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

Percentage Accurate: 100.0% → 100.0%

Time: 9.4s

Alternatives: 10

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, 1.0× 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]

Derivation

1. Initial program 100.0%

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

Alternative 2: 86.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x + y\right) - x \cdot y\\ t_1 := -x \cdot y\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+283}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (+ x y) (* x y))) (t_1 (- (* x y))))
   (if (<= t_0 (- INFINITY)) t_1 (if (<= t_0 2e+283) (+ x y) t_1))))

C

double code(double x, double y) {
	double t_0 = (x + y) - (x * y);
	double t_1 = -(x * y);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_0 <= 2e+283) {
		tmp = x + y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = (x + y) - (x * y);
	double t_1 = -(x * y);
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else if (t_0 <= 2e+283) {
		tmp = x + y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x + y) - (x * y)
	t_1 = -(x * y)
	tmp = 0
	if t_0 <= -math.inf:
		tmp = t_1
	elif t_0 <= 2e+283:
		tmp = x + y
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x + y) - Float64(x * y))
	t_1 = Float64(-Float64(x * y))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_0 <= 2e+283)
		tmp = Float64(x + y);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x + y) - (x * y);
	t_1 = -(x * y);
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = t_1;
	elseif (t_0 <= 2e+283)
		tmp = x + y;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x + y), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-N[(x * y), $MachinePrecision])}, If[LessEqual[t$95$0, (-Infinity)], t$95$1, If[LessEqual[t$95$0, 2e+283], N[(x + y), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 x y) (*.f64 x y)) < -inf.0 or 1.99999999999999991e283 < (-.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. --lowering--.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

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

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

      3. mul-1-neg N/A

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

      4. *-lowering-*.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. neg-lowering-neg.f64 93.3

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

   8. Simplified 93.3%

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

   if -inf.0 < (-.f64 (+.f64 x y) (*.f64 x y)) < 1.99999999999999991e283

   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. accelerator-lowering-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. --lowering--.f64 100.0

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

   5. Simplified 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. Simplified 89.1%

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

      1. *-rgt-identity N/A

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

      2. +-lowering-+.f64 89.1

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

   10. Applied egg-rr 89.1%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x + y\right) - x \cdot y \leq -\infty:\\ \;\;\;\;-x \cdot y\\ \mathbf{elif}\;\left(x + y\right) - x \cdot y \leq 2 \cdot 10^{+283}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 63.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(x + y\right) - x \cdot y \leq -2 \cdot 10^{-234}:\\ \;\;\;\;\mathsf{fma}\left(-y, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-y, x, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (- (+ x y) (* x y)) -2e-234) (fma (- y) x x) (fma (- y) x y)))

C

double code(double x, double y) {
	double tmp;
	if (((x + y) - (x * y)) <= -2e-234) {
		tmp = fma(-y, x, x);
	} else {
		tmp = fma(-y, x, y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x + y) - Float64(x * y)) <= -2e-234)
		tmp = fma(Float64(-y), x, x);
	else
		tmp = fma(Float64(-y), x, y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x + y), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision], -2e-234], N[((-y) * x + x), $MachinePrecision], N[((-y) * x + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 x y) (*.f64 x y)) < -1.9999999999999999e-234

   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. --lowering--.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 67.7

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

   5. Simplified 67.7%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. neg-lowering-neg.f64 67.7

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

   7. Applied egg-rr 67.7%

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

   if -1.9999999999999999e-234 < (-.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. --lowering--.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 58.6

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

   5. Simplified 58.6%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. neg-lowering-neg.f64 58.6

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

   7. Applied egg-rr 58.6%

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

Alternative 4: 63.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(x + y\right) - x \cdot y \leq -2 \cdot 10^{-234}:\\ \;\;\;\;\mathsf{fma}\left(-y, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;y - x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (- (+ x y) (* x y)) -2e-234) (fma (- y) x x) (- y (* x y))))

C

double code(double x, double y) {
	double tmp;
	if (((x + y) - (x * y)) <= -2e-234) {
		tmp = fma(-y, x, x);
	} else {
		tmp = y - (x * y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x + y) - Float64(x * y)) <= -2e-234)
		tmp = fma(Float64(-y), x, x);
	else
		tmp = Float64(y - Float64(x * y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x + y), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision], -2e-234], N[((-y) * x + x), $MachinePrecision], N[(y - N[(x * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 x y) (*.f64 x y)) < -1.9999999999999999e-234

   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. --lowering--.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 67.7

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

   5. Simplified 67.7%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. neg-lowering-neg.f64 67.7

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

   7. Applied egg-rr 67.7%

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

   if -1.9999999999999999e-234 < (-.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. --lowering--.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 58.6

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

   5. Simplified 58.6%

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

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x + y\right) - x \cdot y \leq -2 \cdot 10^{-234}:\\ \;\;\;\;\mathsf{fma}\left(-y, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;y - x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 63.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(x + y\right) - x \cdot y \leq -2 \cdot 10^{-234}:\\ \;\;\;\;x - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;y - x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (- (+ x y) (* x y)) -2e-234) (- x (* x y)) (- y (* x y))))

C

double code(double x, double y) {
	double tmp;
	if (((x + y) - (x * y)) <= -2e-234) {
		tmp = x - (x * y);
	} else {
		tmp = y - (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 (((x + y) - (x * y)) <= (-2d-234)) then
        tmp = x - (x * y)
    else
        tmp = y - (x * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((x + y) - (x * y)) <= -2e-234) {
		tmp = x - (x * y);
	} else {
		tmp = y - (x * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x + y) - (x * y)) <= -2e-234:
		tmp = x - (x * y)
	else:
		tmp = y - (x * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x + y) - Float64(x * y)) <= -2e-234)
		tmp = Float64(x - Float64(x * y));
	else
		tmp = Float64(y - Float64(x * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((x + y) - (x * y)) <= -2e-234)
		tmp = x - (x * y);
	else
		tmp = y - (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x + y), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision], -2e-234], N[(x - N[(x * y), $MachinePrecision]), $MachinePrecision], N[(y - N[(x * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 x y) (*.f