Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1

Percentage Accurate: 87.9% → 99.9%

Time: 7.3s

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: 99.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + 1}\\ t\_0 + t\_0 \cdot \frac{x}{y} \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ x 1.0)))) (+ t_0 (* t_0 (/ x y)))))

C

double code(double x, double y) {
	double t_0 = x / (x + 1.0);
	return t_0 + (t_0 * (x / y));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    t_0 = x / (x + 1.0d0)
    code = t_0 + (t_0 * (x / y))
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (x + 1.0);
	return t_0 + (t_0 * (x / y));
}

Python

def code(x, y):
	t_0 = x / (x + 1.0)
	return t_0 + (t_0 * (x / y))

Julia

function code(x, y)
	t_0 = Float64(x / Float64(x + 1.0))
	return Float64(t_0 + Float64(t_0 * Float64(x / y)))
end

MATLAB

function tmp = code(x, y)
	t_0 = x / (x + 1.0);
	tmp = t_0 + (t_0 * (x / y));
end

Wolfram

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

Derivation

1. Initial program 87.2%

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

   1. *-commutative 87.2%

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

   2. associate-/l* 99.9%

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

4. Applied egg-rr 99.9%

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

   1. *-commutative 99.9%

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

   2. +-commutative 99.9%

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

   3. distribute-lft-in 100.0%

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

   4. *-commutative 100.0%

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

   5. *-un-lft-identity 100.0%

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

6. Applied egg-rr 100.0%

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

7. Final simplification 100.0%

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

Alternative 2: 86.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + 1}\\ \mathbf{if}\;x \leq -1950000:\\ \;\;\;\;1 + \frac{x}{y}\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-83}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ x 1.0))))
   (if (<= x -1950000.0)
     (+ 1.0 (/ x y))
     (if (<= x 2.6e-83) t_0 (+ t_0 (/ x y))))))

C

double code(double x, double y) {
	double t_0 = x / (x + 1.0);
	double tmp;
	if (x <= -1950000.0) {
		tmp = 1.0 + (x / y);
	} else if (x <= 2.6e-83) {
		tmp = t_0;
	} else {
		tmp = t_0 + (x / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / (x + 1.0d0)
    if (x <= (-1950000.0d0)) then
        tmp = 1.0d0 + (x / y)
    else if (x <= 2.6d-83) then
        tmp = t_0
    else
        tmp = t_0 + (x / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (x + 1.0);
	double tmp;
	if (x <= -1950000.0) {
		tmp = 1.0 + (x / y);
	} else if (x <= 2.6e-83) {
		tmp = t_0;
	} else {
		tmp = t_0 + (x / y);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (x + 1.0)
	tmp = 0
	if x <= -1950000.0:
		tmp = 1.0 + (x / y)
	elif x <= 2.6e-83:
		tmp = t_0
	else:
		tmp = t_0 + (x / y)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(x + 1.0))
	tmp = 0.0
	if (x <= -1950000.0)
		tmp = Float64(1.0 + Float64(x / y));
	elseif (x <= 2.6e-83)
		tmp = t_0;
	else
		tmp = Float64(t_0 + Float64(x / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (x + 1.0);
	tmp = 0.0;
	if (x <= -1950000.0)
		tmp = 1.0 + (x / y);
	elseif (x <= 2.6e-83)
		tmp = t_0;
	else
		tmp = t_0 + (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1950000.0], N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.6e-83], t$95$0, N[(t$95$0 + N[(x / y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.95e6

   1. Initial program 71.7%

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

      1. *-commutative 71.7%

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

      2. associate-/l* 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. *-commutative 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-lft-in 100.0%

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

      4. *-commutative 100.0%

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

      5. *-un-lft-identity 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around inf 98.9%

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

   8. Taylor expanded in x around inf 98.7%

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

      1. distribute-rgt-in 98.7%

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

      2. lft-mult-inverse 98.7%

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

      3. associate-*l/ 98.9%

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

      4. *-lft-identity 98.9%

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

   10. Simplified 98.9%

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

   if -1.95e6 < x < 2.60000000000000009e-83

   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 79.4%

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

   if 2.60000000000000009e-83 < x

   1. Initial program 77.6%

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

      1. *-commutative 77.6%

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

      2. associate-/l* 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. *-commutative 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-lft-in 100.0%

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

      4. *-commutative 100.0%

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

      5. *-un-lft-identity 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around inf 86.2%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1950000:\\ \;\;\;\;1 + \frac{x}{y}\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-83}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} + \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.0146\right):\\ \;\;\;\;1 + \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 0.0146))) (+ 1.0 (/ x y)) x))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.0) || !(x <= 0.0146)) {
		tmp = 1.0 + (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 <= 0.0146d0))) then
        tmp = 1.0d0 + (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 <= 0.0146)) {
		tmp = 1.0 + (x / y);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.0) or not (x <= 0.0146):
		tmp = 1.0 + (x / y)
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 0.0146))
		tmp = Float64(1.0 + 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 <= 0.0146)))
		tmp = 1.0 + (x / y);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 0.0146]], $MachinePrecision]], N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 0.0146000000000000001 < x

