Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1

Percentage Accurate: 88.1% → 99.9%

Time: 9.0s

Alternatives: 14

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: 88.1% 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, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{x}{x + 1} \cdot \left(x + y\right)}{y}\\ \mathbf{if}\;x \leq -4.5 \cdot 10^{-17}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-22}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (/ x (+ x 1.0)) (+ x y)) y)))
   (if (<= x -4.5e-17) t_0 (if (<= x 5e-22) (fma (/ x y) x x) t_0))))

C

double code(double x, double y) {
	double t_0 = ((x / (x + 1.0)) * (x + y)) / y;
	double tmp;
	if (x <= -4.5e-17) {
		tmp = t_0;
	} else if (x <= 5e-22) {
		tmp = fma((x / y), x, x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(x / Float64(x + 1.0)) * Float64(x + y)) / y)
	tmp = 0.0
	if (x <= -4.5e-17)
		tmp = t_0;
	elseif (x <= 5e-22)
		tmp = fma(Float64(x / y), x, x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[x, -4.5e-17], t$95$0, If[LessEqual[x, 5e-22], N[(N[(x / y), $MachinePrecision] * x + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.49999999999999978e-17 or 4.99999999999999954e-22 < x

   1. Initial program 84.2%

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

   3. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. unpow2 N/A

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

      5. associate-/l* N/A

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

      6. distribute-rgt-out N/A

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

      7. +-commutative N/A

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

      8. *-lowering-*.f64 N/A

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

      9. /-lowering-/.f64 N/A

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

      10. +-commutative N/A

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

      11. +-lowering-+.f64 N/A

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

      12. +-lowering-+.f64 100.0

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

   5. Simplified 100.0%

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

   if -4.49999999999999978e-17 < x < 4.99999999999999954e-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

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

   5. Simplified 99.9%

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

      1. /-rgt-identity N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. /-lowering-/.f64 99.9

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

   7. Applied egg-rr 99.9%

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

Alternative 2: 85.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1}\\ \mathbf{if}\;t\_0 \leq -400000:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t\_0 \leq 10^{-7}:\\ \;\;\;\;x - x \cdot x\\ \mathbf{elif}\;t\_0 \leq 20:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ 1.0 (/ x y))) (+ x 1.0))))
   (if (<= t_0 -400000.0)
     (/ x y)
     (if (<= t_0 1e-7)
       (- x (* x x))
       (if (<= t_0 20.0) (+ 1.0 (/ -1.0 x)) (/ 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 <= -400000.0) {
		tmp = x / y;
	} else if (t_0 <= 1e-7) {
		tmp = x - (x * x);
	} else if (t_0 <= 20.0) {
		tmp = 1.0 + (-1.0 / x);
	} 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 <= (-400000.0d0)) then
        tmp = x / y
    else if (t_0 <= 1d-7) then
        tmp = x - (x * x)
    else if (t_0 <= 20.0d0) 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 * (1.0 + (x / y))) / (x + 1.0);
	double tmp;
	if (t_0 <= -400000.0) {
		tmp = x / y;
	} else if (t_0 <= 1e-7) {
		tmp = x - (x * x);
	} else if (t_0 <= 20.0) {
		tmp = 1.0 + (-1.0 / x);
	} 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 <= -400000.0:
		tmp = x / y
	elif t_0 <= 1e-7:
		tmp = x - (x * x)
	elif t_0 <= 20.0:
		tmp = 1.0 + (-1.0 / x)
	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 <= -400000.0)
		tmp = Float64(x / y);
	elseif (t_0 <= 1e-7)
		tmp = Float64(x - Float64(x * x));
	elseif (t_0 <= 20.0)
		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 * (1.0 + (x / y))) / (x + 1.0);
	tmp = 0.0;
	if (t_0 <= -400000.0)
		tmp = x / y;
	elseif (t_0 <= 1e-7)
		tmp = x - (x * x);
	elseif (t_0 <= 20.0)
		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[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -400000.0], N[(x / y), $MachinePrecision], If[LessEqual[t$95$0, 1e-7], N[(x - N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 20.0], 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))) < -4e5 or 20 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 82.6%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 81.9

