Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1

Percentage Accurate: 87.9% → 99.8%

Time: 7.2s

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: 87.9% 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: 99.8% 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 87.5%

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

Alternative 2: 74.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + 1}\\ \mathbf{if}\;y \leq -9.6 \cdot 10^{-34}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{-39}:\\ \;\;\;\;\frac{x}{y + \frac{y}{x}}\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-8}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{+65}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ x 1.0))))
   (if (<= y -9.6e-34)
     t_0
     (if (<= y 8.5e-39)
       (/ x (+ y (/ y x)))
       (if (<= y 1.3e-8) x (if (<= y 7.2e+65) (/ x y) t_0))))))

C

double code(double x, double y) {
	double t_0 = x / (x + 1.0);
	double tmp;
	if (y <= -9.6e-34) {
		tmp = t_0;
	} else if (y <= 8.5e-39) {
		tmp = x / (y + (y / x));
	} else if (y <= 1.3e-8) {
		tmp = x;
	} else if (y <= 7.2e+65) {
		tmp = x / y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / (x + 1.0d0)
    if (y <= (-9.6d-34)) then
        tmp = t_0
    else if (y <= 8.5d-39) then
        tmp = x / (y + (y / x))
    else if (y <= 1.3d-8) then
        tmp = x
    else if (y <= 7.2d+65) then
        tmp = x / y
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (x + 1.0);
	double tmp;
	if (y <= -9.6e-34) {
		tmp = t_0;
	} else if (y <= 8.5e-39) {
		tmp = x / (y + (y / x));
	} else if (y <= 1.3e-8) {
		tmp = x;
	} else if (y <= 7.2e+65) {
		tmp = x / y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (x + 1.0)
	tmp = 0
	if y <= -9.6e-34:
		tmp = t_0
	elif y <= 8.5e-39:
		tmp = x / (y + (y / x))
	elif y <= 1.3e-8:
		tmp = x
	elif y <= 7.2e+65:
		tmp = x / y
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(x + 1.0))
	tmp = 0.0
	if (y <= -9.6e-34)
		tmp = t_0;
	elseif (y <= 8.5e-39)
		tmp = Float64(x / Float64(y + Float64(y / x)));
	elseif (y <= 1.3e-8)
		tmp = x;
	elseif (y <= 7.2e+65)
		tmp = Float64(x / y);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (x + 1.0);
	tmp = 0.0;
	if (y <= -9.6e-34)
		tmp = t_0;
	elseif (y <= 8.5e-39)
		tmp = x / (y + (y / x));
	elseif (y <= 1.3e-8)
		tmp = x;
	elseif (y <= 7.2e+65)
		tmp = x / y;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -9.6e-34], t$95$0, If[LessEqual[y, 8.5e-39], N[(x / N[(y + N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.3e-8], x, If[LessEqual[y, 7.2e+65], N[(x / y), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -9.59999999999999965e-34 or 7.19999999999999957e65 < y

