Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1

Percentage Accurate: 88.2% → 99.1%

Time: 3.0s

Alternatives: 14

Speedup: 0.3×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+232}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+182}:\\ \;\;\;\;\frac{x \cdot \frac{y + x}{y}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0))))
   (if (<= t_0 -1e+232)
     (/ x y)
     (if (<= t_0 5e+182) (/ (* x (/ (+ y x) y)) (+ x 1.0)) (/ x y)))))

C

double code(double x, double y) {
	double t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	double tmp;
	if (t_0 <= -1e+232) {
		tmp = x / y;
	} else if (t_0 <= 5e+182) {
		tmp = (x * ((y + x) / y)) / (x + 1.0);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x * ((x / y) + 1.0d0)) / (x + 1.0d0)
    if (t_0 <= (-1d+232)) then
        tmp = x / y
    else if (t_0 <= 5d+182) then
        tmp = (x * ((y + x) / y)) / (x + 1.0d0)
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	double tmp;
	if (t_0 <= -1e+232) {
		tmp = x / y;
	} else if (t_0 <= 5e+182) {
		tmp = (x * ((y + x) / y)) / (x + 1.0);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x * ((x / y) + 1.0)) / (x + 1.0)
	tmp = 0
	if t_0 <= -1e+232:
		tmp = x / y
	elif t_0 <= 5e+182:
		tmp = (x * ((y + x) / y)) / (x + 1.0)
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x * Float64(Float64(x / y) + 1.0)) / Float64(x + 1.0))
	tmp = 0.0
	if (t_0 <= -1e+232)
		tmp = Float64(x / y);
	elseif (t_0 <= 5e+182)
		tmp = Float64(Float64(x * Float64(Float64(y + x) / y)) / Float64(x + 1.0));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	tmp = 0.0;
	if (t_0 <= -1e+232)
		tmp = x / y;
	elseif (t_0 <= 5e+182)
		tmp = (x * ((y + x) / y)) / (x + 1.0);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+232], N[(x / y), $MachinePrecision], If[LessEqual[t$95$0, 5e+182], N[(N[(x * N[(N[(y + x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -1.00000000000000006e232 or 4.99999999999999973e182 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 57.2%

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

   2. Taylor expanded in x around inf

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

      1. lift-/.f64 97.0

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

   4. Applied rewrites 97.0%

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

   if -1.00000000000000006e232 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 4.99999999999999973e182

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-inverses N/A

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

      4. div-add N/A

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

      5. +-commutative N/A

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

      6. *-lft-identity N/A

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

      7. *-rgt-identity N/A

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

      8. lower-/.f64 N/A

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

      9. *-lft-identity N/A

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

      10. *-rgt-identity N/A

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

      11. lower-+.f64 99.9

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

   3. Applied rewrites 99.9%

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

Alternative 2: 99.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+232}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+182}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0))))
   (if (<= t_0 -1e+232) (/ x y) (if (<= t_0 5e+182) t_0 (/ x y)))))

C

double code(double x, double y) {
	double t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	double tmp;
	if (t_0 <= -1e+232) {
		tmp = x / y;
	} else if (t_0 <= 5e+182) {
		tmp = t_0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x * ((x / y) + 1.0d0)) / (x + 1.0d0)
    if (t_0 <= (-1d+232)) then
        tmp = x / y
    else if (t_0 <= 5d+182) then
        tmp = t_0
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	double tmp;
	if (t_0 <= -1e+232) {
		tmp = x / y;
	} else if (t_0 <= 5e+182) {
		tmp = t_0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x * ((x / y) + 1.0)) / (x + 1.0)
	tmp = 0
	if t_0 <= -1e+232:
		tmp = x / y
	elif t_0 <= 5e+182:
		tmp = t_0
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x * Float64(Float64(x / y) + 1.0)) / Float64(x + 1.0))
	tmp = 0.0
	if (t_0 <= -1e+232)
		tmp = Float64(x / y);
	elseif (t_0 <= 5e+182)
		tmp = t_0;
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	tmp = 0.0;
	if (t_0 <= -1e+232)
		tmp = x / y;
	elseif (t_0 <= 5e+182)
		tmp = t_0;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+232], N[(x / y), $MachinePrecision], If[LessEqual[t$95$0, 5e+182], t$95$0, N[(x / y), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -1.00000000000000006e232 or 4.99999999999999973e182 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 57.2%