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

   5. Simplified 81.9%

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

   if -4e5 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 9.9999999999999995e-8

   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

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

   5. Simplified 83.7%

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

   6. Taylor expanded in x around 0

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. mul-1-neg N/A

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

      4. distribute-lft-neg-out N/A

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

      5. unpow2 N/A

         \[\leadsto x + \left(\mathsf{neg}\left(\color{blue}{{x}^{2}}\right)\right) \]

      6. unsub-neg N/A

         \[\leadsto \color{blue}{x - {x}^{2}} \]

      7. --lowering--.f64 N/A

         \[\leadsto \color{blue}{x - {x}^{2}} \]

      8. unpow2 N/A

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

      9. *-lowering-*.f64 82.8

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

   8. Simplified 82.8%

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

   if 9.9999999999999995e-8 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 20

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 94.8%

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

   6. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

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

      3. distribute-neg-frac N/A

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

      4. metadata-eval N/A

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

      5. /-lowering-/.f64 95.0

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

   8. Simplified 95.0%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1} \leq -400000:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;\frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1} \leq 10^{-7}:\\ \;\;\;\;x - x \cdot x\\ \mathbf{elif}\;\frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1} \leq 20:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1}\\ \mathbf{if}\;t\_0 \leq -400000:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t\_0 \leq 10^{-7}:\\ \;\;\;\;x - x \cdot x\\ \mathbf{elif}\;t\_0 \leq 20:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ 1.0 (/ x y))) (+ x 1.0))))
   (if (<= t_0 -400000.0)
     (/ x y)
     (if (<= t_0 1e-7) (- x (* x x)) (if (<= t_0 20.0) 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 <= -400000.0) {
		tmp = x / y;
	} else if (t_0 <= 1e-7) {
		tmp = x - (x * x);
	} else if (t_0 <= 20.0) {
		tmp = 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 <= (-400000.0d0)) then
        tmp = x / y
    else if (t_0 <= 1d-7) then
        tmp = x - (x * x)
    else if (t_0 <= 20.0d0) then
        tmp = 1.0d0
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x * (1.0 + (x / y))) / (x + 1.0);
	double tmp;
	if (t_0 <= -400000.0) {
		tmp = x / y;
	} else if (t_0 <= 1e-7) {
		tmp = x - (x * x);
	} else if (t_0 <= 20.0) {
		tmp = 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 <= -400000.0:
		tmp = x / y
	elif t_0 <= 1e-7:
		tmp = x - (x * x)
	elif t_0 <= 20.0:
		tmp = 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 <= -400000.0)
		tmp = Float64(x / y);
	elseif (t_0 <= 1e-7)
		tmp = Float64(x - Float64(x * x));
	elseif (t_0 <= 20.0)
		tmp = 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 <= -400000.0)
		tmp = x / y;
	elseif (t_0 <= 1e-7)
		tmp = x - (x * x);
	elseif (t_0 <= 20.0)
		tmp = 1.0;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -400000.0], N[(x / y), $MachinePrecision], If[LessEqual[t$95$0, 1e-7], N[(x - N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 20.0], 1.0, N[(x / y), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -4e5 or 20 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 82.6%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 81.9

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

   5. Simplified 81.9%

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

   if -4e5 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 9.9999999999999995e-8

   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

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

   5. Simplified 83.7%

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

   6. Taylor expanded in x around 0

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. mul-1-neg N/A

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

      4. distribute-lft-neg-out N/A

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

      5. unpow2 N/A

         \[\leadsto x + \left(\mathsf{neg}\left(\color{blue}{{x}^{2}}\right)\right) \]

      6. unsub-neg N/A

         \[\leadsto \color{blue}{x - {x}^{2}} \]

      7. --lowering--.f64 N/A

         \[\leadsto \color{blue}{x - {x}^{2}} \]

