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

Percentage Accurate: 100.0% → 100.0%

Time: 8.7s

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

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

FPCore

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

C

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

Julia

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

Wolfram

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: 86.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y + x\right) - y \cdot x\\ t_1 := -y \cdot x\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+298}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+306}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (+ y x) (* y x))) (t_1 (- (* y x))))
   (if (<= t_0 -1e+298) t_1 (if (<= t_0 2e+306) (+ y x) t_1))))

C

double code(double x, double y) {
	double t_0 = (y + x) - (y * x);
	double t_1 = -(y * x);
	double tmp;
	if (t_0 <= -1e+298) {
		tmp = t_1;
	} else if (t_0 <= 2e+306) {
		tmp = y + x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (y + x) - (y * x)
    t_1 = -(y * x)
    if (t_0 <= (-1d+298)) then
        tmp = t_1
    else if (t_0 <= 2d+306) then
        tmp = y + x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (y + x) - (y * x);
	double t_1 = -(y * x);
	double tmp;
	if (t_0 <= -1e+298) {
		tmp = t_1;
	} else if (t_0 <= 2e+306) {
		tmp = y + x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y + x) - (y * x)
	t_1 = -(y * x)
	tmp = 0
	if t_0 <= -1e+298:
		tmp = t_1
	elif t_0 <= 2e+306:
		tmp = y + x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y + x) - Float64(y * x))
	t_1 = Float64(-Float64(y * x))
	tmp = 0.0
	if (t_0 <= -1e+298)
		tmp = t_1;
	elseif (t_0 <= 2e+306)
		tmp = Float64(y + x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y + x) - (y * x);
	t_1 = -(y * x);
	tmp = 0.0;
	if (t_0 <= -1e+298)
		tmp = t_1;
	elseif (t_0 <= 2e+306)
		tmp = y + x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y + x), $MachinePrecision] - N[(y * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-N[(y * x), $MachinePrecision])}, If[LessEqual[t$95$0, -1e+298], t$95$1, If[LessEqual[t$95$0, 2e+306], N[(y + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 x y) (*.f64 x y)) < -9.9999999999999996e297 or 2.00000000000000003e306 < (-.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 97.5

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

   5. Applied rewrites 97.5%

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

   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 97.5

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

   8. Applied rewrites 97.5%

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

   if -9.9999999999999996e297 < (-.f64 (+.f64 x y) (*.f64 x y)) < 2.00000000000000003e306

   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 80.9%

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

      1. *-rgt-identity N/A

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

      2. lower-+.f64 80.9

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

   10. Applied rewrites 80.9%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y + x\right) - y \cdot x \leq -1 \cdot 10^{+298}:\\ \;\;\;\;-y \cdot x\\ \mathbf{elif}\;\left(y + x\right) - y \cdot x \leq 2 \cdot 10^{+306}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;-y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 63.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(y + x\right) - y \cdot x \leq -2 \cdot 10^{-279}:\\ \;\;\;\;\mathsf{fma}\left(-y, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;y - y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (- (+ y x) (* y x)) -2e-279) (fma (- y) x x) (- y (* y x))))

C

double code(double x, double y) {
	double tmp;
	if (((y + x) - (y * x)) <= -2e-279) {
		tmp = fma(-y, x, x);
	} else {
		tmp = y - (y * x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(y + x) - Float64(y * x)) <= -2e-279)
		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[(y + x), $MachinePrecision] - N[(y * x), $MachinePrecision]), $MachinePrecision], -2e-279], 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-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 65.2

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

   5. Applied rewrites 65.2%

      \[\leadsto \color{blue}{x - y \cdot x} \]
   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 65.2

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

   7. Applied rewrites 65.2%

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

   if -2.00000000000000011e-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 63.3

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

   5. Applied rewrites 63.3%

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

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y + x\right) - y \cdot x \leq -2 \cdot 10^{-279}:\\ \;\;\;\;\mathsf{fma}\left(-y, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;y - y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 63.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(y + x\right) - y \cdot x \leq -2 \cdot 10^{-279}:\\ \;\;\;\;x - y \cdot x\\ \mathbf{else}:\\ \;\;\;\;y - y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (- (+ y x) (* y x)) -2e-279) (- x (* y x)) (- y (* y x))))

