Data.Colour.RGBSpace.HSL:hsl from colour-2.3.3, A

Percentage Accurate: 100.0% → 100.0%
Time: 8.0s
Alternatives: 8
Speedup: 1.0×

Specification

?
\[\begin{array}{l} \\ \left(x + y\right) - x \cdot y \end{array} \]
(FPCore (x y) :precision binary64 (- (+ x y) (* x y)))
double code(double x, double y) {
	return (x + y) - (x * y);
}
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x + y) - (x * y)
end function
public static double code(double x, double y) {
	return (x + y) - (x * y);
}
def code(x, y):
	return (x + y) - (x * y)
function code(x, y)
	return Float64(Float64(x + y) - Float64(x * y))
end
function tmp = code(x, y)
	tmp = (x + y) - (x * y);
end
code[x_, y_] := N[(N[(x + y), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]
\begin{array}{l}

\\
\left(x + y\right) - x \cdot y
\end{array}

Sampling outcomes in binary64 precision:

Local Percentage Accuracy vs ?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

Accuracy vs Speed?

Herbie found 8 alternatives:

AlternativeAccuracySpeedup
The accuracy (vertical axis) and speed (horizontal axis) of each alternatives. Up and to the right is better. The red square shows the initial program, and each blue circle shows an alternative.The line shows the best available speed-accuracy tradeoffs.

Initial Program: 100.0% accurate, 1.0× speedup?

\[\begin{array}{l} \\ \left(x + y\right) - x \cdot y \end{array} \]
(FPCore (x y) :precision binary64 (- (+ x y) (* x y)))
double code(double x, double y) {
	return (x + y) - (x * y);
}
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x + y) - (x * y)
end function
public static double code(double x, double y) {
	return (x + y) - (x * y);
}
def code(x, y):
	return (x + y) - (x * y)
function code(x, y)
	return Float64(Float64(x + y) - Float64(x * y))
end
function tmp = code(x, y)
	tmp = (x + y) - (x * y);
end
code[x_, y_] := N[(N[(x + y), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]
\begin{array}{l}

\\
\left(x + y\right) - x \cdot y
\end{array}

Alternative 1: 100.0% accurate, 0.1× speedup?

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \mathsf{fma}\left(x, 1 - y, y\right) \end{array} \]
NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (fma x (- 1.0 y) y))
assert(x < y);
double code(double x, double y) {
	return fma(x, (1.0 - y), y);
}
x, y = sort([x, y])
function code(x, y)
	return fma(x, Float64(1.0 - y), y)
end
NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[(x * N[(1.0 - y), $MachinePrecision] + y), $MachinePrecision]
\begin{array}{l}
[x, y] = \mathsf{sort}([x, y])\\
\\
\mathsf{fma}\left(x, 1 - y, y\right)
\end{array}
Derivation
  1. Initial program 100.0%

    \[\left(x + y\right) - x \cdot y \]
  2. Step-by-step derivation
    1. +-commutative100.0%

      \[\leadsto \color{blue}{\left(y + x\right)} - x \cdot y \]
    2. *-commutative100.0%

      \[\leadsto \left(y + x\right) - \color{blue}{y \cdot x} \]
    3. associate--l+100.0%

      \[\leadsto \color{blue}{y + \left(x - y \cdot x\right)} \]
    4. +-commutative100.0%

      \[\leadsto \color{blue}{\left(x - y \cdot x\right) + y} \]
    5. *-lft-identity100.0%

      \[\leadsto \left(\color{blue}{1 \cdot x} - y \cdot x\right) + y \]
    6. metadata-eval100.0%

      \[\leadsto \left(\color{blue}{\left(--1\right)} \cdot x - y \cdot x\right) + y \]
    7. distribute-rgt-out--100.0%

      \[\leadsto \color{blue}{x \cdot \left(\left(--1\right) - y\right)} + y \]
    8. fma-define100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \left(--1\right) - y, y\right)} \]
    9. metadata-eval100.0%

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{1} - y, y\right) \]
  3. Simplified100.0%

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

Alternative 2: 69.4% accurate, 0.4× speedup?

