Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1

Percentage Accurate: 88.2% → 99.9%

Time: 8.8s

Alternatives: 12

Speedup: 1.1×

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

real(8) function code(x, y)
    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: 88.2% 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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.7%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 99.9

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

4. Applied rewrites 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 92.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1}\\ \mathbf{if}\;t\_0 \leq -5000000000000:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t\_0 \leq 0.0001:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{x}{y}, x\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ 1.0 (/ x y))) (+ x 1.0))))
   (if (<= t_0 -5000000000000.0)
     (/ x y)
     (if (<= t_0 0.0001)
       (fma x (/ x y) x)
       (if (<= t_0 2.0) (/ x (+ x 1.0)) (/ x y))))))

C

double code(double x, double y) {
	double t_0 = (x * (1.0 + (x / y))) / (x + 1.0);
	double tmp;
	if (t_0 <= -5000000000000.0) {
		tmp = x / y;
	} else if (t_0 <= 0.0001) {
		tmp = fma(x, (x / y), x);
	} else if (t_0 <= 2.0) {
		tmp = x / (x + 1.0);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(x * Float64(1.0 + Float64(x / y))) / Float64(x + 1.0))
	tmp = 0.0
	if (t_0 <= -5000000000000.0)
		tmp = Float64(x / y);
	elseif (t_0 <= 0.0001)
		tmp = fma(x, Float64(x / y), x);
	elseif (t_0 <= 2.0)
		tmp = Float64(x / Float64(x + 1.0));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5000000000000.0], N[(x / y), $MachinePrecision], If[LessEqual[t$95$0, 0.0001], N[(x * N[(x / y), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -5e12 or 2 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 72.6%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 89.3

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

   5. Applied rewrites 89.3%

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

   if -5e12 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 1.00000000000000005e-4

   1. Initial program 99.9%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. distribute-rgt-out-- N/A

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

      6. associate-*l/ N/A

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

      7. *-lft-identity N/A

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

      8. *-lft-identity N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 99.4

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 99.3%

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

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

   1. Initial program 100.0%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 98.3

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

   5. Applied rewrites 98.3%

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

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1} \leq -5000000000000:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;\frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1} \leq 0.0001:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{x}{y}, x\right)\\ \mathbf{elif}\;\frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1} \leq 2:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1}\\ \mathbf{if}\;t\_0 \leq -5000000000000:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t\_0 \leq 10^{-22}:\\ \;\;\;\;x - x \cdot x\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;1 \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ 1.0 (/ x y))) (+ x 1.0))))
   (if (<= t_0 -5000000000000.0)
     (/ x y)
     (if (<= t_0 1e-22) (- x (* x x)) (if (<= t_0 2.0) (* 1.0 1.0) (/ x y))))))

C

double code(double x, double y) {
	double t_0 = (x * (1.0 + (x / y))) / (x + 1.0);
	double tmp;
	if (t_0 <= -5000000000000.0) {
		tmp = x / y;
	} else if (t_0 <= 1e-22) {
		tmp = x - (x * x);
	} else if (t_0 <= 2.0) {
		tmp = 1.0 * 1.0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x * (1.0d0 + (x / y))) / (x + 1.0d0)
    if (t_0 <= (-5000000000000.0d0)) then
        tmp = x / y
    else if (t_0 <= 1d-22) then
        tmp = x - (x * x)
    else if (t_0 <= 2.0d0) then
        tmp = 1.0d0 * 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 * (1.0 + (x / y))) / (x + 1.0);
	double tmp;
	if (t_0 <= -5000000000000.0) {
		tmp = x / y;
	} else if (t_0 <= 1e-22) {
		tmp = x - (x * x);
	} else if (t_0 <= 2.0) {
		tmp = 1.0 * 1.0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x * (1.0 + (x / y))) / (x + 1.0)
	tmp = 0
	if t_0 <= -5000000000000.0:
		tmp = x / y
	elif t_0 <= 1e-22:
		tmp = x - (x * x)
	elif t_0 <= 2.0:
		tmp = 1.0 * 1.0
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x * Float64(1.0 + Float64(x / y))) / Float64(x + 1.0))
	tmp = 0.0
	if (t_0 <= -5000000000000.0)
		tmp = Float64(x / y);
	elseif (t_0 <= 1e-22)
		tmp = Float64(x - Float64(x * x));
	elseif (t_0 <= 2.0)
		tmp = Float64(1.0 * 1.0);
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x * (1.0 + (x / y))) / (x + 1.0);
	tmp = 0.0;
	if (t_0 <= -5000000000000.0)
		tmp = x / y;
	elseif (t_0 <= 1e-22)
		tmp = x - (x * x);
	elseif (t_0 <= 2.0)
		tmp = 1.0 * 1.0;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5000000000000.0], N[(x / y), $MachinePrecision], If[LessEqual[t$95$0, 1e-22], N[(x - N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(1.0 * 1.0), $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))) < -5e12 or 2 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 72.6%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 89.3