C

double code(double x, double y) {
	double tmp;
	if (((y + x) - (y * x)) <= -2e-279) {
		tmp = x - (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 (((y + x) - (y * x)) <= (-2d-279)) then
        tmp = x - (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 (((y + x) - (y * x)) <= -2e-279) {
		tmp = x - (y * x);
	} else {
		tmp = y - (y * x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((y + x) - (y * x)) <= -2e-279:
		tmp = x - (y * x)
	else:
		tmp = y - (y * x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(y + x) - Float64(y * x)) <= -2e-279)
		tmp = Float64(x - Float64(y * x));
	else
		tmp = Float64(y - Float64(y * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((y + x) - (y * x)) <= -2e-279)
		tmp = x - (y * x);
	else
		tmp = y - (y * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(y + x), $MachinePrecision] - N[(y * x), $MachinePrecision]), $MachinePrecision], -2e-279], N[(x - N[(y * x), $MachinePrecision]), $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-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 65.2

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

   5. Applied rewrites 65.2%

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

   if -2.00000000000000011e-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 63.3

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

   5. Applied rewrites 63.3%

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

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y + x\right) - y \cdot x \leq -2 \cdot 10^{-279}:\\ \;\;\;\;x - y \cdot x\\ \mathbf{else}:\\ \;\;\;\;y - y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;x - y \cdot x\\ \mathbf{elif}\;x \leq 240000:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;-y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.0) (- x (* y x)) (if (<= x 240000.0) (+ y x) (- (* y x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = x - (y * x);
	} else if (x <= 240000.0) {
		tmp = y + x;
	} else {
		tmp = -(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 <= (-1.0d0)) then
        tmp = x - (y * x)
    else if (x <= 240000.0d0) then
        tmp = y + x
    else
        tmp = -(y * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = x - (y * x);
	} else if (x <= 240000.0) {
		tmp = y + x;
	} else {
		tmp = -(y * x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.0:
		tmp = x - (y * x)
	elif x <= 240000.0:
		tmp = y + x
	else:
		tmp = -(y * x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(x - Float64(y * x));
	elseif (x <= 240000.0)
		tmp = Float64(y + x);
	else
		tmp = Float64(-Float64(y * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = x - (y * x);
	elseif (x <= 240000.0)
		tmp = y + x;
	else
		tmp = -(y * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.0], N[(x - N[(y * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 240000.0], N[(y + x), $MachinePrecision], (-N[(y * x), $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if x < -1

   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 96.9

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

   5. Applied rewrites 96.9%

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

   if -1 < x < 2.4e5

   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 98.7%

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

      1. *-rgt-identity N/A

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

      2. lower-+.f64 98.7

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

   10. Applied rewrites 98.7%

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

   if 2.4e5 < 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 98.8

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

   5. Applied rewrites 98.8%

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

   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 50.0

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

   8. Applied rewrites 50.0%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;x - y \cdot x\\ \mathbf{elif}\;x \leq 240000:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;-y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.8% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ y + x \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(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-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 70.1%

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

   1. *-rgt-identity N/A

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

   2. lower-+.f64 70.1

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

10. Applied rewrites 70.1%

    \[\leadsto \color{blue}{y + x} \]
11. Add Preprocessing

Alternative 7: 2.6% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ -y \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return -y

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := (-y)

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 70.1%

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

   1. *-rgt-identity N/A

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

   2. +-commutative N/A

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

   3. unpow1 N/A

      \[\leadsto x + \color{blue}{{y}^{1}} \]

   4. metadata-eval N/A

      \[\leadsto x + {y}^{\color{blue}{\left(3 - 2\right)}} \]

   5. pow-div N/A

      \[\leadsto x + \color{blue}{\frac{{y}^{3}}{{y}^{2}}} \]

