Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1

Percentage Accurate: 88.4% → 99.9%

Time: 6.9s

Alternatives: 12

Speedup: 1.0×

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: 88.4% 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: 99.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+25} \lor \neg \left(t\_0 \leq 2 \cdot 10^{+235}\right):\\ \;\;\;\;\frac{\frac{x}{1 + x} \cdot x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{y}, x, x\right)}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0))))
   (if (or (<= t_0 -2e+25) (not (<= t_0 2e+235)))
     (/ (* (/ x (+ 1.0 x)) x) y)
     (/ (fma (/ x y) x x) (+ x 1.0)))))

C

double code(double x, double y) {
	double t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	double tmp;
	if ((t_0 <= -2e+25) || !(t_0 <= 2e+235)) {
		tmp = ((x / (1.0 + x)) * x) / y;
	} else {
		tmp = fma((x / y), x, x) / (x + 1.0);
	}
	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+25) || !(t_0 <= 2e+235))
		tmp = Float64(Float64(Float64(x / Float64(1.0 + x)) * x) / y);
	else
		tmp = Float64(fma(Float64(x / y), x, x) / Float64(x + 1.0));
	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[Or[LessEqual[t$95$0, -2e+25], N[Not[LessEqual[t$95$0, 2e+235]], $MachinePrecision]], N[(N[(N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] / y), $MachinePrecision], N[(N[(N[(x / y), $MachinePrecision] * x + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $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))) < -2.00000000000000018e25 or 2.0000000000000001e235 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 63.3%

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

   3. Taylor expanded in y around 0

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. *-rgt-identity N/A

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

      9. lower-fma.f64 82.6

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

   5. Applied rewrites 82.6%

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

   7. Applied rewrites 100.0%

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

   if -2.00000000000000018e25 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 2.0000000000000001e235

   1. Initial program 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

      5. lower-fma.f64 99.9

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

   4. Applied rewrites 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \leq -2 \cdot 10^{+25} \lor \neg \left(\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \leq 2 \cdot 10^{+235}\right):\\ \;\;\;\;\frac{\frac{x}{1 + x} \cdot x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{y}, x, x\right)}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ t_1 := {y}^{-1} \cdot x\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+17}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\frac{x}{x - -1}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+48}:\\ \;\;\;\;\frac{x}{y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0))) (t_1 (* (pow y -1.0) x)))
   (if (<= t_0 -5e+17)
     t_1
     (if (<= t_0 2.0)
       (/ x (- x -1.0))
       (if (<= t_0 2e+48) (* (/ x y) x) t_1)))))

C

double code(double x, double y) {
	double t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	double t_1 = pow(y, -1.0) * x;
	double tmp;
	if (t_0 <= -5e+17) {
		tmp = t_1;
	} else if (t_0 <= 2.0) {
		tmp = x / (x - -1.0);
	} else if (t_0 <= 2e+48) {
		tmp = (x / y) * x;
	} else {
		tmp = t_1;
	}
	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) + 1.0d0)) / (x + 1.0d0)
    t_1 = (y ** (-1.0d0)) * x
    if (t_0 <= (-5d+17)) then
        tmp = t_1
    else if (t_0 <= 2.0d0) then
        tmp = x / (x - (-1.0d0))
    else if (t_0 <= 2d+48) then
        tmp = (x / y) * x
    else
        tmp = t_1
    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 t_1 = Math.pow(y, -1.0) * x;
	double tmp;
	if (t_0 <= -5e+17) {
		tmp = t_1;
	} else if (t_0 <= 2.0) {
		tmp = x / (x - -1.0);
	} else if (t_0 <= 2e+48) {
		tmp = (x / y) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x * ((x / y) + 1.0)) / (x + 1.0)
	t_1 = math.pow(y, -1.0) * x
	tmp = 0
	if t_0 <= -5e+17:
		tmp = t_1
	elif t_0 <= 2.0:
		tmp = x / (x - -1.0)
	elif t_0 <= 2e+48:
		tmp = (x / y) * x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x * Float64(Float64(x / y) + 1.0)) / Float64(x + 1.0))
	t_1 = Float64((y ^ -1.0) * x)
	tmp = 0.0
	if (t_0 <= -5e+17)
		tmp = t_1;
	elseif (t_0 <= 2.0)
		tmp = Float64(x / Float64(x - -1.0));
	elseif (t_0 <= 2e+48)
		tmp = Float64(Float64(x / y) * x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	t_1 = (y ^ -1.0) * x;
	tmp = 0.0;
	if (t_0 <= -5e+17)
		tmp = t_1;
	elseif (t_0 <= 2.0)
		tmp = x / (x - -1.0);
	elseif (t_0 <= 2e+48)
		tmp = (x / y) * x;
	else
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(N[Power[y, -1.0], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+17], t$95$1, If[LessEqual[t$95$0, 2.0], N[(x / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+48], N[(N[(x / y), $MachinePrecision] * x), $MachinePrecision], t$95$1]]]]]

