Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1

Percentage Accurate: 87.3% → 99.6%

Time: 2.7s

Alternatives: 12

Speedup: 0.3×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 87.3% 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: 99.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+225}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{y}, x, 1 \cdot x\right)}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

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 4e+225) (/ (fma (/ x y) x (* 1.0 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 <= -((double) INFINITY)) {
		tmp = x / y;
	} else if (t_0 <= 4e+225) {
		tmp = fma((x / y), x, (1.0 * 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 <= 4e+225)
		tmp = Float64(fma(Float64(x / y), x, Float64(1.0 * 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, 4e+225], N[(N[(N[(x / y), $MachinePrecision] * x + N[(1.0 * x), $MachinePrecision]), $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 3.99999999999999971e225 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 52.8%

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

   2. Taylor expanded in x around inf

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

      1. lift-/.f64 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))) < 3.99999999999999971e225

   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. lower-fma.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lower-*.f64 99.9

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

   3. Applied rewrites 99.9%

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

Alternative 2: 99.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+225}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

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 4e+225) 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 <= 4e+225) {
		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 <= 4e+225) {
		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 <= 4e+225:
		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 <= 4e+225)
		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 <= 4e+225)
		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, 4e+225], 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 3.99999999999999971e225 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 52.8%

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

   2. Taylor expanded in x around inf

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

      1. lift-/.f64 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))) < 3.99999999999999971e225

   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 3: 95.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{x \cdot x}{x - -1}}{y}\\ t_1 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t\_1 \leq -20:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 10^{-12}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{y}, x, x\right)}{1}\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\frac{x}{x - -1}\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+225}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (/ (* x x) (- x -1.0)) y))
        (t_1 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0))))
   (if (<= t_1 (- INFINITY))
     (/ x y)
     (if (<= t_1 -20.0)
       t_0
       (if (<= t_1 1e-12)
         (/ (fma (/ x y) x x) 1.0)
         (if (<= t_1 2.0)
           (/ x (- x -1.0))
           (if (<= t_1 4e+225) t_0 (/ x y))))))))

C

double code(double x, double y) {
	double t_0 = ((x * x) / (x - -1.0)) / y;
	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 <= -20.0) {
		tmp = t_0;
	} else if (t_1 <= 1e-12) {
		tmp = fma((x / y), x, x) / 1.0;
	} else if (t_1 <= 2.0) {
		tmp = x / (x - -1.0);
	} else if (t_1 <= 4e+225) {
		tmp = t_0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(x * x) / Float64(x - -1.0)) / y)
	t_1 = Float64(Float64(x * Float64(Float64(x / y) + 1.0)) / Float64(x + 1.0))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(x / y);
	elseif (t_1 <= -20.0)
		tmp = t_0;
	elseif (t_1 <= 1e-12)
		tmp = Float64(fma(Float64(x / y), x, x) / 1.0);
	elseif (t_1 <= 2.0)
		tmp = Float64(x / Float64(x - -1.0));
	elseif (t_1 <= 4e+225)
		tmp = t_0;
	else
		tmp = Float64(x / y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(x * x), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision] / y), $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, -20.0], t$95$0, If[LessEqual[t$95$1, 1e-12], N[(N[(N[(x / y), $MachinePrecision] * x + x), $MachinePrecision] / 1.0), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(x / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e+225], 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 3.99999999999999971e225 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 52.8%

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

   2. Taylor expanded in x around inf

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

      1. lift-/.f64 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))) < -20 or 2 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 3.99999999999999971e225

   1. Initial program 99.7%

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

   2. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. div-add-rev N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

      10. negate-sub-reverse N/A

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

      11. lower--.f64 84.3

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

   4. Applied rewrites 84.3%

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

   5. Taylor expanded in x around inf

      \[\leadsto \frac{\frac{{x}^{2}}{x - -1}}{y} \]
   6. Step-by-step derivation

      1. pow2 N/A

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

      2. lift-*.f64 82.1

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

   7. Applied rewrites 82.1%

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

   if -20 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 9.9999999999999998e-13

