Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1

Percentage Accurate: 88.1% → 99.9%

Time: 7.5s

Alternatives: 8

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: 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, 1.0× speedup

Math

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

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(x * Float64(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[(x * N[(N[(N[(x / y), $MachinePrecision] + 1.0), $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. Add Preprocessing

Alternative 2: 97.9% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (/ x y) 1.0)))
   (if (or (<= x -1.0) (not (<= x 1.0))) t_0 (* x t_0))))

C

double code(double x, double y) {
	double t_0 = (x / y) + 1.0;
	double tmp;
	if ((x <= -1.0) || !(x <= 1.0)) {
		tmp = t_0;
	} else {
		tmp = x * 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
    if ((x <= (-1.0d0)) .or. (.not. (x <= 1.0d0))) then
        tmp = t_0
    else
        tmp = x * t_0
    end if
    code = tmp
end function

Java

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

Python

def code(x, y):
	t_0 = (x / y) + 1.0
	tmp = 0
	if (x <= -1.0) or not (x <= 1.0):
		tmp = t_0
	else:
		tmp = x * t_0
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x / y) + 1.0;
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 1.0)))
		tmp = t_0;
	else
		tmp = x * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]}, If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], t$95$0, N[(x * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 76.2%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 y around 0 66.0%

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

      1. *-commutative 66.0%

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

      2. +-commutative 66.0%

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

      3. associate-/l* 73.7%

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

      4. *-lft-identity 73.7%

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

      5. associate-*l/ 73.7%

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

      6. unpow2 73.7%

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

      7. +-commutative 73.7%

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

      8. associate-/l* 99.9%

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

      9. *-lft-identity 99.9%

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

      10. associate-*l/ 99.8%

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

      11. distribute-rgt-out 99.8%

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

      12. associate-*l/ 100.0%

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

      13. *-lft-identity 100.0%

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

      14. +-commutative 100.0%

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

   7. Simplified 100.0%

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

      1. *-commutative 100.0%

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

      2. clear-num 100.0%

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

      3. un-div-inv 100.0%

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

      4. +-commutative 100.0%

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

      5. +-commutative 100.0%

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

   9. Applied egg-rr 100.0%

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

   10. Taylor expanded in x around inf 98.5%

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

   11. Taylor expanded in x around 0 98.5%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 78.5%

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

      1. *-commutative 78.5%

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

      2. +-commutative 78.5%

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

      3. associate-/l* 78.5%

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

      4. *-lft-identity 78.5%

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

      5. associate-*l/ 78.6%

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

      6. unpow2 78.6%

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

      7. +-commutative 78.6%

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

      8. associate-/l* 78.6%

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

      9. *-lft-identity 78.6%

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

      10. associate-*l/ 78.6%

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

      11. distribute-rgt-out 78.6%

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

      12. associate-*l/ 78.5%

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

      13. *-lft-identity 78.5%

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

      14. +-commutative 78.5%

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

   7. Simplified 78.5%

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

      1. *-commutative 78.5%

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

      2. clear-num 78.3%

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

      3. un-div-inv 78.4%

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

      4. +-commutative 78.4%

