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

Percentage Accurate: 68.7% → 99.8%

Time: 3.5s

Alternatives: 15

Speedup: 0.8×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 68.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	return ((x / (y + x)) * (y / ((y + x) + 1.0))) / (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 + x)) * (y / ((y + x) + 1.0d0))) / (y + x)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.7%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. times-frac N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. pow2 N/A

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

   13. lower-pow.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 N/A

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

   16. lower-/.f64 N/A

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

   17. lift-+.f64 N/A

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

   18. +-commutative N/A

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

   19. lower-+.f64 87.6

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

3. Applied rewrites 87.6%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-pow.f64 N/A

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

   4. pow2 N/A

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

   5. lift-*.f64 N/A

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

   6. +-commutative N/A

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. lift-+.f64 87.6

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

5. Applied rewrites 87.6%

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

   1. lift-/.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/r* N/A

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

   6. +-commutative N/A

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

   7. +-commutative N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lift-+.f64 99.8

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

7. Applied rewrites 99.8%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. associate-*l/ N/A

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

   10. lower-/.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lift-/.f64 N/A

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

   13. lift-+.f64 N/A

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

   14. lift-+.f64 N/A

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

   15. lift-+.f64 N/A

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

   16. lift-/.f64 N/A

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

   17. lift-+.f64 99.8

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

9. Applied rewrites 99.8%

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

Alternative 2: 71.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{\left(y + x\right) + 1}\\ \mathbf{if}\;x \leq -1.26 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{y + x} \cdot t\_0\\ \mathbf{elif}\;x \leq -3 \cdot 10^{-164}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 + y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (+ (+ y x) 1.0))))
   (if (<= x -1.26e+154)
     (* (/ 1.0 (+ y x)) t_0)
     (if (<= x -3e-164)
       (* (/ x (* (+ y x) (+ y x))) t_0)
       (/ (/ x (+ 1.0 y)) y)))))

C

double code(double x, double y) {
	double t_0 = y / ((y + x) + 1.0);
	double tmp;
	if (x <= -1.26e+154) {
		tmp = (1.0 / (y + x)) * t_0;
	} else if (x <= -3e-164) {
		tmp = (x / ((y + x) * (y + x))) * t_0;
	} else {
		tmp = (x / (1.0 + y)) / y;
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y / ((y + x) + 1.0d0)
    if (x <= (-1.26d+154)) then
        tmp = (1.0d0 / (y + x)) * t_0
    else if (x <= (-3d-164)) then
        tmp = (x / ((y + x) * (y + x))) * t_0
    else
        tmp = (x / (1.0d0 + y)) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y / ((y + x) + 1.0);
	double tmp;
	if (x <= -1.26e+154) {
		tmp = (1.0 / (y + x)) * t_0;
	} else if (x <= -3e-164) {
		tmp = (x / ((y + x) * (y + x))) * t_0;
	} else {
		tmp = (x / (1.0 + y)) / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y / ((y + x) + 1.0)
	tmp = 0
	if x <= -1.26e+154:
		tmp = (1.0 / (y + x)) * t_0
	elif x <= -3e-164:
		tmp = (x / ((y + x) * (y + x))) * t_0
	else:
		tmp = (x / (1.0 + y)) / y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y / Float64(Float64(y + x) + 1.0))
	tmp = 0.0
	if (x <= -1.26e+154)
		tmp = Float64(Float64(1.0 / Float64(y + x)) * t_0);
	elseif (x <= -3e-164)
		tmp = Float64(Float64(x / Float64(Float64(y + x) * Float64(y + x))) * t_0);
	else
		tmp = Float64(Float64(x / Float64(1.0 + y)) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y / ((y + x) + 1.0);
	tmp = 0.0;
	if (x <= -1.26e+154)
		tmp = (1.0 / (y + x)) * t_0;
	elseif (x <= -3e-164)
		tmp = (x / ((y + x) * (y + x))) * t_0;
	else
		tmp = (x / (1.0 + y)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y / N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.26e+154], N[(N[(1.0 / N[(y + x), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[x, -3e-164], N[(N[(x / N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(x / N[(1.0 + y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.26e154

   1. Initial program 57.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. lower-pow.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 80.6

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

   3. Applied rewrites 80.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 80.6

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

   5. Applied rewrites 80.6%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. +-commutative N/A

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

      7. +-commutative N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 99.9

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 88.0%

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

   if -1.26e154 < x < -3.0000000000000001e-164

   1. Initial program 77.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. lower-pow.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 96.8