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

   5. Applied rewrites 89.3%

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

   if -5e12 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 1e-22

   1. Initial program 99.9%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. distribute-rgt-out-- N/A

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

      6. associate-*l/ N/A

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

      7. *-lft-identity N/A

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

      8. *-lft-identity N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 99.4

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 83.2%

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

   if 1e-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. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 86.6%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 85.9%

       \[\leadsto 1 \cdot \color{blue}{1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1} \leq -5000000000000:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;\frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1} \leq 10^{-22}:\\ \;\;\;\;x - x \cdot x\\ \mathbf{elif}\;\frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1} \leq 2:\\ \;\;\;\;1 \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1}\\ \mathbf{if}\;t\_0 \leq -5000000000000:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ 1.0 (/ x y))) (+ x 1.0))))
   (if (<= t_0 -5000000000000.0)
     (/ x y)
     (if (<= t_0 2.0) (/ x (+ x 1.0)) (/ x y)))))

C

double code(double x, double y) {
	double t_0 = (x * (1.0 + (x / y))) / (x + 1.0);
	double tmp;
	if (t_0 <= -5000000000000.0) {
		tmp = x / y;
	} else if (t_0 <= 2.0) {
		tmp = x / (x + 1.0);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x * (1.0d0 + (x / y))) / (x + 1.0d0)
    if (t_0 <= (-5000000000000.0d0)) 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 * (1.0 + (x / y))) / (x + 1.0);
	double tmp;
	if (t_0 <= -5000000000000.0) {
		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 * (1.0 + (x / y))) / (x + 1.0)
	tmp = 0
	if t_0 <= -5000000000000.0:
		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(1.0 + Float64(x / y))) / Float64(x + 1.0))
	tmp = 0.0
	if (t_0 <= -5000000000000.0)
		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 * (1.0 + (x / y))) / (x + 1.0);
	tmp = 0.0;
	if (t_0 <= -5000000000000.0)
		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[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5000000000000.0], 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))) < -5e12 or 2 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 72.6%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 89.3

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

   5. Applied rewrites 89.3%

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

   if -5e12 < (/.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. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 85.2

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

   5. Applied rewrites 85.2%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1} \leq -5000000000000:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;\frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1} \leq 2:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 54.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+57}:\\ \;\;\;\;x \cdot \left(-x\right)\\ \mathbf{elif}\;t\_0 \leq 10^{-22}:\\ \;\;\;\;x \cdot 1\\ \mathbf{else}:\\ \;\;\;\;1 \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ 1.0 (/ x y))) (+ x 1.0))))
   (if (<= t_0 -1e+57) (* x (- x)) (if (<= t_0 1e-22) (* x 1.0) (* 1.0 1.0)))))

C

double code(double x, double y) {
	double t_0 = (x * (1.0 + (x / y))) / (x + 1.0);
	double tmp;
	if (t_0 <= -1e+57) {
		tmp = x * -x;
	} else if (t_0 <= 1e-22) {
		tmp = x * 1.0;
	} else {
		tmp = 1.0 * 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x * (1.0d0 + (x / y))) / (x + 1.0d0)
    if (t_0 <= (-1d+57)) then
        tmp = x * -x
    else if (t_0 <= 1d-22) then
        tmp = x * 1.0d0
    else
        tmp = 1.0d0 * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x * (1.0 + (x / y))) / (x + 1.0);
	double tmp;
	if (t_0 <= -1e+57) {
		tmp = x * -x;
	} else if (t_0 <= 1e-22) {
		tmp = x * 1.0;
	} else {
		tmp = 1.0 * 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x * (1.0 + (x / y))) / (x + 1.0)
	tmp = 0
	if t_0 <= -1e+57:
		tmp = x * -x
	elif t_0 <= 1e-22:
		tmp = x * 1.0
	else:
		tmp = 1.0 * 1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x * Float64(1.0 + Float64(x / y))) / Float64(x + 1.0))
	tmp = 0.0
	if (t_0 <= -1e+57)
		tmp = Float64(x * Float64(-x));
	elseif (t_0 <= 1e-22)
		tmp = Float64(x * 1.0);
	else
		tmp = Float64(1.0 * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x * (1.0 + (x / y))) / (x + 1.0);
	tmp = 0.0;
	if (t_0 <= -1e+57)
		tmp = x * -x;
	elseif (t_0 <= 1e-22)
		tmp = x * 1.0;
	else
		tmp = 1.0 * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+57], N[(x * (-x)), $MachinePrecision], If[LessEqual[t$95$0, 1e-22], N[(x * 1.0), $MachinePrecision], N[(1.0 * 1.0), $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))) < -1.00000000000000005e57

