Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1

Percentage Accurate: 88.3% → 99.9%

Time: 6.3s

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: 88.3% 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 88.6%

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

   1. associate-/l* 99.9%

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

3. Simplified 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 72.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{-13}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{+30} \lor \neg \left(x \leq 5.8 \cdot 10^{+91}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.0)
   (/ x y)
   (if (<= x 2.1e-13)
     x
     (if (or (<= x 3.9e+30) (not (<= x 5.8e+91))) (/ x y) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = x / y;
	} else if (x <= 2.1e-13) {
		tmp = x;
	} else if ((x <= 3.9e+30) || !(x <= 5.8e+91)) {
		tmp = x / y;
	} 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 <= (-1.0d0)) then
        tmp = x / y
    else if (x <= 2.1d-13) then
        tmp = x
    else if ((x <= 3.9d+30) .or. (.not. (x <= 5.8d+91))) then
        tmp = x / y
    else
        tmp = 1.0d0
    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 <= 2.1e-13) {
		tmp = x;
	} else if ((x <= 3.9e+30) || !(x <= 5.8e+91)) {
		tmp = x / y;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.0:
		tmp = x / y
	elif x <= 2.1e-13:
		tmp = x
	elif (x <= 3.9e+30) or not (x <= 5.8e+91):
		tmp = x / y
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(x / y);
	elseif (x <= 2.1e-13)
		tmp = x;
	elseif ((x <= 3.9e+30) || !(x <= 5.8e+91))
		tmp = Float64(x / y);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = x / y;
	elseif (x <= 2.1e-13)
		tmp = x;
	elseif ((x <= 3.9e+30) || ~((x <= 5.8e+91)))
		tmp = x / y;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.0], N[(x / y), $MachinePrecision], If[LessEqual[x, 2.1e-13], x, If[Or[LessEqual[x, 3.9e+30], N[Not[LessEqual[x, 5.8e+91]], $MachinePrecision]], N[(x / y), $MachinePrecision], 1.0]]]

Derivation

1. Split input into 3 regimes

2. if x < -1 or 2.09999999999999989e-13 < x < 3.90000000000000011e30 or 5.80000000000000028e91 < x

   1. Initial program 77.7%

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

   5. Taylor expanded in x around inf 84.1%

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

   if -1 < x < 2.09999999999999989e-13

   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%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 68.6%

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

   if 3.90000000000000011e30 < x < 5.80000000000000028e91

   1. Initial program 84.5%

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

   5. Taylor expanded in y around inf 69.3%

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

   6. Taylor expanded in x around inf 69.3%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{-13}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{+30} \lor \neg \left(x \leq 5.8 \cdot 10^{+91}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.32 \cdot 10^{+22}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 550000:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 8.8 \cdot 10^{+30} \lor \neg \left(x \leq 1.1 \cdot 10^{+92}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.32e+22)
   (/ x y)
   (if (<= x 550000.0)
     (/ x (+ x 1.0))
     (if (or (<= x 8.8e+30) (not (<= x 1.1e+92))) (/ x y) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.32e+22) {
		tmp = x / y;
	} else if (x <= 550000.0) {
		tmp = x / (x + 1.0);
	} else if ((x <= 8.8e+30) || !(x <= 1.1e+92)) {
		tmp = x / y;
	} 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 <= (-1.32d+22)) then
        tmp = x / y
    else if (x <= 550000.0d0) then
        tmp = x / (x + 1.0d0)
    else if ((x <= 8.8d+30) .or. (.not. (x <= 1.1d+92))) then
        tmp = x / y
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.32e+22) {
		tmp = x / y;
	} else if (x <= 550000.0) {
		tmp = x / (x + 1.0);
	} else if ((x <= 8.8e+30) || !(x <= 1.1e+92)) {
		tmp = x / y;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.32e+22:
		tmp = x / y
	elif x <= 550000.0:
		tmp = x / (x + 1.0)
	elif (x <= 8.8e+30) or not (x <= 1.1e+92):
		tmp = x / y
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.32e+22)
		tmp = Float64(x / y);
	elseif (x <= 550000.0)
		tmp = Float64(x / Float64(x + 1.0));
	elseif ((x <= 8.8e+30) || !(x <= 1.1e+92))
		tmp = Float64(x / y);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.32e+22)
		tmp = x / y;
	elseif (x <= 550000.0)
		tmp = x / (x + 1.0);
	elseif ((x <= 8.8e+30) || ~((x <= 1.1e+92)))
		tmp = x / y;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.32e+22], N[(x / y), $MachinePrecision], If[LessEqual[x, 550000.0], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[x, 8.8e+30], N[Not[LessEqual[x, 1.1e+92]], $MachinePrecision]], N[(x / y), $MachinePrecision], 1.0]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.32e22 or 5.5e5 < x < 8.7999999999999999e30 or 1.09999999999999996e92 < x

   1. Initial program 76.4%

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

   5. Taylor expanded in x around inf 88.6%

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

   if -1.32e22 < x < 5.5e5

   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%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around inf 68.8%

