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

Percentage Accurate: 69.1% → 99.8%

Time: 9.3s

Alternatives: 19

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: 69.1% 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.4%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. times-frac N/A

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

   5. associate-*l/ N/A

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

   6. lift-*.f64 N/A

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

   7. times-frac N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lower-/.f64 99.8

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

   15. lift-+.f64 N/A

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

   16. +-commutative N/A

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

   17. lower-+.f64 99.8

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

   18. lift-+.f64 N/A

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

   19. +-commutative N/A

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

   20. lower-+.f64 99.8

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

   21. lift-+.f64 N/A

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

   22. +-commutative N/A

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

   23. lower-+.f64 99.8

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

4. Applied rewrites 99.8%

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

Alternative 2: 67.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{+220}:\\ \;\;\;\;1 \cdot \frac{\frac{y}{1 + \left(y + x\right)}}{y + x}\\ \mathbf{elif}\;x \leq -2 \cdot 10^{+41}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(1 + x, x, \mathsf{fma}\left(2, x, 1\right) \cdot y\right)} \cdot 1\\ \mathbf{elif}\;x \leq -4.2 \cdot 10^{-143}:\\ \;\;\;\;\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + x} \cdot {\left(1 + y\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -6e+220)
   (* 1.0 (/ (/ y (+ 1.0 (+ y x))) (+ y x)))
   (if (<= x -2e+41)
     (* (/ y (fma (+ 1.0 x) x (* (fma 2.0 x 1.0) y))) 1.0)
     (if (<= x -4.2e-143)
       (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0)))
       (* (/ x (+ y x)) (pow (+ 1.0 y) -1.0))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -6e+220) {
		tmp = 1.0 * ((y / (1.0 + (y + x))) / (y + x));
	} else if (x <= -2e+41) {
		tmp = (y / fma((1.0 + x), x, (fma(2.0, x, 1.0) * y))) * 1.0;
	} else if (x <= -4.2e-143) {
		tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
	} else {
		tmp = (x / (y + x)) * pow((1.0 + y), -1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -6e+220)
		tmp = Float64(1.0 * Float64(Float64(y / Float64(1.0 + Float64(y + x))) / Float64(y + x)));
	elseif (x <= -2e+41)
		tmp = Float64(Float64(y / fma(Float64(1.0 + x), x, Float64(fma(2.0, x, 1.0) * y))) * 1.0);
	elseif (x <= -4.2e-143)
		tmp = Float64(Float64(x * y) / Float64(Float64(Float64(x + y) * Float64(x + y)) * Float64(Float64(x + y) + 1.0)));
	else
		tmp = Float64(Float64(x / Float64(y + x)) * (Float64(1.0 + y) ^ -1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -6e+220], N[(1.0 * N[(N[(y / N[(1.0 + N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2e+41], N[(N[(y / N[(N[(1.0 + x), $MachinePrecision] * x + N[(N[(2.0 * x + 1.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[x, -4.2e-143], N[(N[(x * y), $MachinePrecision] / N[(N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(N[(x + y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[Power[N[(1.0 + y), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -6.00000000000000048e220

   1. Initial program 42.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. associate-*l/ N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 99.8

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 99.8

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.8

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

      21. lift-+.f64 N/A

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

      22. +-commutative N/A

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

      23. lower-+.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 81.4%

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

   if -6.00000000000000048e220 < x < -2.00000000000000001e41

   1. Initial program 62.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 N/A

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

      22. lower-/.f64 90.8

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

      23. lift-+.f64 N/A

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

      24. +-commutative N/A

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

   4. Applied rewrites 90.8%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 82.7%

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

   8. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 79.4

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

   10. Applied rewrites 79.4%

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

   if -2.00000000000000001e41 < x < -4.2000000000000002e-143

   1. Initial program 89.2%

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

   if -4.2000000000000002e-143 < x

   1. Initial program 63.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. associate-*l/ N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 99.8

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 99.8

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.8

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

      21. lift-+.f64 N/A

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

      22. +-commutative N/A

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

      23. lower-+.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 62.1

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

   7. Applied rewrites 62.1%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{+220}:\\ \;\;\;\;1 \cdot \frac{\frac{y}{1 + \left(y + x\right)}}{y + x}\\ \mathbf{elif}\;x \leq -2 \cdot 10^{+41}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(1 + x, x, \mathsf{fma}\left(2, x, 1\right) \cdot y\right)} \cdot 1\\ \mathbf{elif}\;x \leq -4.2 \cdot 10^{-143}:\\ \;\;\;\;\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + x} \cdot {\left(1 + y\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y + x}\\ t_1 := 1 + \left(y + x\right)\\ \mathbf{if}\;x \leq -6 \cdot 10^{+220}:\\ \;\;\;\;1 \cdot \frac{\frac{y}{t\_1}}{y + x}\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{-208}:\\ \;\;\;\;y \cdot \frac{t\_0}{t\_1 \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot {\left(1 + y\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ y x))) (t_1 (+ 1.0 (+ y x))))
   (if (<= x -6e+220)
     (* 1.0 (/ (/ y t_1) (+ y x)))
     (if (<= x 6.8e-208)
       (* y (/ t_0 (* t_1 (+ y x))))
       (* t_0 (pow (+ 1.0 y) -1.0))))))

C

double code(double x, double y) {
	double t_0 = x / (y + x);
	double t_1 = 1.0 + (y + x);
	double tmp;
	if (x <= -6e+220) {
		tmp = 1.0 * ((y / t_1) / (y + x));
	} else if (x <= 6.8e-208) {
		tmp = y * (t_0 / (t_1 * (y + x)));
	} else {
		tmp = t_0 * pow((1.0 + y), -1.0);
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = x / (y + x)
    t_1 = 1.0d0 + (y + x)
    if (x <= (-6d+220)) then
        tmp = 1.0d0 * ((y / t_1) / (y + x))
    else if (x <= 6.8d-208) then
        tmp = y * (t_0 / (t_1 * (y + x)))
    else
        tmp = t_0 * ((1.0d0 + y) ** (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (y + x);
	double t_1 = 1.0 + (y + x);
	double tmp;
	if (x <= -6e+220) {
		tmp = 1.0 * ((y / t_1) / (y + x));
	} else if (x <= 6.8e-208) {
		tmp = y * (t_0 / (t_1 * (y + x)));
	} else {
		tmp = t_0 * Math.pow((1.0 + y), -1.0);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (y + x)
	t_1 = 1.0 + (y + x)
	tmp = 0
	if x <= -6e+220:
		tmp = 1.0 * ((y / t_1) / (y + x))
	elif x <= 6.8e-208:
		tmp = y * (t_0 / (t_1 * (y + x)))
	else:
		tmp = t_0 * math.pow((1.0 + y), -1.0)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(y + x))
	t_1 = Float64(1.0 + Float64(y + x))
	tmp = 0.0
	if (x <= -6e+220)
		tmp = Float64(1.0 * Float64(Float64(y / t_1) / Float64(y + x)));
	elseif (x <= 6.8e-208)
		tmp = Float64(y * Float64(t_0 / Float64(t_1 * Float64(y + x))));
	else
		tmp = Float64(t_0 * (Float64(1.0 + y) ^ -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (y + x);
	t_1 = 1.0 + (y + x);
	tmp = 0.0;
	if (x <= -6e+220)
		tmp = 1.0 * ((y / t_1) / (y + x));
	elseif (x <= 6.8e-208)
		tmp = y * (t_0 / (t_1 * (y + x)));
	else
		tmp = t_0 * ((1.0 + y) ^ -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -6e+220], N[(1.0 * N[(N[(y / t$95$1), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6.8e-208], N[(y * N[(t$95$0 / N[(t$95$1 * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[Power[N[(1.0 + y), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.00000000000000048e220