   1. Initial program 89.3%

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

   3. Taylor expanded in y around inf 79.1%

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

      1. +-commutative 79.1%

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

   5. Simplified 79.1%

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

   if -9.59999999999999965e-34 < y < 8.5000000000000005e-39

   1. Initial program 89.9%

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

      1. associate-/l* 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 99.7%

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

      1. +-commutative 99.7%

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

      2. +-commutative 99.7%

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

      3. +-commutative 99.7%

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

   7. Simplified 99.7%

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

      1. clear-num 99.6%

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

      2. un-div-inv 99.8%

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

      3. +-commutative 99.8%

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

      4. div-inv 99.7%

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

      5. div-inv 99.7%

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

      6. distribute-rgt-out 99.7%

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

   9. Applied egg-rr 99.7%

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

   10. Taylor expanded in y around 0 78.4%

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

   11. Taylor expanded in x around inf 78.4%

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

   if 8.5000000000000005e-39 < y < 1.3000000000000001e-8

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 79.8%

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

   if 1.3000000000000001e-8 < y < 7.19999999999999957e65

   1. Initial program 57.6%

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

   3. Taylor expanded in x around inf 67.8%

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

Alternative 3: 93.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{+162} \lor \neg \left(x \leq 1.8 \cdot 10^{+125}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{x + y}{y \cdot \left(x + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -4.8e+162) (not (<= x 1.8e+125)))
   (/ x y)
   (* x (/ (+ x y) (* y (+ x 1.0))))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -4.8e+162) || !(x <= 1.8e+125)) {
		tmp = x / y;
	} else {
		tmp = x * ((x + y) / (y * (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 <= (-4.8d+162)) .or. (.not. (x <= 1.8d+125))) then
        tmp = x / y
    else
        tmp = x * ((x + y) / (y * (x + 1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -4.8e+162) || !(x <= 1.8e+125)) {
		tmp = x / y;
	} else {
		tmp = x * ((x + y) / (y * (x + 1.0)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -4.8e+162) or not (x <= 1.8e+125):
		tmp = x / y
	else:
		tmp = x * ((x + y) / (y * (x + 1.0)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -4.8e+162) || !(x <= 1.8e+125))
		tmp = Float64(x / y);
	else
		tmp = Float64(x * Float64(Float64(x + y) / Float64(y * Float64(x + 1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -4.8e+162) || ~((x <= 1.8e+125)))
		tmp = x / y;
	else
		tmp = x * ((x + y) / (y * (x + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -4.8e+162], N[Not[LessEqual[x, 1.8e+125]], $MachinePrecision]], N[(x / y), $MachinePrecision], N[(x * N[(N[(x + y), $MachinePrecision] / N[(y * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.80000000000000018e162 or 1.8000000000000002e125 < x

   1. Initial program 58.8%

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

   3. Taylor expanded in x around inf 84.9%

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

   if -4.80000000000000018e162 < x < 1.8000000000000002e125

   1. Initial program 97.0%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 0 99.8%

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

      1. +-commutative 99.8%

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

      2. +-commutative 99.8%

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

      3. +-commutative 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in y around 0 99.8%

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

      1. *-lft-identity 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-*l/ 99.7%

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

      4. *-lft-identity 99.7%

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

      5. +-commutative 99.7%

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

      6. associate-*l/ 99.7%

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

      7. distribute-lft-in 99.7%

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

      8. associate-*l/ 99.7%

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

      9. *-lft-identity 99.7%

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

      10. associate-/l/ 96.7%

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

   10. Simplified 96.7%

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

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{+162} \lor \neg \left(x \leq 1.8 \cdot 10^{+125}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{x + y}{y \cdot \left(x + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{-11} \lor \neg \left(x \leq 0.0064\right):\\ \;\;\;\;\frac{x}{y + \frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + \left(\frac{x}{y} - x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -5.5e-11) (not (<= x 0.0064)))
   (/ x (+ y (/ y x)))
   (* x (+ 1.0 (- (/ x y) x)))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -5.5e-11) || !(x <= 0.0064)) {
		tmp = x / (y + (y / x));
	} 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 <= (-5.5d-11)) .or. (.not. (x <= 0.0064d0))) then
        tmp = x / (y + (y / x))
    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 <= -5.5e-11) || !(x <= 0.0064)) {
		tmp = x / (y + (y / x));
	} else {
		tmp = x * (1.0 + ((x / y) - x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -5.5e-11) or not (x <= 0.0064):
		tmp = x / (y + (y / x))
	else:
		tmp = x * (1.0 + ((x / y) - x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -5.5e-11) || !(x <= 0.0064))
		tmp = Float64(x / Float64(y + Float64(y / x)));
	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 <= -5.5e-11) || ~((x <= 0.0064)))
		tmp = x / (y + (y / x));
	else
		tmp = x * (1.0 + ((x / y) - x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -5.5e-11], N[Not[LessEqual[x, 0.0064]], $MachinePrecision]], N[(x / N[(y + N[(y / x), $MachinePrecision]), $MachinePrecision]), $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 < -5.49999999999999975e-11 or 0.00640000000000000031 < x

