Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1

Percentage Accurate: 87.9% → 99.8%

Time: 6.7s

Alternatives: 9

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: 87.9% 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: 99.8% 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 90.4%

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

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

Alternative 2: 98.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.8\right):\\ \;\;\;\;\frac{x}{y} + 1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + x \cdot \left(\frac{1}{y} + -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 0.8)))
   (+ (/ x y) 1.0)
   (* x (+ 1.0 (* x (+ (/ 1.0 y) -1.0))))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.0) || !(x <= 0.8)) {
		tmp = (x / y) + 1.0;
	} else {
		tmp = x * (1.0 + (x * ((1.0 / y) + -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)) .or. (.not. (x <= 0.8d0))) then
        tmp = (x / y) + 1.0d0
    else
        tmp = x * (1.0d0 + (x * ((1.0d0 / y) + (-1.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -1.0) || !(x <= 0.8)) {
		tmp = (x / y) + 1.0;
	} else {
		tmp = x * (1.0 + (x * ((1.0 / y) + -1.0)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.0) or not (x <= 0.8):
		tmp = (x / y) + 1.0
	else:
		tmp = x * (1.0 + (x * ((1.0 / y) + -1.0)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 0.8))
		tmp = Float64(Float64(x / y) + 1.0);
	else
		tmp = Float64(x * Float64(1.0 + Float64(x * Float64(Float64(1.0 / y) + -1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 0.8)))
		tmp = (x / y) + 1.0;
	else
		tmp = x * (1.0 + (x * ((1.0 / y) + -1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 0.8]], $MachinePrecision]], N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision], N[(x * N[(1.0 + N[(x * N[(N[(1.0 / y), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 0.80000000000000004 < x

   1. Initial program 82.4%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 69.9%

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

      1. *-commutative 69.9%

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

      2. +-commutative 69.9%

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

      3. associate-/l* 74.4%

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

      4. *-lft-identity 74.4%

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

      5. associate-*l/ 74.4%

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

      6. unpow2 74.4%

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

      7. +-commutative 74.4%

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

      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. Taylor expanded in x around inf 98.0%

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

   9. Taylor expanded in y around 0 98.0%

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

       1. *-lft-identity 98.0%

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

       2. associate-*l/ 97.8%

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

       3. +-commutative 97.8%

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

       4. distribute-rgt-in 97.8%

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

       5. rgt-mult-inverse 97.9%

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

       6. associate-*r/ 98.0%

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

       7. *-rgt-identity 98.0%

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

   11. Simplified 98.0%

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

   if -1 < x < 0.80000000000000004

   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. Taylor expanded in x around 0 98.4%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.8\right):\\ \;\;\;\;\frac{x}{y} + 1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + x \cdot \left(\frac{1}{y} + -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.7% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 82.4%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 69.9%

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

      1. *-commutative 69.9%

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

      2. +-commutative 69.9%

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

      3. associate-/l* 74.4%

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

      4. *-lft-identity 74.4%

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

      5. associate-*l/ 74.4%

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

      6. unpow2 74.4%

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

      7. +-commutative 74.4%

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

      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. Taylor expanded in x around inf 98.0%

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

   9. Taylor expanded in y around 0 98.0%

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

       1. *-lft-identity 98.0%

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

       2. associate-*l/ 97.8%

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

       3. +-commutative 97.8%

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

       4. distribute-rgt-in 97.8%

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

       5. rgt-mult-inverse 97.9%

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

       6. associate-*r/ 98.0%

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

       7. *-rgt-identity 98.0%

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

   11. Simplified 98.0%

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

   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. Taylor expanded in x around 0 98.4%

