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

Percentage Accurate: 100.0% → 100.0%

Time: 9.0s

Alternatives: 7

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

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

FPCore

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))

C

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

Julia

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

Wolfram

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]

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 y around 0

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

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

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

   2. *-lft-identity N/A

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

   3. unsub-neg N/A

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

   4. mul-1-neg N/A

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

   5. +-commutative N/A

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

   6. associate-+l+ N/A

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

   7. +-commutative N/A

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

   8. +-commutative N/A

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

   9. mul-1-neg N/A

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

   10. *-rgt-identity N/A

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

   11. distribute-rgt-neg-in N/A

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

   12. mul-1-neg N/A

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

   13. distribute-lft-in N/A

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

   14. lower-fma.f64 N/A

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

   15. mul-1-neg N/A

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

   16. sub-neg N/A

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

   17. lower--.f64 100.0

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

8. Applied rewrites 100.0%

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

Alternative 2: 87.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \left(x + y\right) - x \cdot y\\ t_1 := y \cdot \left(-x\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{+308}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 (* y (- x))))
   (if (<= t_0 (- INFINITY)) t_1 (if (<= t_0 1e+308) (+ x y) t_1))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = (x + y) - (x * y);
	double t_1 = y * -x;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_0 <= 1e+308) {
		tmp = x + y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = (x + y) - (x * y);
	double t_1 = y * -x;
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else if (t_0 <= 1e+308) {
		tmp = x + y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = (x + y) - (x * y)
	t_1 = y * -x
	tmp = 0
	if t_0 <= -math.inf:
		tmp = t_1
	elif t_0 <= 1e+308:
		tmp = x + y
	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(y * Float64(-x))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_0 <= 1e+308)
		tmp = Float64(x + y);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = (x + y) - (x * y);
	t_1 = y * -x;
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = t_1;
	elseif (t_0 <= 1e+308)
		tmp = x + y;
	else
		tmp = t_1;
	end
	tmp_2 = 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[(y * (-x)), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], t$95$1, If[LessEqual[t$95$0, 1e+308], N[(x + y), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 x y) (*.f64 x y)) < -inf.0 or 1e308 < (-.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 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. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. mul-1-neg N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 100.0

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

   8. Applied rewrites 100.0%

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

   if -inf.0 < (-.f64 (+.f64 x y) (*.f64 x y)) < 1e308

   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

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

   8. Applied rewrites 88.5%

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

      1. *-rgt-identity N/A

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

      2. lower-+.f64 88.5

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

   10. Applied rewrites 88.5%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x + y\right) - x \cdot y \leq -\infty:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;\left(x + y\right) - x \cdot y \leq 10^{+308}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;\left(x + y\right) - x \cdot y \leq -1 \cdot 10^{-279}:\\ \;\;\;\;\mathsf{fma}\left(-y, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-y, x, y\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= (- (+ x y) (* x y)) -1e-279) (fma (- y) x x) (fma (- y) x y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (((x + y) - (x * y)) <= -1e-279) {
		tmp = fma(-y, x, x);
	} else {
		tmp = fma(-y, x, y);
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x + y) - Float64(x * y)) <= -1e-279)
		tmp = fma(Float64(-y), x, x);
	else
		tmp = fma(Float64(-y), 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[N[(N[(x + y), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision], -1e-279], N[((-y) * x + x), $MachinePrecision], N[((-y) * x + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 x y) (*.f64 x y)) < -1.00000000000000006e-279

   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 62.1

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

   5. Applied rewrites 62.1%

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

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

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

      2. distribute-rgt1-in N/A

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

      3. distribute-lft1-in N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 62.1

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

   7. Applied rewrites 62.1%

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

   if -1.00000000000000006e-279 < (-.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 60.8

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

   5. Applied rewrites 60.8%

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

      1. lift-*.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-lft-neg-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 60.8

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

   7. Applied rewrites 60.8%

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

Alternative 4: 98.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;\left(x + y\right) - x \cdot y \leq -1 \cdot 10^{-279}:\\ \;\;\;\;\mathsf{fma}\left(-y, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;y - x \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= (- (+ x y) (* x y)) -1e-279) (fma (- y) x x) (- y (* x y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (((x + y) - (x * y)) <= -1e-279) {
		tmp = fma(-y, x, x);
	} else {
		tmp = y - (x * y);
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x + y) - Float64(x * y)) <= -1e-279)
		tmp = fma(Float64(-y), x, 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[N[(N[(x + y), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision], -1e-279], N[((-y) * x + x), $MachinePrecision], N[(y - N[(x * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 x y) (*.f64 x y)) < -1.00000000000000006e-279

