Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1

Percentage Accurate: 87.5% → 99.9%

Time: 7.9s

Alternatives: 12

Speedup: 1.0×

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: 87.5% 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

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

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(x / Float64(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[(x / N[(N[(x + 1.0), $MachinePrecision] / N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 87.8%

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

   1. associate-/l* 99.9%

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

3. Simplified 99.9%

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

   1. clear-num 99.8%

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

   2. un-div-inv 99.9%

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

6. Applied egg-rr 99.9%

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

7. Final simplification 99.9%

   \[\leadsto \frac{x}{\frac{x + 1}{1 + \frac{x}{y}}} \]
8. Add Preprocessing

Alternative 2: 85.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{x}{y}\\ \mathbf{if}\;x \leq -110000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -5.5 \cdot 10^{-88}:\\ \;\;\;\;\frac{x \cdot \frac{x}{y}}{x + 1}\\ \mathbf{elif}\;x \leq 20000:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ x y))))
   (if (<= x -110000000.0)
     t_0
     (if (<= x -5.5e-88)
       (/ (* x (/ x y)) (+ x 1.0))
       (if (<= x 20000.0) (/ x (+ x 1.0)) t_0)))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (x / y);
	double tmp;
	if (x <= -110000000.0) {
		tmp = t_0;
	} else if (x <= -5.5e-88) {
		tmp = (x * (x / y)) / (x + 1.0);
	} else if (x <= 20000.0) {
		tmp = x / (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 = 1.0d0 + (x / y)
    if (x <= (-110000000.0d0)) then
        tmp = t_0
    else if (x <= (-5.5d-88)) then
        tmp = (x * (x / y)) / (x + 1.0d0)
    else if (x <= 20000.0d0) then
        tmp = x / (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 = 1.0 + (x / y);
	double tmp;
	if (x <= -110000000.0) {
		tmp = t_0;
	} else if (x <= -5.5e-88) {
		tmp = (x * (x / y)) / (x + 1.0);
	} else if (x <= 20000.0) {
		tmp = x / (x + 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (x / y)
	tmp = 0
	if x <= -110000000.0:
		tmp = t_0
	elif x <= -5.5e-88:
		tmp = (x * (x / y)) / (x + 1.0)
	elif x <= 20000.0:
		tmp = x / (x + 1.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(x / y))
	tmp = 0.0
	if (x <= -110000000.0)
		tmp = t_0;
	elseif (x <= -5.5e-88)
		tmp = Float64(Float64(x * Float64(x / y)) / Float64(x + 1.0));
	elseif (x <= 20000.0)
		tmp = Float64(x / Float64(x + 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + (x / y);
	tmp = 0.0;
	if (x <= -110000000.0)
		tmp = t_0;
	elseif (x <= -5.5e-88)
		tmp = (x * (x / y)) / (x + 1.0);
	elseif (x <= 20000.0)
		tmp = x / (x + 1.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -110000000.0], t$95$0, If[LessEqual[x, -5.5e-88], N[(N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 20000.0], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.1e8 or 2e4 < x