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

   3. Applied rewrites 96.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 96.8

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

   5. Applied rewrites 96.8%

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

   if -3.0000000000000001e-164 < x

   1. Initial program 67.6%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 56.4

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

   4. Applied rewrites 56.4%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-+.f64 57.5

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

   6. Applied rewrites 57.5%

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

Alternative 3: 94.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y + x\right) + 1\\ \mathbf{if}\;x \leq -1.2 \cdot 10^{+154}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{x}, -2, 1\right)}{x} \cdot \frac{y}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x} \cdot y}{\left(y + x\right) \cdot t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (+ y x) 1.0)))
   (if (<= x -1.2e+154)
     (* (/ (fma (/ y x) -2.0 1.0) x) (/ y t_0))
     (/ (* (/ x (+ y x)) y) (* (+ y x) t_0)))))

C

double code(double x, double y) {
	double t_0 = (y + x) + 1.0;
	double tmp;
	if (x <= -1.2e+154) {
		tmp = (fma((y / x), -2.0, 1.0) / x) * (y / t_0);
	} else {
		tmp = ((x / (y + x)) * y) / ((y + x) * t_0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(y + x) + 1.0)
	tmp = 0.0
	if (x <= -1.2e+154)
		tmp = Float64(Float64(fma(Float64(y / x), -2.0, 1.0) / x) * Float64(y / t_0));
	else
		tmp = Float64(Float64(Float64(x / Float64(y + x)) * y) / Float64(Float64(y + x) * t_0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[x, -1.2e+154], N[(N[(N[(N[(y / x), $MachinePrecision] * -2.0 + 1.0), $MachinePrecision] / x), $MachinePrecision] * N[(y / t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.20000000000000007e154

   1. Initial program 57.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. lower-pow.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 80.6

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

   3. Applied rewrites 80.6%

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

   4. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 88.1

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

   6. Applied rewrites 88.1%

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

   if -1.20000000000000007e154 < x

   1. Initial program 70.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. lower-pow.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 88.5

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

   3. Applied rewrites 88.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 88.5

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

   5. Applied rewrites 88.5%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. +-commutative N/A

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

      7. +-commutative N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 99.7

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

   7. Applied rewrites 99.7%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. frac-times N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. lift-+.f64 95.4

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

   9. Applied rewrites 95.4%

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

Alternative 4: 65.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{y + x} \cdot \frac{y}{\left(y + x\right) + 1}\\ \mathbf{if}\;x \leq -4.5 \cdot 10^{+86}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -1.35 \cdot 10^{-116}:\\ \;\;\;\;\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\\ \mathbf{elif}\;x \leq -1.22 \cdot 10^{-166}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 + y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (/ 1.0 (+ y x)) (/ y (+ (+ y x) 1.0)))))
   (if (<= x -4.5e+86)
     t_0
     (if (<= x -1.35e-116)
       (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0)))
       (if (<= x -1.22e-166) t_0 (/ (/ x (+ 1.0 y)) y))))))

