Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1

Percentage Accurate: 88.5% → 99.9%

Time: 4.3s

Alternatives: 10

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

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.5%

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

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 99.9

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

   6. lift-+.f64 N/A

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

   7. add-flip N/A

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

   8. lower--.f64 N/A

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

   9. metadata-eval 99.9

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

   10. lift-+.f64 N/A

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

   11. add-flip N/A

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

   12. lower--.f64 N/A

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

   13. metadata-eval 99.9

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

3. Applied rewrites 99.9%

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

Alternative 2: 99.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ t_1 := \frac{1}{\frac{y}{x}}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+261}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{y}, x, x\right)}{x - -1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 (- INFINITY))
     t_1
     (if (<= t_0 2e+261) (/ (fma (/ x y) x 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 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_0 <= 2e+261) {
		tmp = fma((x / y), x, 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(1.0 / Float64(y / x))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_0 <= 2e+261)
		tmp = Float64(fma(Float64(x / y), x, 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[(1.0 / N[(y / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], t$95$1, If[LessEqual[t$95$0, 2e+261], N[(N[(N[(x / y), $MachinePrecision] * x + x), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

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 1.9999999999999999e261 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 88.5%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-special-/.f64 N/A

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

      4. lower-special-/.f64 88.3

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

      5. lift-+.f64 N/A

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

      6. add-flip N/A

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

      7. lower--.f64 N/A

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

      8. metadata-eval 88.3

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. distribute-lft1-in N/A

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

      13. lower-fma.f64 88.3

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

   3. Applied rewrites 88.3%

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

   4. Taylor expanded in x around inf

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

      1. lower-/.f64 38.6

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

   6. Applied rewrites 38.6%

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

   if -inf.0 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 1.9999999999999999e261

   1. Initial program 88.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. distribute-lft1-in N/A

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

      5. lower-fma.f64 88.5

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

      6. lift-+.f64 N/A

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

      7. add-flip N/A

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

      8. lower--.f64 N/A

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

      9. metadata-eval 88.5

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

   3. Applied rewrites 88.5%

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

Alternative 3: 99.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y + x\right) \cdot \frac{1}{y}\\ \mathbf{if}\;x \leq -1.25 \cdot 10^{+16}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+15}:\\ \;\;\;\;\frac{y + x}{\mathsf{fma}\left(y, x, y\right)} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (+ y x) (/ 1.0 y))))
   (if (<= x -1.25e+16)
     t_0
     (if (<= x 4e+15) (* (/ (+ y x) (fma y x y)) x) t_0))))

C

double code(double x, double y) {
	double t_0 = (y + x) * (1.0 / y);
	double tmp;
	if (x <= -1.25e+16) {
		tmp = t_0;
	} else if (x <= 4e+15) {
		tmp = ((y + x) / fma(y, x, y)) * x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(y + x) * Float64(1.0 / y))
	tmp = 0.0
	if (x <= -1.25e+16)
		tmp = t_0;
	elseif (x <= 4e+15)
		tmp = Float64(Float64(Float64(y + x) / fma(y, x, y)) * x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y + x), $MachinePrecision] * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.25e+16], t$95$0, If[LessEqual[x, 4e+15], N[(N[(N[(y + x), $MachinePrecision] / N[(y * x + y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.25e16 or 4e15 < x

   1. Initial program 88.5%

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

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. add-to-fraction N/A

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

      9. frac-times N/A

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

      10. associate-/l* N/A

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

      11. associate-/l/ N/A

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

      12. lift-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. add-flip N/A

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

      15. *-lft-identity N/A

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

      16. add-flip-rev N/A

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

      17. lower-+.f64 N/A

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

      18. lift-/.f64 N/A

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

      19. associate-/l/ N/A

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

      20. lower-/.f64 N/A

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

      21. lift-+.f64 N/A

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

      22. distribute-rgt-in N/A

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

      23. *-commutative N/A

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

      24. *-lft-identity N/A

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

      25. lower-fma.f64 76.5

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

   3. Applied rewrites 76.5%

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

   4. Taylor expanded in x around inf

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

      1. lower-/.f64 49.4

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

   6. Applied rewrites 49.4%

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

   if -1.25e16 < x < 4e15

   1. Initial program 88.5%

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

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-/.f64 N/A

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

      9. add-to-fraction N/A

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

      10. associate-/l/ N/A

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

      11. lower-/.f64 N/A

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

      12. add-flip N/A

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

      13. *-lft-identity N/A

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

      14. add-flip-rev N/A

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

      15. lower-+.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. distribute-rgt-in N/A

