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

Percentage Accurate: 68.8% → 88.6%

Time: 3.1s

Alternatives: 8

Speedup: 1.4×

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: 88.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -5.8 \cdot 10^{-160}:\\ \;\;\;\;\frac{y}{\left(\left(y + x\right) + 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(2, y, x\right), x, y \cdot y\right)} \cdot x\\ \mathbf{elif}\;x \leq -2.3 \cdot 10^{-218}:\\ \;\;\;\;\frac{y}{\left(1 + x\right) \cdot x}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+22}:\\ \;\;\;\;\frac{x}{\left(1 + y\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -9e+102)
   (/ (/ y x) x)
   (if (<= x -5.8e-160)
     (* (/ y (* (+ (+ y x) 1.0) (fma (fma 2.0 y x) x (* y y)))) x)
     (if (<= x -2.3e-218)
       (/ y (* (+ 1.0 x) x))
       (if (<= x 7.5e+22) (/ x (* (+ 1.0 y) y)) (/ (/ x y) y))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -9e+102) {
		tmp = (y / x) / x;
	} else if (x <= -5.8e-160) {
		tmp = (y / (((y + x) + 1.0) * fma(fma(2.0, y, x), x, (y * y)))) * x;
	} else if (x <= -2.3e-218) {
		tmp = y / ((1.0 + x) * x);
	} else if (x <= 7.5e+22) {
		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 (x <= -9e+102)
		tmp = Float64(Float64(y / x) / x);
	elseif (x <= -5.8e-160)
		tmp = Float64(Float64(y / Float64(Float64(Float64(y + x) + 1.0) * fma(fma(2.0, y, x), x, Float64(y * y)))) * x);
	elseif (x <= -2.3e-218)
		tmp = Float64(y / Float64(Float64(1.0 + x) * x));
	elseif (x <= 7.5e+22)
		tmp = Float64(x / Float64(Float64(1.0 + y) * y));
	else
		tmp = Float64(Float64(x / y) / y);
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -9e+102], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, -5.8e-160], N[(N[(y / N[(N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(2.0 * y + x), $MachinePrecision] * x + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[x, -2.3e-218], N[(y / N[(N[(1.0 + x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 7.5e+22], N[(x / N[(N[(1.0 + y), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -9.00000000000000042e102

   1. Initial program 58.1%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 81.9

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

   4. Applied rewrites 81.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 89.3

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

   6. Applied rewrites 89.3%

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

   if -9.00000000000000042e102 < x < -5.7999999999999998e-160

   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. 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

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 91.0

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

   3. Applied rewrites 91.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   5. Applied rewrites 91.0%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 90.9

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

   8. Applied rewrites 90.9%

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

   if -5.7999999999999998e-160 < x < -2.29999999999999995e-218

   1. Initial program 57.8%

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

   2. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 35.3

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

   4. Applied rewrites 35.3%

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

   if -2.29999999999999995e-218 < x < 7.5000000000000002e22

   1. Initial program 74.9%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 94.4

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

   4. Applied rewrites 94.4%

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

   if 7.5000000000000002e22 < x

   1. Initial program 24.3%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 75.2

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

   4. Applied rewrites 75.2%

      \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \]
   5. 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 95.3

