Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1

Percentage Accurate: 88.0% → 99.9%

Time: 2.9s

Alternatives: 14

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, 0.2× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0)))
        (t_1 (/ x (* (+ (pow x -1.0) 1.0) 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 / ((pow(x, -1.0) + 1.0) * 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 / (((x ** (-1.0d0)) + 1.0d0) * 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 / ((Math.pow(x, -1.0) + 1.0) * 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 / ((math.pow(x, -1.0) + 1.0) * 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((x ^ -1.0) + 1.0) * 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 / (((x ^ -1.0) + 1.0) * 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[(N[Power[x, -1.0], $MachinePrecision] + 1.0), $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. inv-pow N/A

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

      14. lower-pow.f64 67.2

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

   3. Applied rewrites 67.2%

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

   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. inv-pow N/A

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

      7. lift-pow.f64 99.8

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

   6. Applied rewrites 99.8%

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

   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: 98.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \frac{x}{y}}{x + 1}\\ t_1 := x \cdot \left(\frac{x}{y} + 1\right)\\ t_2 := \frac{t\_1}{x + 1}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t\_2 \leq -2000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_2 \leq 0.05:\\ \;\;\;\;\left(\mathsf{fma}\left(-1, x, \frac{x}{y}\right) + 1\right) \cdot x\\ \mathbf{elif}\;t\_2 \leq 2000:\\ \;\;\;\;\frac{t\_1}{x}\\ \mathbf{elif}\;t\_2 \leq 10^{+237}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (/ x y)) (+ x 1.0)))
        (t_1 (* x (+ (/ x y) 1.0)))
        (t_2 (/ t_1 (+ x 1.0))))
   (if (<= t_2 (- INFINITY))
     (/ x y)
     (if (<= t_2 -2000000.0)
       t_0
       (if (<= t_2 0.05)
         (* (+ (fma -1.0 x (/ x y)) 1.0) x)
         (if (<= t_2 2000.0) (/ t_1 x) (if (<= t_2 1e+237) t_0 (/ x y))))))))

C

double code(double x, double y) {
	double t_0 = (x * (x / y)) / (x + 1.0);
	double t_1 = x * ((x / y) + 1.0);
	double t_2 = t_1 / (x + 1.0);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = x / y;
	} else if (t_2 <= -2000000.0) {
		tmp = t_0;
	} else if (t_2 <= 0.05) {
		tmp = (fma(-1.0, x, (x / y)) + 1.0) * x;
	} else if (t_2 <= 2000.0) {
		tmp = t_1 / x;
	} else if (t_2 <= 1e+237) {
		tmp = t_0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(x * Float64(x / y)) / Float64(x + 1.0))
	t_1 = Float64(x * Float64(Float64(x / y) + 1.0))
	t_2 = Float64(t_1 / Float64(x + 1.0))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(x / y);
	elseif (t_2 <= -2000000.0)
		tmp = t_0;
	elseif (t_2 <= 0.05)
		tmp = Float64(Float64(fma(-1.0, x, Float64(x / y)) + 1.0) * x);
	elseif (t_2 <= 2000.0)
		tmp = Float64(t_1 / x);
	elseif (t_2 <= 1e+237)
		tmp = t_0;
	else
		tmp = Float64(x / y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(x / y), $MachinePrecision], If[LessEqual[t$95$2, -2000000.0], t$95$0, If[LessEqual[t$95$2, 0.05], N[(N[(N[(-1.0 * x + N[(x / y), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t$95$2, 2000.0], N[(t$95$1 / x), $MachinePrecision], If[LessEqual[t$95$2, 1e+237], t$95$0, N[(x / y), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -inf.0 or 9.9999999999999994e236 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 53.6%

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

   2. Taylor expanded in x around inf

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

      1. lift-/.f64 99.0

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

   4. Applied rewrites 99.0%

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

   if -inf.0 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -2e6 or 2e3 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 9.9999999999999994e236