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 99.1%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. distribute-rgt-in N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. *-lft-identity 99.1

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

   6. Applied rewrites 99.1%

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

   if 9.9999999999999998e-13 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. negate-sub-reverse N/A

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

      5. lower--.f64 93.3

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

   4. Applied rewrites 93.3%

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

Alternative 4: 95.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{+236}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t\_0 \leq -20:\\ \;\;\;\;\frac{x \cdot x}{\left(x - -1\right) \cdot y}\\ \mathbf{elif}\;t\_0 \leq 10^{-12}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{y}, x, x\right)}{1}\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\frac{x}{x - -1}\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+225}:\\ \;\;\;\;x \cdot \frac{x}{\mathsf{fma}\left(y, x, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0))))
   (if (<= t_0 -4e+236)
     (/ x y)
     (if (<= t_0 -20.0)
       (/ (* x x) (* (- x -1.0) y))
       (if (<= t_0 1e-12)
         (/ (fma (/ x y) x x) 1.0)
         (if (<= t_0 2.0)
           (/ x (- x -1.0))
           (if (<= t_0 4e+225) (* 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 <= -4e+236) {
		tmp = x / y;
	} else if (t_0 <= -20.0) {
		tmp = (x * x) / ((x - -1.0) * y);
	} else if (t_0 <= 1e-12) {
		tmp = fma((x / y), x, x) / 1.0;
	} else if (t_0 <= 2.0) {
		tmp = x / (x - -1.0);
	} else if (t_0 <= 4e+225) {
		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 <= -4e+236)
		tmp = Float64(x / y);
	elseif (t_0 <= -20.0)
		tmp = Float64(Float64(x * x) / Float64(Float64(x - -1.0) * y));
	elseif (t_0 <= 1e-12)
		tmp = Float64(fma(Float64(x / y), x, x) / 1.0);
	elseif (t_0 <= 2.0)
		tmp = Float64(x / Float64(x - -1.0));
	elseif (t_0 <= 4e+225)
		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, -4e+236], N[(x / y), $MachinePrecision], If[LessEqual[t$95$0, -20.0], N[(N[(x * x), $MachinePrecision] / N[(N[(x - -1.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e-12], N[(N[(N[(x / y), $MachinePrecision] * x + x), $MachinePrecision] / 1.0), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(x / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e+225], N[(x * N[(x / N[(y * x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -4.00000000000000021e236 or 3.99999999999999971e225 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 53.8%

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

   2. Taylor expanded in x around inf

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

      1. lift-/.f64 98.3

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

   4. Applied rewrites 98.3%

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

   if -4.00000000000000021e236 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -20

   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

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. negate-sub-reverse N/A

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

      9. lower--.f64 81.0

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

   4. Applied rewrites 81.0%

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

   if -20 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 9.9999999999999998e-13

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 99.1%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. distribute-rgt-in N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. *-lft-identity 99.1

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

   6. Applied rewrites 99.1%

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

   if 9.9999999999999998e-13 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. negate-sub-reverse N/A

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

      5. lower--.f64 93.3

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

   4. Applied rewrites 93.3%

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

   if 2 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 3.99999999999999971e225

   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. lower-fma.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lower-*.f64 99.7

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

   3. Applied rewrites 99.7%

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

   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. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.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. metadata-eval N/A

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

      6. negate-sub N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift--.f64 80.9

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

   6. Applied rewrites 80.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. negate-sub N/A

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

      9. metadata-eval N/A

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

      10. distribute-rgt1-in N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 86.2

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

   8. Applied rewrites 86.2%

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

Alternative 5: 95.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{x}{\mathsf{fma}\left(y, x, y\right)}\\ t_1 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+199}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t\_1 \leq -20:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 10^{-12}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{y}, x, x\right)}{1}\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\frac{x}{x - -1}\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+225}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (/ x (fma y x y))))
        (t_1 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0))))
   (if (<= t_1 -1e+199)
     (/ x y)
     (if (<= t_1 -20.0)
       t_0
       (if (<= t_1 1e-12)
         (/ (fma (/ x y) x x) 1.0)
         (if (<= t_1 2.0)
           (/ x (- x -1.0))
           (if (<= t_1 4e+225) t_0 (/ x y))))))))