   1. Initial program 73.8%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 98.7%

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

   6. Taylor expanded in x around 0 98.8%

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

   if -1.1e8 < x < -5.49999999999999971e-88

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf 67.7%

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

   if -5.49999999999999971e-88 < x < 2e4

   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 81.3%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -110000000:\\ \;\;\;\;1 + \frac{x}{y}\\ \mathbf{elif}\;x \leq -5.5 \cdot 10^{-88}:\\ \;\;\;\;\frac{x \cdot \frac{x}{y}}{x + 1}\\ \mathbf{elif}\;x \leq 20000:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{x}{y}\\ \mathbf{if}\;x \leq -92000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -5.8 \cdot 10^{-88}:\\ \;\;\;\;x \cdot \frac{\frac{x}{y}}{x + 1}\\ \mathbf{elif}\;x \leq 1100:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ x y))))
   (if (<= x -92000000.0)
     t_0
     (if (<= x -5.8e-88)
       (* x (/ (/ x y) (+ x 1.0)))
       (if (<= x 1100.0) (/ x (+ x 1.0)) t_0)))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (x / y);
	double tmp;
	if (x <= -92000000.0) {
		tmp = t_0;
	} else if (x <= -5.8e-88) {
		tmp = x * ((x / y) / (x + 1.0));
	} else if (x <= 1100.0) {
		tmp = x / (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 = 1.0d0 + (x / y)
    if (x <= (-92000000.0d0)) then
        tmp = t_0
    else if (x <= (-5.8d-88)) then
        tmp = x * ((x / y) / (x + 1.0d0))
    else if (x <= 1100.0d0) then
        tmp = x / (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 = 1.0 + (x / y);
	double tmp;
	if (x <= -92000000.0) {
		tmp = t_0;
	} else if (x <= -5.8e-88) {
		tmp = x * ((x / y) / (x + 1.0));
	} else if (x <= 1100.0) {
		tmp = x / (x + 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (x / y)
	tmp = 0
	if x <= -92000000.0:
		tmp = t_0
	elif x <= -5.8e-88:
		tmp = x * ((x / y) / (x + 1.0))
	elif x <= 1100.0:
		tmp = x / (x + 1.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(x / y))
	tmp = 0.0
	if (x <= -92000000.0)
		tmp = t_0;
	elseif (x <= -5.8e-88)
		tmp = Float64(x * Float64(Float64(x / y) / Float64(x + 1.0)));
	elseif (x <= 1100.0)
		tmp = Float64(x / Float64(x + 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + (x / y);
	tmp = 0.0;
	if (x <= -92000000.0)
		tmp = t_0;
	elseif (x <= -5.8e-88)
		tmp = x * ((x / y) / (x + 1.0));
	elseif (x <= 1100.0)
		tmp = x / (x + 1.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -92000000.0], t$95$0, If[LessEqual[x, -5.8e-88], N[(x * N[(N[(x / y), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1100.0], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -9.2e7 or 1100 < x

   1. Initial program 73.8%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 98.7%

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

   6. Taylor expanded in x around 0 98.8%

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

   if -9.2e7 < x < -5.8000000000000003e-88

   1. Initial program 99.7%

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

      1. associate-/l* 99.7%

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

   3. Simplified 99.7%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 67.7%

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

   if -5.8000000000000003e-88 < x < 1100

   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 81.3%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -92000000:\\ \;\;\;\;1 + \frac{x}{y}\\ \mathbf{elif}\;x \leq -5.8 \cdot 10^{-88}:\\ \;\;\;\;x \cdot \frac{\frac{x}{y}}{x + 1}\\ \mathbf{elif}\;x \leq 1100:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{x}{y}\\ \mathbf{if}\;x \leq -5200000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -5.8 \cdot 10^{-88}:\\ \;\;\;\;\frac{x}{y + \frac{y}{x}}\\ \mathbf{elif}\;x \leq 49000:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ x y))))
   (if (<= x -5200000.0)
     t_0
     (if (<= x -5.8e-88)
       (/ x (+ y (/ y x)))
       (if (<= x 49000.0) (/ x (+ x 1.0)) t_0)))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (x / y);
	double tmp;
	if (x <= -5200000.0) {
		tmp = t_0;
	} else if (x <= -5.8e-88) {
		tmp = x / (y + (y / x));
	} else if (x <= 49000.0) {
		tmp = x / (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 = 1.0d0 + (x / y)
    if (x <= (-5200000.0d0)) then
        tmp = t_0
    else if (x <= (-5.8d-88)) then
        tmp = x / (y + (y / x))
    else if (x <= 49000.0d0) then
        tmp = x / (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 = 1.0 + (x / y);
	double tmp;
	if (x <= -5200000.0) {
		tmp = t_0;
	} else if (x <= -5.8e-88) {
		tmp = x / (y + (y / x));
	} else if (x <= 49000.0) {
		tmp = x / (x + 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (x / y)
	tmp = 0
	if x <= -5200000.0:
		tmp = t_0
	elif x <= -5.8e-88:
		tmp = x / (y + (y / x))
	elif x <= 49000.0:
		tmp = x / (x + 1.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(x / y))
	tmp = 0.0
	if (x <= -5200000.0)
		tmp = t_0;
	elseif (x <= -5.8e-88)
		tmp = Float64(x / Float64(y + Float64(y / x)));
	elseif (x <= 49000.0)
		tmp = Float64(x / Float64(x + 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + (x / y);
	tmp = 0.0;
	if (x <= -5200000.0)
		tmp = t_0;
	elseif (x <= -5.8e-88)
		tmp = x / (y + (y / x));
	elseif (x <= 49000.0)
		tmp = x / (x + 1.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5200000.0], t$95$0, If[LessEqual[x, -5.8e-88], N[(x / N[(y + N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 49000.0], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.2e6 or 49000 < x