   1. Initial program 99.7%

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

   2. Taylor expanded in x around inf

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

      1. lift-/.f64 98.0

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

   4. Applied rewrites 98.0%

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

   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. inv-pow N/A

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

      8. lower-pow.f64 98.9

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

   4. Applied rewrites 98.9%

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

   5. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lift-/.f64 98.9

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

   7. Applied rewrites 98.9%

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

   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 4 regimes into one program.
4. Add Preprocessing

Alternative 3: 98.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \frac{x}{y}}{x + 1}\\ t_1 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t\_1 \leq -2000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 0.05:\\ \;\;\;\;\left(\mathsf{fma}\left(-1, x, \frac{x}{y}\right) + 1\right) \cdot x\\ \mathbf{elif}\;t\_1 \leq 2000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{y}, x, x\right)}{x}\\ \mathbf{elif}\;t\_1 \leq 10^{+237}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (/ x y)) (+ x 1.0)))
        (t_1 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0))))
   (if (<= t_1 (- INFINITY))
     (/ x y)
     (if (<= t_1 -2000000.0)
       t_0
       (if (<= t_1 0.05)
         (* (+ (fma -1.0 x (/ x y)) 1.0) x)
         (if (<= t_1 2000.0)
           (/ (fma (/ x y) x x) x)
           (if (<= t_1 1e+237) t_0 (/ x y))))))))

C

double code(double x, double y) {
	double t_0 = (x * (x / y)) / (x + 1.0);
	double t_1 = (x * ((x / y) + 1.0)) / (x + 1.0);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = x / y;
	} else if (t_1 <= -2000000.0) {
		tmp = t_0;
	} else if (t_1 <= 0.05) {
		tmp = (fma(-1.0, x, (x / y)) + 1.0) * x;
	} else if (t_1 <= 2000.0) {
		tmp = fma((x / y), x, x) / x;
	} else if (t_1 <= 1e+237) {
		tmp = t_0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(x * Float64(x / y)) / Float64(x + 1.0))
	t_1 = Float64(Float64(x * Float64(Float64(x / y) + 1.0)) / Float64(x + 1.0))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(x / y);
	elseif (t_1 <= -2000000.0)
		tmp = t_0;
	elseif (t_1 <= 0.05)
		tmp = Float64(Float64(fma(-1.0, x, Float64(x / y)) + 1.0) * x);
	elseif (t_1 <= 2000.0)
		tmp = Float64(fma(Float64(x / y), x, x) / x);
	elseif (t_1 <= 1e+237)
		tmp = t_0;
	else
		tmp = Float64(x / y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(x / y), $MachinePrecision], If[LessEqual[t$95$1, -2000000.0], t$95$0, If[LessEqual[t$95$1, 0.05], N[(N[(N[(-1.0 * x + N[(x / y), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t$95$1, 2000.0], N[(N[(N[(x / y), $MachinePrecision] * x + x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[t$95$1, 1e+237], t$95$0, N[(x / y), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -inf.0 or 9.9999999999999994e236 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 53.6%

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

   2. Taylor expanded in x around inf

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

      1. lift-/.f64 99.0

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

   4. Applied rewrites 99.0%

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

   if -inf.0 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -2e6 or 2e3 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 9.9999999999999994e236

   1. Initial program 99.7%

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

   2. Taylor expanded in x around inf

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

      1. lift-/.f64 98.0

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

   4. Applied rewrites 98.0%

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

   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. inv-pow N/A

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

      8. lower-pow.f64 98.9

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

   4. Applied rewrites 98.9%

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

   5. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lift-/.f64 98.9

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

   7. Applied rewrites 98.9%

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

   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. 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 4 regimes into one program.
4. Add Preprocessing