   6. pow2 N/A

      \[\leadsto x + \frac{{y}^{3}}{\color{blue}{y \cdot y}} \]

   7. sqr-pow N/A

      \[\leadsto x + \frac{\color{blue}{{y}^{\left(\frac{3}{2}\right)} \cdot {y}^{\left(\frac{3}{2}\right)}}}{y \cdot y} \]

   8. pow-prod-down N/A

      \[\leadsto x + \frac{\color{blue}{{\left(y \cdot y\right)}^{\left(\frac{3}{2}\right)}}}{y \cdot y} \]

   9. sqr-neg N/A

      \[\leadsto x + \frac{{\color{blue}{\left(\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}}^{\left(\frac{3}{2}\right)}}{y \cdot y} \]

   10. lift-neg.f64 N/A

       \[\leadsto x + \frac{{\left(\color{blue}{\left(\mathsf{neg}\left(y\right)\right)} \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}^{\left(\frac{3}{2}\right)}}{y \cdot y} \]

   11. lift-neg.f64 N/A

       \[\leadsto x + \frac{{\left(\left(\mathsf{neg}\left(y\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)}^{\left(\frac{3}{2}\right)}}{y \cdot y} \]

   12. pow-prod-down N/A

       \[\leadsto x + \frac{\color{blue}{{\left(\mathsf{neg}\left(y\right)\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\mathsf{neg}\left(y\right)\right)}^{\left(\frac{3}{2}\right)}}}{y \cdot y} \]

   13. sqr-pow N/A

       \[\leadsto x + \frac{\color{blue}{{\left(\mathsf{neg}\left(y\right)\right)}^{3}}}{y \cdot y} \]

   14. +-lft-identity N/A

       \[\leadsto x + \frac{{\left(\mathsf{neg}\left(y\right)\right)}^{3}}{y \cdot \color{blue}{\left(0 + y\right)}} \]

   15. +-commutative N/A

       \[\leadsto x + \frac{{\left(\mathsf{neg}\left(y\right)\right)}^{3}}{y \cdot \color{blue}{\left(y + 0\right)}} \]

   16. distribute-rgt-out N/A

       \[\leadsto x + \frac{{\left(\mathsf{neg}\left(y\right)\right)}^{3}}{\color{blue}{y \cdot y + 0 \cdot y}} \]

   17. +-lft-identity N/A

       \[\leadsto x + \frac{{\left(\mathsf{neg}\left(y\right)\right)}^{3}}{\color{blue}{0 + \left(y \cdot y + 0 \cdot y\right)}} \]

   18. metadata-eval N/A

       \[\leadsto x + \frac{{\left(\mathsf{neg}\left(y\right)\right)}^{3}}{\color{blue}{0 \cdot 0} + \left(y \cdot y + 0 \cdot y\right)} \]

   19. lift-neg.f64 N/A

       \[\leadsto x + \frac{{\color{blue}{\left(\mathsf{neg}\left(y\right)\right)}}^{3}}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)} \]

   20. cube-neg N/A

       \[\leadsto x + \frac{\color{blue}{\mathsf{neg}\left({y}^{3}\right)}}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)} \]

   21. sub0-neg N/A

       \[\leadsto x + \frac{\color{blue}{0 - {y}^{3}}}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)} \]

   22. metadata-eval N/A

       \[\leadsto x + \frac{\color{blue}{{0}^{3}} - {y}^{3}}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)} \]

   23. flip3-- N/A

       \[\leadsto x + \color{blue}{\left(0 - y\right)} \]

   24. neg-sub0 N/A

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

10. Applied rewrites 38.6%

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

11. Taylor expanded in x around 0

    \[\leadsto \color{blue}{-1 \cdot y} \]
12. Step-by-step derivation

    1. mul-1-neg N/A

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

    2. lower-neg.f64 2.7

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

13. Applied rewrites 2.7%

    \[\leadsto \color{blue}{-y} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024216 
(FPCore (x y)
  :name "Data.Colour.RGBSpace.HSL:hsl from colour-2.3.3, A"
  :precision binary64
  (- (+ x y) (* x y)))