   1. Initial program 73.8%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 98.7%

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

   6. Taylor expanded in x around 0 98.8%

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

   if -5.2e6 < x < -5.8000000000000003e-88

   1. Initial program 99.7%

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

      1. associate-/l* 99.7%

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

   3. Simplified 99.7%

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

      1. clear-num 99.7%

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

      2. un-div-inv 99.4%

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

   6. Applied egg-rr 99.4%

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

   7. Taylor expanded in y around 0 67.6%

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

      1. associate-/l* 67.5%

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

      2. +-commutative 67.5%

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

   9. Simplified 67.5%

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

   10. Taylor expanded in y around 0 67.7%

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

       1. unpow2 67.7%

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

       2. +-commutative 67.7%

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

       3. associate-*l/ 67.6%

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

       4. associate-/r/ 67.6%

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

       5. associate-*r/ 67.5%

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

       6. *-lft-identity 67.5%

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

       7. associate-*l/ 67.5%

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

       8. distribute-lft-in 67.5%

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

       9. lft-mult-inverse 67.5%

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

       10. *-rgt-identity 67.5%

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

       11. distribute-rgt-in 67.5%

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

       12. *-lft-identity 67.5%

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

       13. associate-*l/ 67.6%

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

       14. *-lft-identity 67.6%

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

   12. Simplified 67.6%

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

   if -5.8000000000000003e-88 < x < 49000

   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 81.3%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5200000:\\ \;\;\;\;1 + \frac{x}{y}\\ \mathbf{elif}\;x \leq -5.8 \cdot 10^{-88}:\\ \;\;\;\;\frac{x}{y + \frac{y}{x}}\\ \mathbf{elif}\;x \leq 49000:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{x}{y}\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -2.6 \cdot 10^{-89}:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 52000:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ x y))))
   (if (<= x -1.0)
     t_0
     (if (<= x -2.6e-89)
       (* x (/ x y))
       (if (<= x 52000.0) (/ x (+ x 1.0)) t_0)))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (x / y);
	double tmp;
	if (x <= -1.0) {
		tmp = t_0;
	} else if (x <= -2.6e-89) {
		tmp = x * (x / y);
	} else if (x <= 52000.0) {
		tmp = x / (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 = 1.0d0 + (x / y)
    if (x <= (-1.0d0)) then
        tmp = t_0
    else if (x <= (-2.6d-89)) then
        tmp = x * (x / y)
    else if (x <= 52000.0d0) then
        tmp = x / (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 = 1.0 + (x / y);
	double tmp;
	if (x <= -1.0) {
		tmp = t_0;
	} else if (x <= -2.6e-89) {
		tmp = x * (x / y);
	} else if (x <= 52000.0) {
		tmp = x / (x + 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (x / y)
	tmp = 0
	if x <= -1.0:
		tmp = t_0
	elif x <= -2.6e-89:
		tmp = x * (x / y)
	elif x <= 52000.0:
		tmp = x / (x + 1.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(x / y))
	tmp = 0.0
	if (x <= -1.0)
		tmp = t_0;
	elseif (x <= -2.6e-89)
		tmp = Float64(x * Float64(x / y));
	elseif (x <= 52000.0)
		tmp = Float64(x / Float64(x + 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + (x / y);
	tmp = 0.0;
	if (x <= -1.0)
		tmp = t_0;
	elseif (x <= -2.6e-89)
		tmp = x * (x / y);
	elseif (x <= 52000.0)
		tmp = x / (x + 1.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.0], t$95$0, If[LessEqual[x, -2.6e-89], N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 52000.0], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1 or 52000 < x

