Data.Colour.SRGB:transferFunction from colour-2.3.3

Percentage Accurate: 100.0% → 100.0%

Time: 3.9s

Alternatives: 8

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, y + -1, y\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (fma x (+ y -1.0) y))

C

double code(double x, double y) {
	return fma(x, (y + -1.0), y);
}

Julia

function code(x, y)
	return fma(x, Float64(y + -1.0), y)
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. sub-neg 100.0%

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

   2. distribute-rgt1-in 100.0%

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

   3. *-commutative 100.0%

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

   4. associate-+l+ 100.0%

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

   5. *-commutative 100.0%

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

   6. +-commutative 100.0%

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

   7. *-commutative 100.0%

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

   8. neg-mul-1 100.0%

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

   9. distribute-rgt-out 100.0%

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

   10. fma-define 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, y + -1, y\right)} \]
4. Add Preprocessing

5. Final simplification 100.0%

   \[\leadsto \mathsf{fma}\left(x, y + -1, y\right) \]
6. Add Preprocessing

Alternative 2: 62.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.1 \cdot 10^{+260}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq -3.2 \cdot 10^{+241}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq -1:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{-20}:\\ \;\;\;\;-x\\ \mathbf{elif}\;y \leq 2.85 \cdot 10^{+107}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -4.1e+260)
   (* x y)
   (if (<= y -3.2e+241)
     y
     (if (<= y -1.0)
       (* x y)
       (if (<= y 2.8e-20) (- x) (if (<= y 2.85e+107) y (* x y)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -4.1e+260) {
		tmp = x * y;
	} else if (y <= -3.2e+241) {
		tmp = y;
	} else if (y <= -1.0) {
		tmp = x * y;
	} else if (y <= 2.8e-20) {
		tmp = -x;
	} else if (y <= 2.85e+107) {
		tmp = y;
	} 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 (y <= (-4.1d+260)) then
        tmp = x * y
    else if (y <= (-3.2d+241)) then
        tmp = y
    else if (y <= (-1.0d0)) then
        tmp = x * y
    else if (y <= 2.8d-20) then
        tmp = -x
    else if (y <= 2.85d+107) then
        tmp = y
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -4.1e+260) {
		tmp = x * y;
	} else if (y <= -3.2e+241) {
		tmp = y;
	} else if (y <= -1.0) {
		tmp = x * y;
	} else if (y <= 2.8e-20) {
		tmp = -x;
	} else if (y <= 2.85e+107) {
		tmp = y;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -4.1e+260:
		tmp = x * y
	elif y <= -3.2e+241:
		tmp = y
	elif y <= -1.0:
		tmp = x * y
	elif y <= 2.8e-20:
		tmp = -x
	elif y <= 2.85e+107:
		tmp = y
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -4.1e+260)
		tmp = Float64(x * y);
	elseif (y <= -3.2e+241)
		tmp = y;
	elseif (y <= -1.0)
		tmp = Float64(x * y);
	elseif (y <= 2.8e-20)
		tmp = Float64(-x);
	elseif (y <= 2.85e+107)
		tmp = y;
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -4.1e+260)
		tmp = x * y;
	elseif (y <= -3.2e+241)
		tmp = y;
	elseif (y <= -1.0)
		tmp = x * y;
	elseif (y <= 2.8e-20)
		tmp = -x;
	elseif (y <= 2.85e+107)
		tmp = y;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -4.1e+260], N[(x * y), $MachinePrecision], If[LessEqual[y, -3.2e+241], y, If[LessEqual[y, -1.0], N[(x * y), $MachinePrecision], If[LessEqual[y, 2.8e-20], (-x), If[LessEqual[y, 2.85e+107], y, N[(x * y), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.10000000000000025e260 or -3.20000000000000004e241 < y < -1 or 2.84999999999999986e107 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 60.3%

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

   4. Taylor expanded in y around inf 59.0%

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

   if -4.10000000000000025e260 < y < -3.20000000000000004e241 or 2.8000000000000003e-20 < y < 2.84999999999999986e107

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 100.0%

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

      1. associate-+r+ 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-+l+ 100.0%