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

      5. +-commutative 78.4%

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

   9. Applied egg-rr 78.4%

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

   10. Taylor expanded in x around 0 76.5%

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

   11. Taylor expanded in x around 0 98.1%

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

4. Final simplification 98.3%

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

Alternative 3: 87.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -330000000 \lor \neg \left(x \leq 55000000000\right):\\ \;\;\;\;\frac{x}{y} + 1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -330000000.0) (not (<= x 55000000000.0)))
   (+ (/ x y) 1.0)
   (/ x (+ x 1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -330000000.0) || !(x <= 55000000000.0)) {
		tmp = (x / y) + 1.0;
	} else {
		tmp = x / (x + 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 <= (-330000000.0d0)) .or. (.not. (x <= 55000000000.0d0))) then
        tmp = (x / y) + 1.0d0
    else
        tmp = x / (x + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -330000000.0) || !(x <= 55000000000.0)) {
		tmp = (x / y) + 1.0;
	} else {
		tmp = x / (x + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -330000000.0) or not (x <= 55000000000.0):
		tmp = (x / y) + 1.0
	else:
		tmp = x / (x + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -330000000.0) || !(x <= 55000000000.0))
		tmp = Float64(Float64(x / y) + 1.0);
	else
		tmp = Float64(x / Float64(x + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -330000000.0) || ~((x <= 55000000000.0)))
		tmp = (x / y) + 1.0;
	else
		tmp = x / (x + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -330000000.0], N[Not[LessEqual[x, 55000000000.0]], $MachinePrecision]], N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.3e8 or 5.5e10 < x

   1. Initial program 75.5%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 y around 0 65.0%

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

      1. *-commutative 65.0%

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

      2. +-commutative 65.0%

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

      3. associate-/l* 72.9%

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

      4. *-lft-identity 72.9%

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

      5. associate-*l/ 72.8%

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

      6. unpow2 72.8%

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

      7. +-commutative 72.8%

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

      8. associate-/l* 99.9%

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

      9. *-lft-identity 99.9%

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

      10. associate-*l/ 99.8%

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

      11. distribute-rgt-out 99.8%

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

      12. associate-*l/ 100.0%

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

      13. *-lft-identity 100.0%

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

      14. +-commutative 100.0%

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

   7. Simplified 100.0%

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

      1. *-commutative 100.0%

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

      2. clear-num 100.0%

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

      3. un-div-inv 100.0%

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

      4. +-commutative 100.0%

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

      5. +-commutative 100.0%

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

   9. Applied egg-rr 100.0%

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

   10. Taylor expanded in x around inf 99.8%

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

   11. Taylor expanded in x around 0 99.8%

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

   if -3.3e8 < x < 5.5e10

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around inf 78.1%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -330000000 \lor \neg \left(x \leq 55000000000\right):\\ \;\;\;\;\frac{x}{y} + 1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.038\right):\\ \;\;\;\;\frac{x}{y} + 1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 0.038))) (+ (/ x y) 1.0) (* x (- 1.0 x))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.0) || !(x <= 0.038)) {
		tmp = (x / y) + 1.0;
	} 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 <= 0.038d0))) then
        tmp = (x / y) + 1.0d0
    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 <= 0.038)) {
		tmp = (x / y) + 1.0;
	} else {
		tmp = x * (1.0 - x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.0) or not (x <= 0.038):
		tmp = (x / y) + 1.0
	else:
		tmp = x * (1.0 - x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 0.038))
		tmp = Float64(Float64(x / y) + 1.0);
	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 <= 0.038)))
		tmp = (x / y) + 1.0;
	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, 0.038]], $MachinePrecision]], N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision], N[(x * N[(1.0 - x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 0.0379999999999999991 < x