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

   2. Taylor expanded in x around inf

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

      1. lift-/.f64 97.0

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

   4. Applied rewrites 97.0%

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

   if -1.00000000000000006e232 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 4.99999999999999973e182

   1. Initial program 99.9%

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

Alternative 3: 97.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \frac{x}{y}}{x + 1}\\ t_1 := x \cdot \left(\frac{x}{y} + 1\right)\\ t_2 := \frac{t\_1}{x + 1}\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+232}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t\_2 \leq -100000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\left(\left(\left(-x\right) + \frac{x}{y}\right) + 1\right) \cdot x\\ \mathbf{elif}\;t\_2 \leq 100:\\ \;\;\;\;\frac{t\_1}{x}\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+182}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (/ x y)) (+ x 1.0)))
        (t_1 (* x (+ (/ x y) 1.0)))
        (t_2 (/ t_1 (+ x 1.0))))
   (if (<= t_2 -1e+232)
     (/ x y)
     (if (<= t_2 -100000000000.0)
       t_0
       (if (<= t_2 5e-9)
         (* (+ (+ (- x) (/ x y)) 1.0) x)
         (if (<= t_2 100.0) (/ t_1 x) (if (<= t_2 5e+182) t_0 (/ x y))))))))

C

double code(double x, double y) {
	double t_0 = (x * (x / y)) / (x + 1.0);
	double t_1 = x * ((x / y) + 1.0);
	double t_2 = t_1 / (x + 1.0);
	double tmp;
	if (t_2 <= -1e+232) {
		tmp = x / y;
	} else if (t_2 <= -100000000000.0) {
		tmp = t_0;
	} else if (t_2 <= 5e-9) {
		tmp = ((-x + (x / y)) + 1.0) * x;
	} else if (t_2 <= 100.0) {
		tmp = t_1 / x;
	} else if (t_2 <= 5e+182) {
		tmp = t_0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (x * (x / y)) / (x + 1.0d0)
    t_1 = x * ((x / y) + 1.0d0)
    t_2 = t_1 / (x + 1.0d0)
    if (t_2 <= (-1d+232)) then
        tmp = x / y
    else if (t_2 <= (-100000000000.0d0)) then
        tmp = t_0
    else if (t_2 <= 5d-9) then
        tmp = ((-x + (x / y)) + 1.0d0) * x
    else if (t_2 <= 100.0d0) then
        tmp = t_1 / x
    else if (t_2 <= 5d+182) then
        tmp = t_0
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x * (x / y)) / (x + 1.0);
	double t_1 = x * ((x / y) + 1.0);
	double t_2 = t_1 / (x + 1.0);
	double tmp;
	if (t_2 <= -1e+232) {
		tmp = x / y;
	} else if (t_2 <= -100000000000.0) {
		tmp = t_0;
	} else if (t_2 <= 5e-9) {
		tmp = ((-x + (x / y)) + 1.0) * x;
	} else if (t_2 <= 100.0) {
		tmp = t_1 / x;
	} else if (t_2 <= 5e+182) {
		tmp = t_0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x * (x / y)) / (x + 1.0)
	t_1 = x * ((x / y) + 1.0)
	t_2 = t_1 / (x + 1.0)
	tmp = 0
	if t_2 <= -1e+232:
		tmp = x / y
	elif t_2 <= -100000000000.0:
		tmp = t_0
	elif t_2 <= 5e-9:
		tmp = ((-x + (x / y)) + 1.0) * x
	elif t_2 <= 100.0:
		tmp = t_1 / x
	elif t_2 <= 5e+182:
		tmp = t_0
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x * Float64(x / y)) / Float64(x + 1.0))
	t_1 = Float64(x * Float64(Float64(x / y) + 1.0))
	t_2 = Float64(t_1 / Float64(x + 1.0))
	tmp = 0.0
	if (t_2 <= -1e+232)
		tmp = Float64(x / y);
	elseif (t_2 <= -100000000000.0)
		tmp = t_0;
	elseif (t_2 <= 5e-9)
		tmp = Float64(Float64(Float64(Float64(-x) + Float64(x / y)) + 1.0) * x);
	elseif (t_2 <= 100.0)
		tmp = Float64(t_1 / x);
	elseif (t_2 <= 5e+182)
		tmp = t_0;
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x * (x / y)) / (x + 1.0);
	t_1 = x * ((x / y) + 1.0);
	t_2 = t_1 / (x + 1.0);
	tmp = 0.0;
	if (t_2 <= -1e+232)
		tmp = x / y;
	elseif (t_2 <= -100000000000.0)
		tmp = t_0;
	elseif (t_2 <= 5e-9)
		tmp = ((-x + (x / y)) + 1.0) * x;
	elseif (t_2 <= 100.0)
		tmp = t_1 / x;
	elseif (t_2 <= 5e+182)
		tmp = t_0;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+232], N[(x / y), $MachinePrecision], If[LessEqual[t$95$2, -100000000000.0], t$95$0, If[LessEqual[t$95$2, 5e-9], N[(N[(N[((-x) + N[(x / y), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t$95$2, 100.0], N[(t$95$1 / x), $MachinePrecision], If[LessEqual[t$95$2, 5e+182], t$95$0, N[(x / y), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -1.00000000000000006e232 or 4.99999999999999973e182 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 57.2%