   1. Initial program 73.2%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 y around 0 99.7%

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

      1. +-commutative 99.7%

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

      2. +-commutative 99.7%

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

      3. +-commutative 99.7%

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

   7. Simplified 99.7%

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

      1. clear-num 99.6%

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

      2. un-div-inv 99.9%

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

      3. +-commutative 99.9%

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

      4. div-inv 99.8%

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

      5. div-inv 99.7%

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

      6. distribute-rgt-out 99.7%

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

   9. Applied egg-rr 99.7%

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

   10. Taylor expanded in y around 0 60.9%

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

   11. Taylor expanded in x around inf 74.9%

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

   if -5.49999999999999975e-11 < x < 0.00640000000000000031

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 98.9%

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

   4. Taylor expanded in y around inf 99.0%

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

      1. neg-mul-1 99.0%

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

      2. +-commutative 99.0%

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

      3. unsub-neg 99.0%

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

   6. Simplified 99.0%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{-11} \lor \neg \left(x \leq 0.0064\right):\\ \;\;\;\;\frac{x}{y + \frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + \left(\frac{x}{y} - x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -7.2 \cdot 10^{-48}:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 0.7:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.0)
   (/ x y)
   (if (<= x -7.2e-48) (* x (/ x y)) (if (<= x 0.7) x (/ x y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = x / y;
	} else if (x <= -7.2e-48) {
		tmp = x * (x / y);
	} else if (x <= 0.7) {
		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 <= (-7.2d-48)) then
        tmp = x * (x / y)
    else if (x <= 0.7d0) 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 <= -7.2e-48) {
		tmp = x * (x / y);
	} else if (x <= 0.7) {
		tmp = x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.0:
		tmp = x / y
	elif x <= -7.2e-48:
		tmp = x * (x / y)
	elif x <= 0.7:
		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 <= -7.2e-48)
		tmp = Float64(x * Float64(x / y));
	elseif (x <= 0.7)
		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 <= -7.2e-48)
		tmp = x * (x / y);
	elseif (x <= 0.7)
		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, -7.2e-48], N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.7], x, N[(x / y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1 or 0.69999999999999996 < x

   1. Initial program 72.3%

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

   3. Taylor expanded in x around inf 74.0%

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

   if -1 < x < -7.2000000000000003e-48

   1. Initial program 99.6%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 0 72.7%

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

      1. +-commutative 72.7%

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

   7. Simplified 72.7%

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

   8. Taylor expanded in x around 0 62.9%

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

   if -7.2000000000000003e-48 < x < 0.69999999999999996

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 73.0%

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

Alternative 6: 75.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.45 \cdot 10^{+16} \lor \neg \left(x \leq 42000000000000\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -2.45e+16) (not (<= x 42000000000000.0)))
   (/ x y)
   (/ x (+ x 1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -2.45e+16) || !(x <= 42000000000000.0)) {
		tmp = x / y;
	} else {
		tmp = x / (x + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-2.45d+16)) .or. (.not. (x <= 42000000000000.0d0))) then
        tmp = x / y
    else
        tmp = x / (x + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -2.45e+16) || !(x <= 42000000000000.0)) {
		tmp = x / y;
	} else {
		tmp = x / (x + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -2.45e+16) or not (x <= 42000000000000.0):
		tmp = x / y
	else:
		tmp = x / (x + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -2.45e+16) || !(x <= 42000000000000.0))
		tmp = Float64(x / y);
	else
		tmp = Float64(x / Float64(x + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -2.45e+16) || ~((x <= 42000000000000.0)))
		tmp = x / y;
	else
		tmp = x / (x + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -2.45e+16], N[Not[LessEqual[x, 42000000000000.0]], $MachinePrecision]], N[(x / y), $MachinePrecision], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.45e16 or 4.2e13 < x

   1. Initial program 71.1%

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

   3. Taylor expanded in x around inf 76.4%

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

   if -2.45e16 < x < 4.2e13

   1. Initial program 99.8%

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

   3. Taylor expanded in y around inf 69.4%

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

      1. +-commutative 69.4%

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

   5. Simplified 69.4%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.45 \cdot 10^{+16} \lor \neg \left(x \leq 42000000000000\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.7\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 0.7))) (/ x y) x))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.0) || !(x <= 0.7)) {
		tmp = x / y;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-1.0d0)) .or. (.not. (x <= 0.7d0))) then
        tmp = x / y
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -1.0) || !(x <= 0.7)) {
		tmp = x / y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.0) or not (x <= 0.7):
		tmp = x / y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 0.7))
		tmp = Float64(x / y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 0.7)))
		tmp = x / y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 0.7]], $MachinePrecision]], N[(x / y), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 0.69999999999999996 < x

   1. Initial program 72.3%

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

   3. Taylor expanded in x around inf 74.0%

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

   if -1 < x < 0.69999999999999996

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 68.6%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.7\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 38.9% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x y) :precision binary64 x)

C

double code(double x, double y) {
	return x;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x
end function

Java

public static double code(double x, double y) {
	return x;
}

Python

def code(x, y):
	return x

Julia

function code(x, y)
	return x
end

MATLAB

function tmp = code(x, y)
	tmp = x;
end

Wolfram

code[x_, y_] := x

Derivation

1. Initial program 87.5%

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

3. Taylor expanded in x around 0 39.4%

   \[\leadsto \color{blue}{x} \]
4. 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 2024111 
(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)))