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

Percentage Accurate: 68.8% → 93.6%

Time: 4.1s

Alternatives: 16

Speedup: 1.5×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 68.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 93.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \left(y + x\right) + 1\\ \mathbf{if}\;y \leq 1.55 \cdot 10^{-162}:\\ \;\;\;\;\frac{\frac{y}{x}}{t\_0}\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{+163}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(2, y, x\right), x, y \cdot y\right)} \cdot \frac{y}{1 + \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\frac{x}{y}}{y}, -2, {y}^{-1}\right) \cdot x}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (+ y x) 1.0)))
   (if (<= y 1.55e-162)
     (/ (/ y x) t_0)
     (if (<= y 7.2e+163)
       (* (/ x (fma (fma 2.0 y x) x (* y y))) (/ y (+ 1.0 (+ y x))))
       (/ (* (fma (/ (/ x y) y) -2.0 (pow y -1.0)) x) t_0)))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = (y + x) + 1.0;
	double tmp;
	if (y <= 1.55e-162) {
		tmp = (y / x) / t_0;
	} else if (y <= 7.2e+163) {
		tmp = (x / fma(fma(2.0, y, x), x, (y * y))) * (y / (1.0 + (y + x)));
	} else {
		tmp = (fma(((x / y) / y), -2.0, pow(y, -1.0)) * x) / t_0;
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(Float64(y + x) + 1.0)
	tmp = 0.0
	if (y <= 1.55e-162)
		tmp = Float64(Float64(y / x) / t_0);
	elseif (y <= 7.2e+163)
		tmp = Float64(Float64(x / fma(fma(2.0, y, x), x, Float64(y * y))) * Float64(y / Float64(1.0 + Float64(y + x))));
	else
		tmp = Float64(Float64(fma(Float64(Float64(x / y) / y), -2.0, (y ^ -1.0)) * x) / t_0);
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[y, 1.55e-162], N[(N[(y / x), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[y, 7.2e+163], N[(N[(x / N[(N[(2.0 * y + x), $MachinePrecision] * x + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y / N[(1.0 + N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision] * -2.0 + N[Power[y, -1.0], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.5499999999999999e-162

   1. Initial program 64.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. lower-pow.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 68.4

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

   4. Applied rewrites 68.4%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 91.0

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

   7. Applied rewrites 91.0%

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

   if 1.5499999999999999e-162 < y < 7.19999999999999955e163

   1. Initial program 77.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 77.0

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

   5. Applied rewrites 77.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 96.1

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

   7. Applied rewrites 96.1%

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

   if 7.19999999999999955e163 < y

   1. Initial program 61.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. lower-pow.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 61.3

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

   4. Applied rewrites 61.3%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. inv-pow N/A

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

      10. lower-pow.f64 93.7

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

   7. Applied rewrites 93.7%

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

Alternative 2: 93.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \left(y + x\right) + 1\\ \mathbf{if}\;y \leq 1.55 \cdot 10^{-162}:\\ \;\;\;\;\frac{\frac{y}{x}}{t\_0}\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{+163}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(2, y, x\right), x, y \cdot y\right)} \cdot \frac{y}{1 + \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (+ y x) 1.0)))
   (if (<= y 1.55e-162)
     (/ (/ y x) t_0)
     (if (<= y 7.2e+163)
       (* (/ x (fma (fma 2.0 y x) x (* y y))) (/ y (+ 1.0 (+ y x))))
       (/ (/ x y) t_0)))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = (y + x) + 1.0;
	double tmp;
	if (y <= 1.55e-162) {
		tmp = (y / x) / t_0;
	} else if (y <= 7.2e+163) {
		tmp = (x / fma(fma(2.0, y, x), x, (y * y))) * (y / (1.0 + (y + x)));
	} else {
		tmp = (x / y) / t_0;
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(Float64(y + x) + 1.0)
	tmp = 0.0
	if (y <= 1.55e-162)
		tmp = Float64(Float64(y / x) / t_0);
	elseif (y <= 7.2e+163)
		tmp = Float64(Float64(x / fma(fma(2.0, y, x), x, Float64(y * y))) * Float64(y / Float64(1.0 + Float64(y + x))));
	else
		tmp = Float64(Float64(x / y) / t_0);
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[y, 1.55e-162], N[(N[(y / x), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[y, 7.2e+163], N[(N[(x / N[(N[(2.0 * y + x), $MachinePrecision] * x + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y / N[(1.0 + N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.5499999999999999e-162