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

      18. *-commutative N/A

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

      19. *-lft-identity N/A

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

      20. lower-fma.f64 87.7

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

   3. Applied rewrites 87.7%

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

Alternative 4: 98.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y + x\right) \cdot \frac{1}{y}\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 0.85:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y} - x, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (+ y x) (/ 1.0 y))))
   (if (<= x -1.0) t_0 (if (<= x 0.85) (fma (- (/ x y) x) x x) t_0))))

C

double code(double x, double y) {
	double t_0 = (y + x) * (1.0 / y);
	double tmp;
	if (x <= -1.0) {
		tmp = t_0;
	} else if (x <= 0.85) {
		tmp = fma(((x / y) - x), x, x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(y + x) * Float64(1.0 / y))
	tmp = 0.0
	if (x <= -1.0)
		tmp = t_0;
	elseif (x <= 0.85)
		tmp = fma(Float64(Float64(x / y) - x), x, x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y + x), $MachinePrecision] * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.0], t$95$0, If[LessEqual[x, 0.85], N[(N[(N[(x / y), $MachinePrecision] - x), $MachinePrecision] * x + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 0.849999999999999978 < x

   1. Initial program 88.5%

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

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. add-to-fraction N/A

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

      9. frac-times N/A

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

      10. associate-/l* N/A

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

      11. associate-/l/ N/A

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

      12. lift-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. add-flip N/A

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

      15. *-lft-identity N/A

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

      16. add-flip-rev N/A

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

      17. lower-+.f64 N/A

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

      18. lift-/.f64 N/A

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

      19. associate-/l/ N/A

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

      20. lower-/.f64 N/A

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

      21. lift-+.f64 N/A

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

      22. distribute-rgt-in N/A

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

      23. *-commutative N/A

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

      24. *-lft-identity N/A

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

      25. lower-fma.f64 76.5

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

   3. Applied rewrites 76.5%

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

   4. Taylor expanded in x around inf

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

      1. lower-/.f64 49.4

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

   6. Applied rewrites 49.4%

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

   if -1 < x < 0.849999999999999978

   1. Initial program 88.5%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 57.6

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

   4. Applied rewrites 57.6%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f64 57.6

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. sub-flip N/A

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

      10. metadata-eval N/A

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

      11. distribute-rgt-in N/A

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

      12. *-commutative N/A

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

      13. lift-/.f64 N/A

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

      14. mult-flip N/A

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

      15. lift-/.f64 N/A

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

      16. metadata-eval N/A

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

      17. add-flip N/A

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

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

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

      19. *-lft-identity N/A

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

      20. remove-double-neg N/A

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

      21. lower--.f64 57.6

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

   6. Applied rewrites 57.6%

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

Alternative 5: 85.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y + x\right) \cdot \frac{1}{y}\\ \mathbf{if}\;x \leq -14500:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 140000000:\\ \;\;\;\;\frac{x}{x - -1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (+ y x) (/ 1.0 y))))
   (if (<= x -14500.0) t_0 (if (<= x 140000000.0) (/ x (- x -1.0)) t_0))))

C

double code(double x, double y) {
	double t_0 = (y + x) * (1.0 / y);
	double tmp;
	if (x <= -14500.0) {
		tmp = t_0;
	} else if (x <= 140000000.0) {
		tmp = x / (x - -1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (y + x) * (1.0d0 / y)
    if (x <= (-14500.0d0)) then
        tmp = t_0
    else if (x <= 140000000.0d0) then
        tmp = x / (x - (-1.0d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (y + x) * (1.0 / y);
	double tmp;
	if (x <= -14500.0) {
		tmp = t_0;
	} else if (x <= 140000000.0) {
		tmp = x / (x - -1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y + x) * (1.0 / y)
	tmp = 0
	if x <= -14500.0:
		tmp = t_0
	elif x <= 140000000.0:
		tmp = x / (x - -1.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y + x) * Float64(1.0 / y))
	tmp = 0.0
	if (x <= -14500.0)
		tmp = t_0;
	elseif (x <= 140000000.0)
		tmp = Float64(x / Float64(x - -1.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y + x) * (1.0 / y);
	tmp = 0.0;
	if (x <= -14500.0)
		tmp = t_0;
	elseif (x <= 140000000.0)
		tmp = x / (x - -1.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y + x), $MachinePrecision] * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -14500.0], t$95$0, If[LessEqual[x, 140000000.0], N[(x / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -14500 or 1.4e8 < x

