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

Percentage Accurate: 100.0% → 100.0%

Time: 7.4s

Alternatives: 6

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.2× speedup

Math

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

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (fma y (- 1.0 x) x))

C

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

Julia

x, y = sort([x, y])
function code(x, y)
	return fma(y, Float64(1.0 - x), x)
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[(y * N[(1.0 - x), $MachinePrecision] + x), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

   1. distribute-rgt-out-- N/A

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

   2. *-lft-identity N/A

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

   3. cancel-sign-sub-inv N/A

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

   4. +-commutative N/A

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

   5. associate-+r+ N/A

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

   6. *-rgt-identity N/A

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

   7. distribute-lft-neg-out N/A

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

   8. distribute-rgt-neg-out N/A

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

   9. distribute-lft-in N/A

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

   10. mul-1-neg N/A

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

   11. lower-fma.f64 N/A

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

   12. mul-1-neg N/A

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

   13. sub-neg N/A

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

   14. lower--.f64 100.0

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

5. Applied rewrites 100.0%

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

Alternative 2: 84.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \left(x + y\right) - x \cdot y\\ t_1 := -x \cdot y\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+269}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+293}:\\ \;\;\;\;\mathsf{fma}\left(y, 1, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (+ x y) (* x y))) (t_1 (- (* x y))))
   (if (<= t_0 -2e+269) t_1 (if (<= t_0 5e+293) (fma y 1.0 x) t_1))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = (x + y) - (x * y);
	double t_1 = -(x * y);
	double tmp;
	if (t_0 <= -2e+269) {
		tmp = t_1;
	} else if (t_0 <= 5e+293) {
		tmp = fma(y, 1.0, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(Float64(x + y) - Float64(x * y))
	t_1 = Float64(-Float64(x * y))
	tmp = 0.0
	if (t_0 <= -2e+269)
		tmp = t_1;
	elseif (t_0 <= 5e+293)
		tmp = fma(y, 1.0, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(N[(x + y), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-N[(x * y), $MachinePrecision])}, If[LessEqual[t$95$0, -2e+269], t$95$1, If[LessEqual[t$95$0, 5e+293], N[(y * 1.0 + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 x y) (*.f64 x y)) < -2.0000000000000001e269 or 5.00000000000000033e293 < (-.f64 (+.f64 x y) (*.f64 x y))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. distribute-rgt-out-- N/A

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

      2. *-lft-identity N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 95.7

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

   5. Applied rewrites 95.7%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 82.6%

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

   if -2.0000000000000001e269 < (-.f64 (+.f64 x y) (*.f64 x y)) < 5.00000000000000033e293

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. distribute-rgt-out-- N/A

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

      2. *-lft-identity N/A

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

      3. cancel-sign-sub-inv N/A

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

      4. +-commutative N/A

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

      5. associate-+r+ N/A

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

      6. *-rgt-identity N/A

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

      7. distribute-lft-neg-out N/A

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

      8. distribute-rgt-neg-out N/A

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

      9. distribute-lft-in N/A

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

      10. mul-1-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(y, 1, x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 90.6%

      \[\leadsto \mathsf{fma}\left(y, 1, x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x + y\right) - x \cdot y \leq -2 \cdot 10^{+269}:\\ \;\;\;\;-x \cdot y\\ \mathbf{elif}\;\left(x + y\right) - x \cdot y \leq 5 \cdot 10^{+293}:\\ \;\;\;\;\mathsf{fma}\left(y, 1, x\right)\\ \mathbf{else}:\\ \;\;\;\;-x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -112000000000:\\ \;\;\;\;\mathsf{fma}\left(-y, x, x\right)\\ \mathbf{elif}\;x \leq 3.75 \cdot 10^{-121}:\\ \;\;\;\;\mathsf{fma}\left(y, 1, x\right)\\ \mathbf{else}:\\ \;\;\;\;y - x \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -112000000000.0)
   (fma (- y) x x)
   (if (<= x 3.75e-121) (fma y 1.0 x) (- y (* x y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -112000000000.0) {
		tmp = fma(-y, x, x);
	} else if (x <= 3.75e-121) {
		tmp = fma(y, 1.0, x);
	} else {
		tmp = y - (x * y);
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -112000000000.0)
		tmp = fma(Float64(-y), x, x);
	elseif (x <= 3.75e-121)
		tmp = fma(y, 1.0, x);
	else
		tmp = Float64(y - Float64(x * y));
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -112000000000.0], N[((-y) * x + x), $MachinePrecision], If[LessEqual[x, 3.75e-121], N[(y * 1.0 + x), $MachinePrecision], N[(y - N[(x * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.12e11

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. distribute-lft-out-- N/A

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

      2. *-rgt-identity N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   7. Applied rewrites 100.0%

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

   if -1.12e11 < x < 3.75000000000000013e-121

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. distribute-rgt-out-- N/A

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

      2. *-lft-identity N/A

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

      3. cancel-sign-sub-inv N/A

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

      4. +-commutative N/A

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

      5. associate-+r+ N/A

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

      6. *-rgt-identity N/A

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

      7. distribute-lft-neg-out N/A

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

      8. distribute-rgt-neg-out N/A

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

      9. distribute-lft-in N/A

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

      10. mul-1-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(y, 1, x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 99.1%

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

   if 3.75000000000000013e-121 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