C

double code(double x, double y) {
	double t_0 = (1.0 / (y + x)) * (y / ((y + x) + 1.0));
	double tmp;
	if (x <= -4.5e+86) {
		tmp = t_0;
	} else if (x <= -1.35e-116) {
		tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
	} else if (x <= -1.22e-166) {
		tmp = t_0;
	} else {
		tmp = (x / (1.0 + y)) / y;
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 / (y + x)) * (y / ((y + x) + 1.0d0))
    if (x <= (-4.5d+86)) then
        tmp = t_0
    else if (x <= (-1.35d-116)) then
        tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0d0))
    else if (x <= (-1.22d-166)) then
        tmp = t_0
    else
        tmp = (x / (1.0d0 + y)) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (1.0 / (y + x)) * (y / ((y + x) + 1.0));
	double tmp;
	if (x <= -4.5e+86) {
		tmp = t_0;
	} else if (x <= -1.35e-116) {
		tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
	} else if (x <= -1.22e-166) {
		tmp = t_0;
	} else {
		tmp = (x / (1.0 + y)) / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (1.0 / (y + x)) * (y / ((y + x) + 1.0))
	tmp = 0
	if x <= -4.5e+86:
		tmp = t_0
	elif x <= -1.35e-116:
		tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0))
	elif x <= -1.22e-166:
		tmp = t_0
	else:
		tmp = (x / (1.0 + y)) / y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(1.0 / Float64(y + x)) * Float64(y / Float64(Float64(y + x) + 1.0)))
	tmp = 0.0
	if (x <= -4.5e+86)
		tmp = t_0;
	elseif (x <= -1.35e-116)
		tmp = Float64(Float64(x * y) / Float64(Float64(Float64(x + y) * Float64(x + y)) * Float64(Float64(x + y) + 1.0)));
	elseif (x <= -1.22e-166)
		tmp = t_0;
	else
		tmp = Float64(Float64(x / Float64(1.0 + y)) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (1.0 / (y + x)) * (y / ((y + x) + 1.0));
	tmp = 0.0;
	if (x <= -4.5e+86)
		tmp = t_0;
	elseif (x <= -1.35e-116)
		tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
	elseif (x <= -1.22e-166)
		tmp = t_0;
	else
		tmp = (x / (1.0 + y)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(1.0 / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(y / N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4.5e+86], t$95$0, If[LessEqual[x, -1.35e-116], 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], If[LessEqual[x, -1.22e-166], t$95$0, N[(N[(x / N[(1.0 + y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.49999999999999993e86 or -1.35e-116 < x < -1.22e-166

   1. Initial program 60.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. lower-pow.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 86.1

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

   3. Applied rewrites 86.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 86.1

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

   5. Applied rewrites 86.1%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. +-commutative N/A

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

      7. +-commutative N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 99.8

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 72.9%

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

   if -4.49999999999999993e86 < x < -1.35e-116

   1. Initial program 84.2%

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

   if -1.22e-166 < x

   1. Initial program 67.6%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 56.4

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

   4. Applied rewrites 56.4%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-+.f64 57.4

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

   6. Applied rewrites 57.4%

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

Alternative 5: 63.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{y + x} \cdot \frac{y}{\left(y + x\right) + 1}\\ \mathbf{if}\;x \leq -1.42 \cdot 10^{-5}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -1.35 \cdot 10^{-116}:\\ \;\;\;\;\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(y + 1\right)}\\ \mathbf{elif}\;x \leq -1.22 \cdot 10^{-166}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 + y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (/ 1.0 (+ y x)) (/ y (+ (+ y x) 1.0)))))
   (if (<= x -1.42e-5)
     t_0
     (if (<= x -1.35e-116)
       (/ (* x y) (* (* (+ x y) (+ x y)) (+ y 1.0)))
       (if (<= x -1.22e-166) t_0 (/ (/ x (+ 1.0 y)) y))))))

C

double code(double x, double y) {
	double t_0 = (1.0 / (y + x)) * (y / ((y + x) + 1.0));
	double tmp;
	if (x <= -1.42e-5) {
		tmp = t_0;
	} else if (x <= -1.35e-116) {
		tmp = (x * y) / (((x + y) * (x + y)) * (y + 1.0));
	} else if (x <= -1.22e-166) {
		tmp = t_0;
	} else {
		tmp = (x / (1.0 + y)) / y;
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 / (y + x)) * (y / ((y + x) + 1.0d0))
    if (x <= (-1.42d-5)) then
        tmp = t_0
    else if (x <= (-1.35d-116)) then
        tmp = (x * y) / (((x + y) * (x + y)) * (y + 1.0d0))
    else if (x <= (-1.22d-166)) then
        tmp = t_0
    else
        tmp = (x / (1.0d0 + y)) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (1.0 / (y + x)) * (y / ((y + x) + 1.0));
	double tmp;
	if (x <= -1.42e-5) {
		tmp = t_0;
	} else if (x <= -1.35e-116) {
		tmp = (x * y) / (((x + y) * (x + y)) * (y + 1.0));
	} else if (x <= -1.22e-166) {
		tmp = t_0;
	} else {
		tmp = (x / (1.0 + y)) / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (1.0 / (y + x)) * (y / ((y + x) + 1.0))
	tmp = 0
	if x <= -1.42e-5:
		tmp = t_0
	elif x <= -1.35e-116:
		tmp = (x * y) / (((x + y) * (x + y)) * (y + 1.0))
	elif x <= -1.22e-166:
		tmp = t_0
	else:
		tmp = (x / (1.0 + y)) / y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(1.0 / Float64(y + x)) * Float64(y / Float64(Float64(y + x) + 1.0)))
	tmp = 0.0
	if (x <= -1.42e-5)
		tmp = t_0;
	elseif (x <= -1.35e-116)
		tmp = Float64(Float64(x * y) / Float64(Float64(Float64(x + y) * Float64(x + y)) * Float64(y + 1.0)));
	elseif (x <= -1.22e-166)
		tmp = t_0;
	else
		tmp = Float64(Float64(x / Float64(1.0 + y)) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (1.0 / (y + x)) * (y / ((y + x) + 1.0));
	tmp = 0.0;
	if (x <= -1.42e-5)
		tmp = t_0;
	elseif (x <= -1.35e-116)
		tmp = (x * y) / (((x + y) * (x + y)) * (y + 1.0));
	elseif (x <= -1.22e-166)
		tmp = t_0;
	else
		tmp = (x / (1.0 + y)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(1.0 / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(y / N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.42e-5], t$95$0, If[LessEqual[x, -1.35e-116], N[(N[(x * y), $MachinePrecision] / N[(N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.22e-166], t$95$0, N[(N[(x / N[(1.0 + y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.42e-5 or -1.35e-116 < x < -1.22e-166