   1. Initial program 64.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. lower-pow.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 68.4

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

   4. Applied rewrites 68.4%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 91.0

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

   7. Applied rewrites 91.0%

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

   if 1.5499999999999999e-162 < y < 7.19999999999999955e163

   1. Initial program 77.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 77.0

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

   5. Applied rewrites 77.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 96.1

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

   7. Applied rewrites 96.1%

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

   if 7.19999999999999955e163 < y

   1. Initial program 61.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. lower-pow.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 61.3

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

   4. Applied rewrites 61.3%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 93.3

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

   7. Applied rewrites 93.3%

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

Alternative 3: 93.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \left(y + x\right) + 1\\ \mathbf{if}\;y \leq 1.55 \cdot 10^{-162}:\\ \;\;\;\;\frac{\frac{y}{x}}{t\_0}\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{+163}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(2, x, y\right), y, x \cdot x\right)} \cdot \frac{y}{1 + \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (+ y x) 1.0)))
   (if (<= y 1.55e-162)
     (/ (/ y x) t_0)
     (if (<= y 7.2e+163)
       (* (/ x (fma (fma 2.0 x y) y (* x x))) (/ y (+ 1.0 (+ y x))))
       (/ (/ x y) t_0)))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = (y + x) + 1.0;
	double tmp;
	if (y <= 1.55e-162) {
		tmp = (y / x) / t_0;
	} else if (y <= 7.2e+163) {
		tmp = (x / fma(fma(2.0, x, y), y, (x * x))) * (y / (1.0 + (y + x)));
	} else {
		tmp = (x / y) / t_0;
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(Float64(y + x) + 1.0)
	tmp = 0.0
	if (y <= 1.55e-162)
		tmp = Float64(Float64(y / x) / t_0);
	elseif (y <= 7.2e+163)
		tmp = Float64(Float64(x / fma(fma(2.0, x, y), y, Float64(x * x))) * Float64(y / Float64(1.0 + Float64(y + x))));
	else
		tmp = Float64(Float64(x / y) / t_0);
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[y, 1.55e-162], N[(N[(y / x), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[y, 7.2e+163], N[(N[(x / N[(N[(2.0 * x + y), $MachinePrecision] * y + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y / N[(1.0 + N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.5499999999999999e-162

   1. Initial program 64.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. lower-pow.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 68.4

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

   4. Applied rewrites 68.4%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 91.0

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

   7. Applied rewrites 91.0%

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

   if 1.5499999999999999e-162 < y < 7.19999999999999955e163

   1. Initial program 77.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 77.0

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

   5. Applied rewrites 77.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 96.1

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

   7. Applied rewrites 96.1%

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

   8. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 96.0

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

   10. Applied rewrites 96.0%

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

   if 7.19999999999999955e163 < y

   1. Initial program 61.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. lower-pow.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 61.3