   1. Initial program 70.8%

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

      1. *-commutative 70.8%

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

      2. associate-/l* 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. *-commutative 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-lft-in 100.0%

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

      4. *-commutative 100.0%

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

      5. *-un-lft-identity 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around inf 98.5%

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

   8. Taylor expanded in x around inf 97.8%

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

      1. distribute-rgt-in 97.8%

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

      2. lft-mult-inverse 97.8%

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

      3. associate-*l/ 98.1%

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

      4. *-lft-identity 98.1%

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

   10. Simplified 98.1%

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

   if -1 < x < 0.0146000000000000001

   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 74.8%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.0146\right):\\ \;\;\;\;1 + \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -170000 \lor \neg \left(x \leq 196\right):\\ \;\;\;\;1 + \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -170000.0) (not (<= x 196.0))) (+ 1.0 (/ x y)) (/ x (+ x 1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -170000.0) || !(x <= 196.0)) {
		tmp = 1.0 + (x / y);
	} 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 <= (-170000.0d0)) .or. (.not. (x <= 196.0d0))) then
        tmp = 1.0d0 + (x / y)
    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 <= -170000.0) || !(x <= 196.0)) {
		tmp = 1.0 + (x / y);
	} else {
		tmp = x / (x + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -170000.0) or not (x <= 196.0):
		tmp = 1.0 + (x / y)
	else:
		tmp = x / (x + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -170000.0) || !(x <= 196.0))
		tmp = Float64(1.0 + Float64(x / y));
	else
		tmp = Float64(x / Float64(x + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -170000.0) || ~((x <= 196.0)))
		tmp = 1.0 + (x / y);
	else
		tmp = x / (x + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -170000.0], N[Not[LessEqual[x, 196.0]], $MachinePrecision]], N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.7e5 or 196 < x

   1. Initial program 70.8%

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

      1. *-commutative 70.8%

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

      2. associate-/l* 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. *-commutative 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-lft-in 100.0%

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

      4. *-commutative 100.0%

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

      5. *-un-lft-identity 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around inf 98.5%

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

   8. Taylor expanded in x around inf 97.8%

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

      1. distribute-rgt-in 97.8%

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

      2. lft-mult-inverse 97.8%

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

      3. associate-*l/ 98.1%

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

      4. *-lft-identity 98.1%

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

   10. Simplified 98.1%

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

   if -1.7e5 < x < 196

   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 75.3%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -170000 \lor \neg \left(x \leq 196\right):\\ \;\;\;\;1 + \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.112\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 0.112))) (/ x y) x))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.0) || !(x <= 0.112)) {
		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 <= 0.112d0))) 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 <= 0.112)) {
		tmp = x / y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.0) or not (x <= 0.112):
		tmp = x / y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 0.112))
		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 <= 0.112)))
		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, 0.112]], $MachinePrecision]], N[(x / y), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 0.112000000000000002 < x

   1. Initial program 70.8%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 78.6%

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

   if -1 < x < 0.112000000000000002

   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 74.8%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.112\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 48.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{+14}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 20000000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -3.1e+14) 1.0 (if (<= x 20000000.0) x 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -3.1e+14) {
		tmp = 1.0;
	} else if (x <= 20000000.0) {
		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 <= (-3.1d+14)) then
        tmp = 1.0d0
    else if (x <= 20000000.0d0) 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 <= -3.1e+14) {
		tmp = 1.0;
	} else if (x <= 20000000.0) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -3.1e+14:
		tmp = 1.0
	elif x <= 20000000.0:
		tmp = x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -3.1e+14)
		tmp = 1.0;
	elseif (x <= 20000000.0)
		tmp = x;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -3.1e+14)
		tmp = 1.0;
	elseif (x <= 20000000.0)
		tmp = x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -3.1e+14], 1.0, If[LessEqual[x, 20000000.0], x, 1.0]]

Derivation

1. Split input into 2 regimes

2. if x < -3.1e14 or 2e7 < x

   1. Initial program 69.5%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 inf 22.0%

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

   6. Taylor expanded in x around inf 21.6%

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

   if -3.1e14 < x < 2e7

   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 72.5%

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

4. Final simplification 51.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{+14}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 20000000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.2%

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

   1. associate-/l* 99.9%

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

3. Simplified 99.9%

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

5. Final simplification 99.9%

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

Alternative 8: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x}{x + 1} \cdot \left(1 + \frac{x}{y}\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.2%

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

   1. *-commutative 87.2%

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

   2. associate-/l* 99.9%

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

4. Applied egg-rr 99.9%

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

5. Final simplification 99.9%

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

Alternative 9: 14.0% 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 87.2%

   \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
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 51.6%

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

6. Taylor expanded in x around inf 11.2%

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

7. Final simplification 11.2%

   \[\leadsto 1 \]
8. 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 2024066 
(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)))