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

      1. distribute-rgt-out-- N/A

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

      2. *-lft-identity N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 51.5

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

   5. Applied rewrites 51.5%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -112000000000:\\ \;\;\;\;\mathsf{fma}\left(-y, x, x\right)\\ \mathbf{elif}\;x \leq 3.75 \cdot 10^{-121}:\\ \;\;\;\;\mathsf{fma}\left(y, 1, x\right)\\ \mathbf{else}:\\ \;\;\;\;y - x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -112000000000:\\ \;\;\;\;x - x \cdot y\\ \mathbf{elif}\;x \leq 3.75 \cdot 10^{-121}:\\ \;\;\;\;\mathsf{fma}\left(y, 1, x\right)\\ \mathbf{else}:\\ \;\;\;\;y - x \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -112000000000.0)
   (- x (* x y))
   (if (<= x 3.75e-121) (fma y 1.0 x) (- y (* x y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -112000000000.0) {
		tmp = x - (x * y);
	} else if (x <= 3.75e-121) {
		tmp = fma(y, 1.0, x);
	} else {
		tmp = y - (x * y);
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -112000000000.0)
		tmp = Float64(x - Float64(x * y));
	elseif (x <= 3.75e-121)
		tmp = fma(y, 1.0, x);
	else
		tmp = Float64(y - Float64(x * y));
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -112000000000.0], N[(x - N[(x * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.75e-121], N[(y * 1.0 + x), $MachinePrecision], N[(y - N[(x * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.12e11

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. distribute-lft-out-- N/A

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

      2. *-rgt-identity N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -1.12e11 < x < 3.75000000000000013e-121

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. distribute-rgt-out-- N/A

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

      2. *-lft-identity N/A

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

      3. cancel-sign-sub-inv N/A

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

      4. +-commutative N/A

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

      5. associate-+r+ N/A

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

      6. *-rgt-identity N/A

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

      7. distribute-lft-neg-out N/A

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

      8. distribute-rgt-neg-out N/A

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

      9. distribute-lft-in N/A

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

      10. mul-1-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(y, 1, x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 99.1%

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

   if 3.75000000000000013e-121 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

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

      1. distribute-rgt-out-- N/A

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

      2. *-lft-identity N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 51.5

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

   5. Applied rewrites 51.5%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -112000000000:\\ \;\;\;\;x - x \cdot y\\ \mathbf{elif}\;x \leq 3.75 \cdot 10^{-121}:\\ \;\;\;\;\mathsf{fma}\left(y, 1, x\right)\\ \mathbf{else}:\\ \;\;\;\;y - x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -112000000000:\\ \;\;\;\;x - x \cdot y\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+17}:\\ \;\;\;\;\mathsf{fma}\left(y, 1, x\right)\\ \mathbf{else}:\\ \;\;\;\;-x \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -112000000000.0)
   (- x (* x y))
   (if (<= x 2e+17) (fma y 1.0 x) (- (* x y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -112000000000.0) {
		tmp = x - (x * y);
	} else if (x <= 2e+17) {
		tmp = fma(y, 1.0, x);
	} else {
		tmp = -(x * y);
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -112000000000.0)
		tmp = Float64(x - Float64(x * y));
	elseif (x <= 2e+17)
		tmp = fma(y, 1.0, x);
	else
		tmp = Float64(-Float64(x * y));
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -112000000000.0], N[(x - N[(x * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2e+17], N[(y * 1.0 + x), $MachinePrecision], (-N[(x * y), $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if x < -1.12e11

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. distribute-lft-out-- N/A

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

      2. *-rgt-identity N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -1.12e11 < x < 2e17

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. distribute-rgt-out-- N/A

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

      2. *-lft-identity N/A

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

      3. cancel-sign-sub-inv N/A

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

      4. +-commutative N/A

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

      5. associate-+r+ N/A

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

      6. *-rgt-identity N/A

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

      7. distribute-lft-neg-out N/A

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

      8. distribute-rgt-neg-out N/A

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

      9. distribute-lft-in N/A

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

      10. mul-1-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(y, 1, x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 97.7%

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

   if 2e17 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. distribute-rgt-out-- N/A

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

      2. *-lft-identity N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 46.0

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

   5. Applied rewrites 46.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 46.0%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -112000000000:\\ \;\;\;\;x - x \cdot y\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+17}:\\ \;\;\;\;\mathsf{fma}\left(y, 1, x\right)\\ \mathbf{else}:\\ \;\;\;\;-x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.1% accurate, 1.7× speedup

Math

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

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (fma y 1.0 x))

C

assert(x < y);
double code(double x, double y) {
	return fma(y, 1.0, x);
}

Julia

x, y = sort([x, y])
function code(x, y)
	return fma(y, 1.0, x)
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[(y * 1.0 + x), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

   1. distribute-rgt-out-- N/A

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

   2. *-lft-identity N/A

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

   3. cancel-sign-sub-inv N/A

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

   4. +-commutative N/A

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

   5. associate-+r+ N/A

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

   6. *-rgt-identity N/A

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

   7. distribute-lft-neg-out N/A

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

   8. distribute-rgt-neg-out N/A

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

   9. distribute-lft-in N/A

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

   10. mul-1-neg N/A

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

   11. lower-fma.f64 N/A

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

   12. mul-1-neg N/A

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

   13. sub-neg N/A

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

   14. lower--.f64 100.0

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

5. Applied rewrites 100.0%

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

6. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(y, 1, x\right) \]
7. Step-by-step derivation

8. Applied rewrites 78.6%

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

Reproduce

herbie shell --seed 2024221 
(FPCore (x y)
  :name "Data.Colour.RGBSpace.HSL:hsl from colour-2.3.3, A"
  :precision binary64
  (- (+ x y) (* x y)))