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

   4. Applied rewrites 61.3%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 93.3

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

   7. Applied rewrites 93.3%

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

Alternative 4: 90.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \left(y + x\right) + 1\\ \mathbf{if}\;y \leq 7.5 \cdot 10^{-162}:\\ \;\;\;\;\frac{\frac{y}{x}}{t\_0}\\ \mathbf{elif}\;y \leq 2.05 \cdot 10^{-21}:\\ \;\;\;\;x \cdot \frac{y}{\left(1 + x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(2, y, x\right), x, y \cdot y\right)}\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{+82}:\\ \;\;\;\;\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (+ y x) 1.0)))
   (if (<= y 7.5e-162)
     (/ (/ y x) t_0)
     (if (<= y 2.05e-21)
       (* x (/ y (* (+ 1.0 x) (fma (fma 2.0 y x) x (* y y)))))
       (if (<= y 4.8e+82)
         (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0)))
         (/ (/ x y) t_0))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = (y + x) + 1.0;
	double tmp;
	if (y <= 7.5e-162) {
		tmp = (y / x) / t_0;
	} else if (y <= 2.05e-21) {
		tmp = x * (y / ((1.0 + x) * fma(fma(2.0, y, x), x, (y * y))));
	} else if (y <= 4.8e+82) {
		tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
	} else {
		tmp = (x / y) / t_0;
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(Float64(y + x) + 1.0)
	tmp = 0.0
	if (y <= 7.5e-162)
		tmp = Float64(Float64(y / x) / t_0);
	elseif (y <= 2.05e-21)
		tmp = Float64(x * Float64(y / Float64(Float64(1.0 + x) * fma(fma(2.0, y, x), x, Float64(y * y)))));
	elseif (y <= 4.8e+82)
		tmp = Float64(Float64(x * y) / Float64(Float64(Float64(x + y) * Float64(x + y)) * Float64(Float64(x + y) + 1.0)));
	else
		tmp = Float64(Float64(x / y) / t_0);
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[y, 7.5e-162], N[(N[(y / x), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[y, 2.05e-21], N[(x * N[(y / N[(N[(1.0 + x), $MachinePrecision] * N[(N[(2.0 * y + x), $MachinePrecision] * x + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.8e+82], 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 / y), $MachinePrecision] / t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if y < 7.49999999999999972e-162

   1. Initial program 64.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. lower-pow.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 68.3

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

   4. Applied rewrites 68.3%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 90.9

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

   7. Applied rewrites 90.9%

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

   if 7.49999999999999972e-162 < y < 2.04999999999999997e-21

   1. Initial program 79.9%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 79.9

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

   5. Applied rewrites 79.9%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 79.9

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

   8. Applied rewrites 79.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 95.7

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 95.7

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

   10. Applied rewrites 95.7%

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

   if 2.04999999999999997e-21 < y < 4.79999999999999996e82

   1. Initial program 88.9%

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

   if 4.79999999999999996e82 < y

   1. Initial program 61.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. lower-pow.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 68.3

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

   4. Applied rewrites 68.3%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 89.2

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

   7. Applied rewrites 89.2%

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

Alternative 5: 91.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \left(y + x\right) + 1\\ \mathbf{if}\;y \leq 7.5 \cdot 10^{-162}:\\ \;\;\;\;\frac{\frac{y}{x}}{t\_0}\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{+82}:\\ \;\;\;\;x \cdot \frac{y}{\left(1 + \left(y + x\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(2, y, x\right), x, y \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (+ y x) 1.0)))
   (if (<= y 7.5e-162)
     (/ (/ y x) t_0)
     (if (<= y 4.8e+82)
       (* x (/ y (* (+ 1.0 (+ y x)) (fma (fma 2.0 y x) x (* y y)))))
       (/ (/ x y) t_0)))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = (y + x) + 1.0;
	double tmp;
	if (y <= 7.5e-162) {
		tmp = (y / x) / t_0;
	} else if (y <= 4.8e+82) {
		tmp = x * (y / ((1.0 + (y + x)) * fma(fma(2.0, y, x), x, (y * y))));
	} else {
		tmp = (x / y) / t_0;
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(Float64(y + x) + 1.0)
	tmp = 0.0
	if (y <= 7.5e-162)
		tmp = Float64(Float64(y / x) / t_0);
	elseif (y <= 4.8e+82)
		tmp = Float64(x * Float64(y / Float64(Float64(1.0 + Float64(y + x)) * fma(fma(2.0, y, x), x, Float64(y * y)))));
	else
		tmp = Float64(Float64(x / y) / t_0);
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[y, 7.5e-162], N[(N[(y / x), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[y, 4.8e+82], N[(x * N[(y / N[(N[(1.0 + N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * y + x), $MachinePrecision] * x + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 7.49999999999999972e-162