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

      4. mul-1-neg 100.0%

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

      5. unsub-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 73.2%

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

   if -1 < y < 2.8000000000000003e-20

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 78.0%

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

      1. neg-mul-1 78.0%

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

   5. Simplified 78.0%

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

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.1 \cdot 10^{+260}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq -3.2 \cdot 10^{+241}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq -1:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{-20}:\\ \;\;\;\;-x\\ \mathbf{elif}\;y \leq 2.85 \cdot 10^{+107}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.8 \cdot 10^{+261}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq -3.8 \cdot 10^{+239}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq -1.85 \cdot 10^{+34} \lor \neg \left(y \leq 2.1 \cdot 10^{+108}\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;y - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -6.8e+261)
   (* x y)
   (if (<= y -3.8e+239)
     y
     (if (or (<= y -1.85e+34) (not (<= y 2.1e+108))) (* x y) (- y x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -6.8e+261) {
		tmp = x * y;
	} else if (y <= -3.8e+239) {
		tmp = y;
	} else if ((y <= -1.85e+34) || !(y <= 2.1e+108)) {
		tmp = x * y;
	} else {
		tmp = 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 (y <= (-6.8d+261)) then
        tmp = x * y
    else if (y <= (-3.8d+239)) then
        tmp = y
    else if ((y <= (-1.85d+34)) .or. (.not. (y <= 2.1d+108))) then
        tmp = x * y
    else
        tmp = y - x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -6.8e+261) {
		tmp = x * y;
	} else if (y <= -3.8e+239) {
		tmp = y;
	} else if ((y <= -1.85e+34) || !(y <= 2.1e+108)) {
		tmp = x * y;
	} else {
		tmp = y - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -6.8e+261:
		tmp = x * y
	elif y <= -3.8e+239:
		tmp = y
	elif (y <= -1.85e+34) or not (y <= 2.1e+108):
		tmp = x * y
	else:
		tmp = y - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -6.8e+261)
		tmp = Float64(x * y);
	elseif (y <= -3.8e+239)
		tmp = y;
	elseif ((y <= -1.85e+34) || !(y <= 2.1e+108))
		tmp = Float64(x * y);
	else
		tmp = Float64(y - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -6.8e+261)
		tmp = x * y;
	elseif (y <= -3.8e+239)
		tmp = y;
	elseif ((y <= -1.85e+34) || ~((y <= 2.1e+108)))
		tmp = x * y;
	else
		tmp = y - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -6.8e+261], N[(x * y), $MachinePrecision], If[LessEqual[y, -3.8e+239], y, If[Or[LessEqual[y, -1.85e+34], N[Not[LessEqual[y, 2.1e+108]], $MachinePrecision]], N[(x * y), $MachinePrecision], N[(y - x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.8e261 or -3.8000000000000001e239 < y < -1.85000000000000004e34 or 2.1000000000000001e108 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 60.9%

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

   4. Taylor expanded in y around inf 60.9%

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

   if -6.8e261 < y < -3.8000000000000001e239

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 100.0%

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

      1. associate-+r+ 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-+l+ 100.0%

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

      4. mul-1-neg 100.0%

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

      5. unsub-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 100.0%

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

   if -1.85000000000000004e34 < y < 2.1000000000000001e108

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 92.5%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.8 \cdot 10^{+261}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq -3.8 \cdot 10^{+239}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq -1.85 \cdot 10^{+34} \lor \neg \left(y \leq 2.1 \cdot 10^{+108}\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;y - x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1.9 \cdot 10^{-39}\right):\\ \;\;\;\;x \cdot \left(y + -1\right)\\ \mathbf{else}:\\ \;\;\;\;y - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 1.9e-39))) (* x (+ y -1.0)) (- y x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.9e-39)) {
		tmp = x * (y + -1.0);
	} else {
		tmp = 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 <= 1.9d-39))) then
        tmp = x * (y + (-1.0d0))
    else
        tmp = y - x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.9e-39)) {
		tmp = x * (y + -1.0);
	} else {
		tmp = y - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.0) or not (x <= 1.9e-39):
		tmp = x * (y + -1.0)
	else:
		tmp = y - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 1.9e-39))
		tmp = Float64(x * Float64(y + -1.0));
	else
		tmp = Float64(y - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 1.9e-39)))
		tmp = x * (y + -1.0);
	else
		tmp = y - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 1.9e-39]], $MachinePrecision]], N[(x * N[(y + -1.0), $MachinePrecision]), $MachinePrecision], N[(y - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1.9000000000000001e-39 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 98.3%