   1. Initial program 76.2%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 y around 0 66.0%

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

      1. *-commutative 66.0%

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

      2. +-commutative 66.0%

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

      3. associate-/l* 73.7%

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

      4. *-lft-identity 73.7%

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

      5. associate-*l/ 73.7%

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

      6. unpow2 73.7%

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

      7. +-commutative 73.7%

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

      8. associate-/l* 99.9%

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

      9. *-lft-identity 99.9%

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

      10. associate-*l/ 99.8%

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

      11. distribute-rgt-out 99.8%

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

      12. associate-*l/ 100.0%

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

      13. *-lft-identity 100.0%

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

      14. +-commutative 100.0%

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

   7. Simplified 100.0%

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

      1. *-commutative 100.0%

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

      2. clear-num 100.0%

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

      3. un-div-inv 100.0%

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

      4. +-commutative 100.0%

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

      5. +-commutative 100.0%

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

   9. Applied egg-rr 100.0%

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

   10. Taylor expanded in x around inf 98.5%

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

   11. Taylor expanded in x around 0 98.5%

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

   if -1 < x < 0.0379999999999999991

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around inf 77.4%

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

   6. Taylor expanded in x around 0 76.9%

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

      1. neg-mul-1 76.9%

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

      2. sub-neg 76.9%

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

   8. Simplified 76.9%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.038\right):\\ \;\;\;\;\frac{x}{y} + 1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.58\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 0.58))) (/ x y) (* x (- 1.0 x))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.0) || !(x <= 0.58)) {
		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 <= 0.58d0))) 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 <= 0.58)) {
		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 <= 0.58):
		tmp = x / y
	else:
		tmp = x * (1.0 - x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 0.58))
		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 <= 0.58)))
		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, 0.58]], $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 0.57999999999999996 < x

   1. Initial program 76.2%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 75.8%

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

   if -1 < x < 0.57999999999999996

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around inf 77.4%

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

   6. Taylor expanded in x around 0 76.9%

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

      1. neg-mul-1 76.9%

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

      2. sub-neg 76.9%

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

   8. Simplified 76.9%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.58\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 55000000000\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 55000000000.0))) (/ x y) x))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.0) || !(x <= 55000000000.0)) {
		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 <= 55000000000.0d0))) 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 <= 55000000000.0)) {
		tmp = x / y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.0) or not (x <= 55000000000.0):
		tmp = x / y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 55000000000.0))
		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 <= 55000000000.0)))
		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, 55000000000.0]], $MachinePrecision]], N[(x / y), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 5.5e10 < x

   1. Initial program 75.9%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 76.9%

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

   if -1 < x < 5.5e10

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 75.2%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 55000000000\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 49.8% 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 76.2%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 y around 0 66.0%

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

      1. *-commutative 66.0%

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

      2. +-commutative 66.0%

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

      3. associate-/l* 73.7%

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

      4. *-lft-identity 73.7%

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

      5. associate-*l/ 73.7%

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

      6. unpow2 73.7%

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

      7. +-commutative 73.7%

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

      8. associate-/l* 99.9%

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

      9. *-lft-identity 99.9%

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

      10. associate-*l/ 99.8%

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

      11. distribute-rgt-out 99.8%

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

      12. associate-*l/ 100.0%

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

      13. *-lft-identity 100.0%

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

      14. +-commutative 100.0%

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

   7. Simplified 100.0%

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

      1. *-commutative 100.0%

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

      2. clear-num 100.0%

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

      3. un-div-inv 100.0%

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

      4. +-commutative 100.0%

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

      5. +-commutative 100.0%

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

   9. Applied egg-rr 100.0%

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

   10. Taylor expanded in x around inf 98.5%

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

   11. Taylor expanded in x around 0 24.4%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 76.2%

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

Alternative 8: 14.8% 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. 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 y around 0 72.1%

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

   1. *-commutative 72.1%

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

   2. +-commutative 72.1%

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

   3. associate-/l* 76.1%

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

   4. *-lft-identity 76.1%

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

   5. associate-*l/ 76.0%

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

   6. unpow2 76.0%

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

   7. +-commutative 76.0%

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

   8. associate-/l* 89.5%

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

   9. *-lft-identity 89.5%

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

   10. associate-*l/ 89.4%

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

   11. distribute-rgt-out 89.4%

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

   12. associate-*l/ 89.5%

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

   13. *-lft-identity 89.5%

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

   14. +-commutative 89.5%

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

7. Simplified 89.5%

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

   1. *-commutative 89.5%

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

   2. clear-num 89.4%

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

   3. un-div-inv 89.4%

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

   4. +-commutative 89.4%

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

   5. +-commutative 89.4%

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

9. Applied egg-rr 89.4%

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

10. Taylor expanded in x around inf 52.1%

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

11. Taylor expanded in x around 0 14.4%

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

Developer target: 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 2024086 
(FPCore (x y)
  :name "Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1"
  :precision binary64

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

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