   1. Initial program 64.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. lower-pow.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 68.3

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

   4. Applied rewrites 68.3%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 90.9

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

   7. Applied rewrites 90.9%

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

   if 7.49999999999999972e-162 < y < 4.79999999999999996e82

   1. Initial program 84.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 84.0

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

   5. Applied rewrites 84.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 94.6

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-+.f64 94.6

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

   7. Applied rewrites 94.6%

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

   if 4.79999999999999996e82 < y

   1. Initial program 61.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. lower-pow.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 68.3

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

   4. Applied rewrites 68.3%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 89.2

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

   7. Applied rewrites 89.2%

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

Alternative 6: 88.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \left(y + x\right) + 1\\ \mathbf{if}\;y \leq 6.8 \cdot 10^{-154}:\\ \;\;\;\;\frac{\frac{y}{x}}{t\_0}\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{+82}:\\ \;\;\;\;\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (+ y x) 1.0)))
   (if (<= y 6.8e-154)
     (/ (/ y x) t_0)
     (if (<= y 4.8e+82)
       (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0)))
       (/ (/ x y) t_0)))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = (y + x) + 1.0;
	double tmp;
	if (y <= 6.8e-154) {
		tmp = (y / x) / t_0;
	} else if (y <= 4.8e+82) {
		tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
	} else {
		tmp = (x / y) / t_0;
	}
	return tmp;
}

Fortran

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

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (y + x) + 1.0d0
    if (y <= 6.8d-154) then
        tmp = (y / x) / t_0
    else if (y <= 4.8d+82) then
        tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0d0))
    else
        tmp = (x / y) / t_0
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = (y + x) + 1.0;
	double tmp;
	if (y <= 6.8e-154) {
		tmp = (y / x) / t_0;
	} else if (y <= 4.8e+82) {
		tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
	} else {
		tmp = (x / y) / t_0;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = (y + x) + 1.0
	tmp = 0
	if y <= 6.8e-154:
		tmp = (y / x) / t_0
	elif y <= 4.8e+82:
		tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0))
	else:
		tmp = (x / y) / t_0
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(Float64(y + x) + 1.0)
	tmp = 0.0
	if (y <= 6.8e-154)
		tmp = Float64(Float64(y / x) / t_0);
	elseif (y <= 4.8e+82)
		tmp = Float64(Float64(x * y) / Float64(Float64(Float64(x + y) * Float64(x + y)) * Float64(Float64(x + y) + 1.0)));
	else
		tmp = Float64(Float64(x / y) / t_0);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = (y + x) + 1.0;
	tmp = 0.0;
	if (y <= 6.8e-154)
		tmp = (y / x) / t_0;
	elseif (y <= 4.8e+82)
		tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
	else
		tmp = (x / y) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[y, 6.8e-154], N[(N[(y / x), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[y, 4.8e+82], 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 / y), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 6.7999999999999997e-154

   1. Initial program 64.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. lower-pow.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 68.2

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

   4. Applied rewrites 68.2%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 90.4

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

   7. Applied rewrites 90.4%

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

   if 6.7999999999999997e-154 < y < 4.79999999999999996e82

   1. Initial program 84.6%

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

   if 4.79999999999999996e82 < y

   1. Initial program 61.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. lower-pow.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 68.3

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

   4. Applied rewrites 68.3%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 89.2

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

   7. Applied rewrites 89.2%

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

Alternative 7: 86.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \left(y + x\right) + 1\\ \mathbf{if}\;y \leq 6.8 \cdot 10^{-154}:\\ \;\;\;\;\frac{\frac{y}{x}}{t\_0}\\ \mathbf{elif}\;y \leq 0.028:\\ \;\;\;\;\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (+ y x) 1.0)))
   (if (<= y 6.8e-154)
     (/ (/ y x) t_0)
     (if (<= y 0.028)
       (/ (* x y) (* (* (+ x y) (+ x y)) (+ x 1.0)))
       (/ (/ x y) t_0)))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = (y + x) + 1.0;
	double tmp;
	if (y <= 6.8e-154) {
		tmp = (y / x) / t_0;
	} else if (y <= 0.028) {
		tmp = (x * y) / (((x + y) * (x + y)) * (x + 1.0));
	} else {
		tmp = (x / y) / t_0;
	}
	return tmp;
}