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

   if -1 < x < 1.9000000000000001e-39

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 99.3%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1.9 \cdot 10^{-39}\right):\\ \;\;\;\;x \cdot \left(y + -1\right)\\ \mathbf{else}:\\ \;\;\;\;y - x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -225 \lor \neg \left(y \leq 4.5 \cdot 10^{-20}\right):\\ \;\;\;\;y \cdot \left(x + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -225.0) (not (<= y 4.5e-20))) (* y (+ x 1.0)) (- y x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -225.0) || !(y <= 4.5e-20)) {
		tmp = y * (x + 1.0);
	} else {
		tmp = 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 ((y <= (-225.0d0)) .or. (.not. (y <= 4.5d-20))) then
        tmp = y * (x + 1.0d0)
    else
        tmp = y - x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -225.0) || !(y <= 4.5e-20)) {
		tmp = y * (x + 1.0);
	} else {
		tmp = y - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -225.0) or not (y <= 4.5e-20):
		tmp = y * (x + 1.0)
	else:
		tmp = y - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -225.0) || !(y <= 4.5e-20))
		tmp = Float64(y * Float64(x + 1.0));
	else
		tmp = Float64(y - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -225.0) || ~((y <= 4.5e-20)))
		tmp = y * (x + 1.0);
	else
		tmp = y - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -225.0], N[Not[LessEqual[y, 4.5e-20]], $MachinePrecision]], N[(y * N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(y - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -225 or 4.5000000000000001e-20 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 100.0%

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

      1. associate-+r+ 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-+l+ 100.0%

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

      4. mul-1-neg 100.0%

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

      5. unsub-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around inf 98.8%

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

      1. +-commutative 98.8%

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

   8. Simplified 98.8%

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

   if -225 < y < 4.5000000000000001e-20

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 100.0%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -225 \lor \neg \left(y \leq 4.5 \cdot 10^{-20}\right):\\ \;\;\;\;y \cdot \left(x + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y - x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 62.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{-47} \lor \neg \left(x \leq 9 \cdot 10^{-40}\right):\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.4e-47) (not (<= x 9e-40))) (- x) y))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.4e-47) || !(x <= 9e-40)) {
		tmp = -x;
	} else {
		tmp = 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-47)) .or. (.not. (x <= 9d-40))) then
        tmp = -x
    else
        tmp = y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -1.4e-47) || !(x <= 9e-40)) {
		tmp = -x;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.4e-47) or not (x <= 9e-40):
		tmp = -x
	else:
		tmp = y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.4e-47) || !(x <= 9e-40))
		tmp = Float64(-x);
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -1.4e-47) || ~((x <= 9e-40)))
		tmp = -x;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.4e-47], N[Not[LessEqual[x, 9e-40]], $MachinePrecision]], (-x), y]

Derivation

1. Split input into 2 regimes

2. if x < -1.39999999999999996e-47 or 9.0000000000000002e-40 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 51.5%

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

      1. neg-mul-1 51.5%

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

   5. Simplified 51.5%

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

   if -1.39999999999999996e-47 < x < 9.0000000000000002e-40

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 99.9%

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

      1. associate-+r+ 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-+l+ 99.9%

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

      4. mul-1-neg 99.9%

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

      5. unsub-neg 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in x around 0 81.5%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{-47} \lor \neg \left(x \leq 9 \cdot 10^{-40}\right):\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

   \[\leadsto y \cdot \left(x + 1\right) - x \]
4. Add Preprocessing

Alternative 8: 38.6% accurate, 7.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 y)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return y

Julia

function code(x, y)
	return y
end

MATLAB

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

Wolfram

code[x_, y_] := y

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in y around inf 87.2%

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

   1. associate-+r+ 87.2%

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

   2. +-commutative 87.2%

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

   3. associate-+l+ 87.2%

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

   4. mul-1-neg 87.2%

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

   5. unsub-neg 87.2%

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

5. Simplified 87.2%

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

6. Taylor expanded in x around 0 36.0%

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

7. Final simplification 36.0%

   \[\leadsto y \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024115 
(FPCore (x y)
  :name "Data.Colour.SRGB:transferFunction from colour-2.3.3"
  :precision binary64
  (- (* (+ x 1.0) y) x))