Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1

Percentage Accurate: 88.0% → 99.9%

Time: 2.7s

Alternatives: 12

Speedup: 0.3×

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: 88.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ t_1 := \frac{x}{\frac{y}{x} + y}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+147}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 200000000000:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0))) (t_1 (/ x (+ (/ y x) y))))
   (if (<= t_0 -1e+147) t_1 (if (<= t_0 200000000000.0) t_0 t_1))))

C

double code(double x, double y) {
	double t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	double t_1 = x / ((y / x) + y);
	double tmp;
	if (t_0 <= -1e+147) {
		tmp = t_1;
	} else if (t_0 <= 200000000000.0) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (x * ((x / y) + 1.0d0)) / (x + 1.0d0)
    t_1 = x / ((y / x) + y)
    if (t_0 <= (-1d+147)) then
        tmp = t_1
    else if (t_0 <= 200000000000.0d0) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	double t_1 = x / ((y / x) + y);
	double tmp;
	if (t_0 <= -1e+147) {
		tmp = t_1;
	} else if (t_0 <= 200000000000.0) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x * ((x / y) + 1.0)) / (x + 1.0)
	t_1 = x / ((y / x) + y)
	tmp = 0
	if t_0 <= -1e+147:
		tmp = t_1
	elif t_0 <= 200000000000.0:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x * Float64(Float64(x / y) + 1.0)) / Float64(x + 1.0))
	t_1 = Float64(x / Float64(Float64(y / x) + y))
	tmp = 0.0
	if (t_0 <= -1e+147)
		tmp = t_1;
	elseif (t_0 <= 200000000000.0)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	t_1 = x / ((y / x) + y);
	tmp = 0.0;
	if (t_0 <= -1e+147)
		tmp = t_1;
	elseif (t_0 <= 200000000000.0)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[(N[(y / x), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+147], t$95$1, If[LessEqual[t$95$0, 200000000000.0], t$95$0, t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -9.9999999999999998e146 or 2e11 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 67.3%

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

      1. lift-+.f64 N/A

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

      2. *-rgt-identity N/A

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

      3. rgt-mult-inverse N/A

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

      4. distribute-lft-in N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 67.2

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

   3. Applied rewrites 67.2%

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

   4. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lift-/.f64 99.8

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 99.8

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

   9. Applied rewrites 99.8%

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

   if -9.9999999999999998e146 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 2e11

   1. Initial program 99.9%

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

Alternative 2: 99.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ t_1 := \frac{x}{\frac{y}{x} + y}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+147}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 200000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{y}, x, x\right)}{x - -1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0))) (t_1 (/ x (+ (/ y x) y))))
   (if (<= t_0 -1e+147)
     t_1
     (if (<= t_0 200000000000.0) (/ (fma (/ x y) x x) (- x -1.0)) t_1))))

C

double code(double x, double y) {
	double t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	double t_1 = x / ((y / x) + y);
	double tmp;
	if (t_0 <= -1e+147) {
		tmp = t_1;
	} else if (t_0 <= 200000000000.0) {
		tmp = fma((x / y), x, x) / (x - -1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(x * Float64(Float64(x / y) + 1.0)) / Float64(x + 1.0))
	t_1 = Float64(x / Float64(Float64(y / x) + y))
	tmp = 0.0
	if (t_0 <= -1e+147)
		tmp = t_1;
	elseif (t_0 <= 200000000000.0)
		tmp = Float64(fma(Float64(x / y), x, x) / Float64(x - -1.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[(N[(y / x), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+147], t$95$1, If[LessEqual[t$95$0, 200000000000.0], N[(N[(N[(x / y), $MachinePrecision] * x + x), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -9.9999999999999998e146 or 2e11 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 67.3%

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

      1. lift-+.f64 N/A

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

      2. *-rgt-identity N/A

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

      3. rgt-mult-inverse N/A

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

      4. distribute-lft-in N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 67.2

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

   3. Applied rewrites 67.2%

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

   4. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lift-/.f64 99.8

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 99.8

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

   9. Applied rewrites 99.8%

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

   if -9.9999999999999998e146 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 2e11

