Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1

Percentage Accurate: 88.0% → 99.9%

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: 88.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.3%

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

   1. *-commutative 84.3%

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

   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%

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

Alternative 2: 72.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.55 \cdot 10^{+97}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -3 \cdot 10^{+16}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -1 \lor \neg \left(x \leq 0.112\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x - x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.55e+97)
   (/ x y)
   (if (<= x -3e+16)
     1.0
     (if (or (<= x -1.0) (not (<= x 0.112))) (/ x y) (- x (* x x))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.55e+97) {
		tmp = x / y;
	} else if (x <= -3e+16) {
		tmp = 1.0;
	} else if ((x <= -1.0) || !(x <= 0.112)) {
		tmp = x / y;
	} else {
		tmp = x - (x * 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 <= (-1.55d+97)) then
        tmp = x / y
    else if (x <= (-3d+16)) then
        tmp = 1.0d0
    else if ((x <= (-1.0d0)) .or. (.not. (x <= 0.112d0))) then
        tmp = x / y
    else
        tmp = x - (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.55e+97) {
		tmp = x / y;
	} else if (x <= -3e+16) {
		tmp = 1.0;
	} else if ((x <= -1.0) || !(x <= 0.112)) {
		tmp = x / y;
	} else {
		tmp = x - (x * x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.55e+97:
		tmp = x / y
	elif x <= -3e+16:
		tmp = 1.0
	elif (x <= -1.0) or not (x <= 0.112):
		tmp = x / y
	else:
		tmp = x - (x * x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.55e+97)
		tmp = Float64(x / y);
	elseif (x <= -3e+16)
		tmp = 1.0;
	elseif ((x <= -1.0) || !(x <= 0.112))
		tmp = Float64(x / y);
	else
		tmp = Float64(x - Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.55e+97)
		tmp = x / y;
	elseif (x <= -3e+16)
		tmp = 1.0;
	elseif ((x <= -1.0) || ~((x <= 0.112)))
		tmp = x / y;
	else
		tmp = x - (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.55e+97], N[(x / y), $MachinePrecision], If[LessEqual[x, -3e+16], 1.0, If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 0.112]], $MachinePrecision]], N[(x / y), $MachinePrecision], N[(x - N[(x * x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.54999999999999991e97 or -3e16 < x < -1 or 0.112000000000000002 < x

   1. Initial program 63.8%

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

   3. Taylor expanded in x around inf 81.7%

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

   if -1.54999999999999991e97 < x < -3e16

   1. Initial program 95.5%

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

   3. Taylor expanded in x around 0 60.9%

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

   4. Taylor expanded in x around inf 60.9%

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

   if -1 < x < 0.112000000000000002

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 74.6%

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

   4. Taylor expanded in x around 0 74.3%

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

      1. neg-mul-1 74.3%

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

      2. distribute-rgt-in 74.3%

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

      3. distribute-lft-neg-in 74.3%

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

      4. unpow2 74.3%

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

      5. unsub-neg 74.3%

         \[\leadsto \color{blue}{1 \cdot x - {x}^{2}} \]

      6. *-lft-identity 74.3%

         \[\leadsto \color{blue}{x} - {x}^{2} \]

      7. unpow2 74.3%

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

   6. Simplified 74.3%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.55 \cdot 10^{+97}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -3 \cdot 10^{+16}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -1 \lor \neg \left(x \leq 0.112\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x - x \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.55 \cdot 10^{+97}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -6.8 \cdot 10^{+22}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -1 \lor \neg \left(x \leq 0.048\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.55e+97)
   (/ x y)
   (if (<= x -6.8e+22)
     1.0
     (if (or (<= x -1.0) (not (<= x 0.048))) (/ x y) x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.55e+97) {
		tmp = x / y;
	} else if (x <= -6.8e+22) {
		tmp = 1.0;
	} else if ((x <= -1.0) || !(x <= 0.048)) {
		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 <= (-1.55d+97)) then
        tmp = x / y
    else if (x <= (-6.8d+22)) then
        tmp = 1.0d0
    else if ((x <= (-1.0d0)) .or. (.not. (x <= 0.048d0))) 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 <= -1.55e+97) {
		tmp = x / y;
	} else if (x <= -6.8e+22) {
		tmp = 1.0;
	} else if ((x <= -1.0) || !(x <= 0.048)) {
		tmp = x / y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.55e+97:
		tmp = x / y
	elif x <= -6.8e+22:
		tmp = 1.0
	elif (x <= -1.0) or not (x <= 0.048):
		tmp = x / y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.55e+97)
		tmp = Float64(x / y);
	elseif (x <= -6.8e+22)
		tmp = 1.0;
	elseif ((x <= -1.0) || !(x <= 0.048))
		tmp = Float64(x / y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.55e+97)
		tmp = x / y;
	elseif (x <= -6.8e+22)
		tmp = 1.0;
	elseif ((x <= -1.0) || ~((x <= 0.048)))
		tmp = x / y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.55e+97], N[(x / y), $MachinePrecision], If[LessEqual[x, -6.8e+22], 1.0, If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 0.048]], $MachinePrecision]], N[(x / y), $MachinePrecision], x]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.54999999999999991e97 or -6.8e22 < x < -1 or 0.048000000000000001 < x

