Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1

Percentage Accurate: 88.5% → 99.8%

Time: 2.6s

Alternatives: 13

Speedup: 0.8×

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: 99.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+263}:\\ \;\;\;\;\frac{\frac{x}{y} \cdot x + x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0))))
   (if (<= t_0 (- INFINITY))
     (/ x y)
     (if (<= t_0 5e+263) (/ (+ (* (/ x y) x) 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 <= -((double) INFINITY)) {
		tmp = x / y;
	} else if (t_0 <= 5e+263) {
		tmp = (((x / y) * x) + x) / (x + 1.0);
	} else {
		tmp = x / y;
	}
	return tmp;
}

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 <= -Double.POSITIVE_INFINITY) {
		tmp = x / y;
	} else if (t_0 <= 5e+263) {
		tmp = (((x / y) * x) + x) / (x + 1.0);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x * ((x / y) + 1.0)) / (x + 1.0)
	tmp = 0
	if t_0 <= -math.inf:
		tmp = x / y
	elif t_0 <= 5e+263:
		tmp = (((x / y) * x) + 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 <= Float64(-Inf))
		tmp = Float64(x / y);
	elseif (t_0 <= 5e+263)
		tmp = Float64(Float64(Float64(Float64(x / y) * x) + x) / Float64(x + 1.0));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = x / y;
	elseif (t_0 <= 5e+263)
		tmp = (((x / y) * x) + x) / (x + 1.0);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(x / y), $MachinePrecision], If[LessEqual[t$95$0, 5e+263], N[(N[(N[(N[(x / y), $MachinePrecision] * x), $MachinePrecision] + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -inf.0 or 5.00000000000000022e263 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 53.2%

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

   2. Taylor expanded in x around inf

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

      1. lift-/.f64 99.4

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

   4. Applied rewrites 99.4%

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

   if -inf.0 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 5.00000000000000022e263

   1. Initial program 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. distribute-rgt-in N/A

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

      5. *-lft-identity N/A

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

      6. lower-+.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 99.9

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

   3. Applied rewrites 99.9%

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

Alternative 2: 95.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t\_0 \leq -10000:\\ \;\;\;\;\frac{x \cdot \frac{x}{y}}{x + 1}\\ \mathbf{elif}\;t\_0 \leq 0.2:\\ \;\;\;\;\frac{\frac{x}{y} \cdot x + x}{1}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+48}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{y}, x, x\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{x}{\mathsf{fma}\left(y, x, y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0))))
   (if (<= t_0 (- INFINITY))
     (/ x y)
     (if (<= t_0 -10000.0)
       (/ (* x (/ x y)) (+ x 1.0))
       (if (<= t_0 0.2)
         (/ (+ (* (/ x y) x) x) 1.0)
         (if (<= t_0 2e+48)
           (/ (fma (/ x y) x x) x)
           (* x (/ x (fma y x y)))))))))

C

double code(double x, double y) {
	double t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = x / y;
	} else if (t_0 <= -10000.0) {
		tmp = (x * (x / y)) / (x + 1.0);
	} else if (t_0 <= 0.2) {
		tmp = (((x / y) * x) + x) / 1.0;
	} else if (t_0 <= 2e+48) {
		tmp = fma((x / y), x, x) / x;
	} else {
		tmp = x * (x / fma(y, 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 <= Float64(-Inf))
		tmp = Float64(x / y);
	elseif (t_0 <= -10000.0)
		tmp = Float64(Float64(x * Float64(x / y)) / Float64(x + 1.0));
	elseif (t_0 <= 0.2)
		tmp = Float64(Float64(Float64(Float64(x / y) * x) + x) / 1.0);
	elseif (t_0 <= 2e+48)
		tmp = Float64(fma(Float64(x / y), x, x) / x);
	else
		tmp = Float64(x * Float64(x / fma(y, 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, (-Infinity)], N[(x / y), $MachinePrecision], If[LessEqual[t$95$0, -10000.0], N[(N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.2], N[(N[(N[(N[(x / y), $MachinePrecision] * x), $MachinePrecision] + x), $MachinePrecision] / 1.0), $MachinePrecision], If[LessEqual[t$95$0, 2e+48], N[(N[(N[(x / y), $MachinePrecision] * x + x), $MachinePrecision] / x), $MachinePrecision], N[(x * N[(x / N[(y * x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 51.4%