   1. Initial program 73.8%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 98.7%

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

   6. Taylor expanded in x around 0 98.8%

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

   if -1 < x < -2.5999999999999999e-89

   1. Initial program 99.7%

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

      1. associate-/l* 99.7%

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

   3. Simplified 99.7%

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

      1. clear-num 99.7%

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

      2. un-div-inv 99.4%

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

   6. Applied egg-rr 99.4%

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

   7. Taylor expanded in y around 0 67.6%

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

      1. associate-/l* 67.5%

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

      2. +-commutative 67.5%

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

   9. Simplified 67.5%

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

   10. Taylor expanded in x around 0 67.3%

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

       1. div-inv 67.4%

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

       2. associate-/r/ 67.4%

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

   12. Applied egg-rr 67.4%

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

   if -2.5999999999999999e-89 < x < 52000

   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 81.3%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;1 + \frac{x}{y}\\ \mathbf{elif}\;x \leq -2.6 \cdot 10^{-89}:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 52000:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{x}{y}\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -5.8 \cdot 10^{-88}:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-5}:\\ \;\;\;\;x \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ x y))))
   (if (<= x -1.0)
     t_0
     (if (<= x -5.8e-88)
       (* x (/ x y))
       (if (<= x 1.9e-5) (* x (- 1.0 x)) t_0)))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (x / y);
	double tmp;
	if (x <= -1.0) {
		tmp = t_0;
	} else if (x <= -5.8e-88) {
		tmp = x * (x / y);
	} else if (x <= 1.9e-5) {
		tmp = x * (1.0 - x);
	} 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 = 1.0d0 + (x / y)
    if (x <= (-1.0d0)) then
        tmp = t_0
    else if (x <= (-5.8d-88)) then
        tmp = x * (x / y)
    else if (x <= 1.9d-5) then
        tmp = x * (1.0d0 - x)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + (x / y);
	double tmp;
	if (x <= -1.0) {
		tmp = t_0;
	} else if (x <= -5.8e-88) {
		tmp = x * (x / y);
	} else if (x <= 1.9e-5) {
		tmp = x * (1.0 - x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (x / y)
	tmp = 0
	if x <= -1.0:
		tmp = t_0
	elif x <= -5.8e-88:
		tmp = x * (x / y)
	elif x <= 1.9e-5:
		tmp = x * (1.0 - x)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(x / y))
	tmp = 0.0
	if (x <= -1.0)
		tmp = t_0;
	elseif (x <= -5.8e-88)
		tmp = Float64(x * Float64(x / y));
	elseif (x <= 1.9e-5)
		tmp = Float64(x * Float64(1.0 - x));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + (x / y);
	tmp = 0.0;
	if (x <= -1.0)
		tmp = t_0;
	elseif (x <= -5.8e-88)
		tmp = x * (x / y);
	elseif (x <= 1.9e-5)
		tmp = x * (1.0 - x);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.0], t$95$0, If[LessEqual[x, -5.8e-88], N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.9e-5], N[(x * N[(1.0 - x), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1 or 1.9000000000000001e-5 < x