   1. Initial program 42.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. associate-*l/ N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 99.8

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 99.8

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.8

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

      21. lift-+.f64 N/A

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

      22. +-commutative N/A

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

      23. lower-+.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 81.4%

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

   if -6.00000000000000048e220 < x < 6.8e-208

   1. Initial program 70.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. associate-*l/ N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 99.8

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 99.8

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.8

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

      21. lift-+.f64 N/A

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

      22. +-commutative N/A

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

      23. lower-+.f64 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/r* N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 94.3

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

   6. Applied rewrites 94.3%

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

   if 6.8e-208 < x

   1. Initial program 65.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. associate-*l/ N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 99.8

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 99.8

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.8

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

      21. lift-+.f64 N/A

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

      22. +-commutative N/A

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

      23. lower-+.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 46.7

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

   7. Applied rewrites 46.7%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{+220}:\\ \;\;\;\;1 \cdot \frac{\frac{y}{1 + \left(y + x\right)}}{y + x}\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{-208}:\\ \;\;\;\;y \cdot \frac{\frac{x}{y + x}}{\left(1 + \left(y + x\right)\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + x} \cdot {\left(1 + y\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 65.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \left(y + x\right)\\ \mathbf{if}\;x \leq -6 \cdot 10^{+220}:\\ \;\;\;\;1 \cdot \frac{\frac{y}{t\_0}}{y + x}\\ \mathbf{elif}\;x \leq -3.2 \cdot 10^{-129}:\\ \;\;\;\;\frac{y}{t\_0 \cdot \left(y + x\right)} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + x} \cdot {\left(1 + y\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (+ y x))))
   (if (<= x -6e+220)
     (* 1.0 (/ (/ y t_0) (+ y x)))
     (if (<= x -3.2e-129)
       (* (/ y (* t_0 (+ y x))) 1.0)
       (* (/ x (+ y x)) (pow (+ 1.0 y) -1.0))))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (y + x);
	double tmp;
	if (x <= -6e+220) {
		tmp = 1.0 * ((y / t_0) / (y + x));
	} else if (x <= -3.2e-129) {
		tmp = (y / (t_0 * (y + x))) * 1.0;
	} else {
		tmp = (x / (y + x)) * pow((1.0 + y), -1.0);
	}
	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 = 1.0d0 + (y + x)
    if (x <= (-6d+220)) then
        tmp = 1.0d0 * ((y / t_0) / (y + x))
    else if (x <= (-3.2d-129)) then
        tmp = (y / (t_0 * (y + x))) * 1.0d0
    else
        tmp = (x / (y + x)) * ((1.0d0 + y) ** (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + (y + x);
	double tmp;
	if (x <= -6e+220) {
		tmp = 1.0 * ((y / t_0) / (y + x));
	} else if (x <= -3.2e-129) {
		tmp = (y / (t_0 * (y + x))) * 1.0;
	} else {
		tmp = (x / (y + x)) * Math.pow((1.0 + y), -1.0);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (y + x)
	tmp = 0
	if x <= -6e+220:
		tmp = 1.0 * ((y / t_0) / (y + x))
	elif x <= -3.2e-129:
		tmp = (y / (t_0 * (y + x))) * 1.0
	else:
		tmp = (x / (y + x)) * math.pow((1.0 + y), -1.0)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(y + x))
	tmp = 0.0
	if (x <= -6e+220)
		tmp = Float64(1.0 * Float64(Float64(y / t_0) / Float64(y + x)));
	elseif (x <= -3.2e-129)
		tmp = Float64(Float64(y / Float64(t_0 * Float64(y + x))) * 1.0);
	else
		tmp = Float64(Float64(x / Float64(y + x)) * (Float64(1.0 + y) ^ -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + (y + x);
	tmp = 0.0;
	if (x <= -6e+220)
		tmp = 1.0 * ((y / t_0) / (y + x));
	elseif (x <= -3.2e-129)
		tmp = (y / (t_0 * (y + x))) * 1.0;
	else
		tmp = (x / (y + x)) * ((1.0 + y) ^ -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -6e+220], N[(1.0 * N[(N[(y / t$95$0), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -3.2e-129], N[(N[(y / N[(t$95$0 * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[Power[N[(1.0 + y), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.00000000000000048e220

   1. Initial program 42.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. associate-*l/ N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 99.8

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 99.8

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.8

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

      21. lift-+.f64 N/A

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

      22. +-commutative N/A

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

      23. lower-+.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 81.4%

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

   if -6.00000000000000048e220 < x < -3.2000000000000003e-129

   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. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 N/A

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

      22. lower-/.f64 96.0

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

      23. lift-+.f64 N/A

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

      24. +-commutative N/A

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

   4. Applied rewrites 96.0%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 74.4%

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

   if -3.2000000000000003e-129 < x

   1. Initial program 64.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. associate-*l/ N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 99.8

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 99.8

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.8

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

      21. lift-+.f64 N/A

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

      22. +-commutative N/A

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

      23. lower-+.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 62.8

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

   7. Applied rewrites 62.8%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{+220}:\\ \;\;\;\;1 \cdot \frac{\frac{y}{1 + \left(y + x\right)}}{y + x}\\ \mathbf{elif}\;x \leq -3.2 \cdot 10^{-129}:\\ \;\;\;\;\frac{y}{\left(1 + \left(y + x\right)\right) \cdot \left(y + x\right)} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + x} \cdot {\left(1 + y\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 65.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{+220}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -3.2 \cdot 10^{-129}:\\ \;\;\;\;\frac{y}{\left(1 + \left(y + x\right)\right) \cdot \left(y + x\right)} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + x} \cdot {\left(1 + y\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -6e+220)
   (/ (/ y x) x)
   (if (<= x -3.2e-129)
     (* (/ y (* (+ 1.0 (+ y x)) (+ y x))) 1.0)
     (* (/ x (+ y x)) (pow (+ 1.0 y) -1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -6e+220) {
		tmp = (y / x) / x;
	} else if (x <= -3.2e-129) {
		tmp = (y / ((1.0 + (y + x)) * (y + x))) * 1.0;
	} else {
		tmp = (x / (y + x)) * pow((1.0 + y), -1.0);
	}
	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) :: tmp
    if (x <= (-6d+220)) then
        tmp = (y / x) / x
    else if (x <= (-3.2d-129)) then
        tmp = (y / ((1.0d0 + (y + x)) * (y + x))) * 1.0d0
    else
        tmp = (x / (y + x)) * ((1.0d0 + y) ** (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -6e+220) {
		tmp = (y / x) / x;
	} else if (x <= -3.2e-129) {
		tmp = (y / ((1.0 + (y + x)) * (y + x))) * 1.0;
	} else {
		tmp = (x / (y + x)) * Math.pow((1.0 + y), -1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -6e+220:
		tmp = (y / x) / x
	elif x <= -3.2e-129:
		tmp = (y / ((1.0 + (y + x)) * (y + x))) * 1.0
	else:
		tmp = (x / (y + x)) * math.pow((1.0 + y), -1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -6e+220)
		tmp = Float64(Float64(y / x) / x);
	elseif (x <= -3.2e-129)
		tmp = Float64(Float64(y / Float64(Float64(1.0 + Float64(y + x)) * Float64(y + x))) * 1.0);
	else
		tmp = Float64(Float64(x / Float64(y + x)) * (Float64(1.0 + y) ^ -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -6e+220)
		tmp = (y / x) / x;
	elseif (x <= -3.2e-129)
		tmp = (y / ((1.0 + (y + x)) * (y + x))) * 1.0;
	else
		tmp = (x / (y + x)) * ((1.0 + y) ^ -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -6e+220], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, -3.2e-129], N[(N[(y / N[(N[(1.0 + N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[Power[N[(1.0 + y), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.00000000000000048e220