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

   if 8.7999999999999999e30 < x < 1.09999999999999996e92

   1. Initial program 84.5%

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

   5. Taylor expanded in y around inf 69.3%

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

   6. Taylor expanded in x around inf 69.3%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.32 \cdot 10^{+22}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 550000:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 8.8 \cdot 10^{+30} \lor \neg \left(x \leq 1.1 \cdot 10^{+92}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.15 \cdot 10^{+21}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 155000:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 8.8 \cdot 10^{+30}:\\ \;\;\;\;\frac{x + -1}{y}\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{+91}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.15e+21)
   (/ x y)
   (if (<= x 155000.0)
     (/ x (+ x 1.0))
     (if (<= x 8.8e+30) (/ (+ x -1.0) y) (if (<= x 6.2e+91) 1.0 (/ x y))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.15e+21) {
		tmp = x / y;
	} else if (x <= 155000.0) {
		tmp = x / (x + 1.0);
	} else if (x <= 8.8e+30) {
		tmp = (x + -1.0) / y;
	} else if (x <= 6.2e+91) {
		tmp = 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 <= (-1.15d+21)) then
        tmp = x / y
    else if (x <= 155000.0d0) then
        tmp = x / (x + 1.0d0)
    else if (x <= 8.8d+30) then
        tmp = (x + (-1.0d0)) / y
    else if (x <= 6.2d+91) then
        tmp = 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 <= -1.15e+21) {
		tmp = x / y;
	} else if (x <= 155000.0) {
		tmp = x / (x + 1.0);
	} else if (x <= 8.8e+30) {
		tmp = (x + -1.0) / y;
	} else if (x <= 6.2e+91) {
		tmp = 1.0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.15e+21:
		tmp = x / y
	elif x <= 155000.0:
		tmp = x / (x + 1.0)
	elif x <= 8.8e+30:
		tmp = (x + -1.0) / y
	elif x <= 6.2e+91:
		tmp = 1.0
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.15e+21)
		tmp = Float64(x / y);
	elseif (x <= 155000.0)
		tmp = Float64(x / Float64(x + 1.0));
	elseif (x <= 8.8e+30)
		tmp = Float64(Float64(x + -1.0) / y);
	elseif (x <= 6.2e+91)
		tmp = 1.0;
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.15e+21)
		tmp = x / y;
	elseif (x <= 155000.0)
		tmp = x / (x + 1.0);
	elseif (x <= 8.8e+30)
		tmp = (x + -1.0) / y;
	elseif (x <= 6.2e+91)
		tmp = 1.0;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.15e+21], N[(x / y), $MachinePrecision], If[LessEqual[x, 155000.0], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 8.8e+30], N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[x, 6.2e+91], 1.0, N[(x / y), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.15e21 or 6.19999999999999995e91 < x

   1. Initial program 73.6%

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

   5. Taylor expanded in x around inf 89.5%

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

   if -1.15e21 < x < 155000

   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%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around inf 68.8%

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

   if 155000 < x < 8.7999999999999999e30

   1. Initial program 99.7%

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

   5. Taylor expanded in x around inf 99.6%

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

   6. Taylor expanded in y around 0 83.6%

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

   if 8.7999999999999999e30 < x < 6.19999999999999995e91

   1. Initial program 84.5%

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

   5. Taylor expanded in y around inf 69.3%

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

   6. Taylor expanded in x around inf 69.3%

      \[\leadsto \color{blue}{1} \]
3. Recombined 4 regimes into one program.

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.15 \cdot 10^{+21}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 155000:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 8.8 \cdot 10^{+30}:\\ \;\;\;\;\frac{x + -1}{y}\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{+91}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+21} \lor \neg \left(x \leq 112000\right):\\ \;\;\;\;1 + \frac{x + -1}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1e+21) (not (<= x 112000.0)))
   (+ 1.0 (/ (+ x -1.0) y))
   (/ x (+ x 1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1e+21) || !(x <= 112000.0)) {
		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 <= (-1d+21)) .or. (.not. (x <= 112000.0d0))) 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 <= -1e+21) || !(x <= 112000.0)) {
		tmp = 1.0 + ((x + -1.0) / y);
	} else {
		tmp = x / (x + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1e+21) or not (x <= 112000.0):
		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 <= -1e+21) || !(x <= 112000.0))
		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 <= -1e+21) || ~((x <= 112000.0)))
		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, -1e+21], N[Not[LessEqual[x, 112000.0]], $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 < -1e21 or 112000 < x

   1. Initial program 77.1%

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

   5. Taylor expanded in x around inf 100.0%

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

      1. associate--l+ 100.0%

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

      2. +-commutative 100.0%

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

      3. sub-div 100.0%

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

   7. Applied egg-rr 100.0%

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

   if -1e21 < x < 112000

   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%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around inf 68.8%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+21} \lor \neg \left(x \leq 112000\right):\\ \;\;\;\;1 + \frac{x + -1}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 48.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{-13}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.0) 1.0 (if (<= x 2.1e-13) x 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = 1.0;
	} else if (x <= 2.1e-13) {
		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 <= (-1.0d0)) then
        tmp = 1.0d0
    else if (x <= 2.1d-13) 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 <= -1.0) {
		tmp = 1.0;
	} else if (x <= 2.1e-13) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.0:
		tmp = 1.0
	elif x <= 2.1e-13:
		tmp = x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.0)
		tmp = 1.0;
	elseif (x <= 2.1e-13)
		tmp = x;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = 1.0;
	elseif (x <= 2.1e-13)
		tmp = x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.0], 1.0, If[LessEqual[x, 2.1e-13], x, 1.0]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 2.09999999999999989e-13 < x

   1. Initial program 78.3%

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

   5. Taylor expanded in y around inf 20.4%

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

   6. Taylor expanded in x around inf 19.3%

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

   if -1 < x < 2.09999999999999989e-13

   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%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 68.6%

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

4. Final simplification 43.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{-13}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 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 88.6%

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

   1. associate-/l* 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in y around inf 43.9%

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

6. Taylor expanded in x around inf 11.7%

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

7. Final simplification 11.7%

   \[\leadsto 1 \]
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 2024024 
(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)))