   1. Initial program 65.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. lower-pow.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 88.9

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

   3. Applied rewrites 88.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 88.9

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

   5. Applied rewrites 88.9%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. +-commutative N/A

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

      7. +-commutative N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 99.8

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 69.3%

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

   if -1.42e-5 < x < -1.35e-116

   1. Initial program 88.1%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 86.7%

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

   if -1.22e-166 < x

   1. Initial program 67.6%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 56.4

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

   4. Applied rewrites 56.4%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-+.f64 57.4

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

   6. Applied rewrites 57.4%

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

Alternative 6: 63.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{\left(1 + x\right) \cdot x}\\ \mathbf{if}\;x \leq -2 \cdot 10^{+84}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -1.42 \cdot 10^{-5}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -7.5 \cdot 10^{-117}:\\ \;\;\;\;\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(y + 1\right)}\\ \mathbf{elif}\;x \leq -1.22 \cdot 10^{-166}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 + y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (* (+ 1.0 x) x))))
   (if (<= x -2e+84)
     (/ (/ y x) x)
     (if (<= x -1.42e-5)
       t_0
       (if (<= x -7.5e-117)
         (/ (* x y) (* (* (+ x y) (+ x y)) (+ y 1.0)))
         (if (<= x -1.22e-166) t_0 (/ (/ x (+ 1.0 y)) y)))))))

C

double code(double x, double y) {
	double t_0 = y / ((1.0 + x) * x);
	double tmp;
	if (x <= -2e+84) {
		tmp = (y / x) / x;
	} else if (x <= -1.42e-5) {
		tmp = t_0;
	} else if (x <= -7.5e-117) {
		tmp = (x * y) / (((x + y) * (x + y)) * (y + 1.0));
	} else if (x <= -1.22e-166) {
		tmp = t_0;
	} else {
		tmp = (x / (1.0 + y)) / y;
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y / ((1.0d0 + x) * x)
    if (x <= (-2d+84)) then
        tmp = (y / x) / x
    else if (x <= (-1.42d-5)) then
        tmp = t_0
    else if (x <= (-7.5d-117)) then
        tmp = (x * y) / (((x + y) * (x + y)) * (y + 1.0d0))
    else if (x <= (-1.22d-166)) then
        tmp = t_0
    else
        tmp = (x / (1.0d0 + y)) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y / ((1.0 + x) * x);
	double tmp;
	if (x <= -2e+84) {
		tmp = (y / x) / x;
	} else if (x <= -1.42e-5) {
		tmp = t_0;
	} else if (x <= -7.5e-117) {
		tmp = (x * y) / (((x + y) * (x + y)) * (y + 1.0));
	} else if (x <= -1.22e-166) {
		tmp = t_0;
	} else {
		tmp = (x / (1.0 + y)) / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y / ((1.0 + x) * x)
	tmp = 0
	if x <= -2e+84:
		tmp = (y / x) / x
	elif x <= -1.42e-5:
		tmp = t_0
	elif x <= -7.5e-117:
		tmp = (x * y) / (((x + y) * (x + y)) * (y + 1.0))
	elif x <= -1.22e-166:
		tmp = t_0
	else:
		tmp = (x / (1.0 + y)) / y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y / Float64(Float64(1.0 + x) * x))
	tmp = 0.0
	if (x <= -2e+84)
		tmp = Float64(Float64(y / x) / x);
	elseif (x <= -1.42e-5)
		tmp = t_0;
	elseif (x <= -7.5e-117)
		tmp = Float64(Float64(x * y) / Float64(Float64(Float64(x + y) * Float64(x + y)) * Float64(y + 1.0)));
	elseif (x <= -1.22e-166)
		tmp = t_0;
	else
		tmp = Float64(Float64(x / Float64(1.0 + y)) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y / ((1.0 + x) * x);
	tmp = 0.0;
	if (x <= -2e+84)
		tmp = (y / x) / x;
	elseif (x <= -1.42e-5)
		tmp = t_0;
	elseif (x <= -7.5e-117)
		tmp = (x * y) / (((x + y) * (x + y)) * (y + 1.0));
	elseif (x <= -1.22e-166)
		tmp = t_0;
	else
		tmp = (x / (1.0 + y)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y / N[(N[(1.0 + x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2e+84], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, -1.42e-5], t$95$0, If[LessEqual[x, -7.5e-117], N[(N[(x * y), $MachinePrecision] / N[(N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.22e-166], t$95$0, N[(N[(x / N[(1.0 + y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2.00000000000000012e84