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

   2. Taylor expanded in x around inf

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

      1. lift-/.f64 97.0

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

   4. Applied rewrites 97.0%

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

   if -1.00000000000000006e232 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -1e11 or 100 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 4.99999999999999973e182

   1. Initial program 99.7%

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

   2. Taylor expanded in x around inf

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

      1. lift-/.f64 98.4

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

   4. Applied rewrites 98.4%

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

   if -1e11 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 5.0000000000000001e-9

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 98.6

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

   4. Applied rewrites 98.6%

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

   5. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lift-/.f64 98.6

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

   7. Applied rewrites 98.6%

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

   if 5.0000000000000001e-9 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 100

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 92.6%

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

Alternative 4: 97.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \frac{x}{y}}{x + 1}\\ t_1 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+232}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t\_1 \leq -100000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\left(\left(\left(-x\right) + \frac{x}{y}\right) + 1\right) \cdot x\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+182}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (/ x y)) (+ x 1.0)))
        (t_1 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0))))
   (if (<= t_1 -1e+232)
     (/ x y)
     (if (<= t_1 -100000000000.0)
       t_0
       (if (<= t_1 5e-9)
         (* (+ (+ (- x) (/ x y)) 1.0) x)
         (if (<= t_1 2.0)
           (/ x (+ x 1.0))
           (if (<= t_1 5e+182) t_0 (/ x y))))))))

C

double code(double x, double y) {
	double t_0 = (x * (x / y)) / (x + 1.0);
	double t_1 = (x * ((x / y) + 1.0)) / (x + 1.0);
	double tmp;
	if (t_1 <= -1e+232) {
		tmp = x / y;
	} else if (t_1 <= -100000000000.0) {
		tmp = t_0;
	} else if (t_1 <= 5e-9) {
		tmp = ((-x + (x / y)) + 1.0) * x;
	} else if (t_1 <= 2.0) {
		tmp = x / (x + 1.0);
	} else if (t_1 <= 5e+182) {
		tmp = t_0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (x * (x / y)) / (x + 1.0d0)
    t_1 = (x * ((x / y) + 1.0d0)) / (x + 1.0d0)
    if (t_1 <= (-1d+232)) then
        tmp = x / y
    else if (t_1 <= (-100000000000.0d0)) then
        tmp = t_0
    else if (t_1 <= 5d-9) then
        tmp = ((-x + (x / y)) + 1.0d0) * x
    else if (t_1 <= 2.0d0) then
        tmp = x / (x + 1.0d0)
    else if (t_1 <= 5d+182) then
        tmp = t_0
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x * (x / y)) / (x + 1.0);
	double t_1 = (x * ((x / y) + 1.0)) / (x + 1.0);
	double tmp;
	if (t_1 <= -1e+232) {
		tmp = x / y;
	} else if (t_1 <= -100000000000.0) {
		tmp = t_0;
	} else if (t_1 <= 5e-9) {
		tmp = ((-x + (x / y)) + 1.0) * x;
	} else if (t_1 <= 2.0) {
		tmp = x / (x + 1.0);
	} else if (t_1 <= 5e+182) {
		tmp = t_0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x * (x / y)) / (x + 1.0)
	t_1 = (x * ((x / y) + 1.0)) / (x + 1.0)
	tmp = 0
	if t_1 <= -1e+232:
		tmp = x / y
	elif t_1 <= -100000000000.0:
		tmp = t_0
	elif t_1 <= 5e-9:
		tmp = ((-x + (x / y)) + 1.0) * x
	elif t_1 <= 2.0:
		tmp = x / (x + 1.0)
	elif t_1 <= 5e+182:
		tmp = t_0
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x * Float64(x / y)) / Float64(x + 1.0))
	t_1 = Float64(Float64(x * Float64(Float64(x / y) + 1.0)) / Float64(x + 1.0))
	tmp = 0.0
	if (t_1 <= -1e+232)
		tmp = Float64(x / y);
	elseif (t_1 <= -100000000000.0)
		tmp = t_0;
	elseif (t_1 <= 5e-9)
		tmp = Float64(Float64(Float64(Float64(-x) + Float64(x / y)) + 1.0) * x);
	elseif (t_1 <= 2.0)
		tmp = Float64(x / Float64(x + 1.0));
	elseif (t_1 <= 5e+182)
		tmp = t_0;
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x * (x / y)) / (x + 1.0);
	t_1 = (x * ((x / y) + 1.0)) / (x + 1.0);
	tmp = 0.0;
	if (t_1 <= -1e+232)
		tmp = x / y;
	elseif (t_1 <= -100000000000.0)
		tmp = t_0;
	elseif (t_1 <= 5e-9)
		tmp = ((-x + (x / y)) + 1.0) * x;
	elseif (t_1 <= 2.0)
		tmp = x / (x + 1.0);
	elseif (t_1 <= 5e+182)
		tmp = t_0;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+232], N[(x / y), $MachinePrecision], If[LessEqual[t$95$1, -100000000000.0], t$95$0, If[LessEqual[t$95$1, 5e-9], N[(N[(N[((-x) + N[(x / y), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+182], t$95$0, N[(x / y), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -1.00000000000000006e232 or 4.99999999999999973e182 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 57.2%

