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

Percentage Accurate: 100.0% → 100.0%

Time: 5.2s

Alternatives: 8

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 y around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. sub-neg N/A

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

   4. mul-1-neg N/A

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

   5. lower-fma.f64 N/A

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

   6. mul-1-neg N/A

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

   7. sub-neg N/A

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

   8. 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: 86.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y + x\right) - y \cdot x\\ t_1 := \left(-y\right) \cdot x\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+300}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+300}:\\ \;\;\;\;\mathsf{fma}\left(1, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (+ y x) (* y x))) (t_1 (* (- y) x)))
   (if (<= t_0 -2e+300) t_1 (if (<= t_0 2e+300) (fma 1.0 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 <= -2e+300) {
		tmp = t_1;
	} else if (t_0 <= 2e+300) {
		tmp = fma(1.0, 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 <= -2e+300)
		tmp = t_1;
	elseif (t_0 <= 2e+300)
		tmp = fma(1.0, y, x);
	else
		tmp = t_1;
	end
	return 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, -2e+300], t$95$1, If[LessEqual[t$95$0, 2e+300], N[(1.0 * y + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 x y) (*.f64 x y)) < -2.0000000000000001e300 or 2.0000000000000001e300 < (-.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 97.9

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

   5. Applied rewrites 97.9%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 93.8%

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

   if -2.0000000000000001e300 < (-.f64 (+.f64 x y) (*.f64 x y)) < 2.0000000000000001e300

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. sub-neg N/A

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

      8. 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 85.4%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y + x\right) - y \cdot x \leq -2 \cdot 10^{+300}:\\ \;\;\;\;\left(-y\right) \cdot x\\ \mathbf{elif}\;\left(y + x\right) - y \cdot x \leq 2 \cdot 10^{+300}:\\ \;\;\;\;\mathsf{fma}\left(1, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 63.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(y + x\right) - y \cdot x \leq -1 \cdot 10^{-271}:\\ \;\;\;\;\left(1 - y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-y, x, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (- (+ y x) (* y x)) -1e-271) (* (- 1.0 y) x) (fma (- y) x y)))

C

double code(double x, double y) {
	double tmp;
	if (((y + x) - (y * x)) <= -1e-271) {
		tmp = (1.0 - y) * x;
	} else {
		tmp = fma(-y, x, y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(y + x) - Float64(y * x)) <= -1e-271)
		tmp = Float64(Float64(1.0 - y) * x);
	else
		tmp = fma(Float64(-y), x, y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(y + x), $MachinePrecision] - N[(y * x), $MachinePrecision]), $MachinePrecision], -1e-271], N[(N[(1.0 - y), $MachinePrecision] * x), $MachinePrecision], N[((-y) * x + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 x y) (*.f64 x y)) < -9.99999999999999963e-272

   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 63.7

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

   5. Applied rewrites 63.7%

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

   if -9.99999999999999963e-272 < (-.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. *-commutative N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. sub-neg N/A

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

      7. lower--.f64 64.4

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

   5. Applied rewrites 64.4%

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

   7. Applied rewrites 64.5%

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

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y + x\right) - y \cdot x \leq -1 \cdot 10^{-271}:\\ \;\;\;\;\left(1 - y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-y, x, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 63.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(y + x\right) - y \cdot x \leq -1 \cdot 10^{-271}:\\ \;\;\;\;\left(1 - y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(1 - x\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (- (+ y x) (* y x)) -1e-271) (* (- 1.0 y) x) (* (- 1.0 x) y)))

C

double code(double x, double y) {
	double tmp;
	if (((y + x) - (y * x)) <= -1e-271) {
		tmp = (1.0 - y) * x;
	} else {
		tmp = (1.0 - x) * y;
	}
	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)) <= (-1d-271)) then
        tmp = (1.0d0 - y) * x
    else
        tmp = (1.0d0 - x) * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((y + x) - (y * x)) <= -1e-271) {
		tmp = (1.0 - y) * x;
	} else {
		tmp = (1.0 - x) * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((y + x) - (y * x)) <= -1e-271:
		tmp = (1.0 - y) * x
	else:
		tmp = (1.0 - x) * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(y + x) - Float64(y * x)) <= -1e-271)
		tmp = Float64(Float64(1.0 - y) * x);
	else
		tmp = Float64(Float64(1.0 - x) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((y + x) - (y * x)) <= -1e-271)
		tmp = (1.0 - y) * x;
	else
		tmp = (1.0 - x) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(y + x), $MachinePrecision] - N[(y * x), $MachinePrecision]), $MachinePrecision], -1e-271], N[(N[(1.0 - y), $MachinePrecision] * x), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 x y) (*.f64 x y)) < -9.99999999999999963e-272

