Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1

Percentage Accurate: 88.5% → 99.9%

Time: 9.7s

Alternatives: 8

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.5% 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

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

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(x / Float64(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[(x / N[(N[(x + 1.0), $MachinePrecision] / N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 88.2%

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

   1. associate-/l* 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 86.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -860000 \lor \neg \left(x \leq 30000000\right):\\ \;\;\;\;1 + \frac{x + -1}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -860000.0) (not (<= x 30000000.0)))
   (+ 1.0 (/ (+ x -1.0) y))
   (/ x (+ x 1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -860000.0) || !(x <= 30000000.0)) {
		tmp = 1.0 + ((x + -1.0) / 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 <= (-860000.0d0)) .or. (.not. (x <= 30000000.0d0))) then
        tmp = 1.0d0 + ((x + (-1.0d0)) / 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 <= -860000.0) || !(x <= 30000000.0)) {
		tmp = 1.0 + ((x + -1.0) / y);
	} else {
		tmp = x / (x + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -860000.0) or not (x <= 30000000.0):
		tmp = 1.0 + ((x + -1.0) / y)
	else:
		tmp = x / (x + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -860000.0) || !(x <= 30000000.0))
		tmp = Float64(1.0 + Float64(Float64(x + -1.0) / 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 <= -860000.0) || ~((x <= 30000000.0)))
		tmp = 1.0 + ((x + -1.0) / y);
	else
		tmp = x / (x + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -860000.0], N[Not[LessEqual[x, 30000000.0]], $MachinePrecision]], N[(1.0 + N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -8.6e5 or 3e7 < x

   1. Initial program 74.7%

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

      1. distribute-lft-in 74.7%

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

      2. *-rgt-identity 74.7%

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

   3. Applied egg-rr 74.7%

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

   4. Taylor expanded in x around inf 99.5%

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

      1. +-commutative 99.5%

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

      2. associate-+r- 99.5%

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

      3. +-commutative 99.5%

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

      4. associate-+l- 99.5%

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

      5. div-sub 99.5%

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

   6. Simplified 99.5%

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

   if -8.6e5 < x < 3e7

   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%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around inf 82.4%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -860000 \lor \neg \left(x \leq 30000000\right):\\ \;\;\;\;1 + \frac{x + -1}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1}\\ \end{array} \]

Alternative 3: 86.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -190000 \lor \neg \left(x \leq 175000000\right):\\ \;\;\;\;1 + \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -190000.0) (not (<= x 175000000.0)))
   (+ 1.0 (/ x y))
   (/ x (+ x 1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -190000.0) || !(x <= 175000000.0)) {
		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 <= (-190000.0d0)) .or. (.not. (x <= 175000000.0d0))) 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 <= -190000.0) || !(x <= 175000000.0)) {
		tmp = 1.0 + (x / y);
	} else {
		tmp = x / (x + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -190000.0) or not (x <= 175000000.0):
		tmp = 1.0 + (x / y)
	else:
		tmp = x / (x + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -190000.0) || !(x <= 175000000.0))
		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 <= -190000.0) || ~((x <= 175000000.0)))
		tmp = 1.0 + (x / y);
	else
		tmp = x / (x + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -190000.0], N[Not[LessEqual[x, 175000000.0]], $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 < -1.9e5 or 1.75e8 < x

   1. Initial program 74.7%

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

      1. distribute-lft-in 74.7%

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

      2. *-rgt-identity 74.7%

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

   3. Applied egg-rr 74.7%

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

   4. Taylor expanded in x around inf 99.5%

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

      1. +-commutative 99.5%

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

      2. associate-+r- 99.5%

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

      3. +-commutative 99.5%

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

      4. associate-+l- 99.5%

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

      5. div-sub 99.5%

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

   6. Simplified 99.5%

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

   7. Taylor expanded in x around inf 99.0%

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

      1. neg-mul-1 99.0%

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

      2. distribute-neg-frac 99.0%

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

   9. Simplified 99.0%

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

   if -1.9e5 < x < 1.75e8

   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%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around inf 82.4%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -190000 \lor \neg \left(x \leq 175000000\right):\\ \;\;\;\;1 + \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1}\\ \end{array} \]

Alternative 4: 74.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.35 \cdot 10^{+14}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{+35}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.35e+14) (/ x y) (if (<= x 4.5e+35) (/ x (+ x 1.0)) (/ x y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.35e+14) {
		tmp = x / y;
	} else if (x <= 4.5e+35) {
		tmp = x / (x + 1.0);
	} 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 <= (-2.35d+14)) then
        tmp = x / y
    else if (x <= 4.5d+35) then
        tmp = x / (x + 1.0d0)
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -2.35e+14) {
		tmp = x / y;
	} else if (x <= 4.5e+35) {
		tmp = x / (x + 1.0);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.35e+14:
		tmp = x / y
	elif x <= 4.5e+35:
		tmp = x / (x + 1.0)
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.35e+14)
		tmp = Float64(x / y);
	elseif (x <= 4.5e+35)
		tmp = Float64(x / Float64(x + 1.0));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.35e+14)
		tmp = x / y;
	elseif (x <= 4.5e+35)
		tmp = x / (x + 1.0);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.35e+14], N[(x / y), $MachinePrecision], If[LessEqual[x, 4.5e+35], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.35e14 or 4.4999999999999997e35 < x