   1. Initial program 42.1%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 80.3

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

   5. Applied rewrites 80.3%

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

   if -6.00000000000000048e220 < x < -3.2000000000000003e-129

   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. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 N/A

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

      22. lower-/.f64 96.0

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

      23. lift-+.f64 N/A

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

      24. +-commutative N/A

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

   4. Applied rewrites 96.0%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 74.4%

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

   if -3.2000000000000003e-129 < x

   1. Initial program 64.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. associate-*l/ N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 99.8

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 99.8

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.8

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

      21. lift-+.f64 N/A

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

      22. +-commutative N/A

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

      23. lower-+.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 62.8

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

   7. Applied rewrites 62.8%

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

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{+220}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -3.2 \cdot 10^{-129}:\\ \;\;\;\;\frac{y}{\left(1 + \left(y + x\right)\right) \cdot \left(y + x\right)} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + x} \cdot {\left(1 + y\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{+220}:\\ \;\;\;\;\frac{\frac{y - y \cdot \frac{\mathsf{fma}\left(3, y, 1\right)}{x}}{x}}{x}\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-39}:\\ \;\;\;\;\frac{y}{\left(1 + \left(y + x\right)\right) \cdot \left(y + x\right)} \cdot \frac{x}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -6e+220)
   (/ (/ (- y (* y (/ (fma 3.0 y 1.0) x))) x) x)
   (if (<= x 3.6e-39)
     (* (/ y (* (+ 1.0 (+ y x)) (+ y x))) (/ x (+ y x)))
     (/ (/ x y) y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -6e+220) {
		tmp = ((y - (y * (fma(3.0, y, 1.0) / x))) / x) / x;
	} else if (x <= 3.6e-39) {
		tmp = (y / ((1.0 + (y + x)) * (y + x))) * (x / (y + x));
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -6e+220)
		tmp = Float64(Float64(Float64(y - Float64(y * Float64(fma(3.0, y, 1.0) / x))) / x) / x);
	elseif (x <= 3.6e-39)
		tmp = Float64(Float64(y / Float64(Float64(1.0 + Float64(y + x)) * Float64(y + x))) * Float64(x / Float64(y + x)));
	else
		tmp = Float64(Float64(x / y) / y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -6e+220], N[(N[(N[(y - N[(y * N[(N[(3.0 * y + 1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 3.6e-39], N[(N[(y / N[(N[(1.0 + N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.00000000000000048e220

   1. Initial program 42.1%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

      3. lower-/.f64 N/A

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

   5. Applied rewrites 79.7%

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

   if -6.00000000000000048e220 < x < 3.6000000000000001e-39

   1. Initial program 71.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 N/A

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

      22. lower-/.f64 98.2

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

      23. lift-+.f64 N/A

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

      24. +-commutative N/A

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

   4. Applied rewrites 98.2%

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

   if 3.6000000000000001e-39 < x

   1. Initial program 58.8%

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

   3. Taylor expanded in y around inf

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 34.3

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

   5. Applied rewrites 34.3%

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

Alternative 7: 77.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \left(y + x\right)\\ \mathbf{if}\;x \leq -6 \cdot 10^{+220}:\\ \;\;\;\;1 \cdot \frac{\frac{y}{t\_0}}{y + x}\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-39}:\\ \;\;\;\;\frac{y}{t\_0 \cdot \left(y + x\right)} \cdot \frac{x}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (+ y x))))
   (if (<= x -6e+220)
     (* 1.0 (/ (/ y t_0) (+ y x)))
     (if (<= x 3.6e-39)
       (* (/ y (* t_0 (+ y x))) (/ x (+ y x)))
       (/ (/ x y) y)))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (y + x);
	double tmp;
	if (x <= -6e+220) {
		tmp = 1.0 * ((y / t_0) / (y + x));
	} else if (x <= 3.6e-39) {
		tmp = (y / (t_0 * (y + x))) * (x / (y + x));
	} else {
		tmp = (x / 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)
    if (x <= (-6d+220)) then
        tmp = 1.0d0 * ((y / t_0) / (y + x))
    else if (x <= 3.6d-39) then
        tmp = (y / (t_0 * (y + x))) * (x / (y + x))
    else
        tmp = (x / y) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + (y + x);
	double tmp;
	if (x <= -6e+220) {
		tmp = 1.0 * ((y / t_0) / (y + x));
	} else if (x <= 3.6e-39) {
		tmp = (y / (t_0 * (y + x))) * (x / (y + x));
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (y + x)
	tmp = 0
	if x <= -6e+220:
		tmp = 1.0 * ((y / t_0) / (y + x))
	elif x <= 3.6e-39:
		tmp = (y / (t_0 * (y + x))) * (x / (y + x))
	else:
		tmp = (x / y) / y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(y + x))
	tmp = 0.0
	if (x <= -6e+220)
		tmp = Float64(1.0 * Float64(Float64(y / t_0) / Float64(y + x)));
	elseif (x <= 3.6e-39)
		tmp = Float64(Float64(y / Float64(t_0 * Float64(y + x))) * Float64(x / Float64(y + x)));
	else
		tmp = Float64(Float64(x / y) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + (y + x);
	tmp = 0.0;
	if (x <= -6e+220)
		tmp = 1.0 * ((y / t_0) / (y + x));
	elseif (x <= 3.6e-39)
		tmp = (y / (t_0 * (y + x))) * (x / (y + x));
	else
		tmp = (x / y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -6e+220], N[(1.0 * N[(N[(y / t$95$0), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.6e-39], N[(N[(y / N[(t$95$0 * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.00000000000000048e220

   1. Initial program 42.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. associate-*l/ N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 99.8

