / y) * Float64(Float64(y - fma(3.0, x, 1.0)) / y)) / y);
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 1.32e+131], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(y / N[(N[(N[(y + x), $MachinePrecision] - -1.0), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x / y), $MachinePrecision] * N[(N[(y - N[(3.0 * x + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.32e131

   1. Initial program 73.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 97.2

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

      15. lift-+.f64 N/A

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

      16. metadata-eval N/A

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

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

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

      21. lower--.f64 N/A

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

      22. lift-+.f64 N/A

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

      23. +-commutative N/A

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

      24. lower-+.f64 N/A

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

      25. metadata-eval 97.2

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

      26. lift-+.f64 N/A

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

   4. Applied rewrites 97.2%

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

   if 1.32e131 < y

   1. Initial program 52.8%

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

   3. Taylor expanded in y around inf

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

      3. lower-/.f64 N/A

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

   5. Applied rewrites 85.4%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 85.5%

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

Alternative 4: 95.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.32 \cdot 10^{+131}:\\ \;\;\;\;\frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{\frac{y - \mathsf{fma}\left(3, x, 1\right)}{y}}{y}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1.32e+131)
   (* (/ x (+ y x)) (/ y (* (- (+ y x) -1.0) (+ y x))))
   (/ (* x (/ (/ (- y (fma 3.0 x 1.0)) y) y)) y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.32e+131) {
		tmp = (x / (y + x)) * (y / (((y + x) - -1.0) * (y + x)));
	} else {
		tmp = (x * (((y - fma(3.0, x, 1.0)) / y) / y)) / y;
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 1.32e+131)
		tmp = Float64(Float64(x / Float64(y + x)) * Float64(y / Float64(Float64(Float64(y + x) - -1.0) * Float64(y + x))));
	else
		tmp = Float64(Float64(x * Float64(Float64(Float64(y - fma(3.0, x, 1.0)) / y) / y)) / y);
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 1.32e+131], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(y / N[(N[(N[(y + x), $MachinePrecision] - -1.0), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(N[(N[(y - N[(3.0 * x + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.32e131

   1. Initial program 73.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 97.2

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

      15. lift-+.f64 N/A

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

      16. metadata-eval N/A

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

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

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

      21. lower--.f64 N/A

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

      22. lift-+.f64 N/A

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

      23. +-commutative N/A

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

      24. lower-+.f64 N/A

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

      25. metadata-eval 97.2

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

      26. lift-+.f64 N/A

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

   4. Applied rewrites 97.2%

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

   if 1.32e131 < y

   1. Initial program 52.8%

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

   3. Taylor expanded in y around inf

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

      3. lower-/.f64 N/A

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

   5. Applied rewrites 85.4%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 85.5%

      \[\leadsto \frac{\frac{x}{y} \cdot \frac{y - \mathsf{fma}\left(3, x, 1\right)}{y}}{y} \]
   9. Step-by-step derivation

   10. Applied rewrites 85.5%

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

Alternative 5: 95.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.32 \cdot 10^{+131}:\\ \;\;\;\;\frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x, -3 \cdot \frac{x}{y}, x\right)}{y}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1.32e+131)
   (* (/ x (+ y x)) (/ y (* (- (+ y x) -1.0) (+ y x))))
   (/ (/ (fma x (* -3.0 (/ x y)) x) y) y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.32e+131) {
		tmp = (x / (y + x)) * (y / (((y + x) - -1.0) * (y + x)));
	} else {
		tmp = (fma(x, (-3.0 * (x / y)), x) / y) / y;
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 1.32e+131)
		tmp = Float64(Float64(x / Float64(y + x)) * Float64(y / Float64(Float64(Float64(y + x) - -1.0) * Float64(y + x))));
	else
		tmp = Float64(Float64(fma(x, Float64(-3.0 * Float64(x / y)), x) / y) / y);
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 1.32e+131], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(y / N[(N[(N[(y + x), $MachinePrecision] - -1.0), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * N[(-3.0 * N[(x / y), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] / y), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.32e131