Alternative 4: 93.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+40}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t\_0 \leq 0.05:\\ \;\;\;\;\left(\mathsf{fma}\left(-1, x, \frac{x}{y}\right) + 1\right) \cdot x\\ \mathbf{elif}\;t\_0 \leq 2000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{y}, x, x\right)}{x}\\ \mathbf{elif}\;t\_0 \leq 10^{+237}:\\ \;\;\;\;x \cdot \frac{x}{\mathsf{fma}\left(y, x, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0))))
   (if (<= t_0 -5e+40)
     (/ x y)
     (if (<= t_0 0.05)
       (* (+ (fma -1.0 x (/ x y)) 1.0) x)
       (if (<= t_0 2000.0)
         (/ (fma (/ x y) x x) x)
         (if (<= t_0 1e+237) (* x (/ x (fma y x y))) (/ x y)))))))

C

double code(double x, double y) {
	double t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	double tmp;
	if (t_0 <= -5e+40) {
		tmp = x / y;
	} else if (t_0 <= 0.05) {
		tmp = (fma(-1.0, x, (x / y)) + 1.0) * x;
	} else if (t_0 <= 2000.0) {
		tmp = fma((x / y), x, x) / x;
	} else if (t_0 <= 1e+237) {
		tmp = x * (x / fma(y, x, y));
	} else {
		tmp = x / y;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(x * Float64(Float64(x / y) + 1.0)) / Float64(x + 1.0))
	tmp = 0.0
	if (t_0 <= -5e+40)
		tmp = Float64(x / y);
	elseif (t_0 <= 0.05)
		tmp = Float64(Float64(fma(-1.0, x, Float64(x / y)) + 1.0) * x);
	elseif (t_0 <= 2000.0)
		tmp = Float64(fma(Float64(x / y), x, x) / x);
	elseif (t_0 <= 1e+237)
		tmp = Float64(x * Float64(x / fma(y, x, y)));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+40], N[(x / y), $MachinePrecision], If[LessEqual[t$95$0, 0.05], N[(N[(N[(-1.0 * x + N[(x / y), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t$95$0, 2000.0], N[(N[(N[(x / y), $MachinePrecision] * x + x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[t$95$0, 1e+237], N[(x * N[(x / N[(y * x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -5.00000000000000003e40 or 9.9999999999999994e236 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 63.7%

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

   2. Taylor expanded in x around inf

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

      1. lift-/.f64 90.7

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

   4. Applied rewrites 90.7%

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

   if -5.00000000000000003e40 < (/.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. inv-pow N/A

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

      8. lower-pow.f64 96.0

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

   4. Applied rewrites 96.0%

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

   5. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lift-/.f64 96.1

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

   7. Applied rewrites 96.1%

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

   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. 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%

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

   if 2e3 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 9.9999999999999994e236

   1. Initial program 99.7%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 99.7

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

      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 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt1-in N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. pow2 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-fma.f64 87.1

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

   6. Applied rewrites 87.1%

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

Alternative 5: 93.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+40}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t\_0 \leq 0.05:\\ \;\;\;\;\left(\mathsf{fma}\left(-1, x, \frac{x}{y}\right) + 1\right) \cdot x\\ \mathbf{elif}\;t\_0 \leq 2000:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;t\_0 \leq 10^{+237}:\\ \;\;\;\;x \cdot \frac{x}{\mathsf{fma}\left(y, x, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0))))
   (if (<= t_0 -5e+40)
     (/ x y)
     (if (<= t_0 0.05)
       (* (+ (fma -1.0 x (/ x y)) 1.0) x)
       (if (<= t_0 2000.0)
         (/ x (+ x 1.0))
         (if (<= t_0 1e+237) (* x (/ x (fma y x y))) (/ x y)))))))

