Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1

Percentage Accurate: 88.2% → 99.9%

Time: 7.7s

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: 88.2% 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 89.0%

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

   1. clear-num 99.8%

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

   2. un-div-inv 99.9%

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

6. Applied egg-rr 99.9%

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

7. Final simplification 99.9%

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

Alternative 2: 86.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{+61}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -3.1 \cdot 10^{-5}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;x \cdot \left(1 + \left(\frac{x}{y} - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.25e+61)
   (/ x y)
   (if (<= x -3.1e-5)
     (/ x (+ x 1.0))
     (if (<= x 1.0) (* x (+ 1.0 (- (/ x y) x))) (/ x y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.25e+61) {
		tmp = x / y;
	} else if (x <= -3.1e-5) {
		tmp = x / (x + 1.0);
	} else if (x <= 1.0) {
		tmp = x * (1.0 + ((x / y) - 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 <= (-1.25d+61)) then
        tmp = x / y
    else if (x <= (-3.1d-5)) then
        tmp = x / (x + 1.0d0)
    else if (x <= 1.0d0) then
        tmp = x * (1.0d0 + ((x / y) - x))
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.25e+61) {
		tmp = x / y;
	} else if (x <= -3.1e-5) {
		tmp = x / (x + 1.0);
	} else if (x <= 1.0) {
		tmp = x * (1.0 + ((x / y) - x));
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.25e+61:
		tmp = x / y
	elif x <= -3.1e-5:
		tmp = x / (x + 1.0)
	elif x <= 1.0:
		tmp = x * (1.0 + ((x / y) - x))
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.25e+61)
		tmp = Float64(x / y);
	elseif (x <= -3.1e-5)
		tmp = Float64(x / Float64(x + 1.0));
	elseif (x <= 1.0)
		tmp = Float64(x * Float64(1.0 + Float64(Float64(x / y) - x)));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.25e+61)
		tmp = x / y;
	elseif (x <= -3.1e-5)
		tmp = x / (x + 1.0);
	elseif (x <= 1.0)
		tmp = x * (1.0 + ((x / y) - x));
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.25e+61], N[(x / y), $MachinePrecision], If[LessEqual[x, -3.1e-5], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.0], N[(x * N[(1.0 + N[(N[(x / y), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.25000000000000004e61 or 1 < x

   1. Initial program 74.8%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 inf 72.1%

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

   if -1.25000000000000004e61 < x < -3.10000000000000014e-5

   1. Initial program 99.9%

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

      1. associate-/l* 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in y around inf 86.3%

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

   if -3.10000000000000014e-5 < x < 1

   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 99.6%

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

   6. Taylor expanded in y around inf 99.6%

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

      1. neg-mul-1 99.6%

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

      2. +-commutative 99.6%

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

      3. unsub-neg 99.6%

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

   8. Simplified 99.6%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{+61}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -3.1 \cdot 10^{-5}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;x \cdot \left(1 + \left(\frac{x}{y} - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{+63}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -1:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{elif}\;x \leq 0.04:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.4e+63)
   (/ x y)
   (if (<= x -1.0) (+ 1.0 (/ -1.0 x)) (if (<= x 0.04) x (/ x y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.4e+63) {
		tmp = x / y;
	} else if (x <= -1.0) {
		tmp = 1.0 + (-1.0 / x);
	} else if (x <= 0.04) {
		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 <= (-1.4d+63)) then
        tmp = x / y
    else if (x <= (-1.0d0)) then
        tmp = 1.0d0 + ((-1.0d0) / x)
    else if (x <= 0.04d0) 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 <= -1.4e+63) {
		tmp = x / y;
	} else if (x <= -1.0) {
		tmp = 1.0 + (-1.0 / x);
	} else if (x <= 0.04) {
		tmp = x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.4e+63:
		tmp = x / y
	elif x <= -1.0:
		tmp = 1.0 + (-1.0 / x)
	elif x <= 0.04:
		tmp = x
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.4e+63)
		tmp = Float64(x / y);
	elseif (x <= -1.0)
		tmp = Float64(1.0 + Float64(-1.0 / x));
	elseif (x <= 0.04)
		tmp = x;
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.4e+63)
		tmp = x / y;
	elseif (x <= -1.0)
		tmp = 1.0 + (-1.0 / x);
	elseif (x <= 0.04)
		tmp = x;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.4e+63], N[(x / y), $MachinePrecision], If[LessEqual[x, -1.0], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.04], x, N[(x / y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.39999999999999993e63 or 0.0400000000000000008 < x

   1. Initial program 74.8%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 inf 72.1%

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

   if -1.39999999999999993e63 < x < -1

   1. Initial program 100.0%

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

      1. associate-/l* 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in y around inf 85.4%