   1. Initial program 55.9%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 77.7

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

   4. Applied rewrites 77.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 82.4

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

   6. Applied rewrites 82.4%

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

   if -2.00000000000000012e84 < x < -1.42e-5 or -7.50000000000000066e-117 < x < -1.22e-166

   1. Initial program 81.0%

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

   2. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 45.0

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

   4. Applied rewrites 45.0%

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

   if -1.42e-5 < x < -7.50000000000000066e-117

   1. Initial program 88.1%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 86.7%

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

   if -1.22e-166 < x

   1. Initial program 67.6%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 56.4

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

   4. Applied rewrites 56.4%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-+.f64 57.4

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

   6. Applied rewrites 57.4%

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

Alternative 7: 94.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y + x\right) + 1\\ \mathbf{if}\;x \leq -1.4 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{y + x} \cdot \frac{y}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x} \cdot y}{\left(y + x\right) \cdot t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (+ y x) 1.0)))
   (if (<= x -1.4e+154)
     (* (/ 1.0 (+ y x)) (/ y t_0))
     (/ (* (/ x (+ y x)) y) (* (+ y x) t_0)))))

C

double code(double x, double y) {
	double t_0 = (y + x) + 1.0;
	double tmp;
	if (x <= -1.4e+154) {
		tmp = (1.0 / (y + x)) * (y / t_0);
	} else {
		tmp = ((x / (y + x)) * y) / ((y + x) * t_0);
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (y + x) + 1.0d0
    if (x <= (-1.4d+154)) then
        tmp = (1.0d0 / (y + x)) * (y / t_0)
    else
        tmp = ((x / (y + x)) * y) / ((y + x) * t_0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (y + x) + 1.0;
	double tmp;
	if (x <= -1.4e+154) {
		tmp = (1.0 / (y + x)) * (y / t_0);
	} else {
		tmp = ((x / (y + x)) * y) / ((y + x) * t_0);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y + x) + 1.0
	tmp = 0
	if x <= -1.4e+154:
		tmp = (1.0 / (y + x)) * (y / t_0)
	else:
		tmp = ((x / (y + x)) * y) / ((y + x) * t_0)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y + x) + 1.0)
	tmp = 0.0
	if (x <= -1.4e+154)
		tmp = Float64(Float64(1.0 / Float64(y + x)) * Float64(y / t_0));
	else
		tmp = Float64(Float64(Float64(x / Float64(y + x)) * y) / Float64(Float64(y + x) * t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y + x) + 1.0;
	tmp = 0.0;
	if (x <= -1.4e+154)
		tmp = (1.0 / (y + x)) * (y / t_0);
	else
		tmp = ((x / (y + x)) * y) / ((y + x) * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[x, -1.4e+154], N[(N[(1.0 / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(y / t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.4e154

   1. Initial program 57.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. lower-pow.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 80.6

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

   3. Applied rewrites 80.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 80.6

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

   5. Applied rewrites 80.6%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. +-commutative N/A

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

      7. +-commutative N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 99.9

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 88.0%

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

   if -1.4e154 < x

   1. Initial program 70.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. lower-pow.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 88.5