C

double code(double x, double y) {
	double t_0 = x * (x / fma(y, x, y));
	double t_1 = (x * ((x / y) + 1.0)) / (x + 1.0);
	double tmp;
	if (t_1 <= -1e+199) {
		tmp = x / y;
	} else if (t_1 <= -20.0) {
		tmp = t_0;
	} else if (t_1 <= 1e-12) {
		tmp = fma((x / y), x, x) / 1.0;
	} else if (t_1 <= 2.0) {
		tmp = x / (x - -1.0);
	} else if (t_1 <= 4e+225) {
		tmp = t_0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(x * Float64(x / fma(y, x, y)))
	t_1 = Float64(Float64(x * Float64(Float64(x / y) + 1.0)) / Float64(x + 1.0))
	tmp = 0.0
	if (t_1 <= -1e+199)
		tmp = Float64(x / y);
	elseif (t_1 <= -20.0)
		tmp = t_0;
	elseif (t_1 <= 1e-12)
		tmp = Float64(fma(Float64(x / y), x, x) / 1.0);
	elseif (t_1 <= 2.0)
		tmp = Float64(x / Float64(x - -1.0));
	elseif (t_1 <= 4e+225)
		tmp = t_0;
	else
		tmp = Float64(x / y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x * N[(x / N[(y * x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+199], N[(x / y), $MachinePrecision], If[LessEqual[t$95$1, -20.0], t$95$0, If[LessEqual[t$95$1, 1e-12], N[(N[(N[(x / y), $MachinePrecision] * x + x), $MachinePrecision] / 1.0), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(x / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e+225], 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))) < -1.0000000000000001e199 or 3.99999999999999971e225 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 55.0%

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

   2. Taylor expanded in x around inf

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

      1. lift-/.f64 97.4

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

   4. Applied rewrites 97.4%

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

   if -1.0000000000000001e199 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -20 or 2 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 3.99999999999999971e225

   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. lower-fma.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lower-*.f64 99.7

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

   3. Applied rewrites 99.7%

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

   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. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.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. metadata-eval N/A

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

      6. negate-sub N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift--.f64 80.1

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

   6. Applied rewrites 80.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. negate-sub N/A

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

      9. metadata-eval N/A

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

      10. distribute-rgt1-in N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 85.8

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

   8. Applied rewrites 85.8%

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

   if -20 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 9.9999999999999998e-13

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 99.1%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. distribute-rgt-in N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. *-lft-identity 99.1

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

   6. Applied rewrites 99.1%

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

   if 9.9999999999999998e-13 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. negate-sub-reverse N/A

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

      5. lower--.f64 93.3

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

   4. Applied rewrites 93.3%

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

Alternative 6: 90.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{x}{\mathsf{fma}\left(y, x, y\right)}\\ t_1 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+199}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t\_1 \leq -20:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\frac{x}{x - -1}\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+225}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (/ x (fma y x y))))
        (t_1 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0))))
   (if (<= t_1 -1e+199)
     (/ x y)
     (if (<= t_1 -20.0)
       t_0
       (if (<= t_1 2.0) (/ x (- x -1.0)) (if (<= t_1 4e+225) t_0 (/ x y)))))))

