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

Percentage Accurate: 100.0% → 100.0%

Time: 4.8s

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. *-rgt-identity N/A

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

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

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

   4. mul-1-neg N/A

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

   5. +-commutative N/A

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

   6. associate-+r+ N/A

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

   7. +-commutative N/A

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

   8. distribute-lft1-in N/A

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

   9. +-commutative N/A

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

   10. fp-cancel-sign-sub-inv N/A

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

   11. metadata-eval N/A

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

   12. *-lft-identity N/A

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

   13. lower-fma.f64 N/A

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

   14. lower--.f64 100.0

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

5. Applied rewrites 100.0%

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

Alternative 2: 85.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \left(x + y\right) - x \cdot y\\ \mathbf{if}\;t\_0 \leq -\infty \lor \neg \left(t\_0 \leq 2 \cdot 10^{+291}\right):\\ \;\;\;\;\left(-y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, y, x\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (+ x y) (* x y))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 2e+291)))
     (* (- y) x)
     (fma 1.0 y x))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = (x + y) - (x * y);
	double tmp;
	if ((t_0 <= -((double) INFINITY)) || !(t_0 <= 2e+291)) {
		tmp = -y * x;
	} else {
		tmp = fma(1.0, y, x);
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(Float64(x + y) - Float64(x * y))
	tmp = 0.0
	if ((t_0 <= Float64(-Inf)) || !(t_0 <= 2e+291))
		tmp = Float64(Float64(-y) * x);
	else
		tmp = fma(1.0, y, x);
	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]}, If[Or[LessEqual[t$95$0, (-Infinity)], N[Not[LessEqual[t$95$0, 2e+291]], $MachinePrecision]], N[((-y) * x), $MachinePrecision], N[(1.0 * y + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 x y) (*.f64 x y)) < -inf.0 or 1.9999999999999999e291 < (-.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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 97.1%

      \[\leadsto \left(-y\right) \cdot x \]

   if -inf.0 < (-.f64 (+.f64 x y) (*.f64 x y)) < 1.9999999999999999e291

   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-lft-out-- N/A

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

      2. *-rgt-identity N/A

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

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

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. associate-+r+ N/A

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

      7. +-commutative N/A

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

      8. distribute-lft1-in N/A

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

      9. +-commutative N/A

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

      10. fp-cancel-sign-sub-inv N/A

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

      11. metadata-eval N/A

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

      12. *-lft-identity N/A

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

      13. lower-fma.f64 N/A

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

      14. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 86.3%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x + y\right) - x \cdot y \leq -\infty \lor \neg \left(\left(x + y\right) - x \cdot y \leq 2 \cdot 10^{+291}\right):\\ \;\;\;\;\left(-y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -500000:\\ \;\;\;\;\mathsf{fma}\left(-x, y, x\right)\\ \mathbf{elif}\;x \leq 1.02 \cdot 10^{-164}:\\ \;\;\;\;\mathsf{fma}\left(1, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - x\right) \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -500000.0)
   (fma (- x) y x)
   (if (<= x 1.02e-164) (fma 1.0 y x) (* (- 1.0 x) y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -500000.0) {
		tmp = fma(-x, y, x);
	} else if (x <= 1.02e-164) {
		tmp = fma(1.0, y, x);
	} else {
		tmp = (1.0 - x) * y;
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -500000.0)
		tmp = fma(Float64(-x), y, x);
	elseif (x <= 1.02e-164)
		tmp = fma(1.0, y, x);
	else
		tmp = Float64(Float64(1.0 - 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, -500000.0], N[((-x) * y + x), $MachinePrecision], If[LessEqual[x, 1.02e-164], N[(1.0 * y + x), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5e5

