Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1

Percentage Accurate: 88.8% → 99.9%

Time: 5.0s

Alternatives: 11

Speedup: 1.1×

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: 88.8% 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 86.4%

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

Alternative 2: 92.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ \mathbf{if}\;t\_0 \leq -50000000:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t\_0 \leq 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, x, x\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0))))
   (if (<= t_0 -50000000.0)
     (/ x y)
     (if (<= t_0 1e-9)
       (fma (/ x y) x x)
       (if (<= t_0 2.0) (/ x (+ x 1.0)) (/ x y))))))

C

double code(double x, double y) {
	double t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	double tmp;
	if (t_0 <= -50000000.0) {
		tmp = x / y;
	} else if (t_0 <= 1e-9) {
		tmp = fma((x / y), x, x);
	} else if (t_0 <= 2.0) {
		tmp = x / (x + 1.0);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(x * Float64(Float64(x / y) + 1.0)) / Float64(x + 1.0))
	tmp = 0.0
	if (t_0 <= -50000000.0)
		tmp = Float64(x / y);
	elseif (t_0 <= 1e-9)
		tmp = fma(Float64(x / y), x, x);
	elseif (t_0 <= 2.0)
		tmp = Float64(x / Float64(x + 1.0));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -50000000.0], N[(x / y), $MachinePrecision], If[LessEqual[t$95$0, 1e-9], N[(N[(x / y), $MachinePrecision] * x + x), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -5e7 or 2 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 69.5%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 89.3

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

   5. Applied rewrites 89.3%

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

   if -5e7 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 1.00000000000000006e-9

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in y around 0

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

       1. lower-/.f64 99.8

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

   11. Applied rewrites 99.8%

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

   if 1.00000000000000006e-9 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 97.2%

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

Alternative 3: 84.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ \mathbf{if}\;t\_0 \leq -10000:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t\_0 \leq 0.02:\\ \;\;\;\;\left(1 - x\right) \cdot x\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;1 \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0))))
   (if (<= t_0 -10000.0)
     (/ x y)
     (if (<= t_0 0.02)
       (* (- 1.0 x) x)
       (if (<= t_0 2.0) (* 1.0 1.0) (/ x y))))))

C

double code(double x, double y) {
	double t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	double tmp;
	if (t_0 <= -10000.0) {
		tmp = x / y;
	} else if (t_0 <= 0.02) {
		tmp = (1.0 - x) * x;
	} else if (t_0 <= 2.0) {
		tmp = 1.0 * 1.0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x * ((x / y) + 1.0d0)) / (x + 1.0d0)
    if (t_0 <= (-10000.0d0)) then
        tmp = x / y
    else if (t_0 <= 0.02d0) then
        tmp = (1.0d0 - x) * x
    else if (t_0 <= 2.0d0) then
        tmp = 1.0d0 * 1.0d0
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	double tmp;
	if (t_0 <= -10000.0) {
		tmp = x / y;
	} else if (t_0 <= 0.02) {
		tmp = (1.0 - x) * x;
	} else if (t_0 <= 2.0) {
		tmp = 1.0 * 1.0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x * ((x / y) + 1.0)) / (x + 1.0)
	tmp = 0
	if t_0 <= -10000.0:
		tmp = x / y
	elif t_0 <= 0.02:
		tmp = (1.0 - x) * x
	elif t_0 <= 2.0:
		tmp = 1.0 * 1.0
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x * Float64(Float64(x / y) + 1.0)) / Float64(x + 1.0))
	tmp = 0.0
	if (t_0 <= -10000.0)
		tmp = Float64(x / y);
	elseif (t_0 <= 0.02)
		tmp = Float64(Float64(1.0 - x) * x);
	elseif (t_0 <= 2.0)
		tmp = Float64(1.0 * 1.0);
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	tmp = 0.0;
	if (t_0 <= -10000.0)
		tmp = x / y;
	elseif (t_0 <= 0.02)
		tmp = (1.0 - x) * x;
	elseif (t_0 <= 2.0)
		tmp = 1.0 * 1.0;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -10000.0], N[(x / y), $MachinePrecision], If[LessEqual[t$95$0, 0.02], N[(N[(1.0 - x), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(1.0 * 1.0), $MachinePrecision], N[(x / y), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -1e4 or 2 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 69.8%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 88.6

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

   5. Applied rewrites 88.6%

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

   if -1e4 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 0.0200000000000000004

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 99.4

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in y around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower--.f64 85.4

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

   8. Applied rewrites 85.4%

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

   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. 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 x around inf