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))) < -5e17 or 2.00000000000000009e48 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 68.5%

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

   3. Taylor expanded in y around 0

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. *-rgt-identity N/A

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

      9. lower-fma.f64 84.0

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

   5. Applied rewrites 84.0%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{1}{y} \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 87.4%

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

   if -5e17 < (/.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. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. rgt-mult-inverse N/A

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

      3. *-rgt-identity N/A

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

      4. distribute-lft-in N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

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

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

      8. *-rgt-identity N/A

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

      9. distribute-lft-neg-out N/A

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

      10. rgt-mult-inverse N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 89.4

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

   5. Applied rewrites 89.4%

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

   if 2 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 2.00000000000000009e48

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. *-rgt-identity N/A

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

      9. lower-fma.f64 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 76.7%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \leq -5 \cdot 10^{+17}:\\ \;\;\;\;{y}^{-1} \cdot x\\ \mathbf{elif}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \leq 2:\\ \;\;\;\;\frac{x}{x - -1}\\ \mathbf{elif}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \leq 2 \cdot 10^{+48}:\\ \;\;\;\;\frac{x}{y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;{y}^{-1} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+17} \lor \neg \left(t\_0 \leq 50000000000\right):\\ \;\;\;\;{y}^{-1} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0))))
   (if (or (<= t_0 -5e+17) (not (<= t_0 50000000000.0)))
     (* (pow y -1.0) x)
     (/ x (- x -1.0)))))

C

double code(double x, double y) {
	double t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	double tmp;
	if ((t_0 <= -5e+17) || !(t_0 <= 50000000000.0)) {
		tmp = pow(y, -1.0) * x;
	} else {
		tmp = x / (x - -1.0);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x * ((x / y) + 1.0d0)) / (x + 1.0d0)
    if ((t_0 <= (-5d+17)) .or. (.not. (t_0 <= 50000000000.0d0))) then
        tmp = (y ** (-1.0d0)) * x
    else
        tmp = x / (x - (-1.0d0))
    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 <= -5e+17) || !(t_0 <= 50000000000.0)) {
		tmp = Math.pow(y, -1.0) * x;
	} else {
		tmp = x / (x - -1.0);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x * ((x / y) + 1.0)) / (x + 1.0)
	tmp = 0
	if (t_0 <= -5e+17) or not (t_0 <= 50000000000.0):
		tmp = math.pow(y, -1.0) * x
	else:
		tmp = x / (x - -1.0)
	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 <= -5e+17) || !(t_0 <= 50000000000.0))
		tmp = Float64((y ^ -1.0) * x);
	else
		tmp = Float64(x / Float64(x - -1.0));
	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 <= -5e+17) || ~((t_0 <= 50000000000.0)))
		tmp = (y ^ -1.0) * x;
	else
		tmp = x / (x - -1.0);
	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[Or[LessEqual[t$95$0, -5e+17], N[Not[LessEqual[t$95$0, 50000000000.0]], $MachinePrecision]], N[(N[Power[y, -1.0], $MachinePrecision] * x), $MachinePrecision], N[(x / N[(x - -1.0), $MachinePrecision]), $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))) < -5e17 or 5e10 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 70.5%

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

   3. Taylor expanded in y around 0

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. *-rgt-identity N/A

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

      9. lower-fma.f64 85.0

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

   5. Applied rewrites 85.0%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{1}{y} \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 83.6%