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

   2. Taylor expanded in x around inf

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

      1. lift-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   if -inf.0 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -1e4

   1. Initial program 99.8%

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

   2. Taylor expanded in x around inf

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

      1. lift-/.f64 98.4

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

   4. Applied rewrites 98.4%

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

   if -1e4 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 0.20000000000000001

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 98.5%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. *-commutative N/A

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

      5. distribute-lft1-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-/.f64 98.5

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

   6. Applied rewrites 98.5%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate-*l/ N/A

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

      4. unpow2 N/A

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

      5. lower-+.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*l/ N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 98.5

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

   8. Applied rewrites 98.5%

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

   if 0.20000000000000001 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 2.00000000000000009e48

   1. Initial program 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. distribute-rgt-in N/A

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

      5. *-lft-identity N/A

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

      6. lower-+.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 99.9

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in x around inf

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

   6. Applied rewrites 90.4%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-/.f64 90.4

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

   8. Applied rewrites 90.4%

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

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

   1. Initial program 70.5%

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

   2. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. lower-fma.f64 72.7

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

   4. Applied rewrites 72.7%

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

      1. *-commutative 72.7

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

      2. distribute-lft1-in 72.7

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-fma.f64 87.1

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

   6. Applied rewrites 87.1%

      \[\leadsto \color{blue}{x \cdot \frac{x}{\mathsf{fma}\left(y, x, y\right)}} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 3: 95.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ t_1 := \mathsf{fma}\left(\frac{x}{y}, x, x\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t\_0 \leq -10000:\\ \;\;\;\;\frac{x \cdot \frac{x}{y}}{x + 1}\\ \mathbf{elif}\;t\_0 \leq 0.2:\\ \;\;\;\;\frac{t\_1}{1}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+48}:\\ \;\;\;\;\frac{t\_1}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{x}{\mathsf{fma}\left(y, x, y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0))) (t_1 (fma (/ x y) x x)))
   (if (<= t_0 (- INFINITY))
     (/ x y)
     (if (<= t_0 -10000.0)
       (/ (* x (/ x y)) (+ x 1.0))
       (if (<= t_0 0.2)
         (/ t_1 1.0)
         (if (<= t_0 2e+48) (/ t_1 x) (* x (/ x (fma y x y)))))))))

C

double code(double x, double y) {
	double t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	double t_1 = fma((x / y), x, x);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = x / y;
	} else if (t_0 <= -10000.0) {
		tmp = (x * (x / y)) / (x + 1.0);
	} else if (t_0 <= 0.2) {
		tmp = t_1 / 1.0;
	} else if (t_0 <= 2e+48) {
		tmp = t_1 / x;
	} else {
		tmp = x * (x / fma(y, x, y));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(x * Float64(Float64(x / y) + 1.0)) / Float64(x + 1.0))
	t_1 = fma(Float64(x / y), x, x)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(x / y);
	elseif (t_0 <= -10000.0)
		tmp = Float64(Float64(x * Float64(x / y)) / Float64(x + 1.0));
	elseif (t_0 <= 0.2)
		tmp = Float64(t_1 / 1.0);
	elseif (t_0 <= 2e+48)
		tmp = Float64(t_1 / x);
	else
		tmp = Float64(x * Float64(x / fma(y, 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]}, Block[{t$95$1 = N[(N[(x / y), $MachinePrecision] * x + x), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(x / y), $MachinePrecision], If[LessEqual[t$95$0, -10000.0], N[(N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.2], N[(t$95$1 / 1.0), $MachinePrecision], If[LessEqual[t$95$0, 2e+48], N[(t$95$1 / x), $MachinePrecision], N[(x * N[(x / N[(y * x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 51.4%

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

   2. Taylor expanded in x around inf

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

      1. lift-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   if -inf.0 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -1e4

   1. Initial program 99.8%

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

   2. Taylor expanded in x around inf

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

      1. lift-/.f64 98.4

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

   4. Applied rewrites 98.4%

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

   if -1e4 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 0.20000000000000001

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 98.5%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. *-commutative N/A

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

      5. distribute-lft1-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-/.f64 98.5

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

   6. Applied rewrites 98.5%

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

   if 0.20000000000000001 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 2.00000000000000009e48