   1. Initial program 72.9%

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

      1. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 80.0%

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

   if -2.35e14 < x < 4.4999999999999997e35

   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%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around inf 81.1%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.35 \cdot 10^{+14}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{+35}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 5: 74.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -720000000000:\\ \;\;\;\;\frac{x + -1}{y}\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{+32}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -720000000000.0)
   (/ (+ x -1.0) y)
   (if (<= x 6.2e+32) (/ x (+ x 1.0)) (/ x y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -720000000000.0) {
		tmp = (x + -1.0) / y;
	} else if (x <= 6.2e+32) {
		tmp = x / (x + 1.0);
	} 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 <= (-720000000000.0d0)) then
        tmp = (x + (-1.0d0)) / y
    else if (x <= 6.2d+32) then
        tmp = x / (x + 1.0d0)
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -720000000000.0) {
		tmp = (x + -1.0) / y;
	} else if (x <= 6.2e+32) {
		tmp = x / (x + 1.0);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -720000000000.0:
		tmp = (x + -1.0) / y
	elif x <= 6.2e+32:
		tmp = x / (x + 1.0)
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -720000000000.0)
		tmp = Float64(Float64(x + -1.0) / y);
	elseif (x <= 6.2e+32)
		tmp = Float64(x / Float64(x + 1.0));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -720000000000.0)
		tmp = (x + -1.0) / y;
	elseif (x <= 6.2e+32)
		tmp = x / (x + 1.0);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -720000000000.0], N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[x, 6.2e+32], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.2e11

   1. Initial program 73.1%

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

      1. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 99.7%

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

   5. Taylor expanded in y around 0 80.7%

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

   if -7.2e11 < x < 6.19999999999999986e32

   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%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around inf 81.5%

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

   if 6.19999999999999986e32 < x

   1. Initial program 73.8%

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

      1. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 77.9%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -720000000000:\\ \;\;\;\;\frac{x + -1}{y}\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{+32}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 6: 73.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-17}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.0) (/ x y) (if (<= x 4e-17) x (/ x y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = x / y;
	} else if (x <= 4e-17) {
		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 <= (-1.0d0)) then
        tmp = x / y
    else if (x <= 4d-17) 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 <= -1.0) {
		tmp = x / y;
	} else if (x <= 4e-17) {
		tmp = x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.0:
		tmp = x / y
	elif x <= 4e-17:
		tmp = 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 <= 4e-17)
		tmp = 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 <= 4e-17)
		tmp = 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, 4e-17], x, N[(x / y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 4.00000000000000029e-17 < x

   1. Initial program 75.4%

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

      1. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 75.0%

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

   if -1 < x < 4.00000000000000029e-17

   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%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 82.6%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-17}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 7: 49.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2100000:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-17}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2100000.0) 1.0 (if (<= x 4e-17) x 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2100000.0) {
		tmp = 1.0;
	} else if (x <= 4e-17) {
		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 <= (-2100000.0d0)) then
        tmp = 1.0d0
    else if (x <= 4d-17) 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 <= -2100000.0) {
		tmp = 1.0;
	} else if (x <= 4e-17) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2100000.0:
		tmp = 1.0
	elif x <= 4e-17:
		tmp = x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2100000.0)
		tmp = 1.0;
	elseif (x <= 4e-17)
		tmp = x;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2100000.0)
		tmp = 1.0;
	elseif (x <= 4e-17)
		tmp = x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2100000.0], 1.0, If[LessEqual[x, 4e-17], x, 1.0]]

Derivation

1. Split input into 2 regimes

2. if x < -2.1e6 or 4.00000000000000029e-17 < x

   1. Initial program 75.2%

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

      1. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around inf 24.2%

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

   5. Taylor expanded in x around inf 23.5%

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

   if -2.1e6 < x < 4.00000000000000029e-17

   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%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 82.0%

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

4. Final simplification 54.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2100000:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-17}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 8: 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 88.2%

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

   1. associate-/l* 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in y around inf 55.1%

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

5. Taylor expanded in x around inf 13.0%

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

6. Final simplification 13.0%

   \[\leadsto 1 \]

Developer target: 99.9% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023274 
(FPCore (x y)
  :name "Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1"
  :precision binary64

  :herbie-target
  (* (/ x 1.0) (/ (+ (/ x y) 1.0) (+ x 1.0)))

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