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

   if -5e17 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 5e10

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. rgt-mult-inverse N/A

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

      3. *-rgt-identity N/A

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

      4. distribute-lft-in N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

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

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

      8. *-rgt-identity N/A

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

      9. distribute-lft-neg-out N/A

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

      10. rgt-mult-inverse N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 88.2

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

   5. Applied rewrites 88.2%

      \[\leadsto \color{blue}{\frac{x}{x - -1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \leq -5 \cdot 10^{+17} \lor \neg \left(\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \leq 50000000000\right):\\ \;\;\;\;{y}^{-1} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x - -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y + x\right) \cdot {y}^{-1}\\ t_1 := \left(y + x\right) \cdot \frac{x}{y}\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -4.4 \cdot 10^{-239}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 2.15 \cdot 10^{-138}:\\ \;\;\;\;\frac{x}{x - -1}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (+ y x) (pow y -1.0))) (t_1 (* (+ y x) (/ x y))))
   (if (<= x -1.0)
     t_0
     (if (<= x -4.4e-239)
       t_1
       (if (<= x 2.15e-138) (/ x (- x -1.0)) (if (<= x 1.0) t_1 t_0))))))

C

double code(double x, double y) {
	double t_0 = (y + x) * pow(y, -1.0);
	double t_1 = (y + x) * (x / y);
	double tmp;
	if (x <= -1.0) {
		tmp = t_0;
	} else if (x <= -4.4e-239) {
		tmp = t_1;
	} else if (x <= 2.15e-138) {
		tmp = x / (x - -1.0);
	} else if (x <= 1.0) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (y + x) * (y ** (-1.0d0))
    t_1 = (y + x) * (x / y)
    if (x <= (-1.0d0)) then
        tmp = t_0
    else if (x <= (-4.4d-239)) then
        tmp = t_1
    else if (x <= 2.15d-138) then
        tmp = x / (x - (-1.0d0))
    else if (x <= 1.0d0) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (y + x) * Math.pow(y, -1.0);
	double t_1 = (y + x) * (x / y);
	double tmp;
	if (x <= -1.0) {
		tmp = t_0;
	} else if (x <= -4.4e-239) {
		tmp = t_1;
	} else if (x <= 2.15e-138) {
		tmp = x / (x - -1.0);
	} else if (x <= 1.0) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y + x) * math.pow(y, -1.0)
	t_1 = (y + x) * (x / y)
	tmp = 0
	if x <= -1.0:
		tmp = t_0
	elif x <= -4.4e-239:
		tmp = t_1
	elif x <= 2.15e-138:
		tmp = x / (x - -1.0)
	elif x <= 1.0:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y + x) * (y ^ -1.0))
	t_1 = Float64(Float64(y + x) * Float64(x / y))
	tmp = 0.0
	if (x <= -1.0)
		tmp = t_0;
	elseif (x <= -4.4e-239)
		tmp = t_1;
	elseif (x <= 2.15e-138)
		tmp = Float64(x / Float64(x - -1.0));
	elseif (x <= 1.0)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y + x) * (y ^ -1.0);
	t_1 = (y + x) * (x / y);
	tmp = 0.0;
	if (x <= -1.0)
		tmp = t_0;
	elseif (x <= -4.4e-239)
		tmp = t_1;
	elseif (x <= 2.15e-138)
		tmp = x / (x - -1.0);
	elseif (x <= 1.0)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y + x), $MachinePrecision] * N[Power[y, -1.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y + x), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.0], t$95$0, If[LessEqual[x, -4.4e-239], t$95$1, If[LessEqual[x, 2.15e-138], N[(x / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.0], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1 or 1 < x

   1. Initial program 74.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 100.0

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 100.0

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

      1. div-add-rev N/A

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

      2. associate-/l/ N/A

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

      3. *-commutative N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. distribute-lft-in N/A

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

      13. *-rgt-identity N/A

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

      14. lower-fma.f64 61.3

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

   7. Applied rewrites 61.3%

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

   9. Applied rewrites 73.1%

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

   10. Taylor expanded in x around inf

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

   12. Applied rewrites 98.5%

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

   if -1 < x < -4.39999999999999965e-239 or 2.15e-138 < x < 1

   1. Initial program 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 99.9

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 99.9

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around 0

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

      1. div-add-rev N/A

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

      2. associate-/l/ N/A

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

      3. *-commutative N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. distribute-lft-in N/A