   1. Initial program 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. distribute-rgt-in N/A

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

      5. *-lft-identity N/A

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

      6. lower-+.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 99.9

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in x around inf

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

   6. Applied rewrites 90.4%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-/.f64 90.4

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

   8. Applied rewrites 90.4%

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

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

   1. Initial program 70.5%

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

   2. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. lower-fma.f64 72.7

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

   4. Applied rewrites 72.7%

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

      1. *-commutative 72.7

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

      2. distribute-lft1-in 72.7

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-fma.f64 87.1

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

   6. Applied rewrites 87.1%

      \[\leadsto \color{blue}{x \cdot \frac{x}{\mathsf{fma}\left(y, x, y\right)}} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 4: 94.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ t_1 := \mathsf{fma}\left(\frac{x}{y}, x, x\right)\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+201}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t\_0 \leq -1 \cdot 10^{-6}:\\ \;\;\;\;\left(y + x\right) \cdot \frac{x}{\left(x - -1\right) \cdot y}\\ \mathbf{elif}\;t\_0 \leq 0.2:\\ \;\;\;\;\frac{t\_1}{1}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+48}:\\ \;\;\;\;\frac{t\_1}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{x}{\mathsf{fma}\left(y, x, y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0))) (t_1 (fma (/ x y) x x)))
   (if (<= t_0 -5e+201)
     (/ x y)
     (if (<= t_0 -1e-6)
       (* (+ y x) (/ x (* (- x -1.0) y)))
       (if (<= t_0 0.2)
         (/ t_1 1.0)
         (if (<= t_0 2e+48) (/ t_1 x) (* x (/ x (fma y x y)))))))))

C

double code(double x, double y) {
	double t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	double t_1 = fma((x / y), x, x);
	double tmp;
	if (t_0 <= -5e+201) {
		tmp = x / y;
	} else if (t_0 <= -1e-6) {
		tmp = (y + x) * (x / ((x - -1.0) * y));
	} else if (t_0 <= 0.2) {
		tmp = t_1 / 1.0;
	} else if (t_0 <= 2e+48) {
		tmp = t_1 / x;
	} else {
		tmp = x * (x / fma(y, x, y));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(x * Float64(Float64(x / y) + 1.0)) / Float64(x + 1.0))
	t_1 = fma(Float64(x / y), x, x)
	tmp = 0.0
	if (t_0 <= -5e+201)
		tmp = Float64(x / y);
	elseif (t_0 <= -1e-6)
		tmp = Float64(Float64(y + x) * Float64(x / Float64(Float64(x - -1.0) * y)));
	elseif (t_0 <= 0.2)
		tmp = Float64(t_1 / 1.0);
	elseif (t_0 <= 2e+48)
		tmp = Float64(t_1 / x);
	else
		tmp = Float64(x * Float64(x / fma(y, 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]}, Block[{t$95$1 = N[(N[(x / y), $MachinePrecision] * x + x), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+201], N[(x / y), $MachinePrecision], If[LessEqual[t$95$0, -1e-6], N[(N[(y + x), $MachinePrecision] * N[(x / N[(N[(x - -1.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.2], N[(t$95$1 / 1.0), $MachinePrecision], If[LessEqual[t$95$0, 2e+48], N[(t$95$1 / x), $MachinePrecision], N[(x * N[(x / N[(y * x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -4.9999999999999995e201

   1. Initial program 55.7%

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

   2. Taylor expanded in x around inf

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

      1. lift-/.f64 96.3

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

   4. Applied rewrites 96.3%

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

   if -4.9999999999999995e201 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -9.99999999999999955e-7

   1. Initial program 99.8%

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

   2. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 80.3%

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

      1. *-commutative 80.3

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

      2. distribute-lft1-in 80.3

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l/ N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift--.f64 79.7

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

   6. Applied rewrites 79.7%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift--.f64 87.0

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

   8. Applied rewrites 87.0%

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

   if -9.99999999999999955e-7 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 0.20000000000000001

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 99.1%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. *-commutative N/A

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

      5. distribute-lft1-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-/.f64 99.1

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

   6. Applied rewrites 99.1%

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

   if 0.20000000000000001 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 2.00000000000000009e48