      8. unpow2 N/A

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

      9. *-lowering-*.f64 82.8

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

   8. Simplified 82.8%

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

   if 9.9999999999999995e-8 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 20

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 94.8%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 94.9%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1} \leq -400000:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;\frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1} \leq 10^{-7}:\\ \;\;\;\;x - x \cdot x\\ \mathbf{elif}\;\frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1} \leq 20:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1}\\ \mathbf{if}\;t\_0 \leq -10000000000000:\\ \;\;\;\;\frac{x + \left(y + -1\right)}{y}\\ \mathbf{elif}\;t\_0 \leq 0.99999999999998:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ 1.0 (/ x y))) (+ x 1.0))))
   (if (<= t_0 -10000000000000.0)
     (/ (+ x (+ y -1.0)) y)
     (if (<= t_0 0.99999999999998) (/ x (+ x 1.0)) (/ (+ x y) y)))))

C

double code(double x, double y) {
	double t_0 = (x * (1.0 + (x / y))) / (x + 1.0);
	double tmp;
	if (t_0 <= -10000000000000.0) {
		tmp = (x + (y + -1.0)) / y;
	} else if (t_0 <= 0.99999999999998) {
		tmp = x / (x + 1.0);
	} else {
		tmp = (x + y) / 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 <= (-10000000000000.0d0)) then
        tmp = (x + (y + (-1.0d0))) / y
    else if (t_0 <= 0.99999999999998d0) then
        tmp = x / (x + 1.0d0)
    else
        tmp = (x + y) / 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 <= -10000000000000.0) {
		tmp = (x + (y + -1.0)) / y;
	} else if (t_0 <= 0.99999999999998) {
		tmp = x / (x + 1.0);
	} else {
		tmp = (x + y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x * (1.0 + (x / y))) / (x + 1.0)
	tmp = 0
	if t_0 <= -10000000000000.0:
		tmp = (x + (y + -1.0)) / y
	elif t_0 <= 0.99999999999998:
		tmp = x / (x + 1.0)
	else:
		tmp = (x + y) / 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 <= -10000000000000.0)
		tmp = Float64(Float64(x + Float64(y + -1.0)) / y);
	elseif (t_0 <= 0.99999999999998)
		tmp = Float64(x / Float64(x + 1.0));
	else
		tmp = Float64(Float64(x + y) / 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 <= -10000000000000.0)
		tmp = (x + (y + -1.0)) / y;
	elseif (t_0 <= 0.99999999999998)
		tmp = x / (x + 1.0);
	else
		tmp = (x + y) / 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, -10000000000000.0], N[(N[(x + N[(y + -1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[t$95$0, 0.99999999999998], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x + y), $MachinePrecision] / 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))) < -1e13

   1. Initial program 83.3%

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

   3. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. unpow2 N/A

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

      5. associate-/l* N/A

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

      6. distribute-rgt-out N/A

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

      7. +-commutative N/A

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

      8. *-lowering-*.f64 N/A

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

      9. /-lowering-/.f64 N/A

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

      10. +-commutative N/A

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

      11. +-lowering-+.f64 N/A

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

      12. +-lowering-+.f64 99.9

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

   5. Simplified 99.9%

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

   6. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. sub-neg N/A

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

      5. remove-double-neg N/A

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

      6. mul-1-neg N/A

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

      7. associate-/l* N/A

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

      8. mul-1-neg N/A

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

      9. associate-*r/ N/A

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

      10. neg-mul-1 N/A

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

      11. distribute-lft-in N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. associate-*l* N/A

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

      16. distribute-rgt-in N/A

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

   8. Simplified 85.1%

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

   if -1e13 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 0.99999999999998002

   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

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

   5. Simplified 83.1%

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

   if 0.99999999999998002 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 87.0%