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := x \cdot \left(-y\right)\\ \mathbf{if}\;x \leq -1.9 \cdot 10^{+207}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -8 \cdot 10^{-102}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]
NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (- y))))
   (if (<= x -1.9e+207) t_0 (if (<= x -8e-102) x (if (<= x 1.0) y t_0)))))
assert(x < y);
double code(double x, double y) {
	double t_0 = x * -y;
	double tmp;
	if (x <= -1.9e+207) {
		tmp = t_0;
	} else if (x <= -8e-102) {
		tmp = x;
	} else if (x <= 1.0) {
		tmp = y;
	} else {
		tmp = t_0;
	}
	return tmp;
}
NOTE: x and y should be sorted in increasing order before calling this function.
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 * -y
    if (x <= (-1.9d+207)) then
        tmp = t_0
    else if (x <= (-8d-102)) then
        tmp = x
    else if (x <= 1.0d0) then
        tmp = y
    else
        tmp = t_0
    end if
    code = tmp
end function
assert x < y;
public static double code(double x, double y) {
	double t_0 = x * -y;
	double tmp;
	if (x <= -1.9e+207) {
		tmp = t_0;
	} else if (x <= -8e-102) {
		tmp = x;
	} else if (x <= 1.0) {
		tmp = y;
	} else {
		tmp = t_0;
	}
	return tmp;
}
[x, y] = sort([x, y])
def code(x, y):
	t_0 = x * -y
	tmp = 0
	if x <= -1.9e+207:
		tmp = t_0
	elif x <= -8e-102:
		tmp = x
	elif x <= 1.0:
		tmp = y
	else:
		tmp = t_0
	return tmp
x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(x * Float64(-y))
	tmp = 0.0
	if (x <= -1.9e+207)
		tmp = t_0;
	elseif (x <= -8e-102)
		tmp = x;
	elseif (x <= 1.0)
		tmp = y;
	else
		tmp = t_0;
	end
	return tmp
end
x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = x * -y;
	tmp = 0.0;
	if (x <= -1.9e+207)
		tmp = t_0;
	elseif (x <= -8e-102)
		tmp = x;
	elseif (x <= 1.0)
		tmp = y;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end
NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(x * (-y)), $MachinePrecision]}, If[LessEqual[x, -1.9e+207], t$95$0, If[LessEqual[x, -8e-102], x, If[LessEqual[x, 1.0], y, t$95$0]]]]
\begin{array}{l}
[x, y] = \mathsf{sort}([x, y])\\
\\
\begin{array}{l}
t_0 := x \cdot \left(-y\right)\\
\mathbf{if}\;x \leq -1.9 \cdot 10^{+207}:\\
\;\;\;\;t\_0\\

\mathbf{elif}\;x \leq -8 \cdot 10^{-102}:\\
\;\;\;\;x\\

\mathbf{elif}\;x \leq 1:\\
\;\;\;\;y\\

\mathbf{else}:\\
\;\;\;\;t\_0\\


\end{array}
\end{array}
Derivation
  1. Split input into 3 regimes
  2. if x < -1.89999999999999993e207 or 1 < x

    1. Initial program 100.0%

      \[\left(x + y\right) - x \cdot y \]
    2. Add Preprocessing
    3. Taylor expanded in y around inf 49.2%

      \[\leadsto \color{blue}{y \cdot \left(1 - x\right)} \]
    4. Taylor expanded in x around inf 48.6%

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot y\right)} \]
    5. Step-by-step derivation
      1. associate-*r*48.6%

        \[\leadsto \color{blue}{\left(-1 \cdot x\right) \cdot y} \]
      2. neg-mul-148.6%

        \[\leadsto \color{blue}{\left(-x\right)} \cdot y \]
      3. *-commutative48.6%

        \[\leadsto \color{blue}{y \cdot \left(-x\right)} \]
    6. Simplified48.6%

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

    if -1.89999999999999993e207 < x < -7.99999999999999946e-102

    1. Initial program 100.0%

      \[\left(x + y\right) - x \cdot y \]
    2. Add Preprocessing
    3. Taylor expanded in x around inf 78.8%

      \[\leadsto \color{blue}{x} - x \cdot y \]
    4. Taylor expanded in y around 0 46.5%

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

    if -7.99999999999999946e-102 < x < 1

    1. Initial program 100.0%

      \[\left(x + y\right) - x \cdot y \]
    2. Add Preprocessing
    3. Taylor expanded in y around inf 68.2%

      \[\leadsto \color{blue}{y \cdot \left(1 - x\right)} \]
    4. Taylor expanded in x around 0 67.7%

      \[\leadsto \color{blue}{y} \]
  3. Recombined 3 regimes into one program.
  4. Final simplification56.8%

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.9 \cdot 10^{+207}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;x \leq -8 \cdot 10^{-102}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \end{array} \]
  5. Add Preprocessing

Alternative 3: 88.6% accurate, 0.5× speedup?