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

   6. Taylor expanded in x around inf 79.5%

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

   if -1 < x < 0.0400000000000000008

   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 80.3%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{+63}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -1:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{elif}\;x \leq 0.04:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{+62}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 6.2:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -5.2e+62) (/ x y) (if (<= x -1.0) 1.0 (if (<= x 6.2) x (/ x y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -5.2e+62) {
		tmp = x / y;
	} else if (x <= -1.0) {
		tmp = 1.0;
	} else if (x <= 6.2) {
		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 <= (-5.2d+62)) then
        tmp = x / y
    else if (x <= (-1.0d0)) then
        tmp = 1.0d0
    else if (x <= 6.2d0) 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 <= -5.2e+62) {
		tmp = x / y;
	} else if (x <= -1.0) {
		tmp = 1.0;
	} else if (x <= 6.2) {
		tmp = x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -5.2e+62:
		tmp = x / y
	elif x <= -1.0:
		tmp = 1.0
	elif x <= 6.2:
		tmp = x
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -5.2e+62)
		tmp = Float64(x / y);
	elseif (x <= -1.0)
		tmp = 1.0;
	elseif (x <= 6.2)
		tmp = x;
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -5.2e+62)
		tmp = x / y;
	elseif (x <= -1.0)
		tmp = 1.0;
	elseif (x <= 6.2)
		tmp = x;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -5.2e+62], N[(x / y), $MachinePrecision], If[LessEqual[x, -1.0], 1.0, If[LessEqual[x, 6.2], x, N[(x / y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.19999999999999968e62 or 6.20000000000000018 < x

   1. Initial program 74.8%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 inf 72.1%

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

   if -5.19999999999999968e62 < x < -1

   1. Initial program 100.0%

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

      1. associate-/l* 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in y around inf 85.4%

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

      1. clear-num 85.3%

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

      2. +-commutative 85.3%

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

      3. associate-/r/ 84.7%

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

   7. Applied egg-rr 84.7%

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

   8. Taylor expanded in x around inf 76.9%

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

   if -1 < x < 6.20000000000000018

   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 80.3%

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

Alternative 5: 73.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+62} \lor \neg \left(x \leq 8.2 \cdot 10^{+93}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.35e+62) (not (<= x 8.2e+93))) (/ x y) (/ x (+ x 1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.35e+62) || !(x <= 8.2e+93)) {
		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 <= (-1.35d+62)) .or. (.not. (x <= 8.2d+93))) 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 <= -1.35e+62) || !(x <= 8.2e+93)) {
		tmp = x / y;
	} else {
		tmp = x / (x + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.35e+62) or not (x <= 8.2e+93):
		tmp = x / y
	else:
		tmp = x / (x + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.35e+62) || !(x <= 8.2e+93))
		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 <= -1.35e+62) || ~((x <= 8.2e+93)))
		tmp = x / y;
	else
		tmp = x / (x + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.35e+62], N[Not[LessEqual[x, 8.2e+93]], $MachinePrecision]], N[(x / y), $MachinePrecision], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.35e62 or 8.2000000000000002e93 < x

   1. Initial program 69.2%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 x around inf 79.0%

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

   if -1.35e62 < x < 8.2000000000000002e93

   1. Initial program 99.3%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 77.7%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+62} \lor \neg \left(x \leq 8.2 \cdot 10^{+93}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 50.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+14}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.0) 1.0 (if (<= x 1.35e+14) x 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = 1.0;
	} else if (x <= 1.35e+14) {
		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 <= 1.35d+14) 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 <= 1.35e+14) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.0:
		tmp = 1.0
	elif x <= 1.35e+14:
		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 <= 1.35e+14)
		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 <= 1.35e+14)
		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, 1.35e+14], x, 1.0]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1.35e14 < x

   1. Initial program 76.7%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 36.3%

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

      1. clear-num 36.3%

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

      2. +-commutative 36.3%

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

      3. associate-/r/ 36.2%

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

   7. Applied egg-rr 36.2%

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

   8. Taylor expanded in x around inf 35.4%

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

   if -1 < x < 1.35e14

   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 78.1%

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

Alternative 7: 99.9% 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 89.0%

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

   1. associate-/l* 99.8%

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

3. Simplified 99.8%

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

5. Final simplification 99.8%

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

Alternative 8: 14.4% 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 89.0%

   \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
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 59.0%

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

   1. clear-num 59.0%

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

   2. +-commutative 59.0%

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

   3. associate-/r/ 59.0%

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

7. Applied egg-rr 59.0%

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

8. Taylor expanded in x around inf 18.5%

   \[\leadsto \color{blue}{1} \]
9. Add Preprocessing

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 2024087 
(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)))