Fortran

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

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (y + x) + 1.0d0
    if (y <= 6.8d-154) then
        tmp = (y / x) / t_0
    else if (y <= 0.028d0) then
        tmp = (x * y) / (((x + y) * (x + y)) * (x + 1.0d0))
    else
        tmp = (x / y) / t_0
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = (y + x) + 1.0;
	double tmp;
	if (y <= 6.8e-154) {
		tmp = (y / x) / t_0;
	} else if (y <= 0.028) {
		tmp = (x * y) / (((x + y) * (x + y)) * (x + 1.0));
	} else {
		tmp = (x / y) / t_0;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = (y + x) + 1.0
	tmp = 0
	if y <= 6.8e-154:
		tmp = (y / x) / t_0
	elif y <= 0.028:
		tmp = (x * y) / (((x + y) * (x + y)) * (x + 1.0))
	else:
		tmp = (x / y) / t_0
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(Float64(y + x) + 1.0)
	tmp = 0.0
	if (y <= 6.8e-154)
		tmp = Float64(Float64(y / x) / t_0);
	elseif (y <= 0.028)
		tmp = Float64(Float64(x * y) / Float64(Float64(Float64(x + y) * Float64(x + y)) * Float64(x + 1.0)));
	else
		tmp = Float64(Float64(x / y) / t_0);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = (y + x) + 1.0;
	tmp = 0.0;
	if (y <= 6.8e-154)
		tmp = (y / x) / t_0;
	elseif (y <= 0.028)
		tmp = (x * y) / (((x + y) * (x + y)) * (x + 1.0));
	else
		tmp = (x / y) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[y, 6.8e-154], N[(N[(y / x), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[y, 0.028], N[(N[(x * y), $MachinePrecision] / N[(N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 6.7999999999999997e-154

   1. Initial program 64.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. lower-pow.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 68.2

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

   4. Applied rewrites 68.2%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 90.4

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

   7. Applied rewrites 90.4%

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

   if 6.7999999999999997e-154 < y < 0.0280000000000000006

   1. Initial program 82.6%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 80.5%

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

   if 0.0280000000000000006 < y

   1. Initial program 67.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. lower-pow.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 73.9

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

   4. Applied rewrites 73.9%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 84.9

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

   7. Applied rewrites 84.9%

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

Alternative 8: 70.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{x}{y \cdot y}\\ \mathbf{if}\;x \leq -180000:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq -1.05 \cdot 10^{-141}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-177}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (* y y))))
   (if (<= x -180000.0)
     (/ y (* x x))
     (if (<= x -1.05e-141) t_0 (if (<= x 7e-177) (/ x y) t_0)))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = x / (y * y);
	double tmp;
	if (x <= -180000.0) {
		tmp = y / (x * x);
	} else if (x <= -1.05e-141) {
		tmp = t_0;
	} else if (x <= 7e-177) {
		tmp = x / y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

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

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / (y * y)
    if (x <= (-180000.0d0)) then
        tmp = y / (x * x)
    else if (x <= (-1.05d-141)) then
        tmp = t_0
    else if (x <= 7d-177) then
        tmp = x / y
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = x / (y * y);
	double tmp;
	if (x <= -180000.0) {
		tmp = y / (x * x);
	} else if (x <= -1.05e-141) {
		tmp = t_0;
	} else if (x <= 7e-177) {
		tmp = x / y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = x / (y * y)
	tmp = 0
	if x <= -180000.0:
		tmp = y / (x * x)
	elif x <= -1.05e-141:
		tmp = t_0
	elif x <= 7e-177:
		tmp = x / y
	else:
		tmp = t_0
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(x / Float64(y * y))
	tmp = 0.0
	if (x <= -180000.0)
		tmp = Float64(y / Float64(x * x));
	elseif (x <= -1.05e-141)
		tmp = t_0;
	elseif (x <= 7e-177)
		tmp = Float64(x / y);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = x / (y * y);
	tmp = 0.0;
	if (x <= -180000.0)
		tmp = y / (x * x);
	elseif (x <= -1.05e-141)
		tmp = t_0;
	elseif (x <= 7e-177)
		tmp = x / y;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -180000.0], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.05e-141], t$95$0, If[LessEqual[x, 7e-177], N[(x / y), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.8e5