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

   3. Applied rewrites 88.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 88.5

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

   5. Applied rewrites 88.5%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. +-commutative N/A

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

      7. +-commutative N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 99.7

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

   7. Applied rewrites 99.7%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. frac-times N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. lift-+.f64 95.4

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

   9. Applied rewrites 95.4%

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

Alternative 8: 99.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	return ((x / (y + x)) / (y + x)) * (y / ((y + x) + 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 / ((y + x) + 1.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.7%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. times-frac N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. pow2 N/A

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

   13. lower-pow.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 N/A

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

   16. lower-/.f64 N/A

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

   17. lift-+.f64 N/A

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

   18. +-commutative N/A

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

   19. lower-+.f64 87.6

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

3. Applied rewrites 87.6%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-pow.f64 N/A

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

   4. pow2 N/A

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

   5. lift-*.f64 N/A

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

   6. +-commutative N/A

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. lift-+.f64 87.6

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

5. Applied rewrites 87.6%

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

   1. lift-/.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/r* N/A

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

   6. +-commutative N/A

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

   7. +-commutative N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lift-+.f64 99.8

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

7. Applied rewrites 99.8%

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

Alternative 9: 58.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+150}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -1.22 \cdot 10^{-166}:\\ \;\;\;\;\frac{y}{\left(1 + x\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 + y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2e+150)
   (/ (/ y x) x)
   (if (<= x -1.22e-166) (/ y (* (+ 1.0 x) x)) (/ (/ x (+ 1.0 y)) y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2e+150) {
		tmp = (y / x) / x;
	} else if (x <= -1.22e-166) {
		tmp = y / ((1.0 + x) * x);
	} else {
		tmp = (x / (1.0 + y)) / y;
	}
	return tmp;
}

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
    real(8) :: tmp
    if (x <= (-2d+150)) then
        tmp = (y / x) / x
    else if (x <= (-1.22d-166)) then
        tmp = y / ((1.0d0 + x) * x)
    else
        tmp = (x / (1.0d0 + y)) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -2e+150) {
		tmp = (y / x) / x;
	} else if (x <= -1.22e-166) {
		tmp = y / ((1.0 + x) * x);
	} else {
		tmp = (x / (1.0 + y)) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2e+150:
		tmp = (y / x) / x
	elif x <= -1.22e-166:
		tmp = y / ((1.0 + x) * x)
	else:
		tmp = (x / (1.0 + y)) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2e+150)
		tmp = Float64(Float64(y / x) / x);
	elseif (x <= -1.22e-166)
		tmp = Float64(y / Float64(Float64(1.0 + x) * x));
	else
		tmp = Float64(Float64(x / Float64(1.0 + y)) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2e+150)
		tmp = (y / x) / x;
	elseif (x <= -1.22e-166)
		tmp = y / ((1.0 + x) * x);
	else
		tmp = (x / (1.0 + y)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2e+150], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, -1.22e-166], N[(y / N[(N[(1.0 + x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(1.0 + y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.99999999999999996e150

   1. Initial program 57.7%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 80.6

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

   4. Applied rewrites 80.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 87.4

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

   6. Applied rewrites 87.4%

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

   if -1.99999999999999996e150 < x < -1.22e-166

   1. Initial program 77.2%

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

   2. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 48.1

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

   4. Applied rewrites 48.1%

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

   if -1.22e-166 < x

   1. Initial program 67.6%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 56.4

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

   4. Applied rewrites 56.4%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-+.f64 57.4

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

   6. Applied rewrites 57.4%

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

Alternative 10: 54.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y \cdot y}\\ \mathbf{if}\;x \leq -0.00032:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq -6.5 \cdot 10^{-175}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{-139}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (* y y))))
   (if (<= x -0.00032)
     (/ y (* x x))
     (if (<= x -6.5e-175) t_0 (if (<= x 3.8e-139) (/ x y) t_0)))))