   1. Initial program 74.5%

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

      1. associate-/l* 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 96.6%

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

   6. Taylor expanded in x around 0 96.7%

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

   if -1 < x < -5.8000000000000003e-88

   1. Initial program 99.7%

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

      1. associate-/l* 99.7%

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

   3. Simplified 99.7%

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

      1. clear-num 99.7%

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

      2. un-div-inv 99.4%

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

   6. Applied egg-rr 99.4%

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

   7. Taylor expanded in y around 0 67.6%

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

      1. associate-/l* 67.5%

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

      2. +-commutative 67.5%

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

   9. Simplified 67.5%

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

   10. Taylor expanded in x around 0 67.3%

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

       1. div-inv 67.4%

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

       2. associate-/r/ 67.4%

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

   12. Applied egg-rr 67.4%

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

   if -5.8000000000000003e-88 < x < 1.9000000000000001e-5

   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 82.6%

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

   4. Taylor expanded in x around 0 82.5%

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

      1. neg-mul-1 82.5%

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

      2. sub-neg 82.5%

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

   6. Simplified 82.5%

      \[\leadsto \color{blue}{x \cdot \left(1 - x\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;1 + \frac{x}{y}\\ \mathbf{elif}\;x \leq -5.8 \cdot 10^{-88}:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-5}:\\ \;\;\;\;x \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-88}:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 0.8:\\ \;\;\;\;x \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.0)
   (/ x y)
   (if (<= x -4e-88) (* x (/ x y)) (if (<= x 0.8) (* x (- 1.0 x)) (/ x y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = x / y;
	} else if (x <= -4e-88) {
		tmp = x * (x / y);
	} else if (x <= 0.8) {
		tmp = x * (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) :: tmp
    if (x <= (-1.0d0)) then
        tmp = x / y
    else if (x <= (-4d-88)) then
        tmp = x * (x / y)
    else if (x <= 0.8d0) then
        tmp = x * (1.0d0 - x)
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = x / y;
	} else if (x <= -4e-88) {
		tmp = x * (x / y);
	} else if (x <= 0.8) {
		tmp = x * (1.0 - x);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.0:
		tmp = x / y
	elif x <= -4e-88:
		tmp = x * (x / y)
	elif x <= 0.8:
		tmp = x * (1.0 - x)
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(x / y);
	elseif (x <= -4e-88)
		tmp = Float64(x * Float64(x / y));
	elseif (x <= 0.8)
		tmp = Float64(x * Float64(1.0 - x));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = x / y;
	elseif (x <= -4e-88)
		tmp = x * (x / y);
	elseif (x <= 0.8)
		tmp = x * (1.0 - x);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.0], N[(x / y), $MachinePrecision], If[LessEqual[x, -4e-88], N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.8], N[(x * N[(1.0 - x), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1 or 0.80000000000000004 < x

   1. Initial program 73.8%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

      1. clear-num 99.7%

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

      2. un-div-inv 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around inf 73.6%

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

   if -1 < x < -3.99999999999999974e-88

   1. Initial program 99.7%

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

      1. associate-/l* 99.7%

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

   3. Simplified 99.7%

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

      1. clear-num 99.7%

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

      2. un-div-inv 99.4%

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

   6. Applied egg-rr 99.4%

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

   7. Taylor expanded in y around 0 67.6%

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

      1. associate-/l* 67.5%

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

      2. +-commutative 67.5%

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

   9. Simplified 67.5%

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

   10. Taylor expanded in x around 0 67.3%

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

       1. div-inv 67.4%

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

       2. associate-/r/ 67.4%

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

   12. Applied egg-rr 67.4%

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

   if -3.99999999999999974e-88 < 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 y around inf 81.3%

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

   4. Taylor expanded in x around 0 80.6%

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

      1. neg-mul-1 80.6%

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

      2. sub-neg 80.6%

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

   6. Simplified 80.6%

      \[\leadsto \color{blue}{x \cdot \left(1 - x\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-88}:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 0.8:\\ \;\;\;\;x \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.9% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 1.0))) (/ x y) (* x (- 1.0 x))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.0)) {
		tmp = x / y;
	} else {
		tmp = x * (1.0 - x);
	}
	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)) .or. (.not. (x <= 1.0d0))) then
        tmp = x / y
    else
        tmp = x * (1.0d0 - x)
    end if
    code = tmp
end function

Java

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

Python

def code(x, y):
	tmp = 0
	if (x <= -1.0) or not (x <= 1.0):
		tmp = x / y
	else:
		tmp = x * (1.0 - x)
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 1.0)))
		tmp = x / y;
	else
		tmp = x * (1.0 - x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], N[(x / y), $MachinePrecision], N[(x * N[(1.0 - x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 73.8%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

      1. clear-num 99.7%

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

      2. un-div-inv 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around inf 73.6%

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

   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 y around inf 73.3%

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

   4. Taylor expanded in x around 0 72.7%

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

      1. neg-mul-1 72.7%

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

      2. sub-neg 72.7%

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

   6. Simplified 72.7%

      \[\leadsto \color{blue}{x \cdot \left(1 - x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.1%

