Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1

Percentage Accurate: 88.6% → 99.8%

Time: 7.3s

Alternatives: 7

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: 88.6% 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 87.9%

   \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
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. Final simplification 99.8%

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

Alternative 2: 73.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{+57}:\\ \;\;\;\;x \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.0)
   (/ x y)
   (if (<= x 1.0) x (if (<= x 2.5e+57) (* 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 <= 1.0) {
		tmp = x;
	} else if (x <= 2.5e+57) {
		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 <= 1.0d0) then
        tmp = x
    else if (x <= 2.5d+57) 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 <= 1.0) {
		tmp = x;
	} else if (x <= 2.5e+57) {
		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 <= 1.0:
		tmp = x
	elif x <= 2.5e+57:
		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 <= 1.0)
		tmp = x;
	elseif (x <= 2.5e+57)
		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 <= 1.0)
		tmp = x;
	elseif (x <= 2.5e+57)
		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, 1.0], x, If[LessEqual[x, 2.5e+57], N[(x * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1 or 2.49999999999999986e57 < x

   1. Initial program 73.9%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 77.8%

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

   if -1 < x < 1

   1. Initial program 99.8%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 0 70.6%

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

   if 1 < x < 2.49999999999999986e57

   1. Initial program 99.9%

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

      1. associate-/l* 99.8%

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

   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 70.2%

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

   6. Taylor expanded in x around inf 67.1%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{+57}:\\ \;\;\;\;x \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.82\right):\\ \;\;\;\;\frac{x}{y} + 1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + \left(\frac{x}{y} - x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 0.82)))
   (+ (/ x y) 1.0)
   (* x (+ 1.0 (- (/ x y) x)))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.0) || !(x <= 0.82)) {
		tmp = (x / y) + 1.0;
	} else {
		tmp = x * (1.0 + ((x / y) - 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 <= 0.82d0))) then
        tmp = (x / y) + 1.0d0
    else
        tmp = x * (1.0d0 + ((x / y) - x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -1.0) || !(x <= 0.82)) {
		tmp = (x / y) + 1.0;
	} else {
		tmp = x * (1.0 + ((x / y) - x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.0) or not (x <= 0.82):
		tmp = (x / y) + 1.0
	else:
		tmp = x * (1.0 + ((x / y) - x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 0.82))
		tmp = Float64(Float64(x / y) + 1.0);
	else
		tmp = Float64(x * Float64(1.0 + Float64(Float64(x / y) - x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 0.82)))
		tmp = (x / y) + 1.0;
	else
		tmp = x * (1.0 + ((x / y) - x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 0.82]], $MachinePrecision]], N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision], N[(x * N[(1.0 + N[(N[(x / y), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 0.819999999999999951 < x

   1. Initial program 76.5%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 54.5%

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

      1. *-commutative 54.5%

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

      2. +-commutative 54.5%

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

      3. associate-/l* 62.4%

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

      4. *-lft-identity 62.4%

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

      5. associate-*l/ 62.3%

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

      6. unpow2 62.3%

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

      7. +-commutative 62.3%

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

      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 98.6%

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

   9. Taylor expanded in y around 0 98.6%

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

       1. *-lft-identity 98.6%

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

       2. associate-*l/ 98.3%

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

       3. +-commutative 98.3%

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

       4. distribute-lft-in 98.3%

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

       5. lft-mult-inverse 98.3%

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

       6. associate-*l/ 98.6%

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

       7. *-lft-identity 98.6%

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

   11. Simplified 98.6%

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

   if -1 < x < 0.819999999999999951

   1. Initial program 99.8%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 0 95.7%

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

   6. Taylor expanded in y around inf 95.7%

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

      1. neg-mul-1 95.7%

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

      2. +-commutative 95.7%

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

      3. unsub-neg 95.7%

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

   8. Simplified 95.7%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.82\right):\\ \;\;\;\;\frac{x}{y} + 1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + \left(\frac{x}{y} - x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.0011\right):\\ \;\;\;\;\frac{x}{y} + 1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 0.0011))) (+ (/ x y) 1.0) x))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.0) || !(x <= 0.0011)) {
		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 <= 0.0011d0))) 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 <= 0.0011)) {
		tmp = (x / y) + 1.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.0) or not (x <= 0.0011):
		tmp = (x / y) + 1.0
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 0.0011))
		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 <= 0.0011)))
		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, 0.0011]], $MachinePrecision]], N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 0.00110000000000000007 < x