C

double code(double x, double y) {
	double t_0 = x / (y * y);
	double tmp;
	if (x <= -0.00032) {
		tmp = y / (x * x);
	} else if (x <= -6.5e-175) {
		tmp = t_0;
	} else if (x <= 3.8e-139) {
		tmp = x / y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / (y * y)
    if (x <= (-0.00032d0)) then
        tmp = y / (x * x)
    else if (x <= (-6.5d-175)) then
        tmp = t_0
    else if (x <= 3.8d-139) then
        tmp = x / y
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (y * y);
	double tmp;
	if (x <= -0.00032) {
		tmp = y / (x * x);
	} else if (x <= -6.5e-175) {
		tmp = t_0;
	} else if (x <= 3.8e-139) {
		tmp = x / y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (y * y)
	tmp = 0
	if x <= -0.00032:
		tmp = y / (x * x)
	elif x <= -6.5e-175:
		tmp = t_0
	elif x <= 3.8e-139:
		tmp = x / y
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(y * y))
	tmp = 0.0
	if (x <= -0.00032)
		tmp = Float64(y / Float64(x * x));
	elseif (x <= -6.5e-175)
		tmp = t_0;
	elseif (x <= 3.8e-139)
		tmp = Float64(x / y);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (y * y);
	tmp = 0.0;
	if (x <= -0.00032)
		tmp = y / (x * x);
	elseif (x <= -6.5e-175)
		tmp = t_0;
	elseif (x <= 3.8e-139)
		tmp = x / y;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.00032], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -6.5e-175], t$95$0, If[LessEqual[x, 3.8e-139], N[(x / y), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.20000000000000026e-4

   1. Initial program 62.2%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 70.2

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

   4. Applied rewrites 70.2%

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

   if -3.20000000000000026e-4 < x < -6.5000000000000005e-175 or 3.80000000000000008e-139 < x

   1. Initial program 74.8%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 37.1

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

   4. Applied rewrites 37.1%

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

   if -6.5000000000000005e-175 < x < 3.80000000000000008e-139

   1. Initial program 63.6%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 85.7

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

   4. Applied rewrites 85.7%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 72.7%

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

Alternative 11: 58.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+150}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -1.22 \cdot 10^{-166}:\\ \;\;\;\;\frac{y}{\left(1 + x\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(1 + y\right) \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2e+150)
   (/ (/ y x) x)
   (if (<= x -1.22e-166) (/ y (* (+ 1.0 x) x)) (/ x (* (+ 1.0 y) y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2e+150) {
		tmp = (y / x) / x;
	} else if (x <= -1.22e-166) {
		tmp = y / ((1.0 + x) * x);
	} else {
		tmp = x / ((1.0 + y) * y);
	}
	return tmp;
}

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
    real(8) :: tmp
    if (x <= (-2d+150)) then
        tmp = (y / x) / x
    else if (x <= (-1.22d-166)) then
        tmp = y / ((1.0d0 + x) * x)
    else
        tmp = x / ((1.0d0 + y) * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -2e+150) {
		tmp = (y / x) / x;
	} else if (x <= -1.22e-166) {
		tmp = y / ((1.0 + x) * x);
	} else {
		tmp = x / ((1.0 + y) * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2e+150:
		tmp = (y / x) / x
	elif x <= -1.22e-166:
		tmp = y / ((1.0 + x) * x)
	else:
		tmp = x / ((1.0 + y) * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2e+150)
		tmp = Float64(Float64(y / x) / x);
	elseif (x <= -1.22e-166)
		tmp = Float64(y / Float64(Float64(1.0 + x) * x));
	else
		tmp = Float64(x / Float64(Float64(1.0 + y) * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2e+150)
		tmp = (y / x) / x;
	elseif (x <= -1.22e-166)
		tmp = y / ((1.0 + x) * x);
	else
		tmp = x / ((1.0 + y) * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2e+150], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, -1.22e-166], N[(y / N[(N[(1.0 + x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[(1.0 + y), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.99999999999999996e150

   1. Initial program 57.7%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 80.6

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

   4. Applied rewrites 80.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 87.4

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

   6. Applied rewrites 87.4%

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

   if -1.99999999999999996e150 < x < -1.22e-166

   1. Initial program 77.2%

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

   2. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 48.1

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

   4. Applied rewrites 48.1%

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

   if -1.22e-166 < x

   1. Initial program 67.6%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 56.4

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

   4. Applied rewrites 56.4%

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

Alternative 12: 47.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y \cdot y}\\ \mathbf{if}\;y \leq -1.6 \cdot 10^{-34}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (* y y))))
   (if (<= y -1.6e-34) t_0 (if (<= y 1.0) (/ x y) t_0))))