C

double code(double x, double y) {
	double t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	double tmp;
	if (t_0 <= -5e+40) {
		tmp = x / y;
	} else if (t_0 <= 0.05) {
		tmp = (fma(-1.0, x, (x / y)) + 1.0) * x;
	} else if (t_0 <= 2000.0) {
		tmp = x / (x + 1.0);
	} else if (t_0 <= 1e+237) {
		tmp = x * (x / fma(y, x, y));
	} else {
		tmp = x / y;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(x * Float64(Float64(x / y) + 1.0)) / Float64(x + 1.0))
	tmp = 0.0
	if (t_0 <= -5e+40)
		tmp = Float64(x / y);
	elseif (t_0 <= 0.05)
		tmp = Float64(Float64(fma(-1.0, x, Float64(x / y)) + 1.0) * x);
	elseif (t_0 <= 2000.0)
		tmp = Float64(x / Float64(x + 1.0));
	elseif (t_0 <= 1e+237)
		tmp = Float64(x * Float64(x / fma(y, x, y)));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+40], N[(x / y), $MachinePrecision], If[LessEqual[t$95$0, 0.05], N[(N[(N[(-1.0 * x + N[(x / y), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t$95$0, 2000.0], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+237], N[(x * N[(x / N[(y * x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -5.00000000000000003e40 or 9.9999999999999994e236 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 63.7%

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

   2. Taylor expanded in x around inf

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

      1. lift-/.f64 90.7

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

   4. Applied rewrites 90.7%

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

   if -5.00000000000000003e40 < (/.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. inv-pow N/A

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

      8. lower-pow.f64 96.0

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

   4. Applied rewrites 96.0%

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

   5. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lift-/.f64 96.1

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

   7. Applied rewrites 96.1%

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

   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 0

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

   4. Applied rewrites 94.5%

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

   if 2e3 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 9.9999999999999994e236

   1. Initial program 99.7%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 99.7

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

      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 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt1-in N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. pow2 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-fma.f64 87.1

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

   6. Applied rewrites 87.1%

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

Alternative 6: 92.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y} + 1\\ t_1 := \frac{x \cdot t\_0}{x + 1}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+49}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-28}:\\ \;\;\;\;t\_0 \cdot x\\ \mathbf{elif}\;t\_1 \leq 2000:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;t\_1 \leq 10^{+237}:\\ \;\;\;\;x \cdot \frac{x}{\mathsf{fma}\left(y, x, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (/ x y) 1.0)) (t_1 (/ (* x t_0) (+ x 1.0))))
   (if (<= t_1 -5e+49)
     (/ x y)
     (if (<= t_1 5e-28)
       (* t_0 x)
       (if (<= t_1 2000.0)
         (/ x (+ x 1.0))
         (if (<= t_1 1e+237) (* x (/ x (fma y x y))) (/ x y)))))))

C

double code(double x, double y) {
	double t_0 = (x / y) + 1.0;
	double t_1 = (x * t_0) / (x + 1.0);
	double tmp;
	if (t_1 <= -5e+49) {
		tmp = x / y;
	} else if (t_1 <= 5e-28) {
		tmp = t_0 * x;
	} else if (t_1 <= 2000.0) {
		tmp = x / (x + 1.0);
	} else if (t_1 <= 1e+237) {
		tmp = x * (x / fma(y, x, y));
	} else {
		tmp = x / y;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(x / y) + 1.0)
	t_1 = Float64(Float64(x * t_0) / Float64(x + 1.0))
	tmp = 0.0
	if (t_1 <= -5e+49)
		tmp = Float64(x / y);
	elseif (t_1 <= 5e-28)
		tmp = Float64(t_0 * x);
	elseif (t_1 <= 2000.0)
		tmp = Float64(x / Float64(x + 1.0));
	elseif (t_1 <= 1e+237)
		tmp = Float64(x * Float64(x / fma(y, x, y)));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * t$95$0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+49], N[(x / y), $MachinePrecision], If[LessEqual[t$95$1, 5e-28], N[(t$95$0 * x), $MachinePrecision], If[LessEqual[t$95$1, 2000.0], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+237], N[(x * N[(x / N[(y * x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -5.0000000000000004e49 or 9.9999999999999994e236 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 63.3%