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;y \leq 1.16 \cdot 10^{-132}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - x\right)\\ \end{array} \end{array} \]
NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y -1.0) (* x (- y)) (if (<= y 1.16e-132) x (* y (- 1.0 x)))))
assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x * -y;
	} else if (y <= 1.16e-132) {
		tmp = x;
	} else {
		tmp = y * (1.0 - x);
	}
	return tmp;
}
NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.0d0)) then
        tmp = x * -y
    else if (y <= 1.16d-132) then
        tmp = x
    else
        tmp = y * (1.0d0 - x)
    end if
    code = tmp
end function
assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x * -y;
	} else if (y <= 1.16e-132) {
		tmp = x;
	} else {
		tmp = y * (1.0 - x);
	}
	return tmp;
}
[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = x * -y
	elif y <= 1.16e-132:
		tmp = x
	else:
		tmp = y * (1.0 - x)
	return tmp
x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = Float64(x * Float64(-y));
	elseif (y <= 1.16e-132)
		tmp = x;
	else
		tmp = Float64(y * Float64(1.0 - x));
	end
	return tmp
end
x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.0)
		tmp = x * -y;
	elseif (y <= 1.16e-132)
		tmp = x;
	else
		tmp = y * (1.0 - x);
	end
	tmp_2 = tmp;
end
NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, -1.0], N[(x * (-y)), $MachinePrecision], If[LessEqual[y, 1.16e-132], x, N[(y * N[(1.0 - x), $MachinePrecision]), $MachinePrecision]]]
\begin{array}{l}
[x, y] = \mathsf{sort}([x, y])\\
\\
\begin{array}{l}
\mathbf{if}\;y \leq -1:\\
\;\;\;\;x \cdot \left(-y\right)\\

\mathbf{elif}\;y \leq 1.16 \cdot 10^{-132}:\\
\;\;\;\;x\\

\mathbf{else}:\\
\;\;\;\;y \cdot \left(1 - x\right)\\


\end{array}
\end{array}
Derivation
  1. Split input into 3 regimes
  2. if y < -1

    1. Initial program 100.0%

      \[\left(x + y\right) - x \cdot y \]
    2. Add Preprocessing
    3. Taylor expanded in y around inf 98.7%

      \[\leadsto \color{blue}{y \cdot \left(1 - x\right)} \]
    4. Taylor expanded in x around inf 54.3%

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot y\right)} \]
    5. Step-by-step derivation
      1. associate-*r*54.3%

        \[\leadsto \color{blue}{\left(-1 \cdot x\right) \cdot y} \]
      2. neg-mul-154.3%

        \[\leadsto \color{blue}{\left(-x\right)} \cdot y \]
      3. *-commutative54.3%

        \[\leadsto \color{blue}{y \cdot \left(-x\right)} \]
    6. Simplified54.3%

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

    if -1 < y < 1.1599999999999999e-132

    1. Initial program 100.0%

      \[\left(x + y\right) - x \cdot y \]
    2. Add Preprocessing
    3. Taylor expanded in x around inf 77.6%

      \[\leadsto \color{blue}{x} - x \cdot y \]
    4. Taylor expanded in y around 0 77.1%

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

    if 1.1599999999999999e-132 < y

    1. Initial program 99.9%

      \[\left(x + y\right) - x \cdot y \]
    2. Add Preprocessing
    3. Taylor expanded in y around inf 82.3%

      \[\leadsto \color{blue}{y \cdot \left(1 - x\right)} \]
  3. Recombined 3 regimes into one program.
  4. Final simplification73.5%

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;y \leq 1.16 \cdot 10^{-132}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - x\right)\\ \end{array} \]
  5. Add Preprocessing

Alternative 4: 89.5% accurate, 0.7× speedup?

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.95 \cdot 10^{-117}:\\ \;\;\;\;x - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - x\right)\\ \end{array} \end{array} \]
NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.95e-117) (- x (* x y)) (* y (- 1.0 x))))
assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.95e-117) {
		tmp = x - (x * y);
	} else {
		tmp = y * (1.0 - x);
	}
	return tmp;
}
NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.95d-117)) then
        tmp = x - (x * y)
    else
        tmp = y * (1.0d0 - x)
    end if
    code = tmp
end function
assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -1.95e-117) {
		tmp = x - (x * y);
	} else {
		tmp = y * (1.0 - x);
	}
	return tmp;
}
[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1.95e-117:
		tmp = x - (x * y)
	else:
		tmp = y * (1.0 - x)
	return tmp
x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.95e-117)
		tmp = Float64(x - Float64(x * y));
	else
		tmp = Float64(y * Float64(1.0 - x));
	end
	return tmp
end
x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.95e-117)
		tmp = x - (x * y);
	else
		tmp = y * (1.0 - x);
	end
	tmp_2 = tmp;
end
NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1.95e-117], N[(x - N[(x * y), $MachinePrecision]), $MachinePrecision], N[(y * N[(1.0 - x), $MachinePrecision]), $MachinePrecision]]
\begin{array}{l}
[x, y] = \mathsf{sort}([x, y])\\
\\
\begin{array}{l}
\mathbf{if}\;x \leq -1.95 \cdot 10^{-117}:\\
\;\;\;\;x - x \cdot y\\