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

   3. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. unpow2 N/A

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

      5. associate-/l* N/A

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

      6. distribute-rgt-out N/A

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

      7. +-commutative N/A

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

      8. *-lowering-*.f64 N/A

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

      9. /-lowering-/.f64 N/A

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

      10. +-commutative N/A

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

      11. +-lowering-+.f64 N/A

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

      12. +-lowering-+.f64 99.9

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

   5. Simplified 99.9%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. frac-times N/A

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

      4. *-lft-identity N/A

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

      5. /-lowering-/.f64 N/A

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

      6. +-lowering-+.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. +-lowering-+.f64 99.9

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

   7. Applied egg-rr 99.9%

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

   8. Taylor expanded in x around inf

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

   10. Simplified 86.8%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1} \leq -10000000000000:\\ \;\;\;\;\frac{x + \left(y + -1\right)}{y}\\ \mathbf{elif}\;\frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1} \leq 0.99999999999998:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1}\\ \mathbf{if}\;t\_0 \leq -10000000000000:\\ \;\;\;\;\frac{x + -1}{y}\\ \mathbf{elif}\;t\_0 \leq 0.99999999999998:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ 1.0 (/ x y))) (+ x 1.0))))
   (if (<= t_0 -10000000000000.0)
     (/ (+ x -1.0) y)
     (if (<= t_0 0.99999999999998) (/ x (+ x 1.0)) (/ (+ x y) y)))))

C

double code(double x, double y) {
	double t_0 = (x * (1.0 + (x / y))) / (x + 1.0);
	double tmp;
	if (t_0 <= -10000000000000.0) {
		tmp = (x + -1.0) / y;
	} else if (t_0 <= 0.99999999999998) {
		tmp = x / (x + 1.0);
	} else {
		tmp = (x + y) / 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 <= (-10000000000000.0d0)) then
        tmp = (x + (-1.0d0)) / y
    else if (t_0 <= 0.99999999999998d0) then
        tmp = x / (x + 1.0d0)
    else
        tmp = (x + y) / 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 <= -10000000000000.0) {
		tmp = (x + -1.0) / y;
	} else if (t_0 <= 0.99999999999998) {
		tmp = x / (x + 1.0);
	} else {
		tmp = (x + y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x * (1.0 + (x / y))) / (x + 1.0)
	tmp = 0
	if t_0 <= -10000000000000.0:
		tmp = (x + -1.0) / y
	elif t_0 <= 0.99999999999998:
		tmp = x / (x + 1.0)
	else:
		tmp = (x + y) / 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 <= -10000000000000.0)
		tmp = Float64(Float64(x + -1.0) / y);
	elseif (t_0 <= 0.99999999999998)
		tmp = Float64(x / Float64(x + 1.0));
	else
		tmp = Float64(Float64(x + y) / 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 <= -10000000000000.0)
		tmp = (x + -1.0) / y;
	elseif (t_0 <= 0.99999999999998)
		tmp = x / (x + 1.0);
	else
		tmp = (x + y) / 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, -10000000000000.0], N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[t$95$0, 0.99999999999998], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x + y), $MachinePrecision] / 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))) < -1e13

   1. Initial program 83.3%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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. accelerator-lowering-fma.f64 N/A

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

      14. /-lowering-/.f64 N/A

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

      15. +-lowering-+.f64 85.0

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

   5. Simplified 85.0%

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

   6. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. +-lowering-+.f64 84.9

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

   8. Simplified 84.9%

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

   if -1e13 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 0.99999999999998002

   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

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

   5. Simplified 83.1%

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

   if 0.99999999999998002 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 87.0%

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

   3. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. unpow2 N/A

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

      5. associate-/l* N/A

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

      6. distribute-rgt-out N/A

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

      7. +-commutative N/A

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

      8. *-lowering-*.f64 N/A

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

      9. /-lowering-/.f64 N/A

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

      10. +-commutative N/A

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

      11. +-lowering-+.f64 N/A

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

      12. +-lowering-+.f64 99.9

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

   5. Simplified 99.9%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. frac-times N/A

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

      4. *-lft-identity N/A

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

      5. /-lowering-/.f64 N/A

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

      6. +-lowering-+.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. +-lowering-+.f64 99.9