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

      13. *-rgt-identity N/A

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

      14. lower-fma.f64 93.7

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

   7. Applied rewrites 93.7%

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

   9. Applied rewrites 95.4%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 94.4%

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

   if -4.39999999999999965e-239 < x < 2.15e-138

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. rgt-mult-inverse N/A

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

      3. *-rgt-identity N/A

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

      4. distribute-lft-in N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

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

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

      8. *-rgt-identity N/A

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

      9. distribute-lft-neg-out N/A

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

      10. rgt-mult-inverse N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 94.4

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

   5. Applied rewrites 94.4%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\left(y + x\right) \cdot {y}^{-1}\\ \mathbf{elif}\;x \leq -4.4 \cdot 10^{-239}:\\ \;\;\;\;\left(y + x\right) \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 2.15 \cdot 10^{-138}:\\ \;\;\;\;\frac{x}{x - -1}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\left(y + x\right) \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) \cdot {y}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.82\right):\\ \;\;\;\;\left(y + x\right) \cdot {y}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y} - x, x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 0.82)))
   (* (+ y x) (pow y -1.0))
   (fma (- (/ x y) x) x x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.0) || !(x <= 0.82)) {
		tmp = (y + x) * pow(y, -1.0);
	} else {
		tmp = fma(((x / y) - x), x, x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 0.82))
		tmp = Float64(Float64(y + x) * (y ^ -1.0));
	else
		tmp = fma(Float64(Float64(x / y) - x), x, x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 0.82]], $MachinePrecision]], N[(N[(y + x), $MachinePrecision] * N[Power[y, -1.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[(x / y), $MachinePrecision] - x), $MachinePrecision] * x + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 0.819999999999999951 < x

   1. Initial program 74.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 100.0

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 100.0

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

      1. div-add-rev N/A

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

      2. associate-/l/ N/A

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

      3. *-commutative N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. distribute-lft-in N/A

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

      13. *-rgt-identity N/A

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

      14. lower-fma.f64 61.3

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

   7. Applied rewrites 61.3%

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

   9. Applied rewrites 73.1%

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

   10. Taylor expanded in x around inf

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

   12. Applied rewrites 98.5%

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

   if -1 < x < 0.819999999999999951

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

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

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

      5. +-commutative N/A

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

      6. remove-double-neg N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 99.3

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 99.4%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 99.4%

       \[\leadsto \mathsf{fma}\left(\frac{x}{y} - x, \color{blue}{x}, x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.82\right):\\ \;\;\;\;\left(y + x\right) \cdot {y}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y} - x, x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 59.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-15}:\\ \;\;\;\;\left(1 - x\right) \cdot x\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\frac{x}{x - -1}\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0))))
   (if (<= t_0 -4e-15)
     (* (- 1.0 x) x)
     (if (<= t_0 2.0) (/ x (- x -1.0)) (* x x)))))

C

double code(double x, double y) {
	double t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	double tmp;
	if (t_0 <= -4e-15) {
		tmp = (1.0 - x) * x;
	} else if (t_0 <= 2.0) {
		tmp = x / (x - -1.0);
	} else {
		tmp = x * 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) :: t_0
    real(8) :: tmp
    t_0 = (x * ((x / y) + 1.0d0)) / (x + 1.0d0)
    if (t_0 <= (-4d-15)) then
        tmp = (1.0d0 - x) * x
    else if (t_0 <= 2.0d0) then
        tmp = x / (x - (-1.0d0))
    else
        tmp = x * x
    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 <= -4e-15) {
		tmp = (1.0 - x) * x;
	} else if (t_0 <= 2.0) {
		tmp = x / (x - -1.0);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x * ((x / y) + 1.0)) / (x + 1.0)
	tmp = 0
	if t_0 <= -4e-15:
		tmp = (1.0 - x) * x
	elif t_0 <= 2.0:
		tmp = x / (x - -1.0)
	else:
		tmp = x * x
	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 <= -4e-15)
		tmp = Float64(Float64(1.0 - x) * x);
	elseif (t_0 <= 2.0)
		tmp = Float64(x / Float64(x - -1.0));
	else
		tmp = Float64(x * x);
	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 <= -4e-15)
		tmp = (1.0 - x) * x;
	elseif (t_0 <= 2.0)
		tmp = x / (x - -1.0);
	else
		tmp = x * x;
	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, -4e-15], N[(N[(1.0 - x), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(x / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], N[(x * x), $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))) < -4.0000000000000003e-15

