Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1

Percentage Accurate: 88.5% → 99.9%

Time: 4.2s

Alternatives: 11

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: 88.5% 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, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.7%

   \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
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. lower-*.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. metadata-eval N/A

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

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

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. lower--.f64 N/A

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

   13. metadata-eval N/A

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

   14. lower-/.f64 99.9

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

   15. lift-+.f64 N/A

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

   16. metadata-eval N/A

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

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

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

   18. metadata-eval N/A

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

   19. metadata-eval N/A

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

   20. metadata-eval N/A

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

   21. lower--.f64 N/A

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

   22. metadata-eval 99.9

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

4. Applied rewrites 99.9%

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

Alternative 2: 92.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} - -1\right)}{x - -1}\\ \mathbf{if}\;t\_0 \leq -40:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, x, y\right)} \cdot x\\ \mathbf{elif}\;t\_0 \leq 0.02:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y} - x, x, x\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\frac{x}{x - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (- (/ x y) -1.0)) (- x -1.0))))
   (if (<= t_0 -40.0)
     (* (/ x (fma y x y)) x)
     (if (<= t_0 0.02)
       (fma (- (/ x y) x) x x)
       (if (<= t_0 2.0) (/ x (- x -1.0)) (* (/ 1.0 y) x))))))

C

double code(double x, double y) {
	double t_0 = (x * ((x / y) - -1.0)) / (x - -1.0);
	double tmp;
	if (t_0 <= -40.0) {
		tmp = (x / fma(y, x, y)) * x;
	} else if (t_0 <= 0.02) {
		tmp = fma(((x / y) - x), x, x);
	} else if (t_0 <= 2.0) {
		tmp = x / (x - -1.0);
	} else {
		tmp = (1.0 / y) * 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 <= -40.0)
		tmp = Float64(Float64(x / fma(y, x, y)) * x);
	elseif (t_0 <= 0.02)
		tmp = fma(Float64(Float64(x / y) - x), x, x);
	elseif (t_0 <= 2.0)
		tmp = Float64(x / Float64(x - -1.0));
	else
		tmp = Float64(Float64(1.0 / y) * x);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * N[(N[(x / y), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -40.0], N[(N[(x / N[(y * x + y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t$95$0, 0.02], N[(N[(N[(x / y), $MachinePrecision] - x), $MachinePrecision] * x + x), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(x / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / y), $MachinePrecision] * x), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -40

   1. Initial program 71.9%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 92.1

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

   5. Applied rewrites 92.1%

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

   if -40 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 0.0200000000000000004

   1. Initial program 99.8%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 99.0

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

   5. Applied rewrites 99.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 99.0%

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

   if 0.0200000000000000004 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 100.0%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. 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. *-rgt-identity N/A

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

      8. rgt-mult-inverse N/A

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

      9. metadata-eval N/A

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

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

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower--.f64 97.4

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

   5. Applied rewrites 97.4%

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

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

   1. Initial program 71.1%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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. lower-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. metadata-eval N/A

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

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

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 100.0

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

      15. lift-+.f64 N/A

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

      16. metadata-eval N/A

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

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

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

      21. lower--.f64 N/A

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

      22. metadata-eval 100.0

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

   4. Applied rewrites 100.0%

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

   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 70.5

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

   7. Applied rewrites 70.5%

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

   9. Applied rewrites 89.8%

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

   10. Taylor expanded in x around inf

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

   12. Applied rewrites 90.5%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\frac{x}{y} - -1\right)}{x - -1} \leq -40:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, x, y\right)} \cdot x\\ \mathbf{elif}\;\frac{x \cdot \left(\frac{x}{y} - -1\right)}{x - -1} \leq 0.02:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y} - x, x, x\right)\\ \mathbf{elif}\;\frac{x \cdot \left(\frac{x}{y} - -1\right)}{x - -1} \leq 2:\\ \;\;\;\;\frac{x}{x - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} - -1\right)}{x - -1}\\ t_1 := \frac{1}{y} \cdot x\\ \mathbf{if}\;t\_0 \leq -40:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0.02:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y} - x, x, x\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\frac{x}{x - -1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (- (/ x y) -1.0)) (- x -1.0))) (t_1 (* (/ 1.0 y) x)))
   (if (<= t_0 -40.0)
     t_1
     (if (<= t_0 0.02)
       (fma (- (/ x y) x) x x)
       (if (<= t_0 2.0) (/ x (- x -1.0)) t_1)))))