\mathbf{else}:\\
\;\;\;\;y \cdot \left(1 - x\right)\\


\end{array}
\end{array}
Derivation
  1. Split input into 2 regimes
  2. if x < -1.94999999999999996e-117

    1. Initial program 100.0%

      \[\left(x + y\right) - x \cdot y \]
    2. Add Preprocessing
    3. Taylor expanded in x around inf 80.1%

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

    if -1.94999999999999996e-117 < x

    1. Initial program 100.0%

      \[\left(x + y\right) - x \cdot y \]
    2. Add Preprocessing
    3. Taylor expanded in y around inf 60.6%

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

Alternative 5: 100.0% accurate, 1.0× speedup?

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \left(x + y\right) - x \cdot y \end{array} \]
NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (- (+ x y) (* x y)))
assert(x < y);
double code(double x, double y) {
	return (x + y) - (x * y);
}
NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x + y) - (x * y)
end function
assert x < y;
public static double code(double x, double y) {
	return (x + y) - (x * y);
}
[x, y] = sort([x, y])
def code(x, y):
	return (x + y) - (x * y)
x, y = sort([x, y])
function code(x, y)
	return Float64(Float64(x + y) - Float64(x * y))
end
x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = (x + y) - (x * y);
end
NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[(N[(x + y), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]
\begin{array}{l}
[x, y] = \mathsf{sort}([x, y])\\
\\
\left(x + y\right) - x \cdot y
\end{array}
Derivation
  1. Initial program 100.0%

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

Alternative 6: 64.3% accurate, 1.2× speedup?

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-101}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]
NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (if (<= x -1e-101) x y))
assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1e-101) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}
NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1d-101)) then
        tmp = x
    else
        tmp = y
    end if
    code = tmp
end function
assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -1e-101) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}
[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1e-101:
		tmp = x
	else:
		tmp = y
	return tmp
x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1e-101)
		tmp = x;
	else
		tmp = y;
	end
	return tmp
end
x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1e-101)
		tmp = x;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end
NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1e-101], x, y]
\begin{array}{l}
[x, y] = \mathsf{sort}([x, y])\\
\\
\begin{array}{l}
\mathbf{if}\;x \leq -1 \cdot 10^{-101}:\\
\;\;\;\;x\\

\mathbf{else}:\\
\;\;\;\;y\\


\end{array}
\end{array}
Derivation
  1. Split input into 2 regimes
  2. if x < -1.00000000000000005e-101

    1. Initial program 100.0%

      \[\left(x + y\right) - x \cdot y \]
    2. Add Preprocessing
    3. Taylor expanded in x around inf 83.5%

      \[\leadsto \color{blue}{x} - x \cdot y \]
    4. Taylor expanded in y around 0 46.1%

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

    if -1.00000000000000005e-101 < x

    1. Initial program 100.0%

      \[\left(x + y\right) - x \cdot y \]
    2. Add Preprocessing
    3. Taylor expanded in y around inf 61.2%

      \[\leadsto \color{blue}{y \cdot \left(1 - x\right)} \]
    4. Taylor expanded in x around 0 46.0%

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

Alternative 7: 38.8% accurate, 7.0× speedup?

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ x \end{array} \]
NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 x)
assert(x < y);
double code(double x, double y) {
	return x;
}
NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x
end function
assert x < y;
public static double code(double x, double y) {
	return x;
}
[x, y] = sort([x, y])
def code(x, y):
	return x
x, y = sort([x, y])
function code(x, y)
	return x
end
x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = x;
end
NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := x
\begin{array}{l}
[x, y] = \mathsf{sort}([x, y])\\
\\
x
\end{array}
Derivation
  1. Initial program 100.0%

    \[\left(x + y\right) - x \cdot y \]
  2. Add Preprocessing
  3. Taylor expanded in x around inf 64.4%

    \[\leadsto \color{blue}{x} - x \cdot y \]
  4. Taylor expanded in y around 0 42.3%

    \[\leadsto \color{blue}{x} \]
  5. Add Preprocessing

Alternative 8: 2.6% accurate, 7.0× speedup?