   1. Initial program 71.7%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. distribute-rgt-out-- N/A

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

      6. associate-*l/ N/A

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

      7. *-lft-identity N/A

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

      8. *-lft-identity N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 28.3

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

   5. Applied rewrites 28.3%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 33.2%

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

   9. Taylor expanded in x around inf

      \[\leadsto -1 \cdot {x}^{\color{blue}{2}} \]
   10. Step-by-step derivation

   11. Applied rewrites 33.3%

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

   if -1.00000000000000005e57 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 1e-22

   1. Initial program 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 82.3%

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

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

   1. Initial program 85.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 87.1%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 43.9%

       \[\leadsto 1 \cdot \color{blue}{1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 57.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1} \leq -1 \cdot 10^{+57}:\\ \;\;\;\;x \cdot \left(-x\right)\\ \mathbf{elif}\;\frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1} \leq 10^{-22}:\\ \;\;\;\;x \cdot 1\\ \mathbf{else}:\\ \;\;\;\;1 \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 55.0% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* x (+ 1.0 (/ x y))) (+ x 1.0)) 1e-22) (- x (* x x)) (* 1.0 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (((x * (1.0 + (x / y))) / (x + 1.0)) <= 1e-22) {
		tmp = x - (x * x);
	} else {
		tmp = 1.0 * 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (((x * (1.0d0 + (x / y))) / (x + 1.0d0)) <= 1d-22) then
        tmp = x - (x * x)
    else
        tmp = 1.0d0 * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((x * (1.0 + (x / y))) / (x + 1.0)) <= 1e-22) {
		tmp = x - (x * x);
	} else {
		tmp = 1.0 * 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x * (1.0 + (x / y))) / (x + 1.0)) <= 1e-22:
		tmp = x - (x * x)
	else:
		tmp = 1.0 * 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x * Float64(1.0 + Float64(x / y))) / Float64(x + 1.0)) <= 1e-22)
		tmp = Float64(x - Float64(x * x));
	else
		tmp = Float64(1.0 * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((x * (1.0 + (x / y))) / (x + 1.0)) <= 1e-22)
		tmp = x - (x * x);
	else
		tmp = 1.0 * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x * N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], 1e-22], N[(x - N[(x * x), $MachinePrecision]), $MachinePrecision], N[(1.0 * 1.0), $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))) < 1e-22

   1. Initial program 90.5%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. distribute-rgt-out-- N/A

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

      6. associate-*l/ N/A

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

      7. *-lft-identity N/A

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

      8. *-lft-identity N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 75.1

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

   5. Applied rewrites 75.1%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 66.0%

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

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

   1. Initial program 85.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 87.1%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 43.9%

       \[\leadsto 1 \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.4%

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

Alternative 7: 50.1% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* x (+ 1.0 (/ x y))) (+ x 1.0)) 1e-22) (* x 1.0) (* 1.0 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (((x * (1.0 + (x / y))) / (x + 1.0)) <= 1e-22) {
		tmp = x * 1.0;
	} else {
		tmp = 1.0 * 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (((x * (1.0d0 + (x / y))) / (x + 1.0d0)) <= 1d-22) then
        tmp = x * 1.0d0
    else
        tmp = 1.0d0 * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((x * (1.0 + (x / y))) / (x + 1.0)) <= 1e-22) {
		tmp = x * 1.0;
	} else {
		tmp = 1.0 * 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x * (1.0 + (x / y))) / (x + 1.0)) <= 1e-22:
		tmp = x * 1.0
	else:
		tmp = 1.0 * 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x * Float64(1.0 + Float64(x / y))) / Float64(x + 1.0)) <= 1e-22)
		tmp = Float64(x * 1.0);
	else
		tmp = Float64(1.0 * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((x * (1.0 + (x / y))) / (x + 1.0)) <= 1e-22)
		tmp = x * 1.0;
	else
		tmp = 1.0 * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x * N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], 1e-22], N[(x * 1.0), $MachinePrecision], N[(1.0 * 1.0), $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))) < 1e-22