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

   2. Taylor expanded in x around inf

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

      1. lift-/.f64 91.1

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

   4. Applied rewrites 91.1%

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

   if -5.0000000000000004e49 < (/.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. 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. inv-pow N/A

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

      8. lower-pow.f64 95.7

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

   4. Applied rewrites 95.7%

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

   5. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lift-/.f64 95.7

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

   7. Applied rewrites 95.7%

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

   8. Taylor expanded in y around 0

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

      1. lift-/.f64 95.4

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

   10. Applied rewrites 95.4%

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

   if 5.0000000000000002e-28 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 2e3

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 91.9%

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

   if 2e3 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 9.9999999999999994e236

   1. Initial program 99.7%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 99.7

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

      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 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt1-in N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. pow2 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-fma.f64 87.1

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

   6. Applied rewrites 87.1%

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

Alternative 7: 92.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y} + 1\\ t_1 := \frac{x \cdot t\_0}{x + 1}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+49}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-28}:\\ \;\;\;\;t\_0 \cdot x\\ \mathbf{elif}\;t\_1 \leq 2000:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;t\_1 \leq 10^{+237}:\\ \;\;\;\;\frac{x \cdot x}{\mathsf{fma}\left(y, x, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (/ x y) 1.0)) (t_1 (/ (* x t_0) (+ x 1.0))))
   (if (<= t_1 -5e+49)
     (/ x y)
     (if (<= t_1 5e-28)
       (* t_0 x)
       (if (<= t_1 2000.0)
         (/ x (+ x 1.0))
         (if (<= t_1 1e+237) (/ (* x x) (fma y x y)) (/ x y)))))))

C

double code(double x, double y) {
	double t_0 = (x / y) + 1.0;
	double t_1 = (x * t_0) / (x + 1.0);
	double tmp;
	if (t_1 <= -5e+49) {
		tmp = x / y;
	} else if (t_1 <= 5e-28) {
		tmp = t_0 * x;
	} else if (t_1 <= 2000.0) {
		tmp = x / (x + 1.0);
	} else if (t_1 <= 1e+237) {
		tmp = (x * x) / fma(y, x, y);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(x / y) + 1.0)
	t_1 = Float64(Float64(x * t_0) / Float64(x + 1.0))
	tmp = 0.0
	if (t_1 <= -5e+49)
		tmp = Float64(x / y);
	elseif (t_1 <= 5e-28)
		tmp = Float64(t_0 * x);
	elseif (t_1 <= 2000.0)
		tmp = Float64(x / Float64(x + 1.0));
	elseif (t_1 <= 1e+237)
		tmp = Float64(Float64(x * x) / fma(y, x, y));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * t$95$0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+49], N[(x / y), $MachinePrecision], If[LessEqual[t$95$1, 5e-28], N[(t$95$0 * x), $MachinePrecision], If[LessEqual[t$95$1, 2000.0], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+237], N[(N[(x * x), $MachinePrecision] / N[(y * x + y), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -5.0000000000000004e49 or 9.9999999999999994e236 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 63.3%

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

   2. Taylor expanded in x around inf

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

      1. lift-/.f64 91.1

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

   4. Applied rewrites 91.1%

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

   if -5.0000000000000004e49 < (/.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. 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. inv-pow N/A

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

      8. lower-pow.f64 95.7

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

   4. Applied rewrites 95.7%

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

   5. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lift-/.f64 95.7

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

   7. Applied rewrites 95.7%

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

   8. Taylor expanded in y around 0

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

      1. lift-/.f64 95.4

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

   10. Applied rewrites 95.4%

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

   if 5.0000000000000002e-28 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 2e3

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 91.9%

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

   if 2e3 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 9.9999999999999994e236