   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 3: 98.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(\frac{x}{y} + 1\right)\\ t_1 := \frac{t\_0}{x + 1}\\ t_2 := \frac{x}{\frac{y}{x} + y}\\ \mathbf{if}\;t\_1 \leq -2000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0.05:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{y} - 1, x, 1\right) \cdot x\\ \mathbf{elif}\;t\_1 \leq 2000:\\ \;\;\;\;\frac{t\_0}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (+ (/ x y) 1.0)))
        (t_1 (/ t_0 (+ x 1.0)))
        (t_2 (/ x (+ (/ y x) y))))
   (if (<= t_1 -2000000.0)
     t_2
     (if (<= t_1 0.05)
       (* (fma (- (/ 1.0 y) 1.0) x 1.0) x)
       (if (<= t_1 2000.0) (/ t_0 x) t_2)))))

C

double code(double x, double y) {
	double t_0 = x * ((x / y) + 1.0);
	double t_1 = t_0 / (x + 1.0);
	double t_2 = x / ((y / x) + y);
	double tmp;
	if (t_1 <= -2000000.0) {
		tmp = t_2;
	} else if (t_1 <= 0.05) {
		tmp = fma(((1.0 / y) - 1.0), x, 1.0) * x;
	} else if (t_1 <= 2000.0) {
		tmp = t_0 / x;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(x * Float64(Float64(x / y) + 1.0))
	t_1 = Float64(t_0 / Float64(x + 1.0))
	t_2 = Float64(x / Float64(Float64(y / x) + y))
	tmp = 0.0
	if (t_1 <= -2000000.0)
		tmp = t_2;
	elseif (t_1 <= 0.05)
		tmp = Float64(fma(Float64(Float64(1.0 / y) - 1.0), x, 1.0) * x);
	elseif (t_1 <= 2000.0)
		tmp = Float64(t_0 / x);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x * N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x / N[(N[(y / x), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2000000.0], t$95$2, If[LessEqual[t$95$1, 0.05], N[(N[(N[(N[(1.0 / y), $MachinePrecision] - 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t$95$1, 2000.0], N[(t$95$0 / x), $MachinePrecision], t$95$2]]]]]]

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))) < -2e6 or 2e3 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 72.6%

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

      1. lift-+.f64 N/A

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

      2. *-rgt-identity N/A

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

      3. rgt-mult-inverse N/A

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

      4. distribute-lft-in N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 72.6

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

   3. Applied rewrites 72.6%

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

   4. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lift-/.f64 99.2

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

   6. Applied rewrites 99.2%

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

   7. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 99.2

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

   9. Applied rewrites 99.2%

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

   if -2e6 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 0.050000000000000003

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 98.9

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

   4. Applied rewrites 98.9%

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

   if 0.050000000000000003 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 2e3

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 95.6%

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

Alternative 4: 98.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(\frac{x}{y} + 1\right)\\ t_1 := \frac{t\_0}{x + 1}\\ t_2 := \frac{x}{\frac{y}{x} + y}\\ \mathbf{if}\;t\_1 \leq -2000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0.05:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{y}, x, x\right)}{1}\\ \mathbf{elif}\;t\_1 \leq 2000:\\ \;\;\;\;\frac{t\_0}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (+ (/ x y) 1.0)))
        (t_1 (/ t_0 (+ x 1.0)))
        (t_2 (/ x (+ (/ y x) y))))
   (if (<= t_1 -2000000.0)
     t_2
     (if (<= t_1 0.05)
       (/ (fma (/ x y) x x) 1.0)
       (if (<= t_1 2000.0) (/ t_0 x) t_2)))))

C

double code(double x, double y) {
	double t_0 = x * ((x / y) + 1.0);
	double t_1 = t_0 / (x + 1.0);
	double t_2 = x / ((y / x) + y);
	double tmp;
	if (t_1 <= -2000000.0) {
		tmp = t_2;
	} else if (t_1 <= 0.05) {
		tmp = fma((x / y), x, x) / 1.0;
	} else if (t_1 <= 2000.0) {
		tmp = t_0 / x;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(x * Float64(Float64(x / y) + 1.0))
	t_1 = Float64(t_0 / Float64(x + 1.0))
	t_2 = Float64(x / Float64(Float64(y / x) + y))
	tmp = 0.0
	if (t_1 <= -2000000.0)
		tmp = t_2;
	elseif (t_1 <= 0.05)
		tmp = Float64(fma(Float64(x / y), x, x) / 1.0);
	elseif (t_1 <= 2000.0)
		tmp = Float64(t_0 / x);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x * N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x / N[(N[(y / x), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2000000.0], t$95$2, If[LessEqual[t$95$1, 0.05], N[(N[(N[(x / y), $MachinePrecision] * x + x), $MachinePrecision] / 1.0), $MachinePrecision], If[LessEqual[t$95$1, 2000.0], N[(t$95$0 / x), $MachinePrecision], t$95$2]]]]]]