   1. Initial program 88.5%

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

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. add-to-fraction N/A

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

      9. frac-times N/A

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

      10. associate-/l* N/A

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

      11. associate-/l/ N/A

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

      12. lift-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. add-flip N/A

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

      15. *-lft-identity N/A

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

      16. add-flip-rev N/A

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

      17. lower-+.f64 N/A

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

      18. lift-/.f64 N/A

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

      19. associate-/l/ N/A

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

      20. lower-/.f64 N/A

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

      21. lift-+.f64 N/A

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

      22. distribute-rgt-in N/A

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

      23. *-commutative N/A

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

      24. *-lft-identity N/A

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

      25. lower-fma.f64 76.5

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

   3. Applied rewrites 76.5%

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

   4. Taylor expanded in x around inf

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

      1. lower-/.f64 49.4

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

   6. Applied rewrites 49.4%

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

   if -14500 < x < 1.4e8

   1. Initial program 88.5%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 50.8

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

   4. Applied rewrites 50.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. metadata-eval N/A

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

      5. lift--.f64 50.8

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

   6. Applied rewrites 50.8%

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

Alternative 6: 85.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ t_1 := \frac{1}{\frac{y}{x}}\\ \mathbf{if}\;t\_0 \leq -50000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\frac{x}{x - -1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 -50000.0) t_1 (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 <= -50000.0) {
		tmp = t_1;
	} else if (t_0 <= 2.0) {
		tmp = x / (x - -1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (x * ((x / y) + 1.0d0)) / (x + 1.0d0)
    t_1 = 1.0d0 / (y / x)
    if (t_0 <= (-50000.0d0)) then
        tmp = t_1
    else if (t_0 <= 2.0d0) then
        tmp = x / (x - (-1.0d0))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	double t_1 = 1.0 / (y / x);
	double tmp;
	if (t_0 <= -50000.0) {
		tmp = t_1;
	} else if (t_0 <= 2.0) {
		tmp = x / (x - -1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x * ((x / y) + 1.0)) / (x + 1.0)
	t_1 = 1.0 / (y / x)
	tmp = 0
	if t_0 <= -50000.0:
		tmp = t_1
	elif 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(1.0 / Float64(y / x))
	tmp = 0.0
	if (t_0 <= -50000.0)
		tmp = t_1;
	elseif (t_0 <= 2.0)
		tmp = Float64(x / Float64(x - -1.0));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	t_1 = 1.0 / (y / x);
	tmp = 0.0;
	if (t_0 <= -50000.0)
		tmp = t_1;
	elseif (t_0 <= 2.0)
		tmp = x / (x - -1.0);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[(y / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -50000.0], t$95$1, If[LessEqual[t$95$0, 2.0], N[(x / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

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))) < -5e4 or 2 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 88.5%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-special-/.f64 N/A

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

      4. lower-special-/.f64 88.3

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

      5. lift-+.f64 N/A

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

      6. add-flip N/A

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

      7. lower--.f64 N/A

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

      8. metadata-eval 88.3

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. distribute-lft1-in N/A

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

      13. lower-fma.f64 88.3

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

   3. Applied rewrites 88.3%

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

   4. Taylor expanded in x around inf

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

      1. lower-/.f64 38.6

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

   6. Applied rewrites 38.6%

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

   if -5e4 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 88.5%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 50.8

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

   4. Applied rewrites 50.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. metadata-eval N/A

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

      5. lift--.f64 50.8

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

   6. Applied rewrites 50.8%

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

Alternative 7: 56.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x - -1}\\ \mathbf{if}\;y \leq -5 \cdot 10^{-310}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{+36}:\\ \;\;\;\;y \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (- x -1.0))))
   (if (<= y -5e-310) t_0 (if (<= y 4.5e+36) (* y (/ x y)) t_0))))