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

   7. Applied egg-rr 99.9%

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

   8. Taylor expanded in x around inf

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

   10. Simplified 86.8%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1} \leq -10000000000000:\\ \;\;\;\;\frac{x + -1}{y}\\ \mathbf{elif}\;\frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1} \leq 0.99999999999998:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1}\\ \mathbf{if}\;t\_0 \leq -10000000000000:\\ \;\;\;\;\frac{x + -1}{y}\\ \mathbf{elif}\;t\_0 \leq 20:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ 1.0 (/ x y))) (+ x 1.0))))
   (if (<= t_0 -10000000000000.0)
     (/ (+ x -1.0) y)
     (if (<= t_0 20.0) (/ 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 <= -10000000000000.0) {
		tmp = (x + -1.0) / y;
	} else if (t_0 <= 20.0) {
		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 <= (-10000000000000.0d0)) then
        tmp = (x + (-1.0d0)) / y
    else if (t_0 <= 20.0d0) 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 <= -10000000000000.0) {
		tmp = (x + -1.0) / y;
	} else if (t_0 <= 20.0) {
		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 <= -10000000000000.0:
		tmp = (x + -1.0) / y
	elif t_0 <= 20.0:
		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 <= -10000000000000.0)
		tmp = Float64(Float64(x + -1.0) / y);
	elseif (t_0 <= 20.0)
		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 <= -10000000000000.0)
		tmp = (x + -1.0) / y;
	elseif (t_0 <= 20.0)
		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, -10000000000000.0], N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[t$95$0, 20.0], 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))) < -1e13

   1. Initial program 83.3%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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. accelerator-lowering-fma.f64 N/A

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

      14. /-lowering-/.f64 N/A

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

      15. +-lowering-+.f64 85.0

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

   5. Simplified 85.0%

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

   6. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. +-lowering-+.f64 84.9

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

   8. Simplified 84.9%

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

   if -1e13 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 20

   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

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

   5. Simplified 85.1%

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

   if 20 < (/.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. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 81.2

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

   5. Simplified 81.2%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1} \leq -10000000000000:\\ \;\;\;\;\frac{x + -1}{y}\\ \mathbf{elif}\;\frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1} \leq 20:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 86.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1}\\ \mathbf{if}\;t\_0 \leq -400000:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t\_0 \leq 20:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ 1.0 (/ x y))) (+ x 1.0))))
   (if (<= t_0 -400000.0) (/ x y) (if (<= t_0 20.0) (/ 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 <= -400000.0) {
		tmp = x / y;
	} else if (t_0 <= 20.0) {
		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 <= (-400000.0d0)) then
        tmp = x / y
    else if (t_0 <= 20.0d0) 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 <= -400000.0) {
		tmp = x / y;
	} else if (t_0 <= 20.0) {
		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 <= -400000.0:
		tmp = x / y
	elif t_0 <= 20.0:
		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 <= -400000.0)
		tmp = Float64(x / y);
	elseif (t_0 <= 20.0)
		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 <= -400000.0)
		tmp = x / y;
	elseif (t_0 <= 20.0)
		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, -400000.0], N[(x / y), $MachinePrecision], If[LessEqual[t$95$0, 20.0], 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))) < -4e5 or 20 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 82.6%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 81.9