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

   2. Taylor expanded in x around inf

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

      1. lift-/.f64 97.0

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

   4. Applied rewrites 97.0%

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

   if -1.00000000000000006e232 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -1e11 or 2 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 4.99999999999999973e182

   1. Initial program 99.7%

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

   2. Taylor expanded in x around inf

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

      1. lift-/.f64 98.0

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

   4. Applied rewrites 98.0%

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

   if -1e11 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 5.0000000000000001e-9

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 98.6

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

   4. Applied rewrites 98.6%

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

   5. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lift-/.f64 98.6

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

   7. Applied rewrites 98.6%

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

   if 5.0000000000000001e-9 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 95.1%

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

Alternative 5: 92.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+16}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t\_0 \leq 10^{-37}:\\ \;\;\;\;\frac{\mathsf{fma}\left(1, y, x\right)}{y} \cdot x\\ \mathbf{elif}\;t\_0 \leq 100:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+182}:\\ \;\;\;\;\frac{x \cdot x}{\mathsf{fma}\left(y, x, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0))))
   (if (<= t_0 -2e+16)
     (/ x y)
     (if (<= t_0 1e-37)
       (* (/ (fma 1.0 y x) y) x)
       (if (<= t_0 100.0)
         (/ x (+ x 1.0))
         (if (<= t_0 5e+182) (/ (* x x) (fma y x y)) (/ x y)))))))

C

double code(double x, double y) {
	double t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	double tmp;
	if (t_0 <= -2e+16) {
		tmp = x / y;
	} else if (t_0 <= 1e-37) {
		tmp = (fma(1.0, y, x) / y) * x;
	} else if (t_0 <= 100.0) {
		tmp = x / (x + 1.0);
	} else if (t_0 <= 5e+182) {
		tmp = (x * x) / fma(y, x, y);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(x * Float64(Float64(x / y) + 1.0)) / Float64(x + 1.0))
	tmp = 0.0
	if (t_0 <= -2e+16)
		tmp = Float64(x / y);
	elseif (t_0 <= 1e-37)
		tmp = Float64(Float64(fma(1.0, y, x) / y) * x);
	elseif (t_0 <= 100.0)
		tmp = Float64(x / Float64(x + 1.0));
	elseif (t_0 <= 5e+182)
		tmp = Float64(Float64(x * x) / fma(y, x, y));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+16], N[(x / y), $MachinePrecision], If[LessEqual[t$95$0, 1e-37], N[(N[(N[(1.0 * y + x), $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t$95$0, 100.0], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+182], N[(N[(x * x), $MachinePrecision] / N[(y * x + y), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -2e16 or 4.99999999999999973e182 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 66.8%