   1. Initial program 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. distribute-rgt-in N/A

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

      5. *-lft-identity N/A

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

      6. lower-+.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 99.9

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in x around inf

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

   6. Applied rewrites 90.4%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-/.f64 90.4

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

   8. Applied rewrites 90.4%

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

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

   1. Initial program 70.5%

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

   2. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. lower-fma.f64 72.7

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

   4. Applied rewrites 72.7%

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

      1. *-commutative 72.7

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

      2. distribute-lft1-in 72.7

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-fma.f64 87.1

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

   6. Applied rewrites 87.1%

      \[\leadsto \color{blue}{x \cdot \frac{x}{\mathsf{fma}\left(y, x, y\right)}} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 5: 94.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y + x\right) \cdot \frac{x}{\left(x - -1\right) \cdot y}\\ t_1 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+201}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-6}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 0.2:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{y}, x, x\right)}{1}\\ \mathbf{elif}\;t\_1 \leq 1.1:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (+ y x) (/ x (* (- x -1.0) y))))
        (t_1 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0))))
   (if (<= t_1 -5e+201)
     (/ x y)
     (if (<= t_1 -1e-6)
       t_0
       (if (<= t_1 0.2)
         (/ (fma (/ x y) x x) 1.0)
         (if (<= t_1 1.1) (/ x (+ x 1.0)) t_0))))))

C

double code(double x, double y) {
	double t_0 = (y + x) * (x / ((x - -1.0) * y));
	double t_1 = (x * ((x / y) + 1.0)) / (x + 1.0);
	double tmp;
	if (t_1 <= -5e+201) {
		tmp = x / y;
	} else if (t_1 <= -1e-6) {
		tmp = t_0;
	} else if (t_1 <= 0.2) {
		tmp = fma((x / y), x, x) / 1.0;
	} else if (t_1 <= 1.1) {
		tmp = x / (x + 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(y + x) * Float64(x / Float64(Float64(x - -1.0) * y)))
	t_1 = Float64(Float64(x * Float64(Float64(x / y) + 1.0)) / Float64(x + 1.0))
	tmp = 0.0
	if (t_1 <= -5e+201)
		tmp = Float64(x / y);
	elseif (t_1 <= -1e-6)
		tmp = t_0;
	elseif (t_1 <= 0.2)
		tmp = Float64(fma(Float64(x / y), x, x) / 1.0);
	elseif (t_1 <= 1.1)
		tmp = Float64(x / Float64(x + 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y + x), $MachinePrecision] * N[(x / N[(N[(x - -1.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+201], N[(x / y), $MachinePrecision], If[LessEqual[t$95$1, -1e-6], t$95$0, If[LessEqual[t$95$1, 0.2], N[(N[(N[(x / y), $MachinePrecision] * x + x), $MachinePrecision] / 1.0), $MachinePrecision], If[LessEqual[t$95$1, 1.1], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -4.9999999999999995e201

   1. Initial program 55.7%

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

   2. Taylor expanded in x around inf

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

      1. lift-/.f64 96.3

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

   4. Applied rewrites 96.3%

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

   if -4.9999999999999995e201 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -9.99999999999999955e-7 or 1.1000000000000001 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 81.6%

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

   2. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 74.8%

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

      1. *-commutative 74.8

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

      2. distribute-lft1-in 74.8

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l/ N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift--.f64 74.0

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

   6. Applied rewrites 74.0%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift--.f64 85.3

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

   8. Applied rewrites 85.3%

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

   if -9.99999999999999955e-7 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 0.20000000000000001

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 99.1%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. *-commutative N/A

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

      5. distribute-lft1-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-/.f64 99.1

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

   6. Applied rewrites 99.1%

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

   if 0.20000000000000001 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 1.1000000000000001

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 95.7%

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

Alternative 6: 86.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+201}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t\_0 \leq -3 \cdot 10^{+80}:\\ \;\;\;\;\frac{x \cdot x}{\mathsf{fma}\left(y, x, y\right)}\\ \mathbf{elif}\;t\_0 \leq -1000:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0))))
   (if (<= t_0 -5e+201)
     (/ x y)
     (if (<= t_0 -3e+80)
       (/ (* x x) (fma y x y))
       (if (<= t_0 -1000.0)
         (/ x y)
         (if (<= t_0 2.0) (/ x (+ x 1.0)) (/ x y)))))))