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

   5. Simplified 81.9%

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

   if -4e5 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 20

   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

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

   5. Simplified 85.8%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1} \leq -400000:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;\frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1} \leq 20:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 55.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+25}:\\ \;\;\;\;x \cdot \left(-x\right)\\ \mathbf{elif}\;t\_0 \leq 10^{-7}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ 1.0 (/ x y))) (+ x 1.0))))
   (if (<= t_0 -5e+25) (* x (- x)) (if (<= t_0 1e-7) x 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 <= -5e+25) {
		tmp = x * -x;
	} else if (t_0 <= 1e-7) {
		tmp = x;
	} else {
		tmp = 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 <= (-5d+25)) then
        tmp = x * -x
    else if (t_0 <= 1d-7) then
        tmp = x
    else
        tmp = 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 <= -5e+25) {
		tmp = x * -x;
	} else if (t_0 <= 1e-7) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x * (1.0 + (x / y))) / (x + 1.0)
	tmp = 0
	if t_0 <= -5e+25:
		tmp = x * -x
	elif t_0 <= 1e-7:
		tmp = x
	else:
		tmp = 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 <= -5e+25)
		tmp = Float64(x * Float64(-x));
	elseif (t_0 <= 1e-7)
		tmp = x;
	else
		tmp = 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 <= -5e+25)
		tmp = x * -x;
	elseif (t_0 <= 1e-7)
		tmp = x;
	else
		tmp = 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, -5e+25], N[(x * (-x)), $MachinePrecision], If[LessEqual[t$95$0, 1e-7], x, 1.0]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -5.00000000000000024e25

   1. Initial program 83.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 1.2%

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

   6. Taylor expanded in x around 0

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. mul-1-neg N/A

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

      4. distribute-lft-neg-out N/A

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

      5. unpow2 N/A

         \[\leadsto x + \left(\mathsf{neg}\left(\color{blue}{{x}^{2}}\right)\right) \]

      6. unsub-neg N/A

         \[\leadsto \color{blue}{x - {x}^{2}} \]

      7. --lowering--.f64 N/A

         \[\leadsto \color{blue}{x - {x}^{2}} \]

      8. unpow2 N/A

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

      9. *-lowering-*.f64 27.0

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

   8. Simplified 27.0%

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

   9. Taylor expanded in x around inf

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

       1. unpow2 N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. *-lowering-*.f64 N/A

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

       5. mul-1-neg N/A

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

       6. neg-lowering-neg.f64 27.3

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

   11. Simplified 27.3%

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

   if -5.00000000000000024e25 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 9.9999999999999995e-8

   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

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

   5. Simplified 80.8%

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

   if 9.9999999999999995e-8 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 87.2%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 32.0%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 32.4%

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

4. Final simplification 52.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1} \leq -5 \cdot 10^{+25}:\\ \;\;\;\;x \cdot \left(-x\right)\\ \mathbf{elif}\;\frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1} \leq 10^{-7}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 55.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1} \leq 10^{-7}:\\ \;\;\;\;x - x \cdot x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* x (+ 1.0 (/ x y))) (+ x 1.0)) 1e-7) (- x (* x x)) 1.0))

C

double code(double x, double y) {
	double tmp;
	if (((x * (1.0 + (x / y))) / (x + 1.0)) <= 1e-7) {
		tmp = x - (x * x);
	} else {
		tmp = 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-7) then
        tmp = x - (x * x)
    else
        tmp = 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-7) {
		tmp = x - (x * x);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x * (1.0 + (x / y))) / (x + 1.0)) <= 1e-7:
		tmp = x - (x * x)
	else:
		tmp = 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-7)
		tmp = Float64(x - Float64(x * x));
	else
		tmp = 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-7)
		tmp = x - (x * x);
	else
		tmp = 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-7], N[(x - N[(x * x), $MachinePrecision]), $MachinePrecision], 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))) < 9.9999999999999995e-8

   1. Initial program 94.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 53.8%

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

   6. Taylor expanded in x around 0

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. mul-1-neg N/A

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

      4. distribute-lft-neg-out N/A

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

      5. unpow2 N/A

         \[\leadsto x + \left(\mathsf{neg}\left(\color{blue}{{x}^{2}}\right)\right) \]

      6. unsub-neg N/A

         \[\leadsto \color{blue}{x - {x}^{2}} \]

      7. --lowering--.f64 N/A

         \[\leadsto \color{blue}{x - {x}^{2}} \]