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

   2. Taylor expanded in x around inf

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

      1. lift-/.f64 89.9

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

   4. Applied rewrites 89.9%

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

   if -2e16 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 1.00000000000000007e-37

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 98.0

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

   4. Applied rewrites 98.0%

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

   5. Taylor expanded in y around 0

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

      1. lift-/.f64 97.7

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

   7. Applied rewrites 97.7%

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

   8. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

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

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. *-rgt-identity N/A

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

      9. lower--.f64 98.0

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

   10. Applied rewrites 98.0%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 97.7%

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

   if 1.00000000000000007e-37 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 100

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 91.3%

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

   if 100 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 4.99999999999999973e182

   1. Initial program 99.7%

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

   2. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. lower-fma.f64 79.4

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

   4. Applied rewrites 79.4%

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

Alternative 6: 86.2% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 76.5%

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

   2. Taylor expanded in x around inf

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

      1. lift-/.f64 73.3

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

   4. Applied rewrites 73.3%

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

   if -1 < x < 1

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 98.1

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

   4. Applied rewrites 98.1%

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

   5. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lift-/.f64 98.2

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

   7. Applied rewrites 98.2%

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

Alternative 7: 85.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 940000:\\ \;\;\;\;\frac{\mathsf{fma}\left(1, y, x\right)}{y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.0)
   (/ x y)
   (if (<= x 940000.0) (* (/ (fma 1.0 y x) y) x) (/ x y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = x / y;
	} else if (x <= 940000.0) {
		tmp = (fma(1.0, y, x) / y) * x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(x / y);
	elseif (x <= 940000.0)
		tmp = Float64(Float64(fma(1.0, y, x) / y) * x);
	else
		tmp = Float64(x / y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.0], N[(x / y), $MachinePrecision], If[LessEqual[x, 940000.0], N[(N[(N[(1.0 * y + x), $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision], N[(x / y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 9.4e5 < x

   1. Initial program 76.3%

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

   2. Taylor expanded in x around inf

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

      1. lift-/.f64 73.7

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

   4. Applied rewrites 73.7%

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

   if -1 < x < 9.4e5

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 97.3

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

   4. Applied rewrites 97.3%

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

   5. Taylor expanded in y around 0

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

      1. lift-/.f64 96.8

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

   7. Applied rewrites 96.8%

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

   8. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

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

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. *-rgt-identity N/A

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

      9. lower--.f64 97.4

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

   10. Applied rewrites 97.4%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 96.9%

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

Alternative 8: 85.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 940000:\\ \;\;\;\;\left(\frac{x}{y} + 1\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.0) (/ x y) (if (<= x 940000.0) (* (+ (/ x y) 1.0) x) (/ x y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = x / y;
	} else if (x <= 940000.0) {
		tmp = ((x / y) + 1.0) * x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = x / y
    else if (x <= 940000.0d0) then
        tmp = ((x / y) + 1.0d0) * x
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = x / y;
	} else if (x <= 940000.0) {
		tmp = ((x / y) + 1.0) * x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.0:
		tmp = x / y
	elif x <= 940000.0:
		tmp = ((x / y) + 1.0) * x
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(x / y);
	elseif (x <= 940000.0)
		tmp = Float64(Float64(Float64(x / y) + 1.0) * x);
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = x / y;
	elseif (x <= 940000.0)
		tmp = ((x / y) + 1.0) * x;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.0], N[(x / y), $MachinePrecision], If[LessEqual[x, 940000.0], N[(N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision], N[(x / y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 9.4e5 < x

   1. Initial program 76.3%

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

   2. Taylor expanded in x around inf

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

      1. lift-/.f64 73.7

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

   4. Applied rewrites 73.7%

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

   if -1 < x < 9.4e5

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 97.3

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

   4. Applied rewrites 97.3%

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

   5. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lift-/.f64 97.4

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

   7. Applied rewrites 97.4%

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

   8. Taylor expanded in y around 0

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

      1. lift-/.f64 96.9

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

   10. Applied rewrites 96.9%

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

Alternative 9: 85.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ \mathbf{if}\;t\_0 \leq -100000000000:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0))))
   (if (<= t_0 -100000000000.0)
     (/ x y)
     (if (<= t_0 2.0) (/ x (+ x 1.0)) (/ x y)))))