   1. Initial program 73.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 97.2

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

      15. lift-+.f64 N/A

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

      16. metadata-eval N/A

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

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

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

      21. lower--.f64 N/A

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

      22. lift-+.f64 N/A

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

      23. +-commutative N/A

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

      24. lower-+.f64 N/A

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

      25. metadata-eval 97.2

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

      26. lift-+.f64 N/A

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

   4. Applied rewrites 97.2%

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

   if 1.32e131 < y

   1. Initial program 52.8%

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

   3. Taylor expanded in y around inf

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

      3. lower-/.f64 N/A

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

   5. Applied rewrites 85.4%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 85.5%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 85.4%

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

   12. Taylor expanded in x around inf

       \[\leadsto \frac{\frac{\mathsf{fma}\left(x, -3 \cdot \frac{x}{y}, x\right)}{y}}{y} \]
   13. Step-by-step derivation

   14. Applied rewrites 85.4%

       \[\leadsto \frac{\frac{\mathsf{fma}\left(x, -3 \cdot \frac{x}{y}, x\right)}{y}}{y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.32 \cdot 10^{+131}:\\ \;\;\;\;\frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x, -3 \cdot \frac{x}{y}, x\right)}{y}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{-158}:\\ \;\;\;\;1 \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y - -1}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -5.2e-158)
   (* 1.0 (/ y (* (- (+ y x) -1.0) (+ y x))))
   (/ (/ x (- y -1.0)) y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -5.2e-158) {
		tmp = 1.0 * (y / (((y + x) - -1.0) * (y + x)));
	} else {
		tmp = (x / (y - -1.0)) / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-5.2d-158)) then
        tmp = 1.0d0 * (y / (((y + x) - (-1.0d0)) * (y + x)))
    else
        tmp = (x / (y - (-1.0d0))) / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -5.2e-158) {
		tmp = 1.0 * (y / (((y + x) - -1.0) * (y + x)));
	} else {
		tmp = (x / (y - -1.0)) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -5.2e-158:
		tmp = 1.0 * (y / (((y + x) - -1.0) * (y + x)))
	else:
		tmp = (x / (y - -1.0)) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -5.2e-158)
		tmp = Float64(1.0 * Float64(y / Float64(Float64(Float64(y + x) - -1.0) * Float64(y + x))));
	else
		tmp = Float64(Float64(x / Float64(y - -1.0)) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -5.2e-158)
		tmp = 1.0 * (y / (((y + x) - -1.0) * (y + x)));
	else
		tmp = (x / (y - -1.0)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -5.2e-158], N[(1.0 * N[(y / N[(N[(N[(y + x), $MachinePrecision] - -1.0), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y - -1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.2000000000000001e-158

   1. Initial program 69.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 93.3

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

      15. lift-+.f64 N/A

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

      16. metadata-eval N/A

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

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

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

      21. lower--.f64 N/A

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

      22. lift-+.f64 N/A

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

      23. +-commutative N/A

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

      24. lower-+.f64 N/A

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

      25. metadata-eval 93.3

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

      26. lift-+.f64 N/A

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

   4. Applied rewrites 93.3%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 75.2%

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

   if -5.2000000000000001e-158 < x

   1. Initial program 70.6%

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

   3. 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

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 60.0

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

   5. Applied rewrites 60.0%

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, y, y\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 60.2%

      \[\leadsto \frac{\frac{x}{y - -1}}{\color{blue}{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 80.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{-74}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+39}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 4e-74)
   (/ y (fma x x x))
   (if (<= y 2e+39) (/ x (fma y y y)) (/ (/ x y) y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 4e-74) {
		tmp = y / fma(x, x, x);
	} else if (y <= 2e+39) {
		tmp = x / fma(y, y, y);
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 4e-74)
		tmp = Float64(y / fma(x, x, x));
	elseif (y <= 2e+39)
		tmp = Float64(x / fma(y, y, y));
	else
		tmp = Float64(Float64(x / y) / y);
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 4e-74], N[(y / N[(x * x + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2e+39], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 3.99999999999999983e-74