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

   6. Applied rewrites 95.3%

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

Alternative 2: 88.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -5.8 \cdot 10^{-160}:\\ \;\;\;\;\frac{y}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)} \cdot x\\ \mathbf{elif}\;x \leq -2.3 \cdot 10^{-218}:\\ \;\;\;\;\frac{y}{\left(1 + x\right) \cdot x}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+22}:\\ \;\;\;\;\frac{x}{\left(1 + y\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -9e+102)
   (/ (/ y x) x)
   (if (<= x -5.8e-160)
     (* (/ y (* (+ (+ y x) 1.0) (* (+ y x) (+ y x)))) x)
     (if (<= x -2.3e-218)
       (/ y (* (+ 1.0 x) x))
       (if (<= x 7.5e+22) (/ x (* (+ 1.0 y) y)) (/ (/ x y) y))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -9e+102) {
		tmp = (y / x) / x;
	} else if (x <= -5.8e-160) {
		tmp = (y / (((y + x) + 1.0) * ((y + x) * (y + x)))) * x;
	} else if (x <= -2.3e-218) {
		tmp = y / ((1.0 + x) * x);
	} else if (x <= 7.5e+22) {
		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 (x <= (-9d+102)) then
        tmp = (y / x) / x
    else if (x <= (-5.8d-160)) then
        tmp = (y / (((y + x) + 1.0d0) * ((y + x) * (y + x)))) * x
    else if (x <= (-2.3d-218)) then
        tmp = y / ((1.0d0 + x) * x)
    else if (x <= 7.5d+22) 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 (x <= -9e+102) {
		tmp = (y / x) / x;
	} else if (x <= -5.8e-160) {
		tmp = (y / (((y + x) + 1.0) * ((y + x) * (y + x)))) * x;
	} else if (x <= -2.3e-218) {
		tmp = y / ((1.0 + x) * x);
	} else if (x <= 7.5e+22) {
		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 x <= -9e+102:
		tmp = (y / x) / x
	elif x <= -5.8e-160:
		tmp = (y / (((y + x) + 1.0) * ((y + x) * (y + x)))) * x
	elif x <= -2.3e-218:
		tmp = y / ((1.0 + x) * x)
	elif x <= 7.5e+22:
		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 (x <= -9e+102)
		tmp = Float64(Float64(y / x) / x);
	elseif (x <= -5.8e-160)
		tmp = Float64(Float64(y / Float64(Float64(Float64(y + x) + 1.0) * Float64(Float64(y + x) * Float64(y + x)))) * x);
	elseif (x <= -2.3e-218)
		tmp = Float64(y / Float64(Float64(1.0 + x) * x));
	elseif (x <= 7.5e+22)
		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 (x <= -9e+102)
		tmp = (y / x) / x;
	elseif (x <= -5.8e-160)
		tmp = (y / (((y + x) + 1.0) * ((y + x) * (y + x)))) * x;
	elseif (x <= -2.3e-218)
		tmp = y / ((1.0 + x) * x);
	elseif (x <= 7.5e+22)
		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[x, -9e+102], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, -5.8e-160], N[(N[(y / N[(N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[x, -2.3e-218], N[(y / N[(N[(1.0 + x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 7.5e+22], N[(x / N[(N[(1.0 + y), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -9.00000000000000042e102

   1. Initial program 58.1%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 81.9

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

   4. Applied rewrites 81.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 89.3

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

   6. Applied rewrites 89.3%

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

   if -9.00000000000000042e102 < x < -5.7999999999999998e-160

   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. 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

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 91.0

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

   3. Applied rewrites 91.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   5. Applied rewrites 91.0%

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

   if -5.7999999999999998e-160 < x < -2.29999999999999995e-218

   1. Initial program 57.8%

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

   2. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 35.3

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

   4. Applied rewrites 35.3%

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

   if -2.29999999999999995e-218 < x < 7.5000000000000002e22

   1. Initial program 74.9%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 94.4

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

   4. Applied rewrites 94.4%

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

   if 7.5000000000000002e22 < x

   1. Initial program 24.3%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 75.2

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

   4. Applied rewrites 75.2%

      \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \]
   5. 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 95.3