C

double code(double x, double y) {
	double t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	double tmp;
	if (t_0 <= -100000000000.0) {
		tmp = x / y;
	} else if (t_0 <= 2.0) {
		tmp = x / (x + 1.0);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x * ((x / y) + 1.0d0)) / (x + 1.0d0)
    if (t_0 <= (-100000000000.0d0)) then
        tmp = x / y
    else if (t_0 <= 2.0d0) then
        tmp = x / (x + 1.0d0)
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	double tmp;
	if (t_0 <= -100000000000.0) {
		tmp = x / y;
	} else if (t_0 <= 2.0) {
		tmp = x / (x + 1.0);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x * ((x / y) + 1.0)) / (x + 1.0)
	tmp = 0
	if t_0 <= -100000000000.0:
		tmp = x / y
	elif t_0 <= 2.0:
		tmp = x / (x + 1.0)
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x * Float64(Float64(x / y) + 1.0)) / Float64(x + 1.0))
	tmp = 0.0
	if (t_0 <= -100000000000.0)
		tmp = Float64(x / y);
	elseif (t_0 <= 2.0)
		tmp = Float64(x / Float64(x + 1.0));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	tmp = 0.0;
	if (t_0 <= -100000000000.0)
		tmp = x / y;
	elseif (t_0 <= 2.0)
		tmp = x / (x + 1.0);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -100000000000.0], N[(x / y), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -1e11 or 2 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 72.2%

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

   2. Taylor expanded in x around inf

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

      1. lift-/.f64 85.3

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

   4. Applied rewrites 85.3%

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

   if -1e11 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 86.8%

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

Alternative 10: 85.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ \mathbf{if}\;t\_0 \leq -100000000000:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-9}:\\ \;\;\;\;x - x \cdot x\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\frac{x}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0))))
   (if (<= t_0 -100000000000.0)
     (/ x y)
     (if (<= t_0 5e-9) (- x (* x x)) (if (<= t_0 2.0) (/ x x) (/ x y))))))

C

double code(double x, double y) {
	double t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	double tmp;
	if (t_0 <= -100000000000.0) {
		tmp = x / y;
	} else if (t_0 <= 5e-9) {
		tmp = x - (x * x);
	} else if (t_0 <= 2.0) {
		tmp = x / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x * ((x / y) + 1.0d0)) / (x + 1.0d0)
    if (t_0 <= (-100000000000.0d0)) then
        tmp = x / y
    else if (t_0 <= 5d-9) then
        tmp = x - (x * x)
    else if (t_0 <= 2.0d0) then
        tmp = x / x
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	double tmp;
	if (t_0 <= -100000000000.0) {
		tmp = x / y;
	} else if (t_0 <= 5e-9) {
		tmp = x - (x * x);
	} else if (t_0 <= 2.0) {
		tmp = x / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x * ((x / y) + 1.0)) / (x + 1.0)
	tmp = 0
	if t_0 <= -100000000000.0:
		tmp = x / y
	elif t_0 <= 5e-9:
		tmp = x - (x * x)
	elif t_0 <= 2.0:
		tmp = x / x
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x * Float64(Float64(x / y) + 1.0)) / Float64(x + 1.0))
	tmp = 0.0
	if (t_0 <= -100000000000.0)
		tmp = Float64(x / y);
	elseif (t_0 <= 5e-9)
		tmp = Float64(x - Float64(x * x));
	elseif (t_0 <= 2.0)
		tmp = Float64(x / x);
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	tmp = 0.0;
	if (t_0 <= -100000000000.0)
		tmp = x / y;
	elseif (t_0 <= 5e-9)
		tmp = x - (x * x);
	elseif (t_0 <= 2.0)
		tmp = x / x;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -100000000000.0], N[(x / y), $MachinePrecision], If[LessEqual[t$95$0, 5e-9], N[(x - N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(x / x), $MachinePrecision], N[(x / y), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -1e11 or 2 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 72.2%

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

   2. Taylor expanded in x around inf

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

      1. lift-/.f64 85.3

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

   4. Applied rewrites 85.3%

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

   if -1e11 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 5.0000000000000001e-9

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 98.6

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

   4. Applied rewrites 98.6%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lift--.f64 98.6

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

   6. Applied rewrites 98.6%

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

   7. Taylor expanded in y around inf

      \[\leadsto x + \color{blue}{-1 \cdot {x}^{2}} \]
   8. Step-by-step derivation

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

         \[\leadsto x - \left(\mathsf{neg}\left(-1\right)\right) \cdot \color{blue}{{x}^{2}} \]