   1. Initial program 67.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 79.5

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

   5. Applied rewrites 79.5%

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

   if -1.8e5 < x < -1.05e-141 or 7.0000000000000003e-177 < x

   1. Initial program 75.6%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 53.8

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

   5. Applied rewrites 53.8%

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

   if -1.05e-141 < x < 7.0000000000000003e-177

   1. Initial program 63.2%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 63.2

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

   5. Applied rewrites 63.2%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-/l/ N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-+.f64 85.8

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

   8. Applied rewrites 85.8%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 74.0%

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

Alternative 9: 78.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.02 \cdot 10^{-194}:\\ \;\;\;\;\frac{y}{\left(1 + x\right) \cdot x}\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{+163}:\\ \;\;\;\;\frac{x}{\left(1 + y\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1.02e-194)
   (/ y (* (+ 1.0 x) x))
   (if (<= y 7.2e+163) (/ x (* (+ 1.0 y) y)) (/ (/ x y) y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.02e-194) {
		tmp = y / ((1.0 + x) * x);
	} else if (y <= 7.2e+163) {
		tmp = x / ((1.0 + y) * y);
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Fortran

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

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.02d-194) then
        tmp = y / ((1.0d0 + x) * x)
    else if (y <= 7.2d+163) then
        tmp = x / ((1.0d0 + y) * y)
    else
        tmp = (x / y) / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 1.02e-194) {
		tmp = y / ((1.0 + x) * x);
	} else if (y <= 7.2e+163) {
		tmp = x / ((1.0 + y) * y);
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 1.02e-194:
		tmp = y / ((1.0 + x) * x)
	elif y <= 7.2e+163:
		tmp = x / ((1.0 + y) * y)
	else:
		tmp = (x / y) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 1.02e-194)
		tmp = Float64(y / Float64(Float64(1.0 + x) * x));
	elseif (y <= 7.2e+163)
		tmp = Float64(x / Float64(Float64(1.0 + y) * y));
	else
		tmp = Float64(Float64(x / y) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.02e-194)
		tmp = y / ((1.0 + x) * x);
	elseif (y <= 7.2e+163)
		tmp = x / ((1.0 + y) * y);
	else
		tmp = (x / y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 1.02e-194], N[(y / N[(N[(1.0 + x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7.2e+163], N[(x / N[(N[(1.0 + y), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.02e-194

   1. Initial program 65.5%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 89.5

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

   5. Applied rewrites 89.5%

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

   if 1.02e-194 < y < 7.19999999999999955e163

   1. Initial program 75.4%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 62.2

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

   5. Applied rewrites 62.2%

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

   if 7.19999999999999955e163 < y

   1. Initial program 61.3%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 84.0

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

   5. Applied rewrites 84.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 93.1

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

   7. Applied rewrites 93.1%

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

Alternative 10: 80.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \left(y + x\right) + 1\\ \mathbf{if}\;y \leq 1.02 \cdot 10^{-194}:\\ \;\;\;\;\frac{\frac{y}{x}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (+ y x) 1.0)))
   (if (<= y 1.02e-194) (/ (/ y x) t_0) (/ (/ x y) t_0))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = (y + x) + 1.0;
	double tmp;
	if (y <= 1.02e-194) {
		tmp = (y / x) / t_0;
	} else {
		tmp = (x / y) / t_0;
	}
	return tmp;
}