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

   6. Applied rewrites 95.3%

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

Alternative 3: 83.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1820:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -5.8 \cdot 10^{-160}:\\ \;\;\;\;\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(y + 1\right)}\\ \mathbf{elif}\;x \leq -2.3 \cdot 10^{-218}:\\ \;\;\;\;\frac{y}{\left(1 + x\right) \cdot x}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+22}:\\ \;\;\;\;\frac{x}{\left(1 + y\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1820.0)
   (/ (/ y x) x)
   (if (<= x -5.8e-160)
     (/ (* x y) (* (* (+ x y) (+ x y)) (+ y 1.0)))
     (if (<= x -2.3e-218)
       (/ y (* (+ 1.0 x) x))
       (if (<= x 7.5e+22) (/ x (* (+ 1.0 y) y)) (/ (/ x y) y))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1820.0) {
		tmp = (y / x) / x;
	} else if (x <= -5.8e-160) {
		tmp = (x * y) / (((x + y) * (x + y)) * (y + 1.0));
	} else if (x <= -2.3e-218) {
		tmp = y / ((1.0 + x) * x);
	} else if (x <= 7.5e+22) {
		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 (x <= (-1820.0d0)) then
        tmp = (y / x) / x
    else if (x <= (-5.8d-160)) then
        tmp = (x * y) / (((x + y) * (x + y)) * (y + 1.0d0))
    else if (x <= (-2.3d-218)) then
        tmp = y / ((1.0d0 + x) * x)
    else if (x <= 7.5d+22) 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 (x <= -1820.0) {
		tmp = (y / x) / x;
	} else if (x <= -5.8e-160) {
		tmp = (x * y) / (((x + y) * (x + y)) * (y + 1.0));
	} else if (x <= -2.3e-218) {
		tmp = y / ((1.0 + x) * x);
	} else if (x <= 7.5e+22) {
		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 x <= -1820.0:
		tmp = (y / x) / x
	elif x <= -5.8e-160:
		tmp = (x * y) / (((x + y) * (x + y)) * (y + 1.0))
	elif x <= -2.3e-218:
		tmp = y / ((1.0 + x) * x)
	elif x <= 7.5e+22:
		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 (x <= -1820.0)
		tmp = Float64(Float64(y / x) / x);
	elseif (x <= -5.8e-160)
		tmp = Float64(Float64(x * y) / Float64(Float64(Float64(x + y) * Float64(x + y)) * Float64(y + 1.0)));
	elseif (x <= -2.3e-218)
		tmp = Float64(y / Float64(Float64(1.0 + x) * x));
	elseif (x <= 7.5e+22)
		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 (x <= -1820.0)
		tmp = (y / x) / x;
	elseif (x <= -5.8e-160)
		tmp = (x * y) / (((x + y) * (x + y)) * (y + 1.0));
	elseif (x <= -2.3e-218)
		tmp = y / ((1.0 + x) * x);
	elseif (x <= 7.5e+22)
		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[x, -1820.0], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, -5.8e-160], N[(N[(x * y), $MachinePrecision] / N[(N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2.3e-218], N[(y / N[(N[(1.0 + x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 7.5e+22], N[(x / N[(N[(1.0 + y), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -1820

   1. Initial program 65.8%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 78.4

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

   4. Applied rewrites 78.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 83.6

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

   6. Applied rewrites 83.6%

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

   if -1820 < x < -5.7999999999999998e-160

   1. Initial program 81.6%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 77.5%

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

   if -5.7999999999999998e-160 < x < -2.29999999999999995e-218

   1. Initial program 57.8%

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

   2. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 35.3

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

   4. Applied rewrites 35.3%

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

   if -2.29999999999999995e-218 < x < 7.5000000000000002e22

   1. Initial program 74.9%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 94.4

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

   4. Applied rewrites 94.4%

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

   if 7.5000000000000002e22 < x

   1. Initial program 24.3%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 75.2

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

   4. Applied rewrites 75.2%

      \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \]
   5. 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 95.3

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

   6. Applied rewrites 95.3%

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

Alternative 4: 78.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -0.00165:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+22}:\\ \;\;\;\;\frac{x}{\left(1 + y\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -0.00165)
   (/ (/ y x) x)
   (if (<= x 7.5e+22) (/ x (* (+ 1.0 y) y)) (/ (/ x y) y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -0.00165) {
		tmp = (y / x) / x;
	} else if (x <= 7.5e+22) {
		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 (x <= (-0.00165d0)) then
        tmp = (y / x) / x
    else if (x <= 7.5d+22) 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 (x <= -0.00165) {
		tmp = (y / x) / x;
	} else if (x <= 7.5e+22) {
		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 x <= -0.00165:
		tmp = (y / x) / x
	elif x <= 7.5e+22:
		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 (x <= -0.00165)
		tmp = Float64(Float64(y / x) / x);
	elseif (x <= 7.5e+22)
		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 (x <= -0.00165)
		tmp = (y / x) / x;
	elseif (x <= 7.5e+22)
		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[x, -0.00165], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 7.5e+22], 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 x < -0.00165