   1. Initial program 90.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 55.8%

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

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

   1. Initial program 85.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 87.1%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 43.9%

       \[\leadsto 1 \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.2%

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

Alternative 8: 98.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 79.2%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. sub-neg N/A

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

      5. distribute-lft-in N/A

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

      6. distribute-rgt-neg-out N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. rgt-mult-inverse N/A

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

      10. neg-mul-1 N/A

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

      11. distribute-rgt-out N/A

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

      12. rgt-mult-inverse N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-+.f64 98.8

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

   5. Applied rewrites 98.8%

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

   if -1 < x < 1

   1. Initial program 99.9%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. distribute-rgt-out-- N/A

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

      6. associate-*l/ N/A

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

      7. *-lft-identity N/A

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

      8. *-lft-identity N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 97.5

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

   5. Applied rewrites 97.5%

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

   7. Applied rewrites 97.5%

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

Alternative 9: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 79.2%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. sub-neg N/A

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

      5. distribute-lft-in N/A

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

      6. distribute-rgt-neg-out N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. rgt-mult-inverse N/A

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

      10. neg-mul-1 N/A

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

      11. distribute-rgt-out N/A

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

      12. rgt-mult-inverse N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-+.f64 98.8

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

   5. Applied rewrites 98.8%

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

   if -1 < x < 1

   1. Initial program 99.9%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. distribute-rgt-out-- N/A

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

      6. associate-*l/ N/A

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

      7. *-lft-identity N/A

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

      8. *-lft-identity N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 97.5

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

   5. Applied rewrites 97.5%

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

Alternative 10: 98.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{x}{y}, 1, 1\right)\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 0.8:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{x}{y} - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (/ x y) 1.0 1.0)))
   (if (<= x -1.0) t_0 (if (<= x 0.8) (fma x (- (/ x y) x) x) t_0))))

C

double code(double x, double y) {
	double t_0 = fma((x / y), 1.0, 1.0);
	double tmp;
	if (x <= -1.0) {
		tmp = t_0;
	} else if (x <= 0.8) {
		tmp = fma(x, ((x / y) - x), x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(Float64(x / y), 1.0, 1.0)
	tmp = 0.0
	if (x <= -1.0)
		tmp = t_0;
	elseif (x <= 0.8)
		tmp = fma(x, Float64(Float64(x / y) - x), x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x / y), $MachinePrecision] * 1.0 + 1.0), $MachinePrecision]}, If[LessEqual[x, -1.0], t$95$0, If[LessEqual[x, 0.8], N[(x * N[(N[(x / y), $MachinePrecision] - x), $MachinePrecision] + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 0.80000000000000004 < x

   1. Initial program 79.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 98.6%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

      5. lower-fma.f64 98.6

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

   9. Applied rewrites 98.6%

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

   if -1 < x < 0.80000000000000004

   1. Initial program 99.9%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. distribute-rgt-out-- N/A

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

      6. associate-*l/ N/A

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

      7. *-lft-identity N/A

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

      8. *-lft-identity N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 97.5

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

   5. Applied rewrites 97.5%

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

Alternative 11: 97.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 79.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 98.6%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

      5. lower-fma.f64 98.6

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

   9. Applied rewrites 98.6%

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

   if -1 < x < 1

   1. Initial program 99.9%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. distribute-rgt-out-- N/A

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

      6. associate-*l/ N/A

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

      7. *-lft-identity N/A

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

      8. *-lft-identity N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 97.5

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

   5. Applied rewrites 97.5%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 97.3%

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

Alternative 12: 14.7% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.7%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 99.9

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

4. Applied rewrites 99.9%

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

5. Taylor expanded in x around inf

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

7. Applied rewrites 54.8%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 18.8%

    \[\leadsto 1 \cdot \color{blue}{1} \]
11. Add Preprocessing

Developer Target 1: 99.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    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 2024220 
(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)))