Fortran

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

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (y + x) + 1.0d0
    if (y <= 1.02d-194) then
        tmp = (y / x) / t_0
    else
        tmp = (x / y) / t_0
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = (y + x) + 1.0;
	double tmp;
	if (y <= 1.02e-194) {
		tmp = (y / x) / t_0;
	} else {
		tmp = (x / y) / t_0;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = (y + x) + 1.0
	tmp = 0
	if y <= 1.02e-194:
		tmp = (y / x) / t_0
	else:
		tmp = (x / y) / t_0
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(Float64(y + x) + 1.0)
	tmp = 0.0
	if (y <= 1.02e-194)
		tmp = Float64(Float64(y / x) / t_0);
	else
		tmp = Float64(Float64(x / y) / t_0);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = (y + x) + 1.0;
	tmp = 0.0;
	if (y <= 1.02e-194)
		tmp = (y / x) / t_0;
	else
		tmp = (x / y) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[y, 1.02e-194], N[(N[(y / x), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 1.02e-194

   1. Initial program 65.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. lower-pow.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 69.5

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

   4. Applied rewrites 69.5%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 93.4

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

   7. Applied rewrites 93.4%

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

   if 1.02e-194 < y

   1. Initial program 70.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. lower-pow.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 75.6

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

   4. Applied rewrites 75.6%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 73.5

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

   7. Applied rewrites 73.5%

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

Alternative 11: 79.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.02 \cdot 10^{-194}:\\ \;\;\;\;\frac{y}{\left(1 + x\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{\left(y + x\right) + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1.02e-194) (/ y (* (+ 1.0 x) x)) (/ (/ x y) (+ (+ y x) 1.0))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.02e-194) {
		tmp = y / ((1.0 + x) * x);
	} else {
		tmp = (x / y) / ((y + x) + 1.0);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.02d-194) then
        tmp = y / ((1.0d0 + x) * x)
    else
        tmp = (x / y) / ((y + x) + 1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 1.02e-194) {
		tmp = y / ((1.0 + x) * x);
	} else {
		tmp = (x / y) / ((y + x) + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 1.02e-194:
		tmp = y / ((1.0 + x) * x)
	else:
		tmp = (x / y) / ((y + x) + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 1.02e-194)
		tmp = Float64(y / Float64(Float64(1.0 + x) * x));
	else
		tmp = Float64(Float64(x / y) / Float64(Float64(y + x) + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.02e-194)
		tmp = y / ((1.0 + x) * x);
	else
		tmp = (x / y) / ((y + x) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 1.02e-194], N[(y / N[(N[(1.0 + x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.02e-194

   1. Initial program 65.5%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 89.5

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

   5. Applied rewrites 89.5%

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

   if 1.02e-194 < y

   1. Initial program 70.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. lower-pow.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 75.6

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

   4. Applied rewrites 75.6%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 73.5

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

   7. Applied rewrites 73.5%

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

Alternative 12: 79.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.02 \cdot 10^{-194}:\\ \;\;\;\;\frac{y}{\left(1 + x\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 + y}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1.02e-194) (/ y (* (+ 1.0 x) x)) (/ (/ x (+ 1.0 y)) y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.02e-194) {
		tmp = y / ((1.0 + x) * x);
	} else {
		tmp = (x / (1.0 + y)) / y;
	}
	return tmp;
}

Fortran

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

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.02d-194) then
        tmp = y / ((1.0d0 + x) * x)
    else
        tmp = (x / (1.0d0 + y)) / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 1.02e-194) {
		tmp = y / ((1.0 + x) * x);
	} else {
		tmp = (x / (1.0 + y)) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 1.02e-194:
		tmp = y / ((1.0 + x) * x)
	else:
		tmp = (x / (1.0 + y)) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 1.02e-194)
		tmp = Float64(y / Float64(Float64(1.0 + x) * x));
	else
		tmp = Float64(Float64(x / Float64(1.0 + y)) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.02e-194)
		tmp = y / ((1.0 + x) * x);
	else
		tmp = (x / (1.0 + y)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 1.02e-194], N[(y / N[(N[(1.0 + x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(1.0 + y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.02e-194