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

   7. Applied rewrites 99.0%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 96.1%

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

Alternative 4: 85.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ \mathbf{if}\;t\_0 \leq -10000 \lor \neg \left(t\_0 \leq 2\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0))))
   (if (or (<= t_0 -10000.0) (not (<= t_0 2.0))) (/ x y) (/ 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 <= -10000.0) || !(t_0 <= 2.0)) {
		tmp = x / y;
	} 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 <= (-10000.0d0)) .or. (.not. (t_0 <= 2.0d0))) then
        tmp = x / y
    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 <= -10000.0) || !(t_0 <= 2.0)) {
		tmp = x / y;
	} 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 <= -10000.0) or not (t_0 <= 2.0):
		tmp = x / y
	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 <= -10000.0) || !(t_0 <= 2.0))
		tmp = Float64(x / y);
	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 <= -10000.0) || ~((t_0 <= 2.0)))
		tmp = x / y;
	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, -10000.0], N[Not[LessEqual[t$95$0, 2.0]], $MachinePrecision]], N[(x / y), $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))) < -1e4 or 2 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 69.8%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 88.6

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

   5. Applied rewrites 88.6%

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

   if -1e4 < (/.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 x around 0

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

   5. Applied rewrites 88.2%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \leq -10000 \lor \neg \left(\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \leq 2\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 52.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ \mathbf{if}\;t\_0 \leq -10000:\\ \;\;\;\;\left(-x\right) \cdot x\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-43}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1 \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0))))
   (if (<= t_0 -10000.0) (* (- x) x) (if (<= t_0 2e-43) x (* 1.0 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 <= -10000.0) {
		tmp = -x * x;
	} else if (t_0 <= 2e-43) {
		tmp = x;
	} else {
		tmp = 1.0 * 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 <= (-10000.0d0)) then
        tmp = -x * x
    else if (t_0 <= 2d-43) then
        tmp = x
    else
        tmp = 1.0d0 * 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 <= -10000.0) {
		tmp = -x * x;
	} else if (t_0 <= 2e-43) {
		tmp = x;
	} else {
		tmp = 1.0 * 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x * ((x / y) + 1.0)) / (x + 1.0)
	tmp = 0
	if t_0 <= -10000.0:
		tmp = -x * x
	elif t_0 <= 2e-43:
		tmp = x
	else:
		tmp = 1.0 * 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 <= -10000.0)
		tmp = Float64(Float64(-x) * x);
	elseif (t_0 <= 2e-43)
		tmp = x;
	else
		tmp = Float64(1.0 * 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 <= -10000.0)
		tmp = -x * x;
	elseif (t_0 <= 2e-43)
		tmp = x;
	else
		tmp = 1.0 * 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[LessEqual[t$95$0, -10000.0], N[((-x) * x), $MachinePrecision], If[LessEqual[t$95$0, 2e-43], x, N[(1.0 * 1.0), $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))) < -1e4

   1. Initial program 63.5%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 26.3

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

   5. Applied rewrites 26.3%

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

   6. Taylor expanded in y around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower--.f64 28.4

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

   8. Applied rewrites 28.4%

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

   9. Taylor expanded in x around inf

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 28.5

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

   11. Applied rewrites 28.5%

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

   if -1e4 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 2.00000000000000015e-43

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 89.4%

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

   if 2.00000000000000015e-43 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 84.3%

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

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

   7. Applied rewrites 85.0%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 32.9%

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

Alternative 6: 54.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \leq 0.02:\\ \;\;\;\;\left(1 - x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;1 \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* x (+ (/ x y) 1.0)) (+ x 1.0)) 0.02)
   (* (- 1.0 x) x)
   (* 1.0 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (((x * ((x / y) + 1.0)) / (x + 1.0)) <= 0.02) {
		tmp = (1.0 - x) * x;
	} else {
		tmp = 1.0 * 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)) <= 0.02d0) then
        tmp = (1.0d0 - x) * x
    else
        tmp = 1.0d0 * 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)) <= 0.02) {
		tmp = (1.0 - x) * x;
	} else {
		tmp = 1.0 * 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x * ((x / y) + 1.0)) / (x + 1.0)) <= 0.02:
		tmp = (1.0 - x) * x
	else:
		tmp = 1.0 * 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)) <= 0.02)
		tmp = Float64(Float64(1.0 - x) * x);
	else
		tmp = Float64(1.0 * 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)) <= 0.02)
		tmp = (1.0 - x) * x;
	else
		tmp = 1.0 * 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], 0.02], N[(N[(1.0 - x), $MachinePrecision] * x), $MachinePrecision], N[(1.0 * 1.0), $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))) < 0.0200000000000000004