C

double code(double x, double y) {
	double t_0 = x * (x / fma(y, x, y));
	double t_1 = (x * ((x / y) + 1.0)) / (x + 1.0);
	double tmp;
	if (t_1 <= -1e+199) {
		tmp = x / y;
	} else if (t_1 <= -20.0) {
		tmp = t_0;
	} else if (t_1 <= 2.0) {
		tmp = x / (x - -1.0);
	} else if (t_1 <= 4e+225) {
		tmp = t_0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(x * Float64(x / fma(y, x, y)))
	t_1 = Float64(Float64(x * Float64(Float64(x / y) + 1.0)) / Float64(x + 1.0))
	tmp = 0.0
	if (t_1 <= -1e+199)
		tmp = Float64(x / y);
	elseif (t_1 <= -20.0)
		tmp = t_0;
	elseif (t_1 <= 2.0)
		tmp = Float64(x / Float64(x - -1.0));
	elseif (t_1 <= 4e+225)
		tmp = t_0;
	else
		tmp = Float64(x / y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x * N[(x / N[(y * x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+199], N[(x / y), $MachinePrecision], If[LessEqual[t$95$1, -20.0], t$95$0, If[LessEqual[t$95$1, 2.0], N[(x / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e+225], t$95$0, N[(x / y), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -1.0000000000000001e199 or 3.99999999999999971e225 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 55.0%

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

   2. Taylor expanded in x around inf

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

      1. lift-/.f64 97.4

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

   4. Applied rewrites 97.4%

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

   if -1.0000000000000001e199 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -20 or 2 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 3.99999999999999971e225

   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. lower-fma.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lower-*.f64 99.7

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

   3. Applied rewrites 99.7%

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

   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. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.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. metadata-eval N/A

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

      6. negate-sub N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift--.f64 80.1

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

   6. Applied rewrites 80.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. negate-sub N/A

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

      9. metadata-eval N/A

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

      10. distribute-rgt1-in N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 85.8

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

   8. Applied rewrites 85.8%

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

   if -20 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. negate-sub-reverse N/A

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

      5. lower--.f64 87.4

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

   4. Applied rewrites 87.4%

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

Alternative 7: 86.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ \mathbf{if}\;t\_0 \leq -20:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\frac{x}{x - -1}\\ \mathbf{elif}\;t\_0 \leq 10^{+85}:\\ \;\;\;\;\frac{x}{y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0))))
   (if (<= t_0 -20.0)
     (/ x y)
     (if (<= t_0 2.0)
       (/ x (- x -1.0))
       (if (<= t_0 1e+85) (* (/ x y) 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 <= -20.0) {
		tmp = x / y;
	} else if (t_0 <= 2.0) {
		tmp = x / (x - -1.0);
	} else if (t_0 <= 1e+85) {
		tmp = (x / y) * 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 <= (-20.0d0)) then
        tmp = x / y
    else if (t_0 <= 2.0d0) then
        tmp = x / (x - (-1.0d0))
    else if (t_0 <= 1d+85) then
        tmp = (x / y) * 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 <= -20.0) {
		tmp = x / y;
	} else if (t_0 <= 2.0) {
		tmp = x / (x - -1.0);
	} else if (t_0 <= 1e+85) {
		tmp = (x / y) * 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 <= -20.0:
		tmp = x / y
	elif t_0 <= 2.0:
		tmp = x / (x - -1.0)
	elif t_0 <= 1e+85:
		tmp = (x / y) * 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 <= -20.0)
		tmp = Float64(x / y);
	elseif (t_0 <= 2.0)
		tmp = Float64(x / Float64(x - -1.0));
	elseif (t_0 <= 1e+85)
		tmp = Float64(Float64(x / y) * 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 <= -20.0)
		tmp = x / y;
	elseif (t_0 <= 2.0)
		tmp = x / (x - -1.0);
	elseif (t_0 <= 1e+85)
		tmp = (x / y) * 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, -20.0], N[(x / y), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(x / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+85], N[(N[(x / y), $MachinePrecision] * x), $MachinePrecision], N[(x / y), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -20 or 1e85 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 68.2%

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

   2. Taylor expanded in x around inf

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

      1. lift-/.f64 87.6

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

   4. Applied rewrites 87.6%

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

   if -20 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. negate-sub-reverse N/A

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

      5. lower--.f64 87.4

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

   4. Applied rewrites 87.4%

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

   if 2 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 1e85

   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. lower-fma.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lower-*.f64 99.7

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

   3. Applied rewrites 99.7%

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

   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. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.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. metadata-eval N/A