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 99.8

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.8

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

      21. lift-+.f64 N/A

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

      22. +-commutative N/A

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

      23. lower-+.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 81.4%

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

   if -6.00000000000000048e220 < x < 3.6000000000000001e-39

   1. Initial program 71.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 N/A

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

      22. lower-/.f64 98.2

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

      23. lift-+.f64 N/A

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

      24. +-commutative N/A

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

   4. Applied rewrites 98.2%

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

   if 3.6000000000000001e-39 < x

   1. Initial program 58.8%

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

   3. Taylor expanded in y around inf

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 34.3

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

   5. Applied rewrites 34.3%

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

Alternative 8: 80.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{+111}:\\ \;\;\;\;1 \cdot \frac{\frac{y}{1 + \left(y + x\right)}}{y + x}\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{+38}:\\ \;\;\;\;x \cdot \frac{\frac{y}{x + y}}{\left(\left(1 + x\right) + y\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.1e+111)
   (* 1.0 (/ (/ y (+ 1.0 (+ y x))) (+ y x)))
   (if (<= x 5.8e+38)
     (* x (/ (/ y (+ x y)) (* (+ (+ 1.0 x) y) (+ x y))))
     (/ (/ x y) y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.1e+111) {
		tmp = 1.0 * ((y / (1.0 + (y + x))) / (y + x));
	} else if (x <= 5.8e+38) {
		tmp = x * ((y / (x + y)) / (((1.0 + x) + y) * (x + y)));
	} else {
		tmp = (x / 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 <= (-1.1d+111)) then
        tmp = 1.0d0 * ((y / (1.0d0 + (y + x))) / (y + x))
    else if (x <= 5.8d+38) then
        tmp = x * ((y / (x + y)) / (((1.0d0 + x) + y) * (x + y)))
    else
        tmp = (x / y) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.1e+111) {
		tmp = 1.0 * ((y / (1.0 + (y + x))) / (y + x));
	} else if (x <= 5.8e+38) {
		tmp = x * ((y / (x + y)) / (((1.0 + x) + y) * (x + y)));
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.1e+111:
		tmp = 1.0 * ((y / (1.0 + (y + x))) / (y + x))
	elif x <= 5.8e+38:
		tmp = x * ((y / (x + y)) / (((1.0 + x) + y) * (x + y)))
	else:
		tmp = (x / y) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.1e+111)
		tmp = Float64(1.0 * Float64(Float64(y / Float64(1.0 + Float64(y + x))) / Float64(y + x)));
	elseif (x <= 5.8e+38)
		tmp = Float64(x * Float64(Float64(y / Float64(x + y)) / Float64(Float64(Float64(1.0 + x) + y) * Float64(x + y))));
	else
		tmp = Float64(Float64(x / y) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.1e+111)
		tmp = 1.0 * ((y / (1.0 + (y + x))) / (y + x));
	elseif (x <= 5.8e+38)
		tmp = x * ((y / (x + y)) / (((1.0 + x) + y) * (x + y)));
	else
		tmp = (x / y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.1e+111], N[(1.0 * N[(N[(y / N[(1.0 + N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.8e+38], N[(x * N[(N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(1.0 + x), $MachinePrecision] + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.09999999999999999e111

   1. Initial program 47.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. associate-*l/ N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 99.8

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 99.8

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.8

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

      21. lift-+.f64 N/A

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

      22. +-commutative N/A

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

      23. lower-+.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 81.3%

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

   if -1.09999999999999999e111 < x < 5.80000000000000013e38

   1. Initial program 75.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. associate-*l/ N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 99.8

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 99.8

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.8

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

      21. lift-+.f64 N/A

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

      22. +-commutative N/A

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

      23. lower-+.f64 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/r* N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 98.0

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

   6. Applied rewrites 98.0%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. lower-*.f64 98.0

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

      15. lift-+.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. associate-+r+ N/A

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

      19. lift-+.f64 N/A

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

      20. lower-+.f64 98.0

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

      21. lift-+.f64 N/A

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

      22. +-commutative N/A

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

      23. lower-+.f64 98.0

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

   8. Applied rewrites 98.0%

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

   if 5.80000000000000013e38 < x

   1. Initial program 46.2%

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

   3. Taylor expanded in y around inf

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 26.7

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

   5. Applied rewrites 26.7%

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

Alternative 9: 83.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.25 \cdot 10^{-19}:\\ \;\;\;\;\frac{y}{\left(1 + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{y + x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{1}{1 + \left(y + x\right)}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.25e-19)
   (* (/ y (* (+ 1.0 x) (+ y x))) (/ x (+ y x)))
   (* x (/ (/ 1.0 (+ 1.0 (+ y x))) (+ y x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.25e-19) {
		tmp = (y / ((1.0 + x) * (y + x))) * (x / (y + x));
	} else {
		tmp = x * ((1.0 / (1.0 + (y + x))) / (y + x));
	}
	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 (y <= 1.25d-19) then
        tmp = (y / ((1.0d0 + x) * (y + x))) * (x / (y + x))
    else
        tmp = x * ((1.0d0 / (1.0d0 + (y + x))) / (y + x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.25e-19) {
		tmp = (y / ((1.0 + x) * (y + x))) * (x / (y + x));
	} else {
		tmp = x * ((1.0 / (1.0 + (y + x))) / (y + x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.25e-19:
		tmp = (y / ((1.0 + x) * (y + x))) * (x / (y + x))
	else:
		tmp = x * ((1.0 / (1.0 + (y + x))) / (y + x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.25e-19)
		tmp = Float64(Float64(y / Float64(Float64(1.0 + x) * Float64(y + x))) * Float64(x / Float64(y + x)));
	else
		tmp = Float64(x * Float64(Float64(1.0 / Float64(1.0 + Float64(y + x))) / Float64(y + x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.25e-19)
		tmp = (y / ((1.0 + x) * (y + x))) * (x / (y + x));
	else
		tmp = x * ((1.0 / (1.0 + (y + x))) / (y + x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.25e-19], N[(N[(y / N[(N[(1.0 + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(1.0 / N[(1.0 + N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.2500000000000001e-19

   1. Initial program 66.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 N/A

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

      22. lower-/.f64 96.8

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

      23. lift-+.f64 N/A

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

      24. +-commutative N/A

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

   4. Applied rewrites 96.8%

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

   5. Taylor expanded in y around 0

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

      1. lower-+.f64 80.4

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

   7. Applied rewrites 80.4%

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

   if 1.2500000000000001e-19 < y

   1. Initial program 65.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. associate-*l/ N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 99.8

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 99.8

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.8

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

      21. lift-+.f64 N/A

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

      22. +-commutative N/A

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

      23. lower-+.f64 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/r* N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 87.8

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

   6. Applied rewrites 88.9%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 86.1%

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

Alternative 10: 63.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{+220}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -1.8 \cdot 10^{-6}:\\ \;\;\;\;\frac{y}{\left(1 + x\right) \cdot \left(y + x\right)} \cdot 1\\ \mathbf{elif}\;x \leq -3.2 \cdot 10^{-129}:\\ \;\;\;\;\frac{y}{\left(1 + y\right) \cdot \left(y + x\right)} \cdot 1\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{+32}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -6e+220)
   (/ (/ y x) x)
   (if (<= x -1.8e-6)
     (* (/ y (* (+ 1.0 x) (+ y x))) 1.0)
     (if (<= x -3.2e-129)
       (* (/ y (* (+ 1.0 y) (+ y x))) 1.0)
       (if (<= x 5.2e+32) (/ x (fma y y y)) (/ (/ x y) y))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -6e+220) {
		tmp = (y / x) / x;
	} else if (x <= -1.8e-6) {
		tmp = (y / ((1.0 + x) * (y + x))) * 1.0;
	} else if (x <= -3.2e-129) {
		tmp = (y / ((1.0 + y) * (y + x))) * 1.0;
	} else if (x <= 5.2e+32) {
		tmp = x / fma(y, y, y);
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -6e+220)
		tmp = Float64(Float64(y / x) / x);
	elseif (x <= -1.8e-6)
		tmp = Float64(Float64(y / Float64(Float64(1.0 + x) * Float64(y + x))) * 1.0);
	elseif (x <= -3.2e-129)
		tmp = Float64(Float64(y / Float64(Float64(1.0 + y) * Float64(y + x))) * 1.0);
	elseif (x <= 5.2e+32)
		tmp = Float64(x / fma(y, y, y));
	else
		tmp = Float64(Float64(x / y) / y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -6e+220], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, -1.8e-6], N[(N[(y / N[(N[(1.0 + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[x, -3.2e-129], N[(N[(y / N[(N[(1.0 + y), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[x, 5.2e+32], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -6.00000000000000048e220