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

   6. Taylor expanded in y around 0 97.9%

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

4. Final simplification 97.9%

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

Alternative 4: 72.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.85 \cdot 10^{+114}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -5.5 \cdot 10^{-6}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1400000000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -3.85e+114)
   (/ x y)
   (if (<= x -5.5e-6) 1.0 (if (<= x 1400000000.0) x (/ x y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -3.85e+114) {
		tmp = x / y;
	} else if (x <= -5.5e-6) {
		tmp = 1.0;
	} else if (x <= 1400000000.0) {
		tmp = 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 <= (-3.85d+114)) then
        tmp = x / y
    else if (x <= (-5.5d-6)) then
        tmp = 1.0d0
    else if (x <= 1400000000.0d0) then
        tmp = x
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -3.85e+114) {
		tmp = x / y;
	} else if (x <= -5.5e-6) {
		tmp = 1.0;
	} else if (x <= 1400000000.0) {
		tmp = x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -3.85e+114:
		tmp = x / y
	elif x <= -5.5e-6:
		tmp = 1.0
	elif x <= 1400000000.0:
		tmp = x
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -3.85e+114)
		tmp = Float64(x / y);
	elseif (x <= -5.5e-6)
		tmp = 1.0;
	elseif (x <= 1400000000.0)
		tmp = x;
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -3.85e+114)
		tmp = x / y;
	elseif (x <= -5.5e-6)
		tmp = 1.0;
	elseif (x <= 1400000000.0)
		tmp = x;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -3.85e+114], N[(x / y), $MachinePrecision], If[LessEqual[x, -5.5e-6], 1.0, If[LessEqual[x, 1400000000.0], x, N[(x / y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.8500000000000001e114 or 1.4e9 < x

   1. Initial program 77.4%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 77.3%

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

   if -3.8500000000000001e114 < x < -5.4999999999999999e-6

   1. Initial program 97.3%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 inf 58.0%

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

   6. Taylor expanded in x around inf 54.8%

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

   7. Taylor expanded in x around 0 54.9%

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

   if -5.4999999999999999e-6 < x < 1.4e9

   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. Taylor expanded in x around 0 80.7%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.85 \cdot 10^{+114}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -5.5 \cdot 10^{-6}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1400000000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.7% 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 82.4%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 69.9%

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

      1. *-commutative 69.9%

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

      2. +-commutative 69.9%

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

      3. associate-/l* 74.4%

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

      4. *-lft-identity 74.4%

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

      5. associate-*l/ 74.4%

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

      6. unpow2 74.4%

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

      7. +-commutative 74.4%

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

      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. Taylor expanded in x around inf 98.0%

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

   9. Taylor expanded in y around 0 98.0%

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

       1. *-lft-identity 98.0%

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

       2. associate-*l/ 97.8%

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

       3. +-commutative 97.8%

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

       4. distribute-rgt-in 97.8%

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

       5. rgt-mult-inverse 97.9%

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

       6. associate-*r/ 98.0%

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

       7. *-rgt-identity 98.0%

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

   11. Simplified 98.0%

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

   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. Taylor expanded in x around 0 98.4%

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

   6. Taylor expanded in y around 0 97.9%

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

   7. Taylor expanded in x around 0 97.9%

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

4. Final simplification 97.9%

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

Alternative 6: 85.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.004\right):\\ \;\;\;\;\frac{x}{y} + 1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 0.004))) (+ (/ x y) 1.0) x))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.0) || !(x <= 0.004)) {
		tmp = (x / y) + 1.0;
	} 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 <= 0.004d0))) then
        tmp = (x / y) + 1.0d0
    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 <= 0.004)) {
		tmp = (x / y) + 1.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.0) or not (x <= 0.004):
		tmp = (x / y) + 1.0
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 0.004))
		tmp = Float64(Float64(x / y) + 1.0);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 0.004)))
		tmp = (x / y) + 1.0;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 0.004]], $MachinePrecision]], N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 0.0040000000000000001 < x

   1. Initial program 82.5%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 70.1%

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

      1. *-commutative 70.1%

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

      2. +-commutative 70.1%

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

      3. associate-/l* 74.6%

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

      4. *-lft-identity 74.6%

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

      5. associate-*l/ 74.5%

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

      6. unpow2 74.5%

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

      7. +-commutative 74.5%

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

      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. Taylor expanded in x around inf 97.4%