   1. Initial program 73.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

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

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

      5. +-commutative N/A

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

      6. remove-double-neg N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 27.0

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

   5. Applied rewrites 27.0%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 19.3%

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

   if -4.0000000000000003e-15 < (/.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. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. rgt-mult-inverse N/A

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

      3. *-rgt-identity N/A

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

      4. distribute-lft-in N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

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

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

      8. *-rgt-identity N/A

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

      9. distribute-lft-neg-out N/A

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

      10. rgt-mult-inverse N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 89.8

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

   5. Applied rewrites 89.8%

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

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

   1. Initial program 70.3%

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

   4. Applied rewrites 14.2%

      \[\leadsto \color{blue}{\frac{{\left(\frac{x}{y} \cdot x\right)}^{3} - {x}^{3}}{\mathsf{fma}\left(x, x - \frac{x}{y} \cdot x, {\left(\frac{x}{y} \cdot x\right)}^{2}\right) \cdot \left(1 + x\right)}} \]

   5. Taylor expanded in y around inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-+.f64 1.5

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

   7. Applied rewrites 1.5%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 30.6%

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

   11. Taylor expanded in x around inf

       \[\leadsto {x}^{2} \]
   12. Step-by-step derivation

   13. Applied rewrites 30.7%

       \[\leadsto x \cdot x \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 47.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \leq 2 \cdot 10^{-17}:\\ \;\;\;\;\left(1 - x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* x (+ (/ x y) 1.0)) (+ x 1.0)) 2e-17) (* (- 1.0 x) x) (* x x)))

C

double code(double x, double y) {
	double tmp;
	if (((x * ((x / y) + 1.0)) / (x + 1.0)) <= 2e-17) {
		tmp = (1.0 - x) * x;
	} else {
		tmp = x * 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)) <= 2d-17) then
        tmp = (1.0d0 - x) * x
    else
        tmp = x * 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)) <= 2e-17) {
		tmp = (1.0 - x) * x;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x * ((x / y) + 1.0)) / (x + 1.0)) <= 2e-17:
		tmp = (1.0 - x) * x
	else:
		tmp = x * 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)) <= 2e-17)
		tmp = Float64(Float64(1.0 - x) * x);
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((x * ((x / y) + 1.0)) / (x + 1.0)) <= 2e-17)
		tmp = (1.0 - x) * x;
	else
		tmp = x * 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], 2e-17], N[(N[(1.0 - x), $MachinePrecision] * x), $MachinePrecision], N[(x * x), $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))) < 2.00000000000000014e-17

   1. Initial program 91.4%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

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

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

      5. +-commutative N/A

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

      6. remove-double-neg N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 76.8

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

   5. Applied rewrites 76.8%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 66.1%

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

   if 2.00000000000000014e-17 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 80.9%

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

   4. Applied rewrites 9.4%

      \[\leadsto \color{blue}{\frac{{\left(\frac{x}{y} \cdot x\right)}^{3} - {x}^{3}}{\mathsf{fma}\left(x, x - \frac{x}{y} \cdot x, {\left(\frac{x}{y} \cdot x\right)}^{2}\right) \cdot \left(1 + x\right)}} \]

   5. Taylor expanded in y around inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-+.f64 1.5

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

   7. Applied rewrites 1.5%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 21.5%

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

   11. Taylor expanded in x around inf

       \[\leadsto {x}^{2} \]
   12. Step-by-step derivation

   13. Applied rewrites 21.6%

       \[\leadsto x \cdot x \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 43.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \leq 0.999:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* x (+ (/ x y) 1.0)) (+ x 1.0)) 0.999) (* 1.0 x) (* x x)))

C

double code(double x, double y) {
	double tmp;
	if (((x * ((x / y) + 1.0)) / (x + 1.0)) <= 0.999) {
		tmp = 1.0 * x;
	} else {
		tmp = x * 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)) <= 0.999d0) then
        tmp = 1.0d0 * x
    else
        tmp = x * 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)) <= 0.999) {
		tmp = 1.0 * x;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x * ((x / y) + 1.0)) / (x + 1.0)) <= 0.999:
		tmp = 1.0 * x
	else:
		tmp = x * 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)) <= 0.999)
		tmp = Float64(1.0 * x);
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((x * ((x / y) + 1.0)) / (x + 1.0)) <= 0.999)
		tmp = 1.0 * x;
	else
		tmp = x * 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], 0.999], N[(1.0 * x), $MachinePrecision], N[(x * x), $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))) < 0.998999999999999999

