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

Percentage Accurate: 100.0% → 100.0%

Time: 5.5s

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

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

FPCore

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

C

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

Julia

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

Wolfram

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. cancel-sign-sub-inv N/A

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

   4. +-commutative N/A

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

   5. associate-+l+ N/A

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

   6. *-lft-identity N/A

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

   7. distribute-rgt-in N/A

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

   8. sub-neg N/A

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

   9. *-commutative N/A

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

   10. sub-neg N/A

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

   11. mul-1-neg N/A

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

   12. lower-fma.f64 N/A

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

   13. mul-1-neg N/A

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

   14. sub-neg N/A

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

   15. 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: 86.8% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (+ x y) (* x y))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 5e+293)))
     (* (- y) x)
     (fma 1.0 y x))))

C

double code(double x, double y) {
	double t_0 = (x + y) - (x * y);
	double tmp;
	if ((t_0 <= -((double) INFINITY)) || !(t_0 <= 5e+293)) {
		tmp = -y * x;
	} else {
		tmp = fma(1.0, y, x);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(x + y) - Float64(x * y))
	tmp = 0.0
	if ((t_0 <= Float64(-Inf)) || !(t_0 <= 5e+293))
		tmp = Float64(Float64(-y) * x);
	else
		tmp = fma(1.0, y, x);
	end
	return tmp
end

Wolfram

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, 5e+293]], $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 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 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. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. sub-neg N/A

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

      7. lower--.f64 97.8

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

   5. Applied rewrites 97.8%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 95.6%

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

   if -inf.0 < (-.f64 (+.f64 x y) (*.f64 x y)) < 5.00000000000000033e293

   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. cancel-sign-sub-inv N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. *-lft-identity N/A

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

      7. distribute-rgt-in N/A

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

      8. sub-neg N/A

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

      9. *-commutative N/A

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

      10. sub-neg N/A

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

      11. mul-1-neg N/A

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

      12. lower-fma.f64 N/A

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

      13. mul-1-neg N/A

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

      14. sub-neg N/A

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

      15. 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 87.9%

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

Alternative 3: 64.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(x + y\right) - x \cdot y \leq -2 \cdot 10^{-249}:\\ \;\;\;\;\mathsf{fma}\left(-y, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;y - y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (- (+ x y) (* x y)) -2e-249) (fma (- y) x x) (- y (* y x))))

C

double code(double x, double y) {
	double tmp;
	if (((x + y) - (x * y)) <= -2e-249) {
		tmp = fma(-y, x, x);
	} else {
		tmp = y - (y * x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x + y) - Float64(x * y)) <= -2e-249)
		tmp = fma(Float64(-y), x, x);
	else
		tmp = Float64(y - Float64(y * x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x + y), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision], -2e-249], N[((-y) * x + x), $MachinePrecision], N[(y - N[(y * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 x y) (*.f64 x y)) < -2.00000000000000011e-249

   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. *-commutative N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. sub-neg N/A

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

      7. lower--.f64 64.4

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

   5. Applied rewrites 64.4%

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

   7. Applied rewrites 64.4%

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

   if -2.00000000000000011e-249 < (-.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 64.6

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

   5. Applied rewrites 64.6%

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

Alternative 4: 64.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(x + y\right) - x \cdot y \leq -2 \cdot 10^{-249}:\\ \;\;\;\;\left(1 - y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;y - y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (- (+ x y) (* x y)) -2e-249) (* (- 1.0 y) x) (- y (* y x))))

C

double code(double x, double y) {
	double tmp;
	if (((x + y) - (x * y)) <= -2e-249) {
		tmp = (1.0 - y) * x;
	} else {
		tmp = y - (y * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (((x + y) - (x * y)) <= (-2d-249)) then
        tmp = (1.0d0 - y) * x
    else
        tmp = y - (y * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((x + y) - (x * y)) <= -2e-249) {
		tmp = (1.0 - y) * x;
	} else {
		tmp = y - (y * x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x + y) - (x * y)) <= -2e-249:
		tmp = (1.0 - y) * x
	else:
		tmp = y - (y * x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x + y) - Float64(x * y)) <= -2e-249)
		tmp = Float64(Float64(1.0 - y) * x);
	else
		tmp = Float64(y - Float64(y * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((x + y) - (x * y)) <= -2e-249)
		tmp = (1.0 - y) * x;
	else
		tmp = y - (y * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x + y), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision], -2e-249], N[(N[(1.0 - y), $MachinePrecision] * x), $MachinePrecision], N[(y - N[(y * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 x y) (*.f64 x y)) < -2.00000000000000011e-249

   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. *-commutative N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. sub-neg N/A

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

      7. lower--.f64 64.4

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

   5. Applied rewrites 64.4%

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

   if -2.00000000000000011e-249 < (-.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 64.6

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

   5. Applied rewrites 64.6%

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

Alternative 5: 99.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -115 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\left(1 - y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -115.0) (not (<= x 1.0))) (* (- 1.0 y) x) (fma 1.0 y x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -115.0) || !(x <= 1.0)) {
		tmp = (1.0 - y) * x;
	} else {
		tmp = fma(1.0, y, x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -115.0) || !(x <= 1.0))
		tmp = Float64(Float64(1.0 - y) * x);
	else
		tmp = fma(1.0, y, x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -115.0], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], N[(N[(1.0 - y), $MachinePrecision] * x), $MachinePrecision], N[(1.0 * y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -115 or 1 < 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. *-commutative N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. sub-neg N/A

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

      7. lower--.f64 98.6

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

   5. Applied rewrites 98.6%

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

   if -115 < x < 1

   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. cancel-sign-sub-inv N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. *-lft-identity N/A

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

      7. distribute-rgt-in N/A

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

      8. sub-neg N/A

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

      9. *-commutative N/A

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

      10. sub-neg N/A

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

      11. mul-1-neg N/A

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

      12. lower-fma.f64 N/A

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

      13. mul-1-neg N/A

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

      14. sub-neg N/A

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

      15. 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%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -115 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\left(1 - y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.1% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, y)
	return fma(1.0, y, x)
end

Wolfram

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. cancel-sign-sub-inv N/A

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

   4. +-commutative N/A

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

   5. associate-+l+ N/A

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

   6. *-lft-identity N/A

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

   7. distribute-rgt-in N/A

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

   8. sub-neg N/A

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

   9. *-commutative N/A

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

   10. sub-neg N/A

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

   11. mul-1-neg N/A

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

   12. lower-fma.f64 N/A

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

   13. mul-1-neg N/A

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

   14. sub-neg N/A

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

   15. 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 72.4%

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

Reproduce

herbie shell --seed 2024307 
(FPCore (x y)
  :name "Data.Colour.RGBSpace.HSL:hsl from colour-2.3.3, A"
  :precision binary64
  (- (+ x y) (* x y)))