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

   9. Taylor expanded in y around 0 97.4%

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

       1. *-lft-identity 97.4%

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

       2. associate-*l/ 97.2%

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

       3. +-commutative 97.2%

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

       4. distribute-rgt-in 97.2%

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

       5. rgt-mult-inverse 97.3%

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

       6. associate-*r/ 97.4%

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

       7. *-rgt-identity 97.4%

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

   11. Simplified 97.4%

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

   if -1 < x < 0.0040000000000000001

   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. Taylor expanded in x around 0 81.8%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.004\right):\\ \;\;\;\;\frac{x}{y} + 1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 86.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -16500000000 \lor \neg \left(x \leq 1850000000\right):\\ \;\;\;\;\frac{x}{y} + 1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -16500000000.0) (not (<= x 1850000000.0)))
   (+ (/ x y) 1.0)
   (/ x (+ x 1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -16500000000.0) || !(x <= 1850000000.0)) {
		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 <= (-16500000000.0d0)) .or. (.not. (x <= 1850000000.0d0))) 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 <= -16500000000.0) || !(x <= 1850000000.0)) {
		tmp = (x / y) + 1.0;
	} else {
		tmp = x / (x + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -16500000000.0) or not (x <= 1850000000.0):
		tmp = (x / y) + 1.0
	else:
		tmp = x / (x + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -16500000000.0) || !(x <= 1850000000.0))
		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 <= -16500000000.0) || ~((x <= 1850000000.0)))
		tmp = (x / y) + 1.0;
	else
		tmp = x / (x + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -16500000000.0], N[Not[LessEqual[x, 1850000000.0]], $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 < -1.65e10 or 1.85e9 < x

   1. Initial program 81.6%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 68.5%

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

      1. *-commutative 68.5%

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

      2. +-commutative 68.5%

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

      3. associate-/l* 73.3%

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

      4. *-lft-identity 73.3%

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

      5. associate-*l/ 73.2%

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

      6. unpow2 73.2%

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

      7. +-commutative 73.2%

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

      8. associate-/l* 100.0%

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

      9. *-lft-identity 100.0%

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

      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. Taylor expanded in x around inf 99.7%

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

   9. Taylor expanded in y around 0 99.7%

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

       1. *-lft-identity 99.7%

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

       2. associate-*l/ 99.5%

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

       3. +-commutative 99.5%

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

       4. distribute-rgt-in 99.5%

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

       5. rgt-mult-inverse 99.6%

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

       6. associate-*r/ 99.7%

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

       7. *-rgt-identity 99.7%

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

   11. Simplified 99.7%

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

   if -1.65e10 < x < 1.85e9

   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. Taylor expanded in y around inf 82.2%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -16500000000 \lor \neg \left(x \leq 1850000000\right):\\ \;\;\;\;\frac{x}{y} + 1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 49.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{-6}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 0.29:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -5.5e-6) 1.0 (if (<= x 0.29) x 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -5.5e-6) {
		tmp = 1.0;
	} else if (x <= 0.29) {
		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 <= (-5.5d-6)) then
        tmp = 1.0d0
    else if (x <= 0.29d0) 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 <= -5.5e-6) {
		tmp = 1.0;
	} else if (x <= 0.29) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -5.5e-6:
		tmp = 1.0
	elif x <= 0.29:
		tmp = x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -5.5e-6)
		tmp = 1.0;
	elseif (x <= 0.29)
		tmp = x;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -5.5e-6)
		tmp = 1.0;
	elseif (x <= 0.29)
		tmp = x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -5.5e-6], 1.0, If[LessEqual[x, 0.29], x, 1.0]]

Derivation

1. Split input into 2 regimes

2. if x < -5.4999999999999999e-6 or 0.28999999999999998 < x

   1. Initial program 82.6%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 inf 32.8%

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

   6. Taylor expanded in x around inf 31.4%

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

   7. Taylor expanded in x around 0 31.4%

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

   if -5.4999999999999999e-6 < x < 0.28999999999999998

   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. Taylor expanded in x around 0 82.5%

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

4. Final simplification 54.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{-6}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 0.29:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 14.4% 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 90.4%

   \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
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 inf 55.6%

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

6. Taylor expanded in x around inf 18.9%

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

7. Taylor expanded in x around 0 18.9%

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

8. Final simplification 18.9%

   \[\leadsto 1 \]
9. Add Preprocessing

Developer target: 99.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024095 
(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)))