C

double code(double x, double y) {
	double t_0 = x / (y * y);
	double tmp;
	if (y <= -1.6e-34) {
		tmp = t_0;
	} else if (y <= 1.0) {
		tmp = x / y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / (y * y)
    if (y <= (-1.6d-34)) then
        tmp = t_0
    else if (y <= 1.0d0) then
        tmp = x / y
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (y * y);
	double tmp;
	if (y <= -1.6e-34) {
		tmp = t_0;
	} else if (y <= 1.0) {
		tmp = x / y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (y * y)
	tmp = 0
	if y <= -1.6e-34:
		tmp = t_0
	elif y <= 1.0:
		tmp = x / y
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(y * y))
	tmp = 0.0
	if (y <= -1.6e-34)
		tmp = t_0;
	elseif (y <= 1.0)
		tmp = Float64(x / y);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (y * y);
	tmp = 0.0;
	if (y <= -1.6e-34)
		tmp = t_0;
	elseif (y <= 1.0)
		tmp = x / y;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.6e-34], t$95$0, If[LessEqual[y, 1.0], N[(x / y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.60000000000000001e-34 or 1 < y

   1. Initial program 64.4%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 67.4

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

   4. Applied rewrites 67.4%

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

   if -1.60000000000000001e-34 < y < 1

   1. Initial program 73.6%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 24.6

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

   4. Applied rewrites 24.6%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 24.1%

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

Alternative 13: 57.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.22 \cdot 10^{-166}:\\ \;\;\;\;\frac{y}{\left(1 + x\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(1 + y\right) \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.22e-166) (/ y (* (+ 1.0 x) x)) (/ x (* (+ 1.0 y) y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.22e-166) {
		tmp = y / ((1.0 + x) * x);
	} else {
		tmp = x / ((1.0 + y) * y);
	}
	return tmp;
}

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
    real(8) :: tmp
    if (x <= (-1.22d-166)) then
        tmp = y / ((1.0d0 + x) * x)
    else
        tmp = x / ((1.0d0 + y) * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.22e-166) {
		tmp = y / ((1.0 + x) * x);
	} else {
		tmp = x / ((1.0 + y) * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.22e-166:
		tmp = y / ((1.0 + x) * x)
	else:
		tmp = x / ((1.0 + y) * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.22e-166)
		tmp = Float64(y / Float64(Float64(1.0 + x) * x));
	else
		tmp = Float64(x / Float64(Float64(1.0 + y) * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.22e-166)
		tmp = y / ((1.0 + x) * x);
	else
		tmp = x / ((1.0 + y) * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.22e-166], N[(y / N[(N[(1.0 + x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[(1.0 + y), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.22e-166

   1. Initial program 70.7%

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

   2. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 59.0

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

   4. Applied rewrites 59.0%

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

   if -1.22e-166 < x

   1. Initial program 67.6%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 56.4

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

   4. Applied rewrites 56.4%

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

Alternative 14: 61.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.00032:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(1 + y\right) \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.00032) (/ y (* x x)) (/ x (* (+ 1.0 y) y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.00032) {
		tmp = y / (x * x);
	} else {
		tmp = x / ((1.0 + y) * y);
	}
	return tmp;
}

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
    real(8) :: tmp
    if (x <= (-0.00032d0)) then
        tmp = y / (x * x)
    else
        tmp = x / ((1.0d0 + y) * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -0.00032) {
		tmp = y / (x * x);
	} else {
		tmp = x / ((1.0 + y) * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -0.00032:
		tmp = y / (x * x)
	else:
		tmp = x / ((1.0 + y) * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.00032)
		tmp = Float64(y / Float64(x * x));
	else
		tmp = Float64(x / Float64(Float64(1.0 + y) * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -0.00032)
		tmp = y / (x * x);
	else
		tmp = x / ((1.0 + y) * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.00032], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[(1.0 + y), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.20000000000000026e-4

   1. Initial program 62.2%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 70.2

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

   4. Applied rewrites 70.2%

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

   if -3.20000000000000026e-4 < x

   1. Initial program 70.9%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 58.2

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

   4. Applied rewrites 58.2%

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

Alternative 15: 26.5% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \frac{x}{y} \end{array} \]

FPCore

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

C

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

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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.7%

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

2. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lower-+.f64 49.2

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

4. Applied rewrites 49.2%

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

5. Taylor expanded in y around 0

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

7. Applied rewrites 26.5%

   \[\leadsto \frac{x}{y} \]
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 2025093 
(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))))