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))) < -2e6 or 2e3 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 72.6%

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

      1. lift-+.f64 N/A

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

      2. *-rgt-identity N/A

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

      3. rgt-mult-inverse N/A

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

      4. distribute-lft-in N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 72.6

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

   3. Applied rewrites 72.6%

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

   4. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lift-/.f64 99.2

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

   6. Applied rewrites 99.2%

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

   7. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 99.2

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

   9. Applied rewrites 99.2%

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

   if -2e6 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 0.050000000000000003

   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%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 98.4%

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

   if 0.050000000000000003 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 2e3

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 95.6%

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

Alternative 5: 98.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\frac{y}{x} + y}\\ t_1 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ t_2 := \mathsf{fma}\left(\frac{x}{y}, x, x\right)\\ \mathbf{if}\;t\_1 \leq -2000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 0.05:\\ \;\;\;\;\frac{t\_2}{1}\\ \mathbf{elif}\;t\_1 \leq 2000:\\ \;\;\;\;\frac{t\_2}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ (/ y x) y)))
        (t_1 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0)))
        (t_2 (fma (/ x y) x x)))
   (if (<= t_1 -2000000.0)
     t_0
     (if (<= t_1 0.05) (/ t_2 1.0) (if (<= t_1 2000.0) (/ t_2 x) t_0)))))

C

double code(double x, double y) {
	double t_0 = x / ((y / x) + y);
	double t_1 = (x * ((x / y) + 1.0)) / (x + 1.0);
	double t_2 = fma((x / y), x, x);
	double tmp;
	if (t_1 <= -2000000.0) {
		tmp = t_0;
	} else if (t_1 <= 0.05) {
		tmp = t_2 / 1.0;
	} else if (t_1 <= 2000.0) {
		tmp = t_2 / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(x / Float64(Float64(y / x) + y))
	t_1 = Float64(Float64(x * Float64(Float64(x / y) + 1.0)) / Float64(x + 1.0))
	t_2 = fma(Float64(x / y), x, x)
	tmp = 0.0
	if (t_1 <= -2000000.0)
		tmp = t_0;
	elseif (t_1 <= 0.05)
		tmp = Float64(t_2 / 1.0);
	elseif (t_1 <= 2000.0)
		tmp = Float64(t_2 / x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(N[(y / x), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x / y), $MachinePrecision] * x + x), $MachinePrecision]}, If[LessEqual[t$95$1, -2000000.0], t$95$0, If[LessEqual[t$95$1, 0.05], N[(t$95$2 / 1.0), $MachinePrecision], If[LessEqual[t$95$1, 2000.0], N[(t$95$2 / x), $MachinePrecision], t$95$0]]]]]]

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))) < -2e6 or 2e3 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 72.6%

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

      1. lift-+.f64 N/A

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

      2. *-rgt-identity N/A

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

      3. rgt-mult-inverse N/A

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

      4. distribute-lft-in N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 72.6

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

   3. Applied rewrites 72.6%

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

   4. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lift-/.f64 99.2

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

   6. Applied rewrites 99.2%

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

   7. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 99.2

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

   9. Applied rewrites 99.2%

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

   if -2e6 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 0.050000000000000003

   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%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 98.4%

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

   if 0.050000000000000003 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 2e3

   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-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%

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

   4. Taylor expanded in x around inf

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

   6. Applied rewrites 95.6%

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

Alternative 6: 98.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ t_1 := \frac{x}{\frac{y}{x} + y}\\ \mathbf{if}\;t\_0 \leq -2000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-28}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{y}, x, x\right)}{1}\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\frac{x}{x - -1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0))) (t_1 (/ x (+ (/ y x) y))))
   (if (<= t_0 -2000000.0)
     t_1
     (if (<= t_0 5e-28)
       (/ (fma (/ x y) x x) 1.0)
       (if (<= t_0 2.0) (/ x (- x -1.0)) t_1)))))