C

double code(double x, double y) {
	double t_0 = (x * ((x / y) - -1.0)) / (x - -1.0);
	double t_1 = (1.0 / y) * x;
	double tmp;
	if (t_0 <= -40.0) {
		tmp = t_1;
	} else if (t_0 <= 0.02) {
		tmp = fma(((x / y) - x), x, x);
	} else if (t_0 <= 2.0) {
		tmp = x / (x - -1.0);
	} 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(Float64(1.0 / y) * x)
	tmp = 0.0
	if (t_0 <= -40.0)
		tmp = t_1;
	elseif (t_0 <= 0.02)
		tmp = fma(Float64(Float64(x / y) - x), x, x);
	elseif (t_0 <= 2.0)
		tmp = Float64(x / Float64(x - -1.0));
	else
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(N[(1.0 / y), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[t$95$0, -40.0], t$95$1, If[LessEqual[t$95$0, 0.02], N[(N[(N[(x / y), $MachinePrecision] - x), $MachinePrecision] * x + x), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(x / N[(x - -1.0), $MachinePrecision]), $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))) < -40 or 2 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 71.6%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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. lower-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. metadata-eval N/A

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

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

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 99.9

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

      15. lift-+.f64 N/A

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

      16. metadata-eval N/A

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

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

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

      21. lower--.f64 N/A

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

      22. metadata-eval 99.9

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

   4. Applied rewrites 99.9%

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

   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 72.0

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

   7. Applied rewrites 72.0%

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

   9. Applied rewrites 92.2%

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

   10. Taylor expanded in x around inf

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

   12. Applied rewrites 89.4%

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

   if -40 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 0.0200000000000000004

   1. Initial program 99.8%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 99.0

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

   5. Applied rewrites 99.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 99.0%

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

   if 0.0200000000000000004 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 100.0%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. 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. *-rgt-identity N/A

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

      8. rgt-mult-inverse N/A

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

      9. metadata-eval N/A

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

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

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower--.f64 97.4

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

   5. Applied rewrites 97.4%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\frac{x}{y} - -1\right)}{x - -1} \leq -40:\\ \;\;\;\;\frac{1}{y} \cdot x\\ \mathbf{elif}\;\frac{x \cdot \left(\frac{x}{y} - -1\right)}{x - -1} \leq 0.02:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y} - x, x, x\right)\\ \mathbf{elif}\;\frac{x \cdot \left(\frac{x}{y} - -1\right)}{x - -1} \leq 2:\\ \;\;\;\;\frac{x}{x - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} - -1\right)}{x - -1}\\ \mathbf{if}\;t\_0 \leq -\infty \lor \neg \left(t\_0 \leq 2 \cdot 10^{+208}\right):\\ \;\;\;\;\frac{1}{y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{y}, x, x\right)}{x - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (- (/ x y) -1.0)) (- x -1.0))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 2e+208)))
     (* (/ 1.0 y) x)
     (/ (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 <= -((double) INFINITY)) || !(t_0 <= 2e+208)) {
		tmp = (1.0 / y) * x;
	} 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 <= Float64(-Inf)) || !(t_0 <= 2e+208))
		tmp = Float64(Float64(1.0 / y) * x);
	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, (-Infinity)], N[Not[LessEqual[t$95$0, 2e+208]], $MachinePrecision]], N[(N[(1.0 / y), $MachinePrecision] * x), $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))) < -inf.0 or 2e208 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 51.8%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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. lower-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. metadata-eval N/A