   1. Initial program 99.7%

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

   2. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. lower-fma.f64 82.2

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

   4. Applied rewrites 82.2%

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

Alternative 8: 90.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y} + 1\\ t_1 := \frac{x \cdot t\_0}{x + 1}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+49}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-28}:\\ \;\;\;\;t\_0 \cdot x\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (/ x y) 1.0)) (t_1 (/ (* x t_0) (+ x 1.0))))
   (if (<= t_1 -5e+49)
     (/ x y)
     (if (<= t_1 5e-28) (* t_0 x) (if (<= t_1 2.0) (/ x (+ x 1.0)) (/ x y))))))

C

double code(double x, double y) {
	double t_0 = (x / y) + 1.0;
	double t_1 = (x * t_0) / (x + 1.0);
	double tmp;
	if (t_1 <= -5e+49) {
		tmp = x / y;
	} else if (t_1 <= 5e-28) {
		tmp = t_0 * x;
	} else if (t_1 <= 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) :: t_1
    real(8) :: tmp
    t_0 = (x / y) + 1.0d0
    t_1 = (x * t_0) / (x + 1.0d0)
    if (t_1 <= (-5d+49)) then
        tmp = x / y
    else if (t_1 <= 5d-28) then
        tmp = t_0 * x
    else if (t_1 <= 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 / y) + 1.0;
	double t_1 = (x * t_0) / (x + 1.0);
	double tmp;
	if (t_1 <= -5e+49) {
		tmp = x / y;
	} else if (t_1 <= 5e-28) {
		tmp = t_0 * x;
	} else if (t_1 <= 2.0) {
		tmp = x / (x + 1.0);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x / y) + 1.0
	t_1 = (x * t_0) / (x + 1.0)
	tmp = 0
	if t_1 <= -5e+49:
		tmp = x / y
	elif t_1 <= 5e-28:
		tmp = t_0 * x
	elif t_1 <= 2.0:
		tmp = x / (x + 1.0)
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x / y) + 1.0)
	t_1 = Float64(Float64(x * t_0) / Float64(x + 1.0))
	tmp = 0.0
	if (t_1 <= -5e+49)
		tmp = Float64(x / y);
	elseif (t_1 <= 5e-28)
		tmp = Float64(t_0 * x);
	elseif (t_1 <= 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 / y) + 1.0;
	t_1 = (x * t_0) / (x + 1.0);
	tmp = 0.0;
	if (t_1 <= -5e+49)
		tmp = x / y;
	elseif (t_1 <= 5e-28)
		tmp = t_0 * x;
	elseif (t_1 <= 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 / y), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * t$95$0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+49], N[(x / y), $MachinePrecision], If[LessEqual[t$95$1, 5e-28], N[(t$95$0 * x), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -5.0000000000000004e49 or 2 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 71.1%

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

   2. Taylor expanded in x around inf

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

      1. lift-/.f64 84.7

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

   4. Applied rewrites 84.7%

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

   if -5.0000000000000004e49 < (/.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. 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. inv-pow N/A

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

      8. lower-pow.f64 95.7

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

   4. Applied rewrites 95.7%

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

   5. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lift-/.f64 95.7

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

   7. Applied rewrites 95.7%

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

   8. Taylor expanded in y around 0

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

      1. lift-/.f64 95.4

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

   10. Applied rewrites 95.4%

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

   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 x around 0

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

   4. Applied rewrites 92.7%

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

Alternative 9: 99.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t\_0 \leq 10^{+237}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0))))
   (if (<= t_0 (- INFINITY)) (/ x y) (if (<= t_0 1e+237) t_0 (/ x y)))))