   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-lft-out-- N/A

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

      2. *-rgt-identity N/A

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

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

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. associate-+r+ N/A

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

      7. +-commutative N/A

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

      8. distribute-lft1-in N/A

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

      9. +-commutative N/A

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

      10. fp-cancel-sign-sub-inv N/A

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

      11. metadata-eval N/A

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

      12. *-lft-identity N/A

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

      13. lower-fma.f64 N/A

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

      14. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 100.0%

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

   if -5e5 < x < 1.02e-164

   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-lft-out-- N/A

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

      2. *-rgt-identity N/A

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

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

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. associate-+r+ N/A

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

      7. +-commutative N/A

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

      8. distribute-lft1-in N/A

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

      9. +-commutative N/A

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

      10. fp-cancel-sign-sub-inv N/A

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

      11. metadata-eval N/A

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

      12. *-lft-identity N/A

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

      13. lower-fma.f64 N/A

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

      14. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 98.9%

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

   if 1.02e-164 < 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 53.5

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

   5. Applied rewrites 53.5%

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

Alternative 4: 99.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -500000:\\ \;\;\;\;\left(1 - y\right) \cdot x\\ \mathbf{elif}\;x \leq 1.02 \cdot 10^{-164}:\\ \;\;\;\;\mathsf{fma}\left(1, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - x\right) \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -500000.0)
   (* (- 1.0 y) x)
   (if (<= x 1.02e-164) (fma 1.0 y x) (* (- 1.0 x) y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -500000.0) {
		tmp = (1.0 - y) * x;
	} else if (x <= 1.02e-164) {
		tmp = fma(1.0, y, x);
	} else {
		tmp = (1.0 - x) * y;
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -500000.0)
		tmp = Float64(Float64(1.0 - y) * x);
	elseif (x <= 1.02e-164)
		tmp = fma(1.0, y, x);
	else
		tmp = Float64(Float64(1.0 - 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, -500000.0], N[(N[(1.0 - y), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[x, 1.02e-164], N[(1.0 * y + x), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5e5

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -5e5 < x < 1.02e-164

   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-lft-out-- N/A

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

      2. *-rgt-identity N/A

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

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

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. associate-+r+ N/A

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

      7. +-commutative N/A

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

      8. distribute-lft1-in N/A

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

      9. +-commutative N/A

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

      10. fp-cancel-sign-sub-inv N/A

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

      11. metadata-eval N/A

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

      12. *-lft-identity N/A

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

      13. lower-fma.f64 N/A

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

      14. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 98.9%

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

   if 1.02e-164 < 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 53.5

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

   5. Applied rewrites 53.5%

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

Alternative 5: 98.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -3400:\\ \;\;\;\;\left(-y\right) \cdot x\\ \mathbf{elif}\;y \leq 6 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(1, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - x\right) \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y -3400.0)
   (* (- y) x)
   (if (<= y 6e-10) (fma 1.0 y x) (* (- 1.0 x) y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= -3400.0) {
		tmp = -y * x;
	} else if (y <= 6e-10) {
		tmp = fma(1.0, y, x);
	} else {
		tmp = (1.0 - x) * y;
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= -3400.0)
		tmp = Float64(Float64(-y) * x);
	elseif (y <= 6e-10)
		tmp = fma(1.0, y, x);
	else
		tmp = Float64(Float64(1.0 - 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[y, -3400.0], N[((-y) * x), $MachinePrecision], If[LessEqual[y, 6e-10], N[(1.0 * y + x), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3400

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 49.8

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

   5. Applied rewrites 49.8%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 48.3%

      \[\leadsto \left(-y\right) \cdot x \]

   if -3400 < y < 6e-10

   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-lft-out-- N/A

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

      2. *-rgt-identity N/A

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

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

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. associate-+r+ N/A

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

      7. +-commutative N/A

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

      8. distribute-lft1-in N/A

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

      9. +-commutative N/A

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

      10. fp-cancel-sign-sub-inv N/A

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

      11. metadata-eval N/A

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

      12. *-lft-identity N/A

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

      13. lower-fma.f64 N/A

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

      14. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 99.6%

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

   if 6e-10 < 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

Alternative 6: 74.8% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. *-rgt-identity N/A

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

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

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

   4. mul-1-neg N/A

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

   5. +-commutative N/A

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

   6. associate-+r+ N/A

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

   7. +-commutative N/A

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

   8. distribute-lft1-in N/A

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

   9. +-commutative N/A

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

   10. fp-cancel-sign-sub-inv N/A

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

   11. metadata-eval N/A

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

   12. *-lft-identity N/A

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

   13. lower-fma.f64 N/A

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

   14. lower--.f64 100.0

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

5. Applied rewrites 100.0%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 76.5%

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

Alternative 7: 39.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

assert(x < y);
double code(double x, double y) {
	return 1.0 * 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 = 1.0d0 * y
end function

Java

assert x < y;
public static double code(double x, double y) {
	return 1.0 * y;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	return 1.0 * y

Julia

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

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = 1.0 * y;
end

Wolfram

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

Derivation

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. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower--.f64 65.1

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

5. Applied rewrites 65.1%

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

6. Taylor expanded in x around 0

   \[\leadsto 1 \cdot y \]
7. Step-by-step derivation

8. Applied rewrites 41.4%

   \[\leadsto 1 \cdot y \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024332 
(FPCore (x y)
  :name "Data.Colour.RGBSpace.HSL:hsl from colour-2.3.3, A"
  :precision binary64
  (- (+ x y) (* x y)))