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

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

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 100.0

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

      15. lift-+.f64 N/A

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

      16. metadata-eval N/A

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

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

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

      21. lower--.f64 N/A

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

      22. metadata-eval 100.0

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

   4. Applied rewrites 100.0%

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

   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 63.4

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

   7. Applied rewrites 63.4%

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

   9. Applied rewrites 92.5%

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

   10. Taylor expanded in x around inf

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

   12. Applied rewrites 99.7%

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

   if -inf.0 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 2e208

   1. Initial program 99.8%

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

      1. lift-*.f64 N/A

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

      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 99.8

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

      6. lift-+.f64 N/A

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

      7. metadata-eval N/A

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

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

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. metadata-eval 99.8

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

   4. Applied rewrites 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\frac{x}{y} - -1\right)}{x - -1} \leq -\infty \lor \neg \left(\frac{x \cdot \left(\frac{x}{y} - -1\right)}{x - -1} \leq 2 \cdot 10^{+208}\right):\\ \;\;\;\;\frac{1}{y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{y}, x, x\right)}{x - -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} - -1\right)}{x - -1}\\ \mathbf{if}\;t\_0 \leq -40 \lor \neg \left(t\_0 \leq 2\right):\\ \;\;\;\;\frac{1}{y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (- (/ x y) -1.0)) (- x -1.0))))
   (if (or (<= t_0 -40.0) (not (<= t_0 2.0)))
     (* (/ 1.0 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 <= -40.0) || !(t_0 <= 2.0)) {
		tmp = (1.0 / y) * 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 <= (-40.0d0)) .or. (.not. (t_0 <= 2.0d0))) then
        tmp = (1.0d0 / y) * 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 <= -40.0) || !(t_0 <= 2.0)) {
		tmp = (1.0 / y) * 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 <= -40.0) or not (t_0 <= 2.0):
		tmp = (1.0 / y) * 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 <= -40.0) || !(t_0 <= 2.0))
		tmp = Float64(Float64(1.0 / y) * 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 <= -40.0) || ~((t_0 <= 2.0)))
		tmp = (1.0 / y) * 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, -40.0], N[Not[LessEqual[t$95$0, 2.0]], $MachinePrecision]], N[(N[(1.0 / y), $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))) < -40 or 2 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 71.6%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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. lower-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. metadata-eval N/A

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

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

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 99.9

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

      15. lift-+.f64 N/A

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

      16. metadata-eval N/A

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

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

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

      21. lower--.f64 N/A

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

      22. metadata-eval 99.9

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

   4. Applied rewrites 99.9%

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

   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 72.0

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

   7. Applied rewrites 72.0%

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

   9. Applied rewrites 92.2%

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

   10. Taylor expanded in x around inf

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

   12. Applied rewrites 89.4%

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

   if -40 < (/.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. *-rgt-identity N/A

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

      8. rgt-mult-inverse N/A

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

      9. metadata-eval N/A

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

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

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower--.f64 88.3

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

   5. Applied rewrites 88.3%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\frac{x}{y} - -1\right)}{x - -1} \leq -40 \lor \neg \left(\frac{x \cdot \left(\frac{x}{y} - -1\right)}{x - -1} \leq 2\right):\\ \;\;\;\;\frac{1}{y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x - -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} - -1\right)}{x - -1}\\ \mathbf{if}\;t\_0 \leq 0.9999999:\\ \;\;\;\;\frac{y + x}{\mathsf{fma}\left(y, x, y\right)} \cdot x\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\frac{x}{x - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (- (/ x y) -1.0)) (- x -1.0))))
   (if (<= t_0 0.9999999)
     (* (/ (+ y x) (fma y x y)) x)
     (if (<= t_0 2.0) (/ x (- x -1.0)) (* (/ 1.0 y) x)))))

C

double code(double x, double y) {
	double t_0 = (x * ((x / y) - -1.0)) / (x - -1.0);
	double tmp;
	if (t_0 <= 0.9999999) {
		tmp = ((y + x) / fma(y, x, y)) * x;
	} else if (t_0 <= 2.0) {
		tmp = x / (x - -1.0);
	} else {
		tmp = (1.0 / y) * 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 <= 0.9999999)
		tmp = Float64(Float64(Float64(y + x) / fma(y, x, y)) * x);
	elseif (t_0 <= 2.0)
		tmp = Float64(x / Float64(x - -1.0));
	else
		tmp = Float64(Float64(1.0 / y) * x);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * N[(N[(x / y), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.9999999], N[(N[(N[(y + x), $MachinePrecision] / N[(y * x + y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(x / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / y), $MachinePrecision] * 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))) < 0.999999900000000053