   1. Initial program 65.5%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 89.5

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

   5. Applied rewrites 89.5%

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

   if 1.02e-194 < y

   1. Initial program 70.6%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 70.6

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

   5. Applied rewrites 70.6%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-/l/ N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-+.f64 69.6

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

   8. Applied rewrites 69.6%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-+.f64 73.1

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

   10. Applied rewrites 73.1%

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

Alternative 13: 76.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.02 \cdot 10^{-194}:\\ \;\;\;\;\frac{y}{\left(1 + x\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(1 + y\right) \cdot y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1.02e-194) (/ y (* (+ 1.0 x) x)) (/ x (* (+ 1.0 y) y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.02e-194) {
		tmp = y / ((1.0 + x) * x);
	} else {
		tmp = x / ((1.0 + y) * y);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.02d-194) then
        tmp = y / ((1.0d0 + x) * x)
    else
        tmp = x / ((1.0d0 + y) * y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 1.02e-194) {
		tmp = y / ((1.0 + x) * x);
	} else {
		tmp = x / ((1.0 + y) * y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 1.02e-194:
		tmp = y / ((1.0 + x) * x)
	else:
		tmp = x / ((1.0 + y) * y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 1.02e-194)
		tmp = Float64(y / Float64(Float64(1.0 + x) * x));
	else
		tmp = Float64(x / Float64(Float64(1.0 + y) * y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.02e-194)
		tmp = y / ((1.0 + x) * x);
	else
		tmp = x / ((1.0 + y) * y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 1.02e-194], N[(y / N[(N[(1.0 + x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[(1.0 + y), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.02e-194

   1. Initial program 65.5%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 89.5

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

   5. Applied rewrites 89.5%

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

   if 1.02e-194 < y

   1. Initial program 70.6%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 69.6

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

   5. Applied rewrites 69.6%

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

Alternative 14: 76.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -180000:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(1 + y\right) \cdot y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -180000.0) (/ y (* x x)) (/ x (* (+ 1.0 y) y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -180000.0) {
		tmp = y / (x * x);
	} else {
		tmp = x / ((1.0 + y) * y);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-180000.0d0)) then
        tmp = y / (x * x)
    else
        tmp = x / ((1.0d0 + y) * y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -180000.0) {
		tmp = y / (x * x);
	} else {
		tmp = x / ((1.0 + y) * y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -180000.0:
		tmp = y / (x * x)
	else:
		tmp = x / ((1.0 + y) * y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -180000.0)
		tmp = Float64(y / Float64(x * x));
	else
		tmp = Float64(x / Float64(Float64(1.0 + y) * y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -180000.0)
		tmp = y / (x * x);
	else
		tmp = x / ((1.0 + y) * y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -180000.0], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[(1.0 + y), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.8e5

   1. Initial program 67.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 79.5

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

   5. Applied rewrites 79.5%

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

   if -1.8e5 < x

   1. Initial program 70.1%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 73.2

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

   5. Applied rewrites 73.2%

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

Alternative 15: 46.3% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < 1

   1. Initial program 70.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 70.0

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

   5. Applied rewrites 70.0%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-/l/ N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-+.f64 24.2

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

   8. Applied rewrites 24.2%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 21.8%

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

   if 1 < y

   1. Initial program 67.2%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 78.4

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

   5. Applied rewrites 78.4%

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

Alternative 16: 26.4% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.8%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 68.8

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

5. Applied rewrites 68.8%

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

6. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. pow2 N/A

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

   3. +-commutative N/A

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

   4. +-commutative N/A

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

   5. associate-/l/ N/A

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

   6. lower-/.f64 N/A

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

   7. *-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. lift-+.f64 48.1

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

8. Applied rewrites 48.1%

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

9. Taylor expanded in y around 0

   \[\leadsto \frac{x}{y} \]
10. Step-by-step derivation

11. Applied rewrites 26.4%

    \[\leadsto \frac{x}{y} \]
12. 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 2025089 
(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))))