   1. Initial program 66.2%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 77.1

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

   4. Applied rewrites 77.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 82.2

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

   6. Applied rewrites 82.2%

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

   if -0.00165 < x < 7.5000000000000002e22

   1. Initial program 75.5%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 73.8

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

   4. Applied rewrites 73.8%

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

   if 7.5000000000000002e22 < x

   1. Initial program 24.3%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 75.2

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

   4. Applied rewrites 75.2%

      \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \]
   5. 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 95.3

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

   6. Applied rewrites 95.3%

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

Alternative 5: 66.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -0.00165:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -0.00165) (/ (/ y x) x) (/ (/ x y) y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -0.00165) {
		tmp = (y / x) / x;
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Fortran

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

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-0.00165d0)) then
        tmp = (y / x) / x
    else
        tmp = (x / y) / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -0.00165) {
		tmp = (y / x) / x;
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -0.00165:
		tmp = (y / x) / x
	else:
		tmp = (x / y) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -0.00165)
		tmp = Float64(Float64(y / x) / x);
	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 (x <= -0.00165)
		tmp = (y / x) / x;
	else
		tmp = (x / y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -0.00165], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.00165

   1. Initial program 66.2%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 77.1

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

   4. Applied rewrites 77.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 82.2

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

   6. Applied rewrites 82.2%

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

   if -0.00165 < x

   1. Initial program 70.7%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 52.9

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

   4. Applied rewrites 52.9%

      \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \]
   5. 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 54.8

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

   6. Applied rewrites 54.8%

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

Alternative 6: 64.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -0.00165:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -0.00165) (/ y (* x x)) (/ (/ x y) y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -0.00165) {
		tmp = y / (x * x);
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Fortran

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

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-0.00165d0)) then
        tmp = y / (x * x)
    else
        tmp = (x / y) / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -0.00165) {
		tmp = y / (x * x);
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -0.00165:
		tmp = y / (x * x)
	else:
		tmp = (x / y) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -0.00165)
		tmp = Float64(y / Float64(x * x));
	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 (x <= -0.00165)
		tmp = y / (x * x);
	else
		tmp = (x / y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -0.00165], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.00165

   1. Initial program 66.2%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 77.1

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

   4. Applied rewrites 77.1%

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

   if -0.00165 < x

   1. Initial program 70.7%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 52.9

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

   4. Applied rewrites 52.9%

      \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \]
   5. 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 54.8

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

   6. Applied rewrites 54.8%

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

Alternative 7: 63.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -0.00165:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -0.00165) (/ y (* x x)) (/ x (* y y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -0.00165) {
		tmp = y / (x * x);
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-0.00165d0)) then
        tmp = y / (x * x)
    else
        tmp = x / (y * y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -0.00165) {
		tmp = y / (x * x);
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -0.00165:
		tmp = y / (x * x)
	else:
		tmp = x / (y * y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -0.00165)
		tmp = Float64(y / Float64(x * x));
	else
		tmp = Float64(x / Float64(y * y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -0.00165)
		tmp = y / (x * x);
	else
		tmp = x / (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -0.00165], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.00165

   1. Initial program 66.2%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 77.1

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

   4. Applied rewrites 77.1%

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

   if -0.00165 < x

   1. Initial program 70.7%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 52.9

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

   4. Applied rewrites 52.9%

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

Alternative 8: 36.9% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 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. Taylor expanded in y around inf

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

   1. lower-/.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 36.9

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

4. Applied rewrites 36.9%

   \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025106 
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, A"
  :precision binary64
  (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0))))