C

double code(double x, double y) {
	double t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = x / y;
	} else if (t_0 <= 1e+237) {
		tmp = t_0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = x / y;
	} else if (t_0 <= 1e+237) {
		tmp = t_0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x * ((x / y) + 1.0)) / (x + 1.0)
	tmp = 0
	if t_0 <= -math.inf:
		tmp = x / y
	elif t_0 <= 1e+237:
		tmp = t_0
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x * Float64(Float64(x / y) + 1.0)) / Float64(x + 1.0))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(x / y);
	elseif (t_0 <= 1e+237)
		tmp = t_0;
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = x / y;
	elseif (t_0 <= 1e+237)
		tmp = t_0;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(x / y), $MachinePrecision], If[LessEqual[t$95$0, 1e+237], t$95$0, N[(x / y), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -inf.0 or 9.9999999999999994e236 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 53.6%

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

   2. Taylor expanded in x around inf

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

      1. lift-/.f64 99.0

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

   4. Applied rewrites 99.0%

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

   if -inf.0 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 9.9999999999999994e236

   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 10: 99.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t\_0 \leq 10^{+237}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{y}, x, x\right)}{x - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0))))
   (if (<= t_0 (- INFINITY))
     (/ x y)
     (if (<= t_0 1e+237) (/ (fma (/ x y) x x) (- x -1.0)) (/ x y)))))

C

double code(double x, double y) {
	double t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = x / y;
	} else if (t_0 <= 1e+237) {
		tmp = fma((x / y), x, x) / (x - -1.0);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(x * Float64(Float64(x / y) + 1.0)) / Float64(x + 1.0))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(x / y);
	elseif (t_0 <= 1e+237)
		tmp = Float64(fma(Float64(x / y), x, x) / Float64(x - -1.0));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(x / y), $MachinePrecision], If[LessEqual[t$95$0, 1e+237], N[(N[(N[(x / y), $MachinePrecision] * x + x), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -inf.0 or 9.9999999999999994e236 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 53.6%

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

   2. Taylor expanded in x around inf

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

      1. lift-/.f64 99.0

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

   4. Applied rewrites 99.0%

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

   if -inf.0 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 9.9999999999999994e236

   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 11: 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 x around 0

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

   4. Applied rewrites 87.1%

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

Alternative 12: 74.5% accurate, 0.4× speedup

Math

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

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) 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 + 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 + 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 + 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 + 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 = Float64(Float64(Float64(-x) + 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 + 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], N[(N[((-x) + 1.0), $MachinePrecision] * x), $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 0.050000000000000003 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 78.5%

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

   2. Taylor expanded in x around inf

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

      1. lift-/.f64 66.4

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

   4. Applied rewrites 66.4%

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

   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

      \[\leadsto \color{blue}{x \cdot \left(1 + x \cdot \left(\frac{1}{y} - 1\right)\right)} \]
   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. inv-pow N/A

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

      8. lower-pow.f64 98.9

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

   4. Applied rewrites 98.9%

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

   5. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lift-/.f64 98.9

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

   7. Applied rewrites 98.9%

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

   8. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 84.5

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

   10. Applied rewrites 84.5%

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

Alternative 13: 43.6% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.0%

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

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. inv-pow N/A

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

   8. lower-pow.f64 57.4

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

4. Applied rewrites 57.4%

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

5. Taylor expanded in y around inf

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

   1. +-commutative N/A

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

   2. lower-+.f64 N/A

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

   3. lower-fma.f64 N/A

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

   4. lift-/.f64 57.4

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

7. Applied rewrites 57.4%

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

8. Taylor expanded in y around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 43.6

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

10. Applied rewrites 43.6%

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

Alternative 14: 39.3% accurate, 34.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 x)

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

def code(x, y):
	return x

Julia

function code(x, y)
	return x
end

MATLAB

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

Wolfram

code[x_, y_] := x

Derivation

1. Initial program 88.0%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 39.3%

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

Developer Target 1: 99.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{x}{1} \cdot \frac{\frac{x}{y} + 1}{x + 1} \end{array} \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

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

  :alt
  (! :herbie-platform default (* (/ x 1) (/ (+ (/ x y) 1) (+ x 1))))

  (/ (* x (+ (/ x y) 1.0)) (+ x 1.0)))