   1. Initial program 42.1%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 80.3

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

   5. Applied rewrites 80.3%

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

   if -6.00000000000000048e220 < x < -1.79999999999999992e-6

   1. Initial program 67.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 N/A

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

      22. lower-/.f64 93.2

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

      23. lift-+.f64 N/A

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

      24. +-commutative N/A

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

   4. Applied rewrites 93.2%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 78.4%

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

   8. Taylor expanded in y around 0

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

      1. lower-+.f64 76.0

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

   10. Applied rewrites 76.0%

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

   if -1.79999999999999992e-6 < x < -3.2000000000000003e-129

   1. Initial program 90.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 N/A

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

      22. lower-/.f64 99.9

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

      23. lift-+.f64 N/A

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

      24. +-commutative N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 68.8%

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

   8. Taylor expanded in x around 0

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

      1. lower-+.f64 68.8

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

   10. Applied rewrites 68.8%

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

   if -3.2000000000000003e-129 < x < 5.2000000000000004e32

   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. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 76.1

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

   5. Applied rewrites 76.1%

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

   if 5.2000000000000004e32 < x

   1. Initial program 47.4%

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

   3. Taylor expanded in y around inf

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 26.8

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

   5. Applied rewrites 26.8%

      \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y}} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 11: 65.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{+220}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -3.2 \cdot 10^{-129}:\\ \;\;\;\;\frac{y}{\left(1 + \left(y + x\right)\right) \cdot \left(y + x\right)} \cdot 1\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{+32}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -6e+220)
   (/ (/ y x) x)
   (if (<= x -3.2e-129)
     (* (/ y (* (+ 1.0 (+ y x)) (+ y x))) 1.0)
     (if (<= x 5.2e+32) (/ x (fma y y y)) (/ (/ x y) y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -6e+220) {
		tmp = (y / x) / x;
	} else if (x <= -3.2e-129) {
		tmp = (y / ((1.0 + (y + x)) * (y + x))) * 1.0;
	} else if (x <= 5.2e+32) {
		tmp = x / fma(y, y, y);
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -6e+220)
		tmp = Float64(Float64(y / x) / x);
	elseif (x <= -3.2e-129)
		tmp = Float64(Float64(y / Float64(Float64(1.0 + Float64(y + x)) * Float64(y + x))) * 1.0);
	elseif (x <= 5.2e+32)
		tmp = Float64(x / fma(y, y, y));
	else
		tmp = Float64(Float64(x / y) / y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -6e+220], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, -3.2e-129], N[(N[(y / N[(N[(1.0 + N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[x, 5.2e+32], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -6.00000000000000048e220

   1. Initial program 42.1%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 80.3

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

   5. Applied rewrites 80.3%

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

   if -6.00000000000000048e220 < x < -3.2000000000000003e-129

   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. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 N/A

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

      22. lower-/.f64 96.0

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

      23. lift-+.f64 N/A

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

      24. +-commutative N/A

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

   4. Applied rewrites 96.0%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 74.4%

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

   if -3.2000000000000003e-129 < x < 5.2000000000000004e32

   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. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 76.1

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

   5. Applied rewrites 76.1%

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

   if 5.2000000000000004e32 < x

   1. Initial program 47.4%

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

   3. Taylor expanded in y around inf

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 26.8

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

   5. Applied rewrites 26.8%

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

Alternative 12: 65.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{+220}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -3.2 \cdot 10^{-129}:\\ \;\;\;\;y \cdot \frac{1}{\left(\left(1 + x\right) + y\right) \cdot \left(x + y\right)}\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{+32}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -6e+220)
   (/ (/ y x) x)
   (if (<= x -3.2e-129)
     (* y (/ 1.0 (* (+ (+ 1.0 x) y) (+ x y))))
     (if (<= x 5.2e+32) (/ x (fma y y y)) (/ (/ x y) y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -6e+220) {
		tmp = (y / x) / x;
	} else if (x <= -3.2e-129) {
		tmp = y * (1.0 / (((1.0 + x) + y) * (x + y)));
	} else if (x <= 5.2e+32) {
		tmp = x / fma(y, y, y);
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -6e+220)
		tmp = Float64(Float64(y / x) / x);
	elseif (x <= -3.2e-129)
		tmp = Float64(y * Float64(1.0 / Float64(Float64(Float64(1.0 + x) + y) * Float64(x + y))));
	elseif (x <= 5.2e+32)
		tmp = Float64(x / fma(y, y, y));
	else
		tmp = Float64(Float64(x / y) / y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -6e+220], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, -3.2e-129], N[(y * N[(1.0 / N[(N[(N[(1.0 + x), $MachinePrecision] + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.2e+32], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -6.00000000000000048e220

   1. Initial program 42.1%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 80.3

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

   5. Applied rewrites 80.3%

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

   if -6.00000000000000048e220 < x < -3.2000000000000003e-129

   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. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 N/A

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

      22. lower-/.f64 96.0

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

      23. lift-+.f64 N/A

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

      24. +-commutative N/A

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

   4. Applied rewrites 96.0%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 74.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 74.4

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

         \[\leadsto y \cdot \frac{1}{\left(1 + \color{blue}{\left(x + y\right)}\right) \cdot \left(y + x\right)} \]

      10. associate-+l+ N/A

          \[\leadsto y \cdot \frac{1}{\color{blue}{\left(\left(1 + x\right) + y\right)} \cdot \left(y + x\right)} \]

      11. lift-+.f64 N/A

          \[\leadsto y \cdot \frac{1}{\left(\color{blue}{\left(1 + x\right)} + y\right) \cdot \left(y + x\right)} \]

      12. lift-+.f64 74.4

          \[\leadsto y \cdot \frac{1}{\color{blue}{\left(\left(1 + x\right) + y\right)} \cdot \left(y + x\right)} \]

      13. lift-+.f64 N/A

          \[\leadsto y \cdot \frac{1}{\left(\left(1 + x\right) + y\right) \cdot \color{blue}{\left(y + x\right)}} \]

      14. +-commutative N/A

          \[\leadsto y \cdot \frac{1}{\left(\left(1 + x\right) + y\right) \cdot \color{blue}{\left(x + y\right)}} \]

      15. lift-+.f64 74.4

          \[\leadsto y \cdot \frac{1}{\left(\left(1 + x\right) + y\right) \cdot \color{blue}{\left(x + y\right)}} \]

   9. Applied rewrites 74.4%

      \[\leadsto \color{blue}{y \cdot \frac{1}{\left(\left(1 + x\right) + y\right) \cdot \left(x + y\right)}} \]

   if -3.2000000000000003e-129 < x < 5.2000000000000004e32

   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. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

      3. distribute-lft-in N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot y + y \cdot 1}} \]

      4. *-rgt-identity N/A

         \[\leadsto \frac{x}{y \cdot y + \color{blue}{y}} \]

      5. lower-fma.f64 76.1

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(y, y, y\right)}} \]

   5. Applied rewrites 76.1%

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, y, y\right)}} \]

   if 5.2000000000000004e32 < x

   1. Initial program 47.4%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot y}} \]

      2. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y}} \]

      4. lower-/.f64 26.8

         \[\leadsto \frac{\color{blue}{\frac{x}{y}}}{y} \]