      8. unpow2 N/A

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

      9. *-lowering-*.f64 62.4

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

   8. Simplified 62.4%

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

   if 9.9999999999999995e-8 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 87.2%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 32.0%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 32.4%

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

4. Final simplification 52.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1} \leq 10^{-7}:\\ \;\;\;\;x - x \cdot x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 99.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + \left(y + -1\right)}{y}\\ \mathbf{if}\;x \leq -6 \cdot 10^{+14}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.12 \cdot 10^{+15}:\\ \;\;\;\;\frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (+ x (+ y -1.0)) y)))
   (if (<= x -6e+14)
     t_0
     (if (<= x 1.12e+15) (/ (* x (+ 1.0 (/ x y))) (+ x 1.0)) t_0))))

C

double code(double x, double y) {
	double t_0 = (x + (y + -1.0)) / y;
	double tmp;
	if (x <= -6e+14) {
		tmp = t_0;
	} else if (x <= 1.12e+15) {
		tmp = (x * (1.0 + (x / y))) / (x + 1.0);
	} else {
		tmp = t_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 + (y + (-1.0d0))) / y
    if (x <= (-6d+14)) then
        tmp = t_0
    else if (x <= 1.12d+15) then
        tmp = (x * (1.0d0 + (x / y))) / (x + 1.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x + (y + -1.0)) / y;
	double tmp;
	if (x <= -6e+14) {
		tmp = t_0;
	} else if (x <= 1.12e+15) {
		tmp = (x * (1.0 + (x / y))) / (x + 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x + (y + -1.0)) / y
	tmp = 0
	if x <= -6e+14:
		tmp = t_0
	elif x <= 1.12e+15:
		tmp = (x * (1.0 + (x / y))) / (x + 1.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x + Float64(y + -1.0)) / y)
	tmp = 0.0
	if (x <= -6e+14)
		tmp = t_0;
	elseif (x <= 1.12e+15)
		tmp = Float64(Float64(x * Float64(1.0 + Float64(x / y))) / Float64(x + 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x + (y + -1.0)) / y;
	tmp = 0.0;
	if (x <= -6e+14)
		tmp = t_0;
	elseif (x <= 1.12e+15)
		tmp = (x * (1.0 + (x / y))) / (x + 1.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x + N[(y + -1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[x, -6e+14], t$95$0, If[LessEqual[x, 1.12e+15], N[(N[(x * N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -6e14 or 1.12e15 < x

   1. Initial program 82.5%

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

   3. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. unpow2 N/A

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

      5. associate-/l* N/A

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

      6. distribute-rgt-out N/A

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

      7. +-commutative N/A

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

      8. *-lowering-*.f64 N/A

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

      9. /-lowering-/.f64 N/A

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

      10. +-commutative N/A

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

      11. +-lowering-+.f64 N/A

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

      12. +-lowering-+.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. sub-neg N/A

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

      5. remove-double-neg N/A

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

      6. mul-1-neg N/A

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

      7. associate-/l* N/A

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

      8. mul-1-neg N/A

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

      9. associate-*r/ N/A

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

      10. neg-mul-1 N/A

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

      11. distribute-lft-in N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. associate-*l* N/A

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

      16. distribute-rgt-in N/A

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

   8. Simplified 100.0%

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

   if -6e14 < x < 1.12e15

   1. Initial program 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{+14}:\\ \;\;\;\;\frac{x + \left(y + -1\right)}{y}\\ \mathbf{elif}\;x \leq 1.12 \cdot 10^{+15}:\\ \;\;\;\;\frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \left(y + -1\right)}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 50.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1} \leq 10^{-7}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* x (+ 1.0 (/ x y))) (+ x 1.0)) 1e-7) x 1.0))

C

double code(double x, double y) {
	double tmp;
	if (((x * (1.0 + (x / y))) / (x + 1.0)) <= 1e-7) {
		tmp = x;
	} else {
		tmp = 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-7) 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 * (1.0 + (x / y))) / (x + 1.0)) <= 1e-7) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x * (1.0 + (x / y))) / (x + 1.0)) <= 1e-7:
		tmp = x
	else:
		tmp = 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-7)
		tmp = x;
	else
		tmp = 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-7)
		tmp = x;
	else
		tmp = 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-7], 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))) < 9.9999999999999995e-8