   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 62.1

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

   5. Applied rewrites 62.1%

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

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

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

      2. distribute-rgt1-in N/A

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

      3. distribute-lft1-in N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 62.1

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

   7. Applied rewrites 62.1%

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

   if -1.00000000000000006e-279 < (-.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 60.8

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

   5. Applied rewrites 60.8%

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

4. Final simplification 61.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x + y\right) - x \cdot y \leq -1 \cdot 10^{-279}:\\ \;\;\;\;\mathsf{fma}\left(-y, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;y - x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;\left(x + y\right) - x \cdot y \leq -1 \cdot 10^{-279}:\\ \;\;\;\;x - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;y - x \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= (- (+ x y) (* x y)) -1e-279) (- x (* x y)) (- y (* x y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (((x + y) - (x * y)) <= -1e-279) {
		tmp = x - (x * y);
	} else {
		tmp = y - (x * y);
	}
	return tmp;
}

Fortran

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 + y) - (x * y)) <= (-1d-279)) then
        tmp = x - (x * y)
    else
        tmp = y - (x * y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (((x + y) - (x * y)) <= -1e-279) {
		tmp = x - (x * y);
	} else {
		tmp = y - (x * y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if ((x + y) - (x * y)) <= -1e-279:
		tmp = x - (x * y)
	else:
		tmp = y - (x * y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x + y) - Float64(x * y)) <= -1e-279)
		tmp = Float64(x - Float64(x * y));
	else
		tmp = Float64(y - Float64(x * y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((x + y) - (x * y)) <= -1e-279)
		tmp = x - (x * y);
	else
		tmp = y - (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[N[(N[(x + y), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision], -1e-279], N[(x - N[(x * y), $MachinePrecision]), $MachinePrecision], N[(y - N[(x * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 x y) (*.f64 x y)) < -1.00000000000000006e-279

   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 62.1

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

   5. Applied rewrites 62.1%

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

   if -1.00000000000000006e-279 < (-.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 60.8

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

   5. Applied rewrites 60.8%

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

4. Final simplification 61.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x + y\right) - x \cdot y \leq -1 \cdot 10^{-279}:\\ \;\;\;\;x - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;y - x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;x - x \cdot y\\ \mathbf{elif}\;x \leq 9.2 \cdot 10^{+17}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.0) (- x (* x y)) (if (<= x 9.2e+17) (+ x y) (* y (- x)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = x - (x * y);
	} else if (x <= 9.2e+17) {
		tmp = x + y;
	} else {
		tmp = y * -x;
	}
	return tmp;
}

Fortran

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 <= (-1.0d0)) then
        tmp = x - (x * y)
    else if (x <= 9.2d+17) then
        tmp = x + y
    else
        tmp = y * -x
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = x - (x * y);
	} else if (x <= 9.2e+17) {
		tmp = x + y;
	} else {
		tmp = y * -x;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1.0:
		tmp = x - (x * y)
	elif x <= 9.2e+17:
		tmp = x + y
	else:
		tmp = y * -x
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(x - Float64(x * y));
	elseif (x <= 9.2e+17)
		tmp = Float64(x + y);
	else
		tmp = Float64(y * Float64(-x));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = x - (x * y);
	elseif (x <= 9.2e+17)
		tmp = x + y;
	else
		tmp = y * -x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1.0], N[(x - N[(x * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 9.2e+17], N[(x + y), $MachinePrecision], N[(y * (-x)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 99.9%

      \[\left(x + y\right) - x \cdot y \]
   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 98.8

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

   5. Applied rewrites 98.8%

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

   if -1 < x < 9.2e17

   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

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

   8. Applied rewrites 99.2%

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

      1. *-rgt-identity N/A

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

      2. lower-+.f64 99.2

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

   10. Applied rewrites 99.2%

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

   if 9.2e17 < x

   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. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. mul-1-neg N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 48.7

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

   8. Applied rewrites 48.7%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;x - x \cdot y\\ \mathbf{elif}\;x \leq 9.2 \cdot 10^{+17}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ x + y \end{array} \]

FPCore

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

C

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

Fortran

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
end function

Java

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

Python

[x, y] = sort([x, y])
def code(x, y):
	return x + y

Julia

x, y = sort([x, y])
function code(x, y)
	return Float64(x + y)
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = x + y;
end

Wolfram

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

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

8. Applied rewrites 77.7%

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

   1. *-rgt-identity N/A

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

   2. lower-+.f64 77.7

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

10. Applied rewrites 77.7%

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

11. Final simplification 77.7%

    \[\leadsto x + y \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024214 
(FPCore (x y)
  :name "Data.Colour.RGBSpace.HSL:hsl from colour-2.3.3, A"
  :precision binary64
  (- (+ x y) (* x y)))