C

double code(double x, double y) {
	double t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	double tmp;
	if (t_0 <= -5e+201) {
		tmp = x / y;
	} else if (t_0 <= -3e+80) {
		tmp = (x * x) / fma(y, x, y);
	} else if (t_0 <= -1000.0) {
		tmp = x / y;
	} 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 <= -5e+201)
		tmp = Float64(x / y);
	elseif (t_0 <= -3e+80)
		tmp = Float64(Float64(x * x) / fma(y, x, y));
	elseif (t_0 <= -1000.0)
		tmp = Float64(x / y);
	elseif (t_0 <= 2.0)
		tmp = Float64(x / Float64(x + 1.0));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

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, -5e+201], N[(x / y), $MachinePrecision], If[LessEqual[t$95$0, -3e+80], N[(N[(x * x), $MachinePrecision] / N[(y * x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -1000.0], N[(x / y), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -4.9999999999999995e201 or -2.99999999999999987e80 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -1e3 or 2 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 70.6%

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

   2. Taylor expanded in x around inf

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

      1. lift-/.f64 85.9

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

   4. Applied rewrites 85.9%

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

   if -4.9999999999999995e201 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -2.99999999999999987e80

   1. Initial program 99.8%

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

   2. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. lower-fma.f64 88.2

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

   4. Applied rewrites 88.2%

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

   if -1e3 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 86.9%

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

Alternative 7: 87.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+201}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t\_0 \leq -1000:\\ \;\;\;\;x \cdot \frac{x}{\mathsf{fma}\left(y, x, y\right)}\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0))))
   (if (<= t_0 -5e+201)
     (/ x y)
     (if (<= t_0 -1000.0)
       (* x (/ x (fma y x y)))
       (if (<= t_0 2.0) (/ x (+ x 1.0)) (/ x y))))))

C

double code(double x, double y) {
	double t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	double tmp;
	if (t_0 <= -5e+201) {
		tmp = x / y;
	} else if (t_0 <= -1000.0) {
		tmp = x * (x / fma(y, x, y));
	} 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 <= -5e+201)
		tmp = Float64(x / y);
	elseif (t_0 <= -1000.0)
		tmp = Float64(x * Float64(x / fma(y, x, y)));
	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, -5e+201], N[(x / y), $MachinePrecision], If[LessEqual[t$95$0, -1000.0], N[(x * N[(x / N[(y * x + y), $MachinePrecision]), $MachinePrecision]), $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))) < -4.9999999999999995e201 or 2 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 67.4%

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

   2. Taylor expanded in x around inf

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

      1. lift-/.f64 88.2

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

   4. Applied rewrites 88.2%

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

   if -4.9999999999999995e201 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -1e3

   1. Initial program 99.8%

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

   2. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. lower-fma.f64 78.1

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

   4. Applied rewrites 78.1%

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

      1. *-commutative 78.1

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

      2. distribute-lft1-in 78.1

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-fma.f64 85.7

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

   6. Applied rewrites 85.7%

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

   if -1e3 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 86.9%

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

Alternative 8: 84.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ \mathbf{if}\;t\_0 \leq -1000:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-5}:\\ \;\;\;\;x\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\frac{x}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0))))
   (if (<= t_0 -1000.0)
     (/ x y)
     (if (<= t_0 5e-5) x (if (<= t_0 2.0) (/ x x) (/ x y))))))

C

double code(double x, double y) {
	double t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	double tmp;
	if (t_0 <= -1000.0) {
		tmp = x / y;
	} else if (t_0 <= 5e-5) {
		tmp = x;
	} else if (t_0 <= 2.0) {
		tmp = x / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x * ((x / y) + 1.0d0)) / (x + 1.0d0)
    if (t_0 <= (-1000.0d0)) then
        tmp = x / y
    else if (t_0 <= 5d-5) then
        tmp = x
    else if (t_0 <= 2.0d0) then
        tmp = x / x
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	double tmp;
	if (t_0 <= -1000.0) {
		tmp = x / y;
	} else if (t_0 <= 5e-5) {
		tmp = x;
	} else if (t_0 <= 2.0) {
		tmp = x / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x * ((x / y) + 1.0)) / (x + 1.0)
	tmp = 0
	if t_0 <= -1000.0:
		tmp = x / y
	elif t_0 <= 5e-5:
		tmp = x
	elif t_0 <= 2.0:
		tmp = x / x
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x * Float64(Float64(x / y) + 1.0)) / Float64(x + 1.0))
	tmp = 0.0
	if (t_0 <= -1000.0)
		tmp = Float64(x / y);
	elseif (t_0 <= 5e-5)
		tmp = x;
	elseif (t_0 <= 2.0)
		tmp = Float64(x / x);
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	tmp = 0.0;
	if (t_0 <= -1000.0)
		tmp = x / y;
	elseif (t_0 <= 5e-5)
		tmp = x;
	elseif (t_0 <= 2.0)
		tmp = x / x;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1000.0], N[(x / y), $MachinePrecision], If[LessEqual[t$95$0, 5e-5], x, If[LessEqual[t$95$0, 2.0], N[(x / x), $MachinePrecision], N[(x / y), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -1e3 or 2 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 73.3%