C

double code(double x, double y) {
	double t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	double t_1 = x / ((y / x) + y);
	double tmp;
	if (t_0 <= -2000000.0) {
		tmp = t_1;
	} else if (t_0 <= 5e-28) {
		tmp = fma((x / y), x, x) / 1.0;
	} else if (t_0 <= 2.0) {
		tmp = x / (x - -1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(x * Float64(Float64(x / y) + 1.0)) / Float64(x + 1.0))
	t_1 = Float64(x / Float64(Float64(y / x) + y))
	tmp = 0.0
	if (t_0 <= -2000000.0)
		tmp = t_1;
	elseif (t_0 <= 5e-28)
		tmp = Float64(fma(Float64(x / y), x, x) / 1.0);
	elseif (t_0 <= 2.0)
		tmp = Float64(x / Float64(x - -1.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[(N[(y / x), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2000000.0], t$95$1, If[LessEqual[t$95$0, 5e-28], N[(N[(N[(x / y), $MachinePrecision] * x + x), $MachinePrecision] / 1.0), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(x / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -2e6 or 2 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 72.7%

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

      1. lift-+.f64 N/A

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

      2. *-rgt-identity N/A

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

      3. rgt-mult-inverse N/A

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

      4. distribute-lft-in N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 72.7

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

   3. Applied rewrites 72.7%

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

   4. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lift-/.f64 99.0

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

   6. Applied rewrites 99.0%

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

   7. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 99.0

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

   9. Applied rewrites 99.0%

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

   if -2e6 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 5.0000000000000002e-28

   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%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 98.8%

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

   if 5.0000000000000002e-28 < (/.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 y around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

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

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower--.f64 92.7

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

   4. Applied rewrites 92.7%

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

Alternative 7: 92.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ t_1 := \frac{x}{\frac{y}{x} + y}\\ \mathbf{if}\;t\_0 \leq -2000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\frac{x}{x - -1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0))) (t_1 (/ x (+ (/ y x) y))))
   (if (<= t_0 -2000000.0) t_1 (if (<= t_0 2.0) (/ x (- x -1.0)) t_1))))

C

double code(double x, double y) {
	double t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	double t_1 = x / ((y / x) + y);
	double tmp;
	if (t_0 <= -2000000.0) {
		tmp = t_1;
	} else if (t_0 <= 2.0) {
		tmp = x / (x - -1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (x * ((x / y) + 1.0d0)) / (x + 1.0d0)
    t_1 = x / ((y / x) + y)
    if (t_0 <= (-2000000.0d0)) then
        tmp = t_1
    else if (t_0 <= 2.0d0) then
        tmp = x / (x - (-1.0d0))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	double t_1 = x / ((y / x) + y);
	double tmp;
	if (t_0 <= -2000000.0) {
		tmp = t_1;
	} else if (t_0 <= 2.0) {
		tmp = x / (x - -1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x * ((x / y) + 1.0)) / (x + 1.0)
	t_1 = x / ((y / x) + y)
	tmp = 0
	if t_0 <= -2000000.0:
		tmp = t_1
	elif t_0 <= 2.0:
		tmp = x / (x - -1.0)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x * Float64(Float64(x / y) + 1.0)) / Float64(x + 1.0))
	t_1 = Float64(x / Float64(Float64(y / x) + y))
	tmp = 0.0
	if (t_0 <= -2000000.0)
		tmp = t_1;
	elseif (t_0 <= 2.0)
		tmp = Float64(x / Float64(x - -1.0));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	t_1 = x / ((y / x) + y);
	tmp = 0.0;
	if (t_0 <= -2000000.0)
		tmp = t_1;
	elseif (t_0 <= 2.0)
		tmp = x / (x - -1.0);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[(N[(y / x), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2000000.0], t$95$1, If[LessEqual[t$95$0, 2.0], N[(x / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -2e6 or 2 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 72.7%