   1. Initial program 91.7%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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. lower-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. metadata-eval N/A

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

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

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 99.9

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

      15. lift-+.f64 N/A

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

      16. metadata-eval N/A

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

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

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

      21. lower--.f64 N/A

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

      22. metadata-eval 99.9

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

   4. Applied rewrites 99.9%

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

   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 77.9

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

   7. Applied rewrites 77.9%

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

   9. Applied rewrites 98.1%

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

   if 0.999999900000000053 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 100.0%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. 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. *-rgt-identity N/A

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

      8. rgt-mult-inverse N/A

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

      9. metadata-eval N/A

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

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

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower--.f64 98.5

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

   5. Applied rewrites 98.5%

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

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

   1. Initial program 71.1%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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. lower-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. metadata-eval N/A

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

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

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 100.0

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

      15. lift-+.f64 N/A

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

      16. metadata-eval N/A

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

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

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

      21. lower--.f64 N/A

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

      22. metadata-eval 100.0

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

   4. Applied rewrites 100.0%

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

   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 70.5

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

   7. Applied rewrites 70.5%

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

   9. Applied rewrites 89.8%

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

   10. Taylor expanded in x around inf

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

   12. Applied rewrites 90.5%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\frac{x}{y} - -1\right)}{x - -1} \leq 0.9999999:\\ \;\;\;\;\frac{y + x}{\mathsf{fma}\left(y, x, y\right)} \cdot x\\ \mathbf{elif}\;\frac{x \cdot \left(\frac{x}{y} - -1\right)}{x - -1} \leq 2:\\ \;\;\;\;\frac{x}{x - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 56.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\frac{x}{y} - -1\right)}{x - -1} \leq -40:\\ \;\;\;\;\left(1 - x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* x (- (/ x y) -1.0)) (- x -1.0)) -40.0)
   (* (- 1.0 x) x)
   (/ x (- x -1.0))))

C

double code(double x, double y) {
	double tmp;
	if (((x * ((x / y) - -1.0)) / (x - -1.0)) <= -40.0) {
		tmp = (1.0 - x) * 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) :: tmp
    if (((x * ((x / y) - (-1.0d0))) / (x - (-1.0d0))) <= (-40.0d0)) then
        tmp = (1.0d0 - x) * x
    else
        tmp = x / (x - (-1.0d0))
    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)) <= -40.0) {
		tmp = (1.0 - x) * x;
	} else {
		tmp = x / (x - -1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x * ((x / y) - -1.0)) / (x - -1.0)) <= -40.0:
		tmp = (1.0 - x) * x
	else:
		tmp = x / (x - -1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x * Float64(Float64(x / y) - -1.0)) / Float64(x - -1.0)) <= -40.0)
		tmp = Float64(Float64(1.0 - x) * x);
	else
		tmp = Float64(x / Float64(x - -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((x * ((x / y) - -1.0)) / (x - -1.0)) <= -40.0)
		tmp = (1.0 - x) * x;
	else
		tmp = x / (x - -1.0);
	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], -40.0], N[(N[(1.0 - x), $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))) < -40

   1. Initial program 71.9%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 22.0

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

   5. Applied rewrites 22.0%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 27.1%

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

   if -40 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 94.4%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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. *-rgt-identity N/A

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

      8. rgt-mult-inverse N/A

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

      9. metadata-eval N/A

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

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

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower--.f64 72.3

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

   5. Applied rewrites 72.3%

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

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\frac{x}{y} - -1\right)}{x - -1} \leq -40:\\ \;\;\;\;\left(1 - x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x - -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 55.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\frac{x}{y} - -1\right)}{x - -1} \leq 0.4:\\ \;\;\;\;\left(1 - x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* x (- (/ x y) -1.0)) (- x -1.0)) 0.4)
   (* (- 1.0 x) x)
   (- 1.0 (/ 1.0 x))))