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

   2. Taylor expanded in x around inf

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

      1. lift-/.f64 83.9

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

   4. Applied rewrites 83.9%

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

   if -1e3 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 5.00000000000000024e-5

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 84.0%

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

   if 5.00000000000000024e-5 < (/.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. 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. lift-/.f64 N/A

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

      4. distribute-rgt-in N/A

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

      5. *-lft-identity N/A

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

      6. lower-+.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 100.0

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in x around inf

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

   6. Applied rewrites 94.0%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 90.0%

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

Alternative 9: 99.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+263}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{y}, x, x\right)}{x - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0))))
   (if (<= t_0 (- INFINITY))
     (/ x y)
     (if (<= t_0 5e+263) (/ (fma (/ x y) x 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 <= -((double) INFINITY)) {
		tmp = x / y;
	} else if (t_0 <= 5e+263) {
		tmp = fma((x / y), x, 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 <= Float64(-Inf))
		tmp = Float64(x / y);
	elseif (t_0 <= 5e+263)
		tmp = Float64(fma(Float64(x / y), x, 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, (-Infinity)], N[(x / y), $MachinePrecision], If[LessEqual[t$95$0, 5e+263], N[(N[(N[(x / y), $MachinePrecision] * x + x), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -inf.0 or 5.00000000000000022e263 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 53.2%

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

   2. Taylor expanded in x around inf

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

      1. lift-/.f64 99.4

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

   4. Applied rewrites 99.4%

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

   if -inf.0 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 5.00000000000000022e263

   1. Initial program 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. distribute-rgt-in N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-/.f64 99.9

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

      8. lift-+.f64 N/A

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

      9. metadata-eval N/A

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

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

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower--.f64 99.9

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

   3. 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. Add Preprocessing

Alternative 10: 85.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ \mathbf{if}\;t\_0 \leq -1000:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0))))
   (if (<= t_0 -1000.0) (/ x y) (if (<= t_0 2.0) (/ x (+ x 1.0)) (/ x y)))))

C

double code(double x, double y) {
	double t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	double tmp;
	if (t_0 <= -1000.0) {
		tmp = x / y;
	} else if (t_0 <= 2.0) {
		tmp = x / (x + 1.0);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x * ((x / y) + 1.0d0)) / (x + 1.0d0)
    if (t_0 <= (-1000.0d0)) then
        tmp = x / y
    else if (t_0 <= 2.0d0) then
        tmp = x / (x + 1.0d0)
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	double tmp;
	if (t_0 <= -1000.0) {
		tmp = x / y;
	} else if (t_0 <= 2.0) {
		tmp = x / (x + 1.0);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x * ((x / y) + 1.0)) / (x + 1.0)
	tmp = 0
	if t_0 <= -1000.0:
		tmp = x / y
	elif t_0 <= 2.0:
		tmp = x / (x + 1.0)
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x * Float64(Float64(x / y) + 1.0)) / Float64(x + 1.0))
	tmp = 0.0
	if (t_0 <= -1000.0)
		tmp = Float64(x / y);
	elseif (t_0 <= 2.0)
		tmp = Float64(x / Float64(x + 1.0));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	tmp = 0.0;
	if (t_0 <= -1000.0)
		tmp = x / y;
	elseif (t_0 <= 2.0)
		tmp = x / (x + 1.0);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1000.0], N[(x / y), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -1e3 or 2 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 73.3%

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

   2. Taylor expanded in x around inf

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

      1. lift-/.f64 83.9

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

   4. Applied rewrites 83.9%

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

   if -1e3 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 86.9%

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

Alternative 11: 74.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ \mathbf{if}\;t\_0 \leq -1000:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t\_0 \leq 0.9999999999994632:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0))))
   (if (<= t_0 -1000.0) (/ x y) (if (<= t_0 0.9999999999994632) x (/ x y)))))