   1. Initial program 77.2%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 55.9%

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

      1. *-commutative 55.9%

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

      2. +-commutative 55.9%

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

      3. associate-/l* 63.5%

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

      4. *-lft-identity 63.5%

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

      5. associate-*l/ 63.4%

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

      6. unpow2 63.4%

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

      7. +-commutative 63.4%

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

      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 96.1%

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

   9. Taylor expanded in y around 0 96.1%

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

       1. *-lft-identity 96.1%

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

       2. associate-*l/ 95.9%

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

       3. +-commutative 95.9%

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

       4. distribute-lft-in 95.9%

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

       5. lft-mult-inverse 95.9%

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

       6. associate-*l/ 96.1%

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

       7. *-lft-identity 96.1%

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

   11. Simplified 96.1%

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

   if -1 < x < 0.00110000000000000007

   1. Initial program 99.8%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 0 72.7%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.0011\right):\\ \;\;\;\;\frac{x}{y} + 1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -14200000 \lor \neg \left(x \leq 5.2 \cdot 10^{+14}\right):\\ \;\;\;\;\frac{x}{y} + 1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -14200000.0) (not (<= x 5.2e+14)))
   (+ (/ x y) 1.0)
   (/ x (+ x 1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -14200000.0) || !(x <= 5.2e+14)) {
		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 <= (-14200000.0d0)) .or. (.not. (x <= 5.2d+14))) 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 <= -14200000.0) || !(x <= 5.2e+14)) {
		tmp = (x / y) + 1.0;
	} else {
		tmp = x / (x + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -14200000.0) or not (x <= 5.2e+14):
		tmp = (x / y) + 1.0
	else:
		tmp = x / (x + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -14200000.0) || !(x <= 5.2e+14))
		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 <= -14200000.0) || ~((x <= 5.2e+14)))
		tmp = (x / y) + 1.0;
	else
		tmp = x / (x + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -14200000.0], N[Not[LessEqual[x, 5.2e+14]], $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 < -1.42e7 or 5.2e14 < 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 53.8%

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

      1. *-commutative 53.8%

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

      2. +-commutative 53.8%

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

      3. associate-/l* 61.8%

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

      4. *-lft-identity 61.8%

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

      5. associate-*l/ 61.8%

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

      6. unpow2 61.8%

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

      7. +-commutative 61.8%

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

      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. Taylor expanded in x around inf 98.9%

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

   9. Taylor expanded in y around 0 98.9%

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

       1. *-lft-identity 98.9%

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

       2. associate-*l/ 98.6%

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

       3. +-commutative 98.6%

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

       4. distribute-lft-in 98.6%

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

       5. lft-mult-inverse 98.6%

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

       6. associate-*l/ 98.9%

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

       7. *-lft-identity 98.9%

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

   11. Simplified 98.9%

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

   if -1.42e7 < x < 5.2e14

   1. Initial program 99.8%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 72.2%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -14200000 \lor \neg \left(x \leq 5.2 \cdot 10^{+14}\right):\\ \;\;\;\;\frac{x}{y} + 1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 2.8 \cdot 10^{+26}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 2.8e+26))) (/ x y) x))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.0) || !(x <= 2.8e+26)) {
		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 <= 2.8d+26))) 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 <= 2.8e+26)) {
		tmp = x / y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.0) or not (x <= 2.8e+26):
		tmp = x / y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 2.8e+26))
		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 <= 2.8e+26)))
		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, 2.8e+26]], $MachinePrecision]], N[(x / y), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 2.8e26 < x

   1. Initial program 75.8%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 75.6%

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

   if -1 < x < 2.8e26

   1. Initial program 99.8%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 0 68.7%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 2.8 \cdot 10^{+26}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 38.7% accurate, 11.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 x)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return x

Julia

function code(x, y)
	return x
end

MATLAB

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

Wolfram

code[x_, y_] := x

Derivation

1. Initial program 87.9%

   \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
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 0 36.5%

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

6. Final simplification 36.5%

   \[\leadsto x \]
7. 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 2024072 
(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)))