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

Alternative 9: 74.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1.9 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 1.9e-5))) (/ x y) x))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.9e-5)) {
		tmp = x / y;
	} else {
		tmp = x;
	}
	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)) .or. (.not. (x <= 1.9d-5))) then
        tmp = x / y
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.9e-5)) {
		tmp = x / y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.0) or not (x <= 1.9e-5):
		tmp = x / y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 1.9e-5))
		tmp = Float64(x / y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 1.9e-5)))
		tmp = x / y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 1.9e-5]], $MachinePrecision]], N[(x / y), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1.9000000000000001e-5 < x

   1. Initial program 74.5%

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

      1. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

      1. clear-num 99.7%

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

      2. un-div-inv 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around inf 72.0%

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

   if -1 < x < 1.9000000000000001e-5

   1. Initial program 99.9%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

      1. clear-num 99.9%

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

      2. un-div-inv 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in x around 0 73.5%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1.9 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 48.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x -1.0) 1.0 (if (<= x 1.0) x 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = 1.0;
	} else if (x <= 1.0) {
		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)) then
        tmp = 1.0d0
    else if (x <= 1.0d0) 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) {
		tmp = 1.0;
	} else if (x <= 1.0) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.0:
		tmp = 1.0
	elif x <= 1.0:
		tmp = x
	else:
		tmp = 1.0
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = 1.0;
	elseif (x <= 1.0)
		tmp = x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.0], 1.0, If[LessEqual[x, 1.0], x, 1.0]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 73.8%

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

   3. Taylor expanded in y around inf 27.0%

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

      1. +-commutative 27.0%

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

      2. expm1-log1p-u 10.1%

         \[\leadsto \frac{x}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x + 1\right)\right)}} \]

      3. expm1-undefine 10.1%

         \[\leadsto \frac{x}{\color{blue}{e^{\mathsf{log1p}\left(x + 1\right)} - 1}} \]

   5. Applied egg-rr 10.1%

      \[\leadsto \frac{x}{\color{blue}{e^{\mathsf{log1p}\left(x + 1\right)} - 1}} \]
   6. Step-by-step derivation

      1. sub-neg 10.1%

         \[\leadsto \frac{x}{\color{blue}{e^{\mathsf{log1p}\left(x + 1\right)} + \left(-1\right)}} \]

      2. log1p-undefine 10.1%

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

      3. rem-exp-log 27.0%

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

      4. +-commutative 27.0%

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

      5. associate-+r+ 27.0%

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

      6. metadata-eval 27.0%

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

      7. metadata-eval 27.0%

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

   7. Simplified 27.0%

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

   8. Taylor expanded in x around inf 26.6%

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

   if -1 < x < 1

   1. Initial program 99.9%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

      1. clear-num 99.9%

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

      2. un-div-inv 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in x around 0 72.1%

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

Alternative 11: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	return x * ((1.0 + (x / y)) / (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)) / (x + 1.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.8%

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

   1. associate-/l* 99.9%

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

3. Simplified 99.9%

   \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}} \]
4. Add Preprocessing

5. Final simplification 99.9%

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

Alternative 12: 14.1% accurate, 11.0× 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 87.8%

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

3. Taylor expanded in y around inf 51.8%

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

   1. +-commutative 51.8%

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

   2. expm1-log1p-u 44.0%

      \[\leadsto \frac{x}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x + 1\right)\right)}} \]

   3. expm1-undefine 43.9%

      \[\leadsto \frac{x}{\color{blue}{e^{\mathsf{log1p}\left(x + 1\right)} - 1}} \]

5. Applied egg-rr 43.9%

   \[\leadsto \frac{x}{\color{blue}{e^{\mathsf{log1p}\left(x + 1\right)} - 1}} \]
6. Step-by-step derivation

   1. sub-neg 43.9%

      \[\leadsto \frac{x}{\color{blue}{e^{\mathsf{log1p}\left(x + 1\right)} + \left(-1\right)}} \]

   2. log1p-undefine 43.9%

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

   3. rem-exp-log 51.8%

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

   4. +-commutative 51.8%

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

   5. associate-+r+ 51.8%

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

   6. metadata-eval 51.8%

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

   7. metadata-eval 51.8%

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

7. Simplified 51.8%

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

8. Taylor expanded in x around inf 14.5%

   \[\leadsto \color{blue}{1} \]
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 2024137 
(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)))