   5. Applied rewrites 26.8%

      \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 13: 64.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.05 \cdot 10^{-153}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{+154}:\\ \;\;\;\;x \cdot \frac{1}{\left(\left(1 + x\right) + y\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.05e-153)
   (/ y (fma x x x))
   (if (<= y 8.5e+154)
     (* x (/ 1.0 (* (+ (+ 1.0 x) y) (+ x y))))
     (/ (/ x y) y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.05e-153) {
		tmp = y / fma(x, x, x);
	} else if (y <= 8.5e+154) {
		tmp = x * (1.0 / (((1.0 + x) + y) * (x + y)));
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.05e-153)
		tmp = Float64(y / fma(x, x, x));
	elseif (y <= 8.5e+154)
		tmp = Float64(x * Float64(1.0 / Float64(Float64(Float64(1.0 + x) + y) * Float64(x + y))));
	else
		tmp = Float64(Float64(x / y) / y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.05e-153], N[(y / N[(x * x + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8.5e+154], N[(x * N[(1.0 / N[(N[(N[(1.0 + x), $MachinePrecision] + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.05e-153

   1. Initial program 66.4%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{y}{x \cdot \color{blue}{\left(x + 1\right)}} \]

      3. distribute-lft-in N/A

         \[\leadsto \frac{y}{\color{blue}{x \cdot x + x \cdot 1}} \]

      4. *-rgt-identity N/A

         \[\leadsto \frac{y}{x \cdot x + \color{blue}{x}} \]

      5. lower-fma.f64 53.1

         \[\leadsto \frac{y}{\color{blue}{\mathsf{fma}\left(x, x, x\right)}} \]

   5. Applied rewrites 53.1%

      \[\leadsto \color{blue}{\frac{y}{\mathsf{fma}\left(x, x, x\right)}} \]

   if 2.05e-153 < y < 8.5000000000000002e154

   1. Initial program 69.4%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      5. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{x \cdot \frac{y}{\left(x + y\right) + 1}}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot \frac{y}{\left(x + y\right) + 1}}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y}} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{x + y}} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{x + y}} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y} \]

      11. +-commutative N/A

          \[\leadsto \frac{x}{\color{blue}{y + x}} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y} \]

      12. lower-+.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{y + x}} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1}}{x + y}} \]

      14. lower-/.f64 99.8

          \[\leadsto \frac{x}{y + x} \cdot \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1}}}{x + y} \]

      15. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{\color{blue}{\left(x + y\right) + 1}}}{x + y} \]

      16. +-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{\color{blue}{1 + \left(x + y\right)}}}{x + y} \]

      17. lower-+.f64 99.8

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{\color{blue}{1 + \left(x + y\right)}}}{x + y} \]

      18. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{1 + \color{blue}{\left(x + y\right)}}}{x + y} \]

      19. +-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{1 + \color{blue}{\left(y + x\right)}}}{x + y} \]

      20. lower-+.f64 99.8

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{1 + \color{blue}{\left(y + x\right)}}}{x + y} \]

      21. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{1 + \left(y + x\right)}}{\color{blue}{x + y}} \]

      22. +-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{1 + \left(y + x\right)}}{\color{blue}{y + x}} \]

      23. lower-+.f64 99.8

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{1 + \left(y + x\right)}}{\color{blue}{y + x}} \]

   4. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\frac{x}{y + x} \cdot \frac{\frac{y}{1 + \left(y + x\right)}}{y + x}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y + x} \cdot \frac{\frac{y}{1 + \left(y + x\right)}}{y + x}} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y + x}} \cdot \frac{\frac{y}{1 + \left(y + x\right)}}{y + x} \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{x \cdot \frac{\frac{y}{1 + \left(y + x\right)}}{y + x}}{y + x}} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{x \cdot \color{blue}{\frac{\frac{y}{1 + \left(y + x\right)}}{y + x}}}{y + x} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{x \cdot \frac{\color{blue}{\frac{y}{1 + \left(y + x\right)}}}{y + x}}{y + x} \]

      6. associate-/r* N/A

         \[\leadsto \frac{x \cdot \color{blue}{\frac{y}{\left(1 + \left(y + x\right)\right) \cdot \left(y + x\right)}}}{y + x} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot \frac{y}{\color{blue}{\left(1 + \left(y + x\right)\right) \cdot \left(y + x\right)}}}{y + x} \]

      8. lift-/.f64 N/A

         \[\leadsto \frac{x \cdot \color{blue}{\frac{y}{\left(1 + \left(y + x\right)\right) \cdot \left(y + x\right)}}}{y + x} \]

      9. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{\frac{y}{\left(1 + \left(y + x\right)\right) \cdot \left(y + x\right)}}{y + x}} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{x \cdot \frac{\frac{y}{\left(1 + \left(y + x\right)\right) \cdot \left(y + x\right)}}{y + x}} \]

      11. lower-/.f64 93.1

          \[\leadsto x \cdot \color{blue}{\frac{\frac{y}{\left(1 + \left(y + x\right)\right) \cdot \left(y + x\right)}}{y + x}} \]

   6. Applied rewrites 93.2%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{\frac{y}{y + x}}{1 + \left(y + x\right)}}{y + x}} \]

   7. Taylor expanded in x around 0

      \[\leadsto x \cdot \frac{\frac{\color{blue}{1}}{1 + \left(y + x\right)}}{y + x} \]
   8. Step-by-step derivation

   9. Applied rewrites 79.9%

      \[\leadsto x \cdot \frac{\frac{\color{blue}{1}}{1 + \left(y + x\right)}}{y + x} \]
   10. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto x \cdot \color{blue}{\frac{\frac{1}{1 + \left(y + x\right)}}{y + x}} \]

       2. lift-/.f64 N/A

          \[\leadsto x \cdot \frac{\color{blue}{\frac{1}{1 + \left(y + x\right)}}}{y + x} \]

       3. associate-/l/ N/A

          \[\leadsto x \cdot \color{blue}{\frac{1}{\left(1 + \left(y + x\right)\right) \cdot \left(y + x\right)}} \]

       4. lift-*.f64 N/A

          \[\leadsto x \cdot \frac{1}{\color{blue}{\left(1 + \left(y + x\right)\right) \cdot \left(y + x\right)}} \]

       5. lower-/.f64 79.9

          \[\leadsto x \cdot \color{blue}{\frac{1}{\left(1 + \left(y + x\right)\right) \cdot \left(y + x\right)}} \]

       6. lift-+.f64 N/A

          \[\leadsto x \cdot \frac{1}{\color{blue}{\left(1 + \left(y + x\right)\right)} \cdot \left(y + x\right)} \]

       7. lift-+.f64 N/A

          \[\leadsto x \cdot \frac{1}{\left(1 + \color{blue}{\left(y + x\right)}\right) \cdot \left(y + x\right)} \]

       8. +-commutative N/A

          \[\leadsto x \cdot \frac{1}{\left(1 + \color{blue}{\left(x + y\right)}\right) \cdot \left(y + x\right)} \]

       9. associate-+l+ N/A

          \[\leadsto x \cdot \frac{1}{\color{blue}{\left(\left(1 + x\right) + y\right)} \cdot \left(y + x\right)} \]

       10. lift-+.f64 N/A

           \[\leadsto x \cdot \frac{1}{\left(\color{blue}{\left(1 + x\right)} + y\right) \cdot \left(y + x\right)} \]

       11. lift-+.f64 79.9

           \[\leadsto x \cdot \frac{1}{\color{blue}{\left(\left(1 + x\right) + y\right)} \cdot \left(y + x\right)} \]