   1. Initial program 70.4%

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

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 62.1

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

   5. Applied rewrites 62.1%

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

   if 3.99999999999999983e-74 < y < 1.99999999999999988e39

   1. Initial program 89.3%

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

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 61.7

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

   5. Applied rewrites 61.7%

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

   if 1.99999999999999988e39 < y

   1. Initial program 60.5%

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

   3. Taylor expanded in y around inf

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 85.1

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

   5. Applied rewrites 85.1%

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

Alternative 8: 82.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{-74}:\\ \;\;\;\;\frac{\frac{y}{1 + x}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 + y}}{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 4e-74) (/ (/ y (+ 1.0 x)) (+ y x)) (/ (/ x (+ 1.0 y)) (+ y x))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 4e-74) {
		tmp = (y / (1.0 + x)) / (y + x);
	} else {
		tmp = (x / (1.0 + y)) / (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 (y <= 4d-74) then
        tmp = (y / (1.0d0 + x)) / (y + x)
    else
        tmp = (x / (1.0d0 + y)) / (y + x)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 4e-74) {
		tmp = (y / (1.0 + x)) / (y + x);
	} else {
		tmp = (x / (1.0 + y)) / (y + x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 4e-74:
		tmp = (y / (1.0 + x)) / (y + x)
	else:
		tmp = (x / (1.0 + y)) / (y + x)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 4e-74)
		tmp = Float64(Float64(y / Float64(1.0 + x)) / Float64(y + x));
	else
		tmp = Float64(Float64(x / Float64(1.0 + y)) / Float64(y + x));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 4e-74)
		tmp = (y / (1.0 + x)) / (y + x);
	else
		tmp = (x / (1.0 + y)) / (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[y, 4e-74], N[(N[(y / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(1.0 + y), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.99999999999999983e-74

   1. Initial program 70.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 96.6

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

      15. lift-+.f64 N/A

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

      16. metadata-eval N/A

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

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

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

      21. lower--.f64 N/A

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

      22. lift-+.f64 N/A

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

      23. +-commutative N/A

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

      24. lower-+.f64 N/A

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

      25. metadata-eval 96.6

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

      26. lift-+.f64 N/A

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

   4. Applied rewrites 96.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*l/ N/A

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

      9. lower-/.f64 N/A

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 63.2

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

   9. Applied rewrites 63.2%

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

   if 3.99999999999999983e-74 < y

   1. Initial program 69.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 89.7

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

      15. lift-+.f64 N/A

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

      16. metadata-eval N/A

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

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

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

      21. lower--.f64 N/A

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

      22. lift-+.f64 N/A

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

      23. +-commutative N/A

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

      24. lower-+.f64 N/A

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

      25. metadata-eval 89.7

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

      26. lift-+.f64 N/A

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

   4. Applied rewrites 89.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*l/ N/A

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

      9. lower-/.f64 N/A

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 78.0

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

   9. Applied rewrites 78.0%

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

Alternative 9: 80.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{-74}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 + y}}{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 4e-74) (/ y (fma x x x)) (/ (/ x (+ 1.0 y)) (+ y x))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 4e-74) {
		tmp = y / fma(x, x, x);
	} else {
		tmp = (x / (1.0 + y)) / (y + x);
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 4e-74)
		tmp = Float64(y / fma(x, x, x));
	else
		tmp = Float64(Float64(x / Float64(1.0 + y)) / Float64(y + x));
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 4e-74], N[(y / N[(x * x + x), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(1.0 + y), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.99999999999999983e-74

   1. Initial program 70.4%

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

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 62.1

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

   5. Applied rewrites 62.1%

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

   if 3.99999999999999983e-74 < y

   1. Initial program 69.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 89.7

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

      15. lift-+.f64 N/A

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

      16. metadata-eval N/A

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

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

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

      21. lower--.f64 N/A

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

      22. lift-+.f64 N/A

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

      23. +-commutative N/A

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

      24. lower-+.f64 N/A

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

      25. metadata-eval 89.7

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

      26. lift-+.f64 N/A

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

   4. Applied rewrites 89.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*l/ N/A

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

      9. lower-/.f64 N/A

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 78.0

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

   9. Applied rewrites 78.0%

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

Alternative 10: 80.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{-74}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y - -1}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 4e-74) (/ y (fma x x x)) (/ (/ x (- y -1.0)) y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 4e-74) {
		tmp = y / fma(x, x, x);
	} else {
		tmp = (x / (y - -1.0)) / y;
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 4e-74)
		tmp = Float64(y / fma(x, x, x));
	else
		tmp = Float64(Float64(x / Float64(y - -1.0)) / y);
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 4e-74], N[(y / N[(x * x + x), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y - -1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.99999999999999983e-74