   1. Initial program 88.1%

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

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 75.6

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

   5. Applied rewrites 75.6%

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

   6. Taylor expanded in y around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower--.f64 66.9

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

   8. Applied rewrites 66.9%

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

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

   1. Initial program 83.2%

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

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

   7. Applied rewrites 90.3%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 34.4%

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

Alternative 7: 47.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \leq 2 \cdot 10^{-43}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1 \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* x (+ (/ x y) 1.0)) (+ x 1.0)) 2e-43) x (* 1.0 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (((x * ((x / y) + 1.0)) / (x + 1.0)) <= 2e-43) {
		tmp = x;
	} else {
		tmp = 1.0 * 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)) <= 2d-43) then
        tmp = x
    else
        tmp = 1.0d0 * 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)) <= 2e-43) {
		tmp = x;
	} else {
		tmp = 1.0 * 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x * ((x / y) + 1.0)) / (x + 1.0)) <= 2e-43:
		tmp = x
	else:
		tmp = 1.0 * 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)) <= 2e-43)
		tmp = x;
	else
		tmp = Float64(1.0 * 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)) <= 2e-43)
		tmp = x;
	else
		tmp = 1.0 * 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], 2e-43], x, N[(1.0 * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 2.00000000000000015e-43

   1. Initial program 87.6%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 60.4%

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

   if 2.00000000000000015e-43 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 84.3%

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

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

   7. Applied rewrites 85.0%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 32.9%

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

Alternative 8: 99.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{+27} \lor \neg \left(x \leq 2.25 \cdot 10^{+15}\right):\\ \;\;\;\;\left(\frac{x}{y} - -1\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{y}, x, x\right)}{x - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -5.5e+27) (not (<= x 2.25e+15)))
   (* (- (/ x y) -1.0) 1.0)
   (/ (fma (/ x y) x x) (- x -1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -5.5e+27) || !(x <= 2.25e+15)) {
		tmp = ((x / y) - -1.0) * 1.0;
	} else {
		tmp = fma((x / y), x, x) / (x - -1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -5.5e+27) || !(x <= 2.25e+15))
		tmp = Float64(Float64(Float64(x / y) - -1.0) * 1.0);
	else
		tmp = Float64(fma(Float64(x / y), x, x) / Float64(x - -1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -5.5e+27], N[Not[LessEqual[x, 2.25e+15]], $MachinePrecision]], N[(N[(N[(x / y), $MachinePrecision] - -1.0), $MachinePrecision] * 1.0), $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 x < -5.49999999999999966e27 or 2.25e15 < x

   1. Initial program 72.2%

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

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

   7. Applied rewrites 100.0%

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

   if -5.49999999999999966e27 < x < 2.25e15

   1. Initial program 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

      5. lower-fma.f64 99.9

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

      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 99.9

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

   4. Applied rewrites 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{+27} \lor \neg \left(x \leq 2.25 \cdot 10^{+15}\right):\\ \;\;\;\;\left(\frac{x}{y} - -1\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{y}, x, x\right)}{x - -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.8\right):\\ \;\;\;\;\left(\frac{x}{y} - -1\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y} - x, x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 0.8)))
   (* (- (/ x y) -1.0) 1.0)
   (fma (- (/ x y) x) x x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.0) || !(x <= 0.8)) {
		tmp = ((x / y) - -1.0) * 1.0;
	} else {
		tmp = fma(((x / y) - x), x, x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 0.8))
		tmp = Float64(Float64(Float64(x / y) - -1.0) * 1.0);
	else
		tmp = fma(Float64(Float64(x / y) - x), x, x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 0.8]], $MachinePrecision]], N[(N[(N[(x / y), $MachinePrecision] - -1.0), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[(N[(x / y), $MachinePrecision] - x), $MachinePrecision] * x + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 0.80000000000000004 < x

   1. Initial program 73.5%

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

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

   7. Applied rewrites 99.6%

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

   if -1 < x < 0.80000000000000004

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 98.0

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

   5. Applied rewrites 98.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 98.0%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.8\right):\\ \;\;\;\;\left(\frac{x}{y} - -1\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y} - x, x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 97.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 73.5%

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

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

   7. Applied rewrites 99.6%

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

   if -1 < x < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 98.0

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

   5. Applied rewrites 98.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 98.0%

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

   9. Taylor expanded in y around 0

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

       1. lower-/.f64 97.8

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

   11. Applied rewrites 97.8%

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

4. Final simplification 98.7%

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

Alternative 11: 38.4% accurate, 34.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 x)

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

def code(x, y):
	return x

Julia

function code(x, y)
	return x
end

MATLAB

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

Wolfram

code[x_, y_] := x

Derivation

1. Initial program 86.4%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 39.1%

   \[\leadsto \color{blue}{x} \]
6. 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 2025026 
(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)))