   1. Initial program 63.8%

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

   3. Taylor expanded in x around inf 81.7%

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

   if -1.54999999999999991e97 < x < -6.8e22

   1. Initial program 95.5%

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

   3. Taylor expanded in x around 0 60.9%

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

   4. Taylor expanded in x around inf 60.9%

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

   if -1 < x < 0.048000000000000001

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 73.5%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.55 \cdot 10^{+97}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -6.8 \cdot 10^{+22}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -1 \lor \neg \left(x \leq 0.048\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.2% accurate, 0.5× speedup

Math

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

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 (/ 1.0 y)))))))

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 + (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 <= 0.8d0))) then
        tmp = (x / y) + 1.0d0
    else
        tmp = x * (1.0d0 + (x * ((-1.0d0) + (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 <= 0.8)) {
		tmp = (x / y) + 1.0;
	} else {
		tmp = x * (1.0 + (x * (-1.0 + (1.0 / y))));
	}
	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 + (1.0 / y))))
	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(-1.0 + Float64(1.0 / y)))));
	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 + (1.0 / y))));
	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[(-1.0 + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 0.80000000000000004 < x

   1. Initial program 68.5%

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

      1. *-commutative 68.5%

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

      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. Taylor expanded in x around inf 98.5%

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

   if -1 < x < 0.80000000000000004

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 99.0%

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

4. Final simplification 98.7%

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

Alternative 5: 98.0% 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 68.5%

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

      1. *-commutative 68.5%

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

      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. Taylor expanded in x around inf 98.5%

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

   if -1 < x < 1

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      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. Taylor expanded in x around 0 98.2%

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

4. Final simplification 98.4%

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

Alternative 6: 86.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -150 \lor \neg \left(x \leq 0.27\right):\\ \;\;\;\;\frac{x}{y} + 1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -150.0) (not (<= x 0.27))) (+ (/ x y) 1.0) (/ x (+ x 1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -150.0) || !(x <= 0.27)) {
		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 <= (-150.0d0)) .or. (.not. (x <= 0.27d0))) 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 <= -150.0) || !(x <= 0.27)) {
		tmp = (x / y) + 1.0;
	} else {
		tmp = x / (x + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -150.0) or not (x <= 0.27):
		tmp = (x / y) + 1.0
	else:
		tmp = x / (x + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -150.0) || !(x <= 0.27))
		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 <= -150.0) || ~((x <= 0.27)))
		tmp = (x / y) + 1.0;
	else
		tmp = x / (x + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -150.0], N[Not[LessEqual[x, 0.27]], $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 < -150 or 0.27000000000000002 < x

   1. Initial program 68.7%

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

      1. *-commutative 68.7%

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

      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. Taylor expanded in x around inf 97.9%

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

   if -150 < x < 0.27000000000000002

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 74.6%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -150 \lor \neg \left(x \leq 0.27\right):\\ \;\;\;\;\frac{x}{y} + 1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.55 \cdot 10^{+97} \lor \neg \left(x \leq 380000000000\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.55e+97) (not (<= x 380000000000.0)))
   (/ x y)
   (/ x (+ x 1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.55e+97) || !(x <= 380000000000.0)) {
		tmp = 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 <= (-1.55d+97)) .or. (.not. (x <= 380000000000.0d0))) then
        tmp = 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 <= -1.55e+97) || !(x <= 380000000000.0)) {
		tmp = x / y;
	} else {
		tmp = x / (x + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.55e+97) or not (x <= 380000000000.0):
		tmp = x / y
	else:
		tmp = x / (x + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.55e+97) || !(x <= 380000000000.0))
		tmp = 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 <= -1.55e+97) || ~((x <= 380000000000.0)))
		tmp = x / y;
	else
		tmp = x / (x + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.55e+97], N[Not[LessEqual[x, 380000000000.0]], $MachinePrecision]], N[(x / y), $MachinePrecision], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.54999999999999991e97 or 3.8e11 < x

   1. Initial program 62.0%

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

   3. Taylor expanded in x around inf 83.9%

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

   if -1.54999999999999991e97 < x < 3.8e11

   1. Initial program 99.3%

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

   3. Taylor expanded in x around 0 71.0%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.55 \cdot 10^{+97} \lor \neg \left(x \leq 380000000000\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 49.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{+16}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.65e+16) 1.0 (if (<= x 1.0) x 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.65e+16) {
		tmp = 1.0;
	} else if (x <= 1.0) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.65d+16)) then
        tmp = 1.0d0
    else if (x <= 1.0d0) then
        tmp = x
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.65e+16) {
		tmp = 1.0;
	} else if (x <= 1.0) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.65e+16:
		tmp = 1.0
	elif x <= 1.0:
		tmp = x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.65e+16)
		tmp = 1.0;
	elseif (x <= 1.0)
		tmp = x;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.65e+16)
		tmp = 1.0;
	elseif (x <= 1.0)
		tmp = x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.65e+16], 1.0, If[LessEqual[x, 1.0], x, 1.0]]

Derivation

1. Split input into 2 regimes

2. if x < -1.65e16 or 1 < x

   1. Initial program 68.0%

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

   3. Taylor expanded in x around 0 25.9%

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

   4. Taylor expanded in x around inf 25.7%

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

   if -1.65e16 < x < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 71.9%

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

Alternative 9: 14.5% 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 84.3%

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

3. Taylor expanded in x around 0 50.0%

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

4. Taylor expanded in x around inf 14.6%

   \[\leadsto \color{blue}{1} \]
5. 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 2024097 
(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)))