   1. Initial program 70.4%

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

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 62.1

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

   5. Applied rewrites 62.1%

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

   if 3.99999999999999983e-74 < y

   1. Initial program 69.6%

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

   3. 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

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 74.3

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

   5. Applied rewrites 74.3%

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, y, y\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 77.6%

      \[\leadsto \frac{\frac{x}{y - -1}}{\color{blue}{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 78.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{-74}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 4e-74) (/ y (fma x x x)) (/ x (fma y y y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 4e-74) {
		tmp = y / fma(x, x, x);
	} else {
		tmp = x / fma(y, y, y);
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 4e-74)
		tmp = Float64(y / fma(x, x, x));
	else
		tmp = Float64(x / fma(y, y, y));
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 4e-74], N[(y / N[(x * x + x), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.99999999999999983e-74

   1. Initial program 70.4%

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

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 62.1

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

   5. Applied rewrites 62.1%

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

   if 3.99999999999999983e-74 < y

   1. Initial program 69.6%

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

   3. 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

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 74.3

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

   5. Applied rewrites 74.3%

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

Alternative 12: 76.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -950000000000:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -950000000000.0) (/ y (* x x)) (/ x (fma y y y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -950000000000.0) {
		tmp = y / (x * x);
	} else {
		tmp = x / fma(y, y, y);
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -950000000000.0)
		tmp = Float64(y / Float64(x * x));
	else
		tmp = Float64(x / fma(y, 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, -950000000000.0], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -9.5e11

   1. Initial program 61.4%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 76.0

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

   5. Applied rewrites 76.0%

      \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x}} \]
   6. Step-by-step derivation

   7. Applied rewrites 77.5%

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

   if -9.5e11 < 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. 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

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 59.9

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

   5. Applied rewrites 59.9%

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

Alternative 13: 64.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -950000000000:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -950000000000.0) (/ y (* x x)) (/ x (* y y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -950000000000.0) {
		tmp = y / (x * x);
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-950000000000.0d0)) then
        tmp = y / (x * x)
    else
        tmp = x / (y * y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -950000000000.0) {
		tmp = y / (x * x);
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -950000000000.0:
		tmp = y / (x * x)
	else:
		tmp = x / (y * y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -950000000000.0)
		tmp = Float64(y / Float64(x * x));
	else
		tmp = Float64(x / Float64(y * y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -950000000000.0)
		tmp = y / (x * x);
	else
		tmp = x / (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -950000000000.0], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -9.5e11

   1. Initial program 61.4%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 76.0

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

   5. Applied rewrites 76.0%

      \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x}} \]
   6. Step-by-step derivation

   7. Applied rewrites 77.5%

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

   if -9.5e11 < 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

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 96.0

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

      15. lift-+.f64 N/A

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

      16. metadata-eval N/A

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

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

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

      21. lower--.f64 N/A

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

      22. lift-+.f64 N/A

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

      23. +-commutative N/A

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

      24. lower-+.f64 N/A

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

      25. metadata-eval 96.0

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

      26. lift-+.f64 N/A

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

   4. Applied rewrites 96.0%

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

   5. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 41.6

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

   7. Applied rewrites 41.6%

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

Alternative 14: 36.4% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.1%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. times-frac N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-+.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 94.2

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

   15. lift-+.f64 N/A

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

   16. metadata-eval N/A

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

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

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

   18. metadata-eval N/A

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

   19. metadata-eval N/A

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

   20. metadata-eval N/A

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

   21. lower--.f64 N/A

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

   22. lift-+.f64 N/A

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

   23. +-commutative N/A

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

   24. lower-+.f64 N/A

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

   25. metadata-eval 94.2

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

   26. lift-+.f64 N/A

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

4. Applied rewrites 94.2%

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

5. Taylor expanded in y around inf

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

   1. lower-/.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 37.1

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

7. Applied rewrites 37.1%

   \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \]
8. 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 2025008 
(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))))