   1. Initial program 91.5%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

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

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

      5. +-commutative N/A

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

      6. remove-double-neg N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 76.3

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

   5. Applied rewrites 76.3%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 76.3%

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

   9. Taylor expanded in x around 0

      \[\leadsto 1 \cdot x \]
   10. Step-by-step derivation

   11. Applied rewrites 61.3%

       \[\leadsto 1 \cdot x \]

   if 0.998999999999999999 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 80.7%

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

   4. Applied rewrites 9.5%

      \[\leadsto \color{blue}{\frac{{\left(\frac{x}{y} \cdot x\right)}^{3} - {x}^{3}}{\mathsf{fma}\left(x, x - \frac{x}{y} \cdot x, {\left(\frac{x}{y} \cdot x\right)}^{2}\right) \cdot \left(1 + x\right)}} \]

   5. Taylor expanded in y around inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-+.f64 1.5

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

   7. Applied rewrites 1.5%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 21.6%

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

   11. Taylor expanded in x around inf

       \[\leadsto {x}^{2} \]
   12. Step-by-step derivation

   13. Applied rewrites 21.7%

       \[\leadsto x \cdot x \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.8%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 99.9

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-+.f64 99.9

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 99.9

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

4. Applied rewrites 99.9%

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

Alternative 10: 38.7% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ 1 \cdot x \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.8%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

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

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

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

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

   5. +-commutative N/A

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

   6. remove-double-neg N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. lower--.f64 N/A

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

   10. lower-/.f64 58.4

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

5. Applied rewrites 58.4%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 58.5%

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

9. Taylor expanded in x around 0

   \[\leadsto 1 \cdot x \]
10. Step-by-step derivation

11. Applied rewrites 41.4%

    \[\leadsto 1 \cdot x \]
12. Add Preprocessing

Alternative 11: 3.9% accurate, 11.3× 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 Float64(-x)
end

MATLAB

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

Wolfram

code[x_, y_] := (-x)

Derivation

1. Initial program 87.8%

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

4. Applied rewrites 9.5%

   \[\leadsto \color{blue}{\frac{{\left(\frac{x}{y} \cdot x\right)}^{3} - {x}^{3}}{\mathsf{fma}\left(x, x - \frac{x}{y} \cdot x, {\left(\frac{x}{y} \cdot x\right)}^{2}\right) \cdot \left(1 + x\right)}} \]

5. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 3.9

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

7. Applied rewrites 3.9%

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

Alternative 12: 3.2% accurate, 34.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 -1.0)

C

double code(double x, double y) {
	return -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 = -1.0d0
end function

Java

public static double code(double x, double y) {
	return -1.0;
}

Python

def code(x, y):
	return -1.0

Julia

function code(x, y)
	return -1.0
end

MATLAB

function tmp = code(x, y)
	tmp = -1.0;
end

Wolfram

code[x_, y_] := -1.0

Derivation

1. Initial program 87.8%

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

4. Applied rewrites 9.5%

   \[\leadsto \color{blue}{\frac{{\left(\frac{x}{y} \cdot x\right)}^{3} - {x}^{3}}{\mathsf{fma}\left(x, x - \frac{x}{y} \cdot x, {\left(\frac{x}{y} \cdot x\right)}^{2}\right) \cdot \left(1 + x\right)}} \]

5. Taylor expanded in y around inf

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

   1. associate-*r/ N/A

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

   2. mul-1-neg N/A

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

   3. lower-/.f64 N/A

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

   4. lower-neg.f64 N/A

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

   5. lower-+.f64 3.4

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

7. Applied rewrites 3.4%

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

8. Taylor expanded in x around inf

   \[\leadsto -1 \]
9. Step-by-step derivation

10. Applied rewrites 3.1%

    \[\leadsto -1 \]
11. Add Preprocessing

Developer Target 1: 99.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{x}{1} \cdot \frac{\frac{x}{y} + 1}{x + 1} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024363 
(FPCore (x y)
  :name "Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1"
  :precision binary64

  :alt
  (! :herbie-platform default (* (/ x 1) (/ (+ (/ x y) 1) (+ x 1))))

  (/ (* x (+ (/ x y) 1.0)) (+ x 1.0)))