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

      6. negate-sub N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift--.f64 70.4

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

   6. Applied rewrites 70.4%

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

   7. Taylor expanded in x around 0

      \[\leadsto \frac{{x}^{2}}{\color{blue}{y}} \]
   8. Step-by-step derivation

      1. pow2 N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lift-/.f64 30.7

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

   9. Applied rewrites 30.7%

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

Alternative 8: 85.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ \mathbf{if}\;t\_0 \leq -20:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\frac{x}{x - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0))))
   (if (<= t_0 -20.0) (/ 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 <= -20.0) {
		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 <= (-20.0d0)) 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 <= -20.0) {
		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 <= -20.0:
		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 <= -20.0)
		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 <= -20.0)
		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, -20.0], 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))) < -20 or 2 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 71.4%

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

   2. Taylor expanded in x around inf

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

      1. lift-/.f64 85.0

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

   4. Applied rewrites 85.0%

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

   if -20 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. negate-sub-reverse N/A

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

      5. lower--.f64 87.4

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

   4. Applied rewrites 87.4%

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

Alternative 9: 84.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ \mathbf{if}\;t\_0 \leq -20:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-22}:\\ \;\;\;\;x\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;1 - \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0))))
   (if (<= t_0 -20.0)
     (/ x y)
     (if (<= t_0 2e-22) x (if (<= t_0 2.0) (- 1.0 (/ 1.0 x)) (/ x y))))))

C

double code(double x, double y) {
	double t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	double tmp;
	if (t_0 <= -20.0) {
		tmp = x / y;
	} else if (t_0 <= 2e-22) {
		tmp = x;
	} else if (t_0 <= 2.0) {
		tmp = 1.0 - (1.0 / x);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x * ((x / y) + 1.0d0)) / (x + 1.0d0)
    if (t_0 <= (-20.0d0)) then
        tmp = x / y
    else if (t_0 <= 2d-22) then
        tmp = x
    else if (t_0 <= 2.0d0) then
        tmp = 1.0d0 - (1.0d0 / x)
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	double tmp;
	if (t_0 <= -20.0) {
		tmp = x / y;
	} else if (t_0 <= 2e-22) {
		tmp = x;
	} else if (t_0 <= 2.0) {
		tmp = 1.0 - (1.0 / x);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x * ((x / y) + 1.0)) / (x + 1.0)
	tmp = 0
	if t_0 <= -20.0:
		tmp = x / y
	elif t_0 <= 2e-22:
		tmp = x
	elif t_0 <= 2.0:
		tmp = 1.0 - (1.0 / x)
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x * Float64(Float64(x / y) + 1.0)) / Float64(x + 1.0))
	tmp = 0.0
	if (t_0 <= -20.0)
		tmp = Float64(x / y);
	elseif (t_0 <= 2e-22)
		tmp = x;
	elseif (t_0 <= 2.0)
		tmp = Float64(1.0 - Float64(1.0 / x));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	tmp = 0.0;
	if (t_0 <= -20.0)
		tmp = x / y;
	elseif (t_0 <= 2e-22)
		tmp = x;
	elseif (t_0 <= 2.0)
		tmp = 1.0 - (1.0 / x);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -20.0], N[(x / y), $MachinePrecision], If[LessEqual[t$95$0, 2e-22], x, If[LessEqual[t$95$0, 2.0], N[(1.0 - N[(1.0 / x), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -20 or 2 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 71.4%

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

   2. Taylor expanded in x around inf

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

      1. lift-/.f64 85.0

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

   4. Applied rewrites 85.0%

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

   if -20 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 2.0000000000000001e-22

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 85.4%

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

   if 2.0000000000000001e-22 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. negate-sub-reverse N/A