       12. lift-+.f64 N/A

           \[\leadsto x \cdot \frac{1}{\left(\left(1 + x\right) + y\right) \cdot \color{blue}{\left(y + x\right)}} \]

       13. +-commutative N/A

           \[\leadsto x \cdot \frac{1}{\left(\left(1 + x\right) + y\right) \cdot \color{blue}{\left(x + y\right)}} \]

       14. lift-+.f64 79.9

           \[\leadsto x \cdot \frac{1}{\left(\left(1 + x\right) + y\right) \cdot \color{blue}{\left(x + y\right)}} \]

   11. Applied rewrites 79.9%

       \[\leadsto x \cdot \color{blue}{\frac{1}{\left(\left(1 + x\right) + y\right) \cdot \left(x + y\right)}} \]

   if 8.5000000000000002e154 < y

   1. Initial program 61.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot y}} \]

      2. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y}} \]

      4. lower-/.f64 91.3

         \[\leadsto \frac{\color{blue}{\frac{x}{y}}}{y} \]

   5. Applied rewrites 91.3%

      \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 61.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{+220}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-128}:\\ \;\;\;\;\frac{y}{\left(1 + x\right) \cdot \left(y + x\right)} \cdot 1\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{+32}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -6e+220)
   (/ (/ y x) x)
   (if (<= x -2e-128)
     (* (/ y (* (+ 1.0 x) (+ y x))) 1.0)
     (if (<= x 5.2e+32) (/ x (fma y y y)) (/ (/ x y) y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -6e+220) {
		tmp = (y / x) / x;
	} else if (x <= -2e-128) {
		tmp = (y / ((1.0 + x) * (y + x))) * 1.0;
	} else if (x <= 5.2e+32) {
		tmp = x / fma(y, y, y);
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -6e+220)
		tmp = Float64(Float64(y / x) / x);
	elseif (x <= -2e-128)
		tmp = Float64(Float64(y / Float64(Float64(1.0 + x) * Float64(y + x))) * 1.0);
	elseif (x <= 5.2e+32)
		tmp = Float64(x / fma(y, y, y));
	else
		tmp = Float64(Float64(x / y) / y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -6e+220], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, -2e-128], N[(N[(y / N[(N[(1.0 + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[x, 5.2e+32], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -6.00000000000000048e220

   1. Initial program 42.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{y}{\color{blue}{x \cdot x}} \]

      2. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x}} \]

      4. lower-/.f64 80.3

         \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{x} \]

   5. Applied rewrites 80.3%

      \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x}} \]

   if -6.00000000000000048e220 < x < -2.00000000000000011e-128

   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. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      6. associate-*l* N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      7. *-commutative N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right) \cdot \left(x + y\right)}} \]

      8. times-frac N/A

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \cdot \frac{x}{x + y}} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \cdot \frac{x}{x + y}} \]

      10. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \cdot \frac{x}{x + y} \]

      11. *-commutative N/A

          \[\leadsto \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \cdot \frac{x}{x + y} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \cdot \frac{x}{x + y} \]

      13. lift-+.f64 N/A

          \[\leadsto \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right)} \cdot \left(x + y\right)} \cdot \frac{x}{x + y} \]

      14. +-commutative N/A

          \[\leadsto \frac{y}{\color{blue}{\left(1 + \left(x + y\right)\right)} \cdot \left(x + y\right)} \cdot \frac{x}{x + y} \]

      15. lower-+.f64 N/A

          \[\leadsto \frac{y}{\color{blue}{\left(1 + \left(x + y\right)\right)} \cdot \left(x + y\right)} \cdot \frac{x}{x + y} \]

      16. lift-+.f64 N/A

          \[\leadsto \frac{y}{\left(1 + \color{blue}{\left(x + y\right)}\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + y} \]

      17. +-commutative N/A

          \[\leadsto \frac{y}{\left(1 + \color{blue}{\left(y + x\right)}\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + y} \]

      18. lower-+.f64 N/A

          \[\leadsto \frac{y}{\left(1 + \color{blue}{\left(y + x\right)}\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + y} \]

      19. lift-+.f64 N/A

          \[\leadsto \frac{y}{\left(1 + \left(y + x\right)\right) \cdot \color{blue}{\left(x + y\right)}} \cdot \frac{x}{x + y} \]

      20. +-commutative N/A

          \[\leadsto \frac{y}{\left(1 + \left(y + x\right)\right) \cdot \color{blue}{\left(y + x\right)}} \cdot \frac{x}{x + y} \]

      21. lower-+.f64 N/A

          \[\leadsto \frac{y}{\left(1 + \left(y + x\right)\right) \cdot \color{blue}{\left(y + x\right)}} \cdot \frac{x}{x + y} \]

      22. lower-/.f64 96.0

          \[\leadsto \frac{y}{\left(1 + \left(y + x\right)\right) \cdot \left(y + x\right)} \cdot \color{blue}{\frac{x}{x + y}} \]

      23. lift-+.f64 N/A

          \[\leadsto \frac{y}{\left(1 + \left(y + x\right)\right) \cdot \left(y + x\right)} \cdot \frac{x}{\color{blue}{x + y}} \]

      24. +-commutative N/A

          \[\leadsto \frac{y}{\left(1 + \left(y + x\right)\right) \cdot \left(y + x\right)} \cdot \frac{x}{\color{blue}{y + x}} \]

   4. Applied rewrites 96.0%

      \[\leadsto \color{blue}{\frac{y}{\left(1 + \left(y + x\right)\right) \cdot \left(y + x\right)} \cdot \frac{x}{y + x}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{y}{\left(1 + \left(y + x\right)\right) \cdot \left(y + x\right)} \cdot \color{blue}{1} \]
   6. Step-by-step derivation

   7. Applied rewrites 74.4%

      \[\leadsto \frac{y}{\left(1 + \left(y + x\right)\right) \cdot \left(y + x\right)} \cdot \color{blue}{1} \]

   8. Taylor expanded in y around 0

      \[\leadsto \frac{y}{\color{blue}{\left(1 + x\right)} \cdot \left(y + x\right)} \cdot 1 \]
   9. Step-by-step derivation

      1. lower-+.f64 58.1

         \[\leadsto \frac{y}{\color{blue}{\left(1 + x\right)} \cdot \left(y + x\right)} \cdot 1 \]

   10. Applied rewrites 58.1%

       \[\leadsto \frac{y}{\color{blue}{\left(1 + x\right)} \cdot \left(y + x\right)} \cdot 1 \]

   if -2.00000000000000011e-128 < x < 5.2000000000000004e32

   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. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

      3. distribute-lft-in N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot y + y \cdot 1}} \]

      4. *-rgt-identity N/A

         \[\leadsto \frac{x}{y \cdot y + \color{blue}{y}} \]

      5. lower-fma.f64 76.1

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(y, y, y\right)}} \]

   5. Applied rewrites 76.1%

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, y, y\right)}} \]

   if 5.2000000000000004e32 < x

   1. Initial program 47.4%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot y}} \]

      2. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y}} \]

      4. lower-/.f64 26.8

         \[\leadsto \frac{\color{blue}{\frac{x}{y}}}{y} \]