   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 63.7

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

   5. Applied rewrites 63.7%

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

   if -9.99999999999999963e-272 < (-.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. *-commutative N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. sub-neg N/A

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

      7. lower--.f64 64.4

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

   5. Applied rewrites 64.4%

      \[\leadsto \color{blue}{\left(1 - x\right) \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y + x\right) - y \cdot x \leq -1 \cdot 10^{-271}:\\ \;\;\;\;\left(1 - y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(1 - x\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 38.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(y + x\right) - y \cdot x \leq -1 \cdot 10^{-271}:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;1 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (- (+ y x) (* y x)) -1e-271) (* 1.0 x) (* 1.0 y)))

C

double code(double x, double y) {
	double tmp;
	if (((y + x) - (y * x)) <= -1e-271) {
		tmp = 1.0 * x;
	} else {
		tmp = 1.0 * y;
	}
	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)) <= (-1d-271)) then
        tmp = 1.0d0 * x
    else
        tmp = 1.0d0 * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((y + x) - (y * x)) <= -1e-271) {
		tmp = 1.0 * x;
	} else {
		tmp = 1.0 * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((y + x) - (y * x)) <= -1e-271:
		tmp = 1.0 * x
	else:
		tmp = 1.0 * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(y + x) - Float64(y * x)) <= -1e-271)
		tmp = Float64(1.0 * x);
	else
		tmp = Float64(1.0 * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((y + x) - (y * x)) <= -1e-271)
		tmp = 1.0 * x;
	else
		tmp = 1.0 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(y + x), $MachinePrecision] - N[(y * x), $MachinePrecision]), $MachinePrecision], -1e-271], N[(1.0 * x), $MachinePrecision], N[(1.0 * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 x y) (*.f64 x y)) < -9.99999999999999963e-272

   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 63.7

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

   5. Applied rewrites 63.7%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 38.3%

      \[\leadsto 1 \cdot x \]

   if -9.99999999999999963e-272 < (-.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. *-commutative N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. sub-neg N/A

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

      7. lower--.f64 64.4

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

   5. Applied rewrites 64.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 33.5%

      \[\leadsto 1 \cdot y \]
3. Recombined 2 regimes into one program.

4. Final simplification 35.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y + x\right) - y \cdot x \leq -1 \cdot 10^{-271}:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;1 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.38 \cdot 10^{+14}:\\ \;\;\;\;\left(-y\right) \cdot x\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\mathsf{fma}\left(1, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - x\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.38e+14)
   (* (- y) x)
   (if (<= y 1.0) (fma 1.0 y x) (* (- 1.0 x) y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.38e+14) {
		tmp = -y * x;
	} else if (y <= 1.0) {
		tmp = fma(1.0, y, x);
	} else {
		tmp = (1.0 - x) * y;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.38e+14)
		tmp = Float64(Float64(-y) * x);
	elseif (y <= 1.0)
		tmp = fma(1.0, y, x);
	else
		tmp = Float64(Float64(1.0 - x) * y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.38e+14], N[((-y) * x), $MachinePrecision], If[LessEqual[y, 1.0], N[(1.0 * y + x), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.38e14

   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 60.6

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

   5. Applied rewrites 60.6%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 60.6%

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

   if -1.38e14 < y < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. sub-neg N/A

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

      8. 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 99.1%

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

   if 1 < 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. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. sub-neg N/A

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

      7. lower--.f64 98.9

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

   5. Applied rewrites 98.9%

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

Alternative 7: 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 y around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. sub-neg N/A

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

   4. mul-1-neg N/A

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

   5. lower-fma.f64 N/A

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

   6. mul-1-neg N/A

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

   7. sub-neg N/A

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

   8. 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 70.7%

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

Alternative 8: 39.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 1 \cdot y \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 * y
end function

Java

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

Python

def code(x, y):
	return 1.0 * y

Julia

function code(x, y)
	return Float64(1.0 * y)
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 * y;
end

Wolfram

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. sub-neg N/A

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

   3. mul-1-neg N/A

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

   4. lower-*.f64 N/A

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

   5. mul-1-neg N/A

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

   6. sub-neg N/A

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

   7. lower--.f64 64.3

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

5. Applied rewrites 64.3%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 35.7%

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

Reproduce

herbie shell --seed 2024270 
(FPCore (x y)
  :name "Data.Colour.RGBSpace.HSL:hsl from colour-2.3.3, A"
  :precision binary64
  (- (+ x y) (* x y)))