      2. metadata-eval N/A

         \[\leadsto x - 1 \cdot {x}^{2} \]

      3. metadata-eval N/A

         \[\leadsto x - {1}^{2} \cdot {x}^{2} \]

      4. unpow-prod-down N/A

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

      5. *-commutative N/A

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

      6. *-rgt-identity N/A

         \[\leadsto x - {x}^{2} \]

      7. lower--.f64 N/A

         \[\leadsto x - {x}^{\color{blue}{2}} \]

      8. pow2 N/A

         \[\leadsto x - x \cdot x \]

      9. lift-*.f64 84.1

         \[\leadsto x - x \cdot x \]

   9. Applied rewrites 84.1%

      \[\leadsto x - \color{blue}{x \cdot x} \]

   if 5.0000000000000001e-9 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 95.1%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 89.4%

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

Alternative 11: 73.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ \mathbf{if}\;t\_0 \leq -100000000000:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-9}:\\ \;\;\;\;x - x \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0))))
   (if (<= t_0 -100000000000.0)
     (/ x y)
     (if (<= t_0 5e-9) (- x (* x x)) (/ x y)))))

C

double code(double x, double y) {
	double t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	double tmp;
	if (t_0 <= -100000000000.0) {
		tmp = x / y;
	} else if (t_0 <= 5e-9) {
		tmp = x - (x * x);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x * ((x / y) + 1.0d0)) / (x + 1.0d0)
    if (t_0 <= (-100000000000.0d0)) then
        tmp = x / y
    else if (t_0 <= 5d-9) then
        tmp = x - (x * x)
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	double tmp;
	if (t_0 <= -100000000000.0) {
		tmp = x / y;
	} else if (t_0 <= 5e-9) {
		tmp = x - (x * x);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x * ((x / y) + 1.0)) / (x + 1.0)
	tmp = 0
	if t_0 <= -100000000000.0:
		tmp = x / y
	elif t_0 <= 5e-9:
		tmp = x - (x * x)
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x * Float64(Float64(x / y) + 1.0)) / Float64(x + 1.0))
	tmp = 0.0
	if (t_0 <= -100000000000.0)
		tmp = Float64(x / y);
	elseif (t_0 <= 5e-9)
		tmp = Float64(x - Float64(x * x));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	tmp = 0.0;
	if (t_0 <= -100000000000.0)
		tmp = x / y;
	elseif (t_0 <= 5e-9)
		tmp = x - (x * x);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -100000000000.0], N[(x / y), $MachinePrecision], If[LessEqual[t$95$0, 5e-9], N[(x - N[(x * x), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -1e11 or 5.0000000000000001e-9 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 78.9%

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

   2. Taylor expanded in x around inf

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

      1. lift-/.f64 65.9

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

   4. Applied rewrites 65.9%

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

   if -1e11 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 5.0000000000000001e-9

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 98.6

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

   4. Applied rewrites 98.6%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lift--.f64 98.6

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

   6. Applied rewrites 98.6%

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

   7. Taylor expanded in y around inf

      \[\leadsto x + \color{blue}{-1 \cdot {x}^{2}} \]
   8. Step-by-step derivation

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

         \[\leadsto x - \left(\mathsf{neg}\left(-1\right)\right) \cdot \color{blue}{{x}^{2}} \]

      2. metadata-eval N/A

         \[\leadsto x - 1 \cdot {x}^{2} \]

      3. metadata-eval N/A

         \[\leadsto x - {1}^{2} \cdot {x}^{2} \]

      4. unpow-prod-down N/A

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

      5. *-commutative N/A

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

      6. *-rgt-identity N/A

         \[\leadsto x - {x}^{2} \]

      7. lower--.f64 N/A

         \[\leadsto x - {x}^{\color{blue}{2}} \]

      8. pow2 N/A

         \[\leadsto x - x \cdot x \]

      9. lift-*.f64 84.1

         \[\leadsto x - x \cdot x \]

   9. Applied rewrites 84.1%

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

Alternative 12: 44.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \leq -5:\\ \;\;\;\;\left(-x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* x (+ (/ x y) 1.0)) (+ x 1.0)) -5.0) (* (- x) x) x))