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ 0 \end{array} \]
NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 0.0)
assert(x < y);
double code(double x, double y) {
	return 0.0;
}
NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.0d0
end function
assert x < y;
public static double code(double x, double y) {
	return 0.0;
}
[x, y] = sort([x, y])
def code(x, y):
	return 0.0
x, y = sort([x, y])
function code(x, y)
	return 0.0
end
x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = 0.0;
end
NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := 0.0
\begin{array}{l}
[x, y] = \mathsf{sort}([x, y])\\
\\
0
\end{array}
Derivation
  1. Initial program 100.0%

    \[\left(x + y\right) - x \cdot y \]
  2. Add Preprocessing
  3. Taylor expanded in y around inf 59.5%

    \[\leadsto \color{blue}{y \cdot \left(1 - x\right)} \]
  4. Taylor expanded in x around inf 24.9%

    \[\leadsto \color{blue}{-1 \cdot \left(x \cdot y\right)} \]
  5. Step-by-step derivation
    1. associate-*r*24.9%

      \[\leadsto \color{blue}{\left(-1 \cdot x\right) \cdot y} \]
    2. neg-mul-124.9%

      \[\leadsto \color{blue}{\left(-x\right)} \cdot y \]
    3. *-commutative24.9%

      \[\leadsto \color{blue}{y \cdot \left(-x\right)} \]
  6. Simplified24.9%

    \[\leadsto \color{blue}{y \cdot \left(-x\right)} \]
  7. Step-by-step derivation
    1. add-log-exp15.9%

      \[\leadsto \color{blue}{\log \left(e^{y \cdot \left(-x\right)}\right)} \]
    2. add-sqr-sqrt15.9%

      \[\leadsto \log \color{blue}{\left(\sqrt{e^{y \cdot \left(-x\right)}} \cdot \sqrt{e^{y \cdot \left(-x\right)}}\right)} \]
    3. sqrt-unprod15.9%

      \[\leadsto \log \color{blue}{\left(\sqrt{e^{y \cdot \left(-x\right)} \cdot e^{y \cdot \left(-x\right)}}\right)} \]
    4. exp-prod15.9%

      \[\leadsto \log \left(\sqrt{e^{y \cdot \left(-x\right)} \cdot \color{blue}{{\left(e^{y}\right)}^{\left(-x\right)}}}\right) \]
    5. add-sqr-sqrt7.9%

      \[\leadsto \log \left(\sqrt{e^{y \cdot \left(-x\right)} \cdot {\left(e^{y}\right)}^{\left(-\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right)}}\right) \]
    6. sqrt-unprod8.2%

      \[\leadsto \log \left(\sqrt{e^{y \cdot \left(-x\right)} \cdot {\left(e^{y}\right)}^{\left(-\color{blue}{\sqrt{x \cdot x}}\right)}}\right) \]
    7. sqr-neg8.2%

      \[\leadsto \log \left(\sqrt{e^{y \cdot \left(-x\right)} \cdot {\left(e^{y}\right)}^{\left(-\sqrt{\color{blue}{\left(-x\right) \cdot \left(-x\right)}}\right)}}\right) \]
    8. sqrt-unprod2.1%

      \[\leadsto \log \left(\sqrt{e^{y \cdot \left(-x\right)} \cdot {\left(e^{y}\right)}^{\left(-\color{blue}{\sqrt{-x} \cdot \sqrt{-x}}\right)}}\right) \]
    9. add-sqr-sqrt2.3%

      \[\leadsto \log \left(\sqrt{e^{y \cdot \left(-x\right)} \cdot {\left(e^{y}\right)}^{\left(-\color{blue}{\left(-x\right)}\right)}}\right) \]
    10. pow-flip2.3%

      \[\leadsto \log \left(\sqrt{e^{y \cdot \left(-x\right)} \cdot \color{blue}{\frac{1}{{\left(e^{y}\right)}^{\left(-x\right)}}}}\right) \]
    11. exp-prod2.0%

      \[\leadsto \log \left(\sqrt{e^{y \cdot \left(-x\right)} \cdot \frac{1}{\color{blue}{e^{y \cdot \left(-x\right)}}}}\right) \]
    12. rgt-mult-inverse2.8%

      \[\leadsto \log \left(\sqrt{\color{blue}{1}}\right) \]
    13. metadata-eval2.8%

      \[\leadsto \log \color{blue}{1} \]
    14. metadata-eval2.8%

      \[\leadsto \color{blue}{0} \]
  8. Applied egg-rr2.8%

    \[\leadsto \color{blue}{0} \]
  9. Add Preprocessing

Reproduce

?
herbie shell --seed 2024191 
(FPCore (x y)
  :name "Data.Colour.RGBSpace.HSL:hsl from colour-2.3.3, A"
  :precision binary64
  (- (+ x y) (* x y)))