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

      5. lower--.f64 91.8

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

   4. Applied rewrites 91.8%

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

   5. Taylor expanded in x around inf

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 84.3

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

   7. Applied rewrites 84.3%

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

Alternative 10: 83.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ \mathbf{if}\;t\_0 \leq -20:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-22}:\\ \;\;\;\;x\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+42}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0))))
   (if (<= t_0 -20.0)
     (/ x y)
     (if (<= t_0 2e-22) x (if (<= t_0 4e+42) 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 <= -20.0) {
		tmp = x / y;
	} else if (t_0 <= 2e-22) {
		tmp = x;
	} else if (t_0 <= 4e+42) {
		tmp = 1.0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x * ((x / y) + 1.0d0)) / (x + 1.0d0)
    if (t_0 <= (-20.0d0)) then
        tmp = x / y
    else if (t_0 <= 2d-22) then
        tmp = x
    else if (t_0 <= 4d+42) then
        tmp = 1.0d0
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	double tmp;
	if (t_0 <= -20.0) {
		tmp = x / y;
	} else if (t_0 <= 2e-22) {
		tmp = x;
	} else if (t_0 <= 4e+42) {
		tmp = 1.0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x * ((x / y) + 1.0)) / (x + 1.0)
	tmp = 0
	if t_0 <= -20.0:
		tmp = x / y
	elif t_0 <= 2e-22:
		tmp = x
	elif t_0 <= 4e+42:
		tmp = 1.0
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x * Float64(Float64(x / y) + 1.0)) / Float64(x + 1.0))
	tmp = 0.0
	if (t_0 <= -20.0)
		tmp = Float64(x / y);
	elseif (t_0 <= 2e-22)
		tmp = x;
	elseif (t_0 <= 4e+42)
		tmp = 1.0;
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	tmp = 0.0;
	if (t_0 <= -20.0)
		tmp = x / y;
	elseif (t_0 <= 2e-22)
		tmp = x;
	elseif (t_0 <= 4e+42)
		tmp = 1.0;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -20.0], N[(x / y), $MachinePrecision], If[LessEqual[t$95$0, 2e-22], x, If[LessEqual[t$95$0, 4e+42], 1.0, N[(x / y), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -20 or 4.00000000000000018e42 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 69.8%

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

   2. Taylor expanded in x around inf

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

      1. lift-/.f64 86.6

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

   4. Applied rewrites 86.6%

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

   if -20 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 2.0000000000000001e-22

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 85.4%

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

   if 2.0000000000000001e-22 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 4.00000000000000018e42

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. negate-sub-reverse N/A

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

      5. lower--.f64 79.1

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

   4. Applied rewrites 79.1%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 72.2%

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

Alternative 11: 49.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \leq 2 \cdot 10^{-22}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* x (+ (/ x y) 1.0)) (+ x 1.0)) 2e-22) x 1.0))

C

double code(double x, double y) {
	double tmp;
	if (((x * ((x / y) + 1.0)) / (x + 1.0)) <= 2e-22) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (((x * ((x / y) + 1.0d0)) / (x + 1.0d0)) <= 2d-22) then
        tmp = x
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((x * ((x / y) + 1.0)) / (x + 1.0)) <= 2e-22) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x * ((x / y) + 1.0)) / (x + 1.0)) <= 2e-22:
		tmp = x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x * Float64(Float64(x / y) + 1.0)) / Float64(x + 1.0)) <= 2e-22)
		tmp = x;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((x * ((x / y) + 1.0)) / (x + 1.0)) <= 2e-22)
		tmp = x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x * N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], 2e-22], x, 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 2.0000000000000001e-22

   1. Initial program 90.6%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 57.5%

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

   if 2.0000000000000001e-22 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 81.5%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. negate-sub-reverse N/A

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

      5. lower--.f64 37.2

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

   4. Applied rewrites 37.2%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 34.3%

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

Alternative 12: 38.7% accurate, 16.1× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 x)

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

def code(x, y):
	return x

Julia

function code(x, y)
	return x
end

MATLAB

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

Wolfram

code[x_, y_] := x

Derivation

1. Initial program 87.3%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 38.7%

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

Reproduce

herbie shell --seed 2025110 
(FPCore (x y)
  :name "Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1"
  :precision binary64
  (/ (* x (+ (/ x y) 1.0)) (+ x 1.0)))