C

double code(double x, double y) {
	double t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	double tmp;
	if (t_0 <= -1000.0) {
		tmp = x / y;
	} else if (t_0 <= 0.9999999999994632) {
		tmp = x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x * ((x / y) + 1.0d0)) / (x + 1.0d0)
    if (t_0 <= (-1000.0d0)) then
        tmp = x / y
    else if (t_0 <= 0.9999999999994632d0) then
        tmp = x
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	double tmp;
	if (t_0 <= -1000.0) {
		tmp = x / y;
	} else if (t_0 <= 0.9999999999994632) {
		tmp = x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x * ((x / y) + 1.0)) / (x + 1.0)
	tmp = 0
	if t_0 <= -1000.0:
		tmp = x / y
	elif t_0 <= 0.9999999999994632:
		tmp = x
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x * Float64(Float64(x / y) + 1.0)) / Float64(x + 1.0))
	tmp = 0.0
	if (t_0 <= -1000.0)
		tmp = Float64(x / y);
	elseif (t_0 <= 0.9999999999994632)
		tmp = x;
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	tmp = 0.0;
	if (t_0 <= -1000.0)
		tmp = x / y;
	elseif (t_0 <= 0.9999999999994632)
		tmp = x;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1000.0], N[(x / y), $MachinePrecision], If[LessEqual[t$95$0, 0.9999999999994632], x, N[(x / y), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -1e3 or 0.999999999999463207 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 78.8%

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

   2. Taylor expanded in x around inf

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

      1. lift-/.f64 67.7

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

   4. Applied rewrites 67.7%

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

   if -1e3 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 0.999999999999463207

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 82.1%

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

Alternative 12: 91.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + 1}\\ \mathbf{if}\;y \leq -3.8 \cdot 10^{+72}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{+135}:\\ \;\;\;\;\left(y + x\right) \cdot \frac{x}{\left(x - -1\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ x 1.0))))
   (if (<= y -3.8e+72)
     t_0
     (if (<= y 1.6e+135) (* (+ y x) (/ x (* (- x -1.0) y))) t_0))))

C

double code(double x, double y) {
	double t_0 = x / (x + 1.0);
	double tmp;
	if (y <= -3.8e+72) {
		tmp = t_0;
	} else if (y <= 1.6e+135) {
		tmp = (y + x) * (x / ((x - -1.0) * 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 <= (-3.8d+72)) then
        tmp = t_0
    else if (y <= 1.6d+135) then
        tmp = (y + x) * (x / ((x - (-1.0d0)) * 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 <= -3.8e+72) {
		tmp = t_0;
	} else if (y <= 1.6e+135) {
		tmp = (y + x) * (x / ((x - -1.0) * y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (x + 1.0)
	tmp = 0
	if y <= -3.8e+72:
		tmp = t_0
	elif y <= 1.6e+135:
		tmp = (y + x) * (x / ((x - -1.0) * 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 <= -3.8e+72)
		tmp = t_0;
	elseif (y <= 1.6e+135)
		tmp = Float64(Float64(y + x) * Float64(x / Float64(Float64(x - -1.0) * 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 <= -3.8e+72)
		tmp = t_0;
	elseif (y <= 1.6e+135)
		tmp = (y + x) * (x / ((x - -1.0) * 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, -3.8e+72], t$95$0, If[LessEqual[y, 1.6e+135], N[(N[(y + x), $MachinePrecision] * N[(x / N[(N[(x - -1.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.80000000000000006e72 or 1.59999999999999987e135 < y

   1. Initial program 92.3%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 82.0%

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

   if -3.80000000000000006e72 < y < 1.59999999999999987e135

   1. Initial program 86.6%

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

   2. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 77.6%

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

      1. *-commutative 77.6

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

      2. distribute-lft1-in 77.6

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l/ N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift--.f64 77.4

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

   6. Applied rewrites 77.4%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift--.f64 95.8

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

   8. Applied rewrites 95.8%

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

Alternative 13: 39.8% accurate, 34.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 x)

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

def code(x, y):
	return x

Julia

function code(x, y)
	return x
end

MATLAB

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

Wolfram

code[x_, y_] := x

Derivation

1. Initial program 88.5%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 39.8%

   \[\leadsto \color{blue}{x} \]
5. 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 2025105 
(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)))