C

double code(double x, double y) {
	double tmp;
	if (((x * ((x / y) + 1.0)) / (x + 1.0)) <= -5.0) {
		tmp = -x * x;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (((x * ((x / y) + 1.0d0)) / (x + 1.0d0)) <= (-5.0d0)) then
        tmp = -x * x
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((x * ((x / y) + 1.0)) / (x + 1.0)) <= -5.0) {
		tmp = -x * x;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x * ((x / y) + 1.0)) / (x + 1.0)) <= -5.0:
		tmp = -x * x
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x * Float64(Float64(x / y) + 1.0)) / Float64(x + 1.0)) <= -5.0)
		tmp = Float64(Float64(-x) * x);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((x * ((x / y) + 1.0)) / (x + 1.0)) <= -5.0)
		tmp = -x * x;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x * N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], -5.0], N[((-x) * x), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -5

   1. Initial program 73.5%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 31.0

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

   4. Applied rewrites 31.0%

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

   5. Taylor expanded in y around inf

      \[\leadsto \left(1 + -1 \cdot x\right) \cdot x \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(-1 \cdot x + 1\right) \cdot x \]

      2. lower-+.f64 N/A

         \[\leadsto \left(-1 \cdot x + 1\right) \cdot x \]

      3. mul-1-neg N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(x\right)\right) + 1\right) \cdot x \]

      4. lower-neg.f64 26.3

         \[\leadsto \left(\left(-x\right) + 1\right) \cdot x \]

   7. Applied rewrites 26.3%

      \[\leadsto \left(\left(-x\right) + 1\right) \cdot x \]

   8. Taylor expanded in x around inf

      \[\leadsto \left(-1 \cdot x\right) \cdot x \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot x \]

      2. lift-neg.f64 26.4

         \[\leadsto \left(-x\right) \cdot x \]

   10. Applied rewrites 26.4%

       \[\leadsto \left(-x\right) \cdot x \]

   if -5 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 92.4%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 49.3%

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

Alternative 13: 43.4% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ x - x \cdot x \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- x (* x x)))

C

double code(double x, double y) {
	return x - (x * 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 - (x * x)
end function

Java

public static double code(double x, double y) {
	return x - (x * x);
}

Python

def code(x, y):
	return x - (x * x)

Julia

function code(x, y)
	return Float64(x - Float64(x * x))
end

MATLAB

function tmp = code(x, y)
	tmp = x - (x * x);
end

Wolfram

code[x_, y_] := N[(x - N[(x * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 88.2%

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

2. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lower--.f64 N/A

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

   7. lower-/.f64 56.6

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

4. Applied rewrites 56.6%

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

   1. lift-*.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. *-commutative N/A

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

   8. distribute-lft-in N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lift-/.f64 N/A

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

   14. lift--.f64 56.6

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

6. Applied rewrites 56.6%

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

7. Taylor expanded in y around inf

   \[\leadsto x + \color{blue}{-1 \cdot {x}^{2}} \]
8. Step-by-step derivation

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

      \[\leadsto x - \left(\mathsf{neg}\left(-1\right)\right) \cdot \color{blue}{{x}^{2}} \]

   2. metadata-eval N/A

      \[\leadsto x - 1 \cdot {x}^{2} \]

   3. metadata-eval N/A

      \[\leadsto x - {1}^{2} \cdot {x}^{2} \]

   4. unpow-prod-down N/A

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

   5. *-commutative N/A

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

   6. *-rgt-identity N/A

      \[\leadsto x - {x}^{2} \]

   7. lower--.f64 N/A

      \[\leadsto x - {x}^{\color{blue}{2}} \]

   8. pow2 N/A

      \[\leadsto x - x \cdot x \]

   9. lift-*.f64 43.4

      \[\leadsto x - x \cdot x \]

9. Applied rewrites 43.4%

   \[\leadsto x - \color{blue}{x \cdot x} \]
10. Add Preprocessing

Alternative 14: 39.0% accurate, 34.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x y) :precision binary64 x)

C

double code(double x, double y) {
	return 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
end function

Java

public static double code(double x, double y) {
	return x;
}

Python

def code(x, y):
	return x

Julia

function code(x, y)
	return x
end

MATLAB

function tmp = code(x, y)
	tmp = x;
end

Wolfram

code[x_, y_] := x

Derivation

1. Initial program 88.2%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 39.0%

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

Reproduce

herbie shell --seed 2025101 
(FPCore (x y)
  :name "Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1"
  :precision binary64
  (/ (* x (+ (/ x y) 1.0)) (+ x 1.0)))