Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1

Percentage Accurate: 88.5% → 99.9%

Time: 7.3s

Alternatives: 9

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.5%

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

   1. *-commutative 85.5%

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

   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. Add Preprocessing

Alternative 2: 98.2% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-1.0d0)) .or. (.not. (x <= 1.0d0))) then
        tmp = (x / y) + 1.0d0
    else
        tmp = x + (x * ((x / y) - x))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 69.7%

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

      1. *-commutative 69.7%

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

      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 97.0%

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

      1. *-rgt-identity 97.0%

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

   7. Applied egg-rr 97.0%

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

   if -1 < 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 97.8%

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

   6. Taylor expanded in y around inf 97.8%

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

      1. neg-mul-1 97.8%

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

      2. +-commutative 97.8%

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

      3. unsub-neg 97.8%

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

   8. Simplified 97.8%

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

      1. +-commutative 97.8%

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

      2. distribute-lft-in 97.9%

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

      3. *-rgt-identity 97.9%

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

   10. Applied egg-rr 97.9%

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

4. Final simplification 97.5%

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

Alternative 3: 98.2% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-1.0d0)) .or. (.not. (x <= 1.0d0))) then
        tmp = (x / y) + 1.0d0
    else
        tmp = x * (1.0d0 + ((x / y) - x))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 69.7%

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

      1. *-commutative 69.7%

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

      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 97.0%

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

      1. *-rgt-identity 97.0%

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

   7. Applied egg-rr 97.0%

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

   if -1 < 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 97.8%

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

   6. Taylor expanded in y around inf 97.8%

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

      1. neg-mul-1 97.8%

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

      2. +-commutative 97.8%

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

      3. unsub-neg 97.8%

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

   8. Simplified 97.8%

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

4. Final simplification 97.5%

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

Alternative 4: 72.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+109}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -7 \cdot 10^{-5}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{-9}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1e+109)
   (/ x y)
   (if (<= x -7e-5) 1.0 (if (<= x 1.85e-9) x (/ x y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1e+109) {
		tmp = x / y;
	} else if (x <= -7e-5) {
		tmp = 1.0;
	} else if (x <= 1.85e-9) {
		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 <= (-1d+109)) then
        tmp = x / y
    else if (x <= (-7d-5)) then
        tmp = 1.0d0
    else if (x <= 1.85d-9) 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 <= -1e+109) {
		tmp = x / y;
	} else if (x <= -7e-5) {
		tmp = 1.0;
	} else if (x <= 1.85e-9) {
		tmp = x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1e+109:
		tmp = x / y
	elif x <= -7e-5:
		tmp = 1.0
	elif x <= 1.85e-9:
		tmp = x
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1e+109)
		tmp = Float64(x / y);
	elseif (x <= -7e-5)
		tmp = 1.0;
	elseif (x <= 1.85e-9)
		tmp = x;
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1e+109)
		tmp = x / y;
	elseif (x <= -7e-5)
		tmp = 1.0;
	elseif (x <= 1.85e-9)
		tmp = x;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1e+109], N[(x / y), $MachinePrecision], If[LessEqual[x, -7e-5], 1.0, If[LessEqual[x, 1.85e-9], x, N[(x / y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -9.99999999999999982e108 or 1.85e-9 < x