C

double code(double x, double y) {
	double tmp;
	if (((x * ((x / y) - -1.0)) / (x - -1.0)) <= 0.4) {
		tmp = (1.0 - x) * x;
	} else {
		tmp = 1.0 - (1.0 / 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))) <= 0.4d0) then
        tmp = (1.0d0 - x) * x
    else
        tmp = 1.0d0 - (1.0d0 / 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)) <= 0.4) {
		tmp = (1.0 - x) * x;
	} else {
		tmp = 1.0 - (1.0 / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x * ((x / y) - -1.0)) / (x - -1.0)) <= 0.4:
		tmp = (1.0 - x) * x
	else:
		tmp = 1.0 - (1.0 / 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)) <= 0.4)
		tmp = Float64(Float64(1.0 - x) * x);
	else
		tmp = Float64(1.0 - Float64(1.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((x * ((x / y) - -1.0)) / (x - -1.0)) <= 0.4)
		tmp = (1.0 - x) * x;
	else
		tmp = 1.0 - (1.0 / 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], 0.4], N[(N[(1.0 - x), $MachinePrecision] * x), $MachinePrecision], N[(1.0 - N[(1.0 / x), $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))) < 0.40000000000000002

   1. Initial program 91.6%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 75.8

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

   5. Applied rewrites 75.8%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 67.3%

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

   if 0.40000000000000002 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 85.4%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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. *-rgt-identity N/A

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

      8. rgt-mult-inverse N/A

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

      9. metadata-eval N/A

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

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

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower--.f64 50.4

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

   5. Applied rewrites 50.4%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 49.2%

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

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\frac{x}{y} - -1\right)}{x - -1} \leq 0.4:\\ \;\;\;\;\left(1 - x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 46.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\frac{x}{y} - -1\right)}{x - -1} \leq -40:\\ \;\;\;\;\left(1 - x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - 1, x, 1\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* x (- (/ x y) -1.0)) (- x -1.0)) -40.0)
   (* (- 1.0 x) x)
   (* (fma (- x 1.0) x 1.0) x)))

C

double code(double x, double y) {
	double tmp;
	if (((x * ((x / y) - -1.0)) / (x - -1.0)) <= -40.0) {
		tmp = (1.0 - x) * x;
	} else {
		tmp = fma((x - 1.0), x, 1.0) * 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)) <= -40.0)
		tmp = Float64(Float64(1.0 - x) * x);
	else
		tmp = Float64(fma(Float64(x - 1.0), x, 1.0) * x);
	end
	return 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], -40.0], N[(N[(1.0 - x), $MachinePrecision] * x), $MachinePrecision], N[(N[(N[(x - 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision] * 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))) < -40

   1. Initial program 71.9%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 22.0

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

   5. Applied rewrites 22.0%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 27.1%

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

   if -40 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 94.4%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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. *-rgt-identity N/A

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

      8. rgt-mult-inverse N/A

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

      9. metadata-eval N/A

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

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

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower--.f64 72.3

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

   5. Applied rewrites 72.3%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 19.7%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 55.6%

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

4. Final simplification 49.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\frac{x}{y} - -1\right)}{x - -1} \leq -40:\\ \;\;\;\;\left(1 - x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - 1, x, 1\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 42.7% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.7%

   \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
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. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lower--.f64 N/A

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

   7. lower-/.f64 56.1

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

5. Applied rewrites 56.1%

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

6. Taylor expanded in y around inf

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

8. Applied rewrites 47.3%

   \[\leadsto \left(1 - x\right) \cdot x \]
9. Add Preprocessing

Alternative 11: 39.0% 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 89.7%

   \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
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. lower-*.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. metadata-eval N/A

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

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

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. lower--.f64 N/A

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

   13. metadata-eval N/A

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

   14. lower-/.f64 99.9

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

   15. lift-+.f64 N/A

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

   16. metadata-eval N/A

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

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

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

   18. metadata-eval N/A

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

   19. metadata-eval N/A

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

   20. metadata-eval N/A

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

   21. lower--.f64 N/A

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

   22. metadata-eval 99.9

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

4. Applied rewrites 99.9%

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

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 74.4

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

7. Applied rewrites 74.4%

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

9. Applied rewrites 91.8%

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

10. Taylor expanded in x around 0

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

12. Applied rewrites 42.8%

    \[\leadsto 1 \cdot x \]

13. Final simplification 42.8%

    \[\leadsto 1 \cdot x \]
14. 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 2025017 
(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)))