   5. Applied rewrites 26.8%

      \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 15: 59.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6.2 \cdot 10^{-142}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{elif}\;y \leq 450000000000:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 6.2e-142)
   (/ y (fma x x x))
   (if (<= y 450000000000.0) (/ x (fma y y y)) (/ (/ x y) y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 6.2e-142) {
		tmp = y / fma(x, x, x);
	} else if (y <= 450000000000.0) {
		tmp = x / fma(y, y, y);
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 6.2e-142)
		tmp = Float64(y / fma(x, x, x));
	elseif (y <= 450000000000.0)
		tmp = Float64(x / fma(y, y, y));
	else
		tmp = Float64(Float64(x / y) / y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 6.2e-142], N[(y / N[(x * x + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 450000000000.0], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 6.2e-142

   1. Initial program 66.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{y}{x \cdot \color{blue}{\left(x + 1\right)}} \]

      3. distribute-lft-in N/A

         \[\leadsto \frac{y}{\color{blue}{x \cdot x + x \cdot 1}} \]

      4. *-rgt-identity N/A

         \[\leadsto \frac{y}{x \cdot x + \color{blue}{x}} \]

      5. lower-fma.f64 53.1

         \[\leadsto \frac{y}{\color{blue}{\mathsf{fma}\left(x, x, x\right)}} \]

   5. Applied rewrites 53.1%

      \[\leadsto \color{blue}{\frac{y}{\mathsf{fma}\left(x, x, x\right)}} \]

   if 6.2e-142 < y < 4.5e11

   1. Initial program 76.2%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

      3. distribute-lft-in N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot y + y \cdot 1}} \]

      4. *-rgt-identity N/A

         \[\leadsto \frac{x}{y \cdot y + \color{blue}{y}} \]

      5. lower-fma.f64 56.1

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(y, y, y\right)}} \]

   5. Applied rewrites 56.1%

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, y, y\right)}} \]

   if 4.5e11 < y

   1. Initial program 63.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot y}} \]

      2. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y}} \]

      4. lower-/.f64 77.2

         \[\leadsto \frac{\color{blue}{\frac{x}{y}}}{y} \]

   5. Applied rewrites 77.2%

      \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 59.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6.2 \cdot 10^{-142}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 6.2e-142) (/ y (fma x x x)) (/ x (fma y y y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 6.2e-142) {
		tmp = y / fma(x, x, x);
	} else {
		tmp = x / fma(y, y, y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 6.2e-142)
		tmp = Float64(y / fma(x, x, x));
	else
		tmp = Float64(x / fma(y, y, y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 6.2e-142], N[(y / N[(x * x + x), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.2e-142

   1. Initial program 66.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{y}{x \cdot \color{blue}{\left(x + 1\right)}} \]

      3. distribute-lft-in N/A

         \[\leadsto \frac{y}{\color{blue}{x \cdot x + x \cdot 1}} \]

      4. *-rgt-identity N/A

         \[\leadsto \frac{y}{x \cdot x + \color{blue}{x}} \]

      5. lower-fma.f64 53.1

         \[\leadsto \frac{y}{\color{blue}{\mathsf{fma}\left(x, x, x\right)}} \]

   5. Applied rewrites 53.1%

      \[\leadsto \color{blue}{\frac{y}{\mathsf{fma}\left(x, x, x\right)}} \]

   if 6.2e-142 < y

   1. Initial program 66.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

      3. distribute-lft-in N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot y + y \cdot 1}} \]

      4. *-rgt-identity N/A

         \[\leadsto \frac{x}{y \cdot y + \color{blue}{y}} \]

      5. lower-fma.f64 68.4

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(y, y, y\right)}} \]

   5. Applied rewrites 68.4%

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, y, y\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 61.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -700:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -700.0) (/ y (* x x)) (/ x (fma y y y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -700.0) {
		tmp = y / (x * x);
	} else {
		tmp = x / fma(y, y, y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -700.0)
		tmp = Float64(y / Float64(x * x));
	else
		tmp = Float64(x / fma(y, y, y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -700.0], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -700

   1. Initial program 59.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{y}{\color{blue}{x \cdot x}} \]

      2. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x}} \]

      4. lower-/.f64 71.6

         \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{x} \]

   5. Applied rewrites 71.6%

      \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x}} \]
   6. Step-by-step derivation

   7. Applied rewrites 70.4%

      \[\leadsto \frac{y}{\color{blue}{x \cdot x}} \]

   if -700 < x

   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. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

      3. distribute-lft-in N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot y + y \cdot 1}} \]

      4. *-rgt-identity N/A

         \[\leadsto \frac{x}{y \cdot y + \color{blue}{y}} \]

      5. lower-fma.f64 61.8

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(y, y, y\right)}} \]

   5. Applied rewrites 61.8%

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, y, y\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 48.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.6 \cdot 10^{+18}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.6e+18) (/ y (* x x)) (/ x (* y y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.6e+18) {
		tmp = y / (x * x);
	} else {
		tmp = x / (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 (y <= 2.6d+18) then
        tmp = y / (x * x)
    else
        tmp = x / (y * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.6e+18) {
		tmp = y / (x * x);
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.6e+18:
		tmp = y / (x * x)
	else:
		tmp = x / (y * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.6e+18)
		tmp = Float64(y / Float64(x * x));
	else
		tmp = Float64(x / Float64(y * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.6e+18)
		tmp = y / (x * x);
	else
		tmp = x / (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.6e+18], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.6e18

   1. Initial program 67.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{y}{\color{blue}{x \cdot x}} \]

      2. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x}} \]

      4. lower-/.f64 34.1

         \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{x} \]

   5. Applied rewrites 34.1%

      \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x}} \]
   6. Step-by-step derivation

   7. Applied rewrites 35.1%

      \[\leadsto \frac{y}{\color{blue}{x \cdot x}} \]

   if 2.6e18 < y

   1. Initial program 63.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

      3. distribute-lft-in N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot y + y \cdot 1}} \]

      4. *-rgt-identity N/A

         \[\leadsto \frac{x}{y \cdot y + \color{blue}{y}} \]

      5. lower-fma.f64 73.1

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(y, y, y\right)}} \]

   5. Applied rewrites 73.1%

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, y, y\right)}} \]

   6. Taylor expanded in y around inf

      \[\leadsto \frac{x}{{y}^{\color{blue}{2}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 73.1%

      \[\leadsto \frac{x}{y \cdot \color{blue}{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 36.2% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \frac{x}{y \cdot y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ x (* y y)))

C

double code(double x, double y) {
	return x / (y * 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 * y)
end function

Java

public static double code(double x, double y) {
	return x / (y * y);
}

Python

def code(x, y):
	return x / (y * y)

Julia

function code(x, y)
	return Float64(x / Float64(y * y))
end

MATLAB

function tmp = code(x, y)
	tmp = x / (y * y);
end

Wolfram

code[x_, y_] := N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.4%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

   2. +-commutative N/A

      \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

   3. distribute-lft-in N/A

      \[\leadsto \frac{x}{\color{blue}{y \cdot y + y \cdot 1}} \]

   4. *-rgt-identity N/A

      \[\leadsto \frac{x}{y \cdot y + \color{blue}{y}} \]

   5. lower-fma.f64 52.8

      \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(y, y, y\right)}} \]

5. Applied rewrites 52.8%

   \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, y, y\right)}} \]

6. Taylor expanded in y around inf

   \[\leadsto \frac{x}{{y}^{\color{blue}{2}}} \]
7. Step-by-step derivation

8. Applied rewrites 40.4%

   \[\leadsto \frac{x}{y \cdot \color{blue}{y}} \]
9. 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 2024352 
(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))))