   1. Initial program 63.3%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 81.7%

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

   if -9.99999999999999982e108 < x < -6.9999999999999994e-5

   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 y around inf 68.8%

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

      1. +-commutative 68.8%

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

   7. Simplified 68.8%

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

   8. Taylor expanded in x around inf 62.0%

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

   if -6.9999999999999994e-5 < x < 1.85e-9

   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 78.4%

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

Alternative 5: 86.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -290000 \lor \neg \left(x \leq 245000\right):\\ \;\;\;\;\frac{x}{y} + 1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -290000.0) (not (<= x 245000.0)))
   (+ (/ x y) 1.0)
   (/ x (+ x 1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -290000.0) || !(x <= 245000.0)) {
		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 <= (-290000.0d0)) .or. (.not. (x <= 245000.0d0))) 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 <= -290000.0) || !(x <= 245000.0)) {
		tmp = (x / y) + 1.0;
	} else {
		tmp = x / (x + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -290000.0) or not (x <= 245000.0):
		tmp = (x / y) + 1.0
	else:
		tmp = x / (x + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -290000.0) || !(x <= 245000.0))
		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 <= -290000.0) || ~((x <= 245000.0)))
		tmp = (x / y) + 1.0;
	else
		tmp = x / (x + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -290000.0], N[Not[LessEqual[x, 245000.0]], $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 < -2.9e5 or 245000 < x

   1. Initial program 68.9%

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

      1. *-commutative 68.9%

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

      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 98.8%

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

      1. *-rgt-identity 98.8%

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

   7. Applied egg-rr 98.8%

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

   if -2.9e5 < x < 245000

   1. Initial program 99.9%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in y around inf 76.9%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -290000 \lor \neg \left(x \leq 245000\right):\\ \;\;\;\;\frac{x}{y} + 1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1.85 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{x}{y} + 1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 1.85e-9))) (+ (/ x y) 1.0) x))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.85e-9)) {
		tmp = (x / y) + 1.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-1.0d0)) .or. (.not. (x <= 1.85d-9))) then
        tmp = (x / y) + 1.0d0
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.85e-9)) {
		tmp = (x / y) + 1.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.0) or not (x <= 1.85e-9):
		tmp = (x / y) + 1.0
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 1.85e-9))
		tmp = Float64(Float64(x / y) + 1.0);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 1.85e-9)))
		tmp = (x / y) + 1.0;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 1.85e-9]], $MachinePrecision]], N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1.85e-9 < x

   1. Initial program 70.4%

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

      1. *-commutative 70.4%

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

      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 95.1%

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

      1. *-rgt-identity 95.1%

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

   7. Applied egg-rr 95.1%

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

   if -1 < x < 1.85e-9

   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 77.9%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1.85 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{x}{y} + 1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 49.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{-5}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 0.98:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x -7e-5) 1.0 (if (<= x 0.98) x 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -7e-5) {
		tmp = 1.0;
	} else if (x <= 0.98) {
		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 <= (-7d-5)) then
        tmp = 1.0d0
    else if (x <= 0.98d0) 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 <= -7e-5) {
		tmp = 1.0;
	} else if (x <= 0.98) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -7e-5:
		tmp = 1.0
	elif x <= 0.98:
		tmp = x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -7e-5)
		tmp = 1.0;
	elseif (x <= 0.98)
		tmp = x;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -7e-5)
		tmp = 1.0;
	elseif (x <= 0.98)
		tmp = x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -7e-5], 1.0, If[LessEqual[x, 0.98], x, 1.0]]

Derivation

1. Split input into 2 regimes

2. if x < -6.9999999999999994e-5 or 0.97999999999999998 < x

   1. Initial program 70.1%

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

      1. *-commutative 70.1%

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

      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 y around inf 27.8%

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

      1. +-commutative 27.8%

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

   7. Simplified 27.8%

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

   8. Taylor expanded in x around inf 25.9%

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

   if -6.9999999999999994e-5 < x < 0.97999999999999998

   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 77.3%

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

Alternative 8: 99.9% accurate, 1.0× speedup

Math

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

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

Derivation

1. Initial program 85.5%

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

Alternative 9: 14.7% 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 85.5%

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

   1. *-commutative 85.5%

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

   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 y around inf 53.4%

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

   1. +-commutative 53.4%

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

7. Simplified 53.4%

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

8. Taylor expanded in x around inf 14.5%

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

Developer Target 1: 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 2024135 
(FPCore (x y)
  :name "Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1"
  :precision binary64

  :alt
  (! :herbie-platform default (* (/ x 1) (/ (+ (/ x y) 1) (+ x 1))))

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