   1. Initial program 94.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 53.7%

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

   if 9.9999999999999995e-8 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 87.2%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 32.0%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 32.4%

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

4. Final simplification 46.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1} \leq 10^{-7}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 98.4% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (+ x (+ y -1.0)) y)))
   (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 = (x + (y + -1.0)) / y;
	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 = Float64(Float64(x + Float64(y + -1.0)) / y)
	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[(x + N[(y + -1.0), $MachinePrecision]), $MachinePrecision] / y), $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 83.2%

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

   3. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. unpow2 N/A

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

      5. associate-/l* N/A

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

      6. distribute-rgt-out N/A

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

      7. +-commutative N/A

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

      8. *-lowering-*.f64 N/A

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

      9. /-lowering-/.f64 N/A

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

      10. +-commutative N/A

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

      11. +-lowering-+.f64 N/A

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

      12. +-lowering-+.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. sub-neg N/A

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

      5. remove-double-neg N/A

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

      6. mul-1-neg N/A

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

      7. associate-/l* N/A

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

      8. mul-1-neg N/A

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

      9. associate-*r/ N/A

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

      10. neg-mul-1 N/A

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

      11. distribute-lft-in N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. associate-*l* N/A

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

      16. distribute-rgt-in N/A

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

   8. Simplified 99.7%

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

   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. accelerator-lowering-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. --lowering--.f64 N/A

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

      10. /-lowering-/.f64 98.4

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

   5. Simplified 98.4%

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

Alternative 13: 98.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + \left(y + -1\right)}{y}\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (+ x (+ y -1.0)) y)))
   (if (<= x -1.0) t_0 (if (<= x 1.25) (fma (/ x y) x x) t_0))))

C

double code(double x, double y) {
	double t_0 = (x + (y + -1.0)) / y;
	double tmp;
	if (x <= -1.0) {
		tmp = t_0;
	} else if (x <= 1.25) {
		tmp = fma((x / y), x, x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(x + Float64(y + -1.0)) / y)
	tmp = 0.0
	if (x <= -1.0)
		tmp = t_0;
	elseif (x <= 1.25)
		tmp = fma(Float64(x / y), x, x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x + N[(y + -1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[x, -1.0], t$95$0, If[LessEqual[x, 1.25], N[(N[(x / y), $MachinePrecision] * x + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1.25 < x

   1. Initial program 83.2%

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

   3. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. unpow2 N/A

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

      5. associate-/l* N/A

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

      6. distribute-rgt-out N/A

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

      7. +-commutative N/A

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

      8. *-lowering-*.f64 N/A

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

      9. /-lowering-/.f64 N/A

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

      10. +-commutative N/A

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

      11. +-lowering-+.f64 N/A

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

      12. +-lowering-+.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. sub-neg N/A

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

      5. remove-double-neg N/A

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

      6. mul-1-neg N/A

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

      7. associate-/l* N/A

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

      8. mul-1-neg N/A

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

      9. associate-*r/ N/A

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

      10. neg-mul-1 N/A

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

      11. distribute-lft-in N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. associate-*l* N/A

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

      16. distribute-rgt-in N/A

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

   8. Simplified 99.7%

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

   if -1 < x < 1.25

   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

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

   5. Simplified 97.8%

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

      1. /-rgt-identity N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. /-lowering-/.f64 97.8

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

   7. Applied egg-rr 97.8%

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

Alternative 14: 14.6% accurate, 34.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 91.8%

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

3. Taylor expanded in x around 0

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

5. Simplified 46.8%

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

6. Taylor expanded in x around inf

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

8. Simplified 12.3%

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

Developer Target 1: 99.9% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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 2024198 
(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)))