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

      1. lift-+.f64 N/A

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

      2. *-rgt-identity N/A

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

      3. rgt-mult-inverse N/A

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

      4. distribute-lft-in N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 72.7

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

   3. Applied rewrites 72.7%

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

   4. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lift-/.f64 99.0

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

   6. Applied rewrites 99.0%

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

   7. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 99.0

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

   9. Applied rewrites 99.0%

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

   if -2e6 < (/.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 y around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

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

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower--.f64 87.1

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

   4. Applied rewrites 87.1%

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

Alternative 8: 85.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ \mathbf{if}\;t\_0 \leq -2000000:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\frac{x}{x - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0))))
   (if (<= t_0 -2000000.0)
     (/ 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 <= -2000000.0) {
		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 <= (-2000000.0d0)) 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 <= -2000000.0) {
		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 <= -2000000.0:
		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 <= -2000000.0)
		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 <= -2000000.0)
		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, -2000000.0], 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))) < -2e6 or 2 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 72.7%

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

   2. Taylor expanded in x around inf

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

      1. lift-/.f64 83.3

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

   4. Applied rewrites 83.3%

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

   if -2e6 < (/.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 y around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

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

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower--.f64 87.1

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

   4. Applied rewrites 87.1%

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

Alternative 9: 84.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ \mathbf{if}\;t\_0 \leq -2000000:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t\_0 \leq 0.05:\\ \;\;\;\;x\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;1 - \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0))))
   (if (<= t_0 -2000000.0)
     (/ x y)
     (if (<= t_0 0.05) x (if (<= t_0 2.0) (- 1.0 (/ 1.0 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 <= -2000000.0) {
		tmp = x / y;
	} else if (t_0 <= 0.05) {
		tmp = x;
	} else if (t_0 <= 2.0) {
		tmp = 1.0 - (1.0 / 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 <= (-2000000.0d0)) then
        tmp = x / y
    else if (t_0 <= 0.05d0) then
        tmp = x
    else if (t_0 <= 2.0d0) then
        tmp = 1.0d0 - (1.0d0 / 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 <= -2000000.0) {
		tmp = x / y;
	} else if (t_0 <= 0.05) {
		tmp = x;
	} else if (t_0 <= 2.0) {
		tmp = 1.0 - (1.0 / 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 <= -2000000.0:
		tmp = x / y
	elif t_0 <= 0.05:
		tmp = x
	elif t_0 <= 2.0:
		tmp = 1.0 - (1.0 / 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 <= -2000000.0)
		tmp = Float64(x / y);
	elseif (t_0 <= 0.05)
		tmp = x;
	elseif (t_0 <= 2.0)
		tmp = Float64(1.0 - Float64(1.0 / 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 <= -2000000.0)
		tmp = x / y;
	elseif (t_0 <= 0.05)
		tmp = x;
	elseif (t_0 <= 2.0)
		tmp = 1.0 - (1.0 / 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, -2000000.0], N[(x / y), $MachinePrecision], If[LessEqual[t$95$0, 0.05], x, If[LessEqual[t$95$0, 2.0], N[(1.0 - N[(1.0 / x), $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))) < -2e6 or 2 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 72.7%

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

   2. Taylor expanded in x around inf

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

      1. lift-/.f64 83.3

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

   4. Applied rewrites 83.3%

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

   if -2e6 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 0.050000000000000003

   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 0.050000000000000003 < (/.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

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

      2. *-rgt-identity N/A

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

      3. rgt-mult-inverse N/A

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

      4. distribute-lft-in N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 99.9

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lift-/.f64 95.4

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

   6. Applied rewrites 95.4%

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

   7. Taylor expanded in x around inf

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

      1. lower--.f64 N/A

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

      2. lift-/.f64 93.6

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

   9. Applied rewrites 93.6%

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

Alternative 10: 84.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ \mathbf{if}\;t\_0 \leq -2000000:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t\_0 \leq 0.0001:\\ \;\;\;\;x\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0))))
   (if (<= t_0 -2000000.0)
     (/ x y)
     (if (<= t_0 0.0001) x (if (<= t_0 2.0) 1.0 (/ x y))))))