C

double code(double x, double y) {
	double t_0 = x / (x - -1.0);
	double tmp;
	if (y <= -5e-310) {
		tmp = t_0;
	} else if (y <= 4.5e+36) {
		tmp = y * (x / y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / (x - (-1.0d0))
    if (y <= (-5d-310)) then
        tmp = t_0
    else if (y <= 4.5d+36) then
        tmp = y * (x / y)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (x - -1.0);
	double tmp;
	if (y <= -5e-310) {
		tmp = t_0;
	} else if (y <= 4.5e+36) {
		tmp = y * (x / y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (x - -1.0)
	tmp = 0
	if y <= -5e-310:
		tmp = t_0
	elif y <= 4.5e+36:
		tmp = y * (x / y)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(x - -1.0))
	tmp = 0.0
	if (y <= -5e-310)
		tmp = t_0;
	elseif (y <= 4.5e+36)
		tmp = Float64(y * Float64(x / y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (x - -1.0);
	tmp = 0.0;
	if (y <= -5e-310)
		tmp = t_0;
	elseif (y <= 4.5e+36)
		tmp = y * (x / y);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -5e-310], t$95$0, If[LessEqual[y, 4.5e+36], N[(y * N[(x / y), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.999999999999985e-310 or 4.49999999999999997e36 < y

   1. Initial program 88.5%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 50.8

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

   4. Applied rewrites 50.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. metadata-eval N/A

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

      5. lift--.f64 50.8

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

   6. Applied rewrites 50.8%

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

   if -4.999999999999985e-310 < y < 4.49999999999999997e36

   1. Initial program 88.5%

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

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. add-to-fraction N/A

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

      9. frac-times N/A

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

      10. associate-/l* N/A

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

      11. associate-/l/ N/A

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

      12. lift-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. add-flip N/A

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

      15. *-lft-identity N/A

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

      16. add-flip-rev N/A

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

      17. lower-+.f64 N/A

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

      18. lift-/.f64 N/A

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

      19. associate-/l/ N/A

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

      20. lower-/.f64 N/A

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

      21. lift-+.f64 N/A

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

      22. distribute-rgt-in N/A

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

      23. *-commutative N/A

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

      24. *-lft-identity N/A

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

      25. lower-fma.f64 76.5

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

   3. Applied rewrites 76.5%

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

   4. Taylor expanded in x around inf

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

      1. lower-/.f64 49.4

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

   6. Applied rewrites 49.4%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 14.1%

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

   10. Taylor expanded in x around 0

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

       1. lower-/.f64 33.5

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

   12. Applied rewrites 33.5%

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

Alternative 8: 50.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \leq 5 \cdot 10^{-11}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* x (+ (/ x y) 1.0)) (+ x 1.0)) 5e-11) x (- 1.0 (/ 1.0 x))))

C

double code(double x, double y) {
	double tmp;
	if (((x * ((x / y) + 1.0)) / (x + 1.0)) <= 5e-11) {
		tmp = 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)) <= 5d-11) then
        tmp = 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)) <= 5e-11) {
		tmp = 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)) <= 5e-11:
		tmp = 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)) <= 5e-11)
		tmp = 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)) <= 5e-11)
		tmp = 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], 5e-11], x, 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))) < 5.00000000000000018e-11

   1. Initial program 88.5%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 39.5%

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

   if 5.00000000000000018e-11 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 88.5%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 50.8

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

   4. Applied rewrites 50.8%

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

   5. Taylor expanded in x around inf

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 13.4

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

   7. Applied rewrites 13.4%

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

Alternative 9: 50.4% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.5%

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

2. Taylor expanded in y around inf

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

   1. lower-/.f64 N/A

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

   2. lower-+.f64 50.8

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

4. Applied rewrites 50.8%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. add-flip N/A

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

   4. metadata-eval N/A

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

   5. lift--.f64 50.8

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

6. Applied rewrites 50.8%

   \[\leadsto \color{blue}{\frac{x}{x - -1}} \]
7. Add Preprocessing

Alternative 10: 39.5% accurate, 16.1× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 x)

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

def code(x, y):
	return x

Julia

function code(x, y)
	return x
end

MATLAB

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

Wolfram

code[x_, y_] := x

Derivation

1. Initial program 88.5%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 39.5%

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

Reproduce

herbie shell --seed 2025150 
(FPCore (x y)
  :name "Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1"
  :precision binary64
  (/ (* x (+ (/ x y) 1.0)) (+ x 1.0)))