C

double code(double x, double y) {
	double t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	double tmp;
	if (t_0 <= -2000000.0) {
		tmp = x / y;
	} else if (t_0 <= 0.0001) {
		tmp = x;
	} else if (t_0 <= 2.0) {
		tmp = 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 <= (-2000000.0d0)) then
        tmp = x / y
    else if (t_0 <= 0.0001d0) then
        tmp = x
    else if (t_0 <= 2.0d0) then
        tmp = 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 <= -2000000.0) {
		tmp = x / y;
	} else if (t_0 <= 0.0001) {
		tmp = x;
	} else if (t_0 <= 2.0) {
		tmp = 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 <= -2000000.0:
		tmp = x / y
	elif t_0 <= 0.0001:
		tmp = x
	elif t_0 <= 2.0:
		tmp = 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 <= -2000000.0)
		tmp = Float64(x / y);
	elseif (t_0 <= 0.0001)
		tmp = x;
	elseif (t_0 <= 2.0)
		tmp = 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 <= -2000000.0)
		tmp = x / y;
	elseif (t_0 <= 0.0001)
		tmp = x;
	elseif (t_0 <= 2.0)
		tmp = 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, -2000000.0], N[(x / y), $MachinePrecision], If[LessEqual[t$95$0, 0.0001], x, If[LessEqual[t$95$0, 2.0], 1.0, 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))) < -2e6 or 2 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 72.7%

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

   2. Taylor expanded in x around inf

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

      1. lift-/.f64 83.3

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

   4. Applied rewrites 83.3%

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

   if -2e6 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 1.00000000000000005e-4

   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 84.2%

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

   if 1.00000000000000005e-4 < (/.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

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

      2. *-rgt-identity N/A

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

      3. rgt-mult-inverse N/A

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

      4. distribute-lft-in N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 99.9

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lift-/.f64 95.0

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

   6. Applied rewrites 95.0%

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

   7. Taylor expanded in x around inf

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

   9. Applied rewrites 90.8%

      \[\leadsto 1 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 50.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \leq 0.0001:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* x (+ (/ x y) 1.0)) (+ x 1.0)) 0.0001) x 1.0))

C

double code(double x, double y) {
	double tmp;
	if (((x * ((x / y) + 1.0)) / (x + 1.0)) <= 0.0001) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (((x * ((x / y) + 1.0d0)) / (x + 1.0d0)) <= 0.0001d0) then
        tmp = x
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((x * ((x / y) + 1.0)) / (x + 1.0)) <= 0.0001) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x * ((x / y) + 1.0)) / (x + 1.0)) <= 0.0001:
		tmp = x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x * Float64(Float64(x / y) + 1.0)) / Float64(x + 1.0)) <= 0.0001)
		tmp = x;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((x * ((x / y) + 1.0)) / (x + 1.0)) <= 0.0001)
		tmp = x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x * N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], 0.0001], x, 1.0]

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))) < 1.00000000000000005e-4

   1. Initial program 91.7%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 58.0%

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

   if 1.00000000000000005e-4 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 81.0%

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

      1. lift-+.f64 N/A

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

      2. *-rgt-identity N/A

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

      3. rgt-mult-inverse N/A

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

      4. distribute-lft-in N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 81.0

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

   3. Applied rewrites 81.0%

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

   4. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lift-/.f64 36.1

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

   6. Applied rewrites 36.1%

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

   7. Taylor expanded in x around inf

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

   9. Applied rewrites 34.9%

      \[\leadsto 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 13.8% accurate, 16.1× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 1.0)

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 88.0%

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

   1. lift-+.f64 N/A

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

   2. *-rgt-identity N/A

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

   3. rgt-mult-inverse N/A

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

   4. distribute-lft-in N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. +-commutative N/A

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

   8. metadata-eval N/A

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

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

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. lower--.f64 N/A

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

   13. lower-/.f64 87.9

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

3. Applied rewrites 87.9%

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

4. Taylor expanded in y around inf

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 N/A

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

   4. lift-/.f64 50.2

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

6. Applied rewrites 50.2%

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

7. Taylor expanded in x around inf

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

9. Applied rewrites 13.8%

   \[\leadsto 1 \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025106 
(FPCore (x y)
  :name "Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1"
  :precision binary64
  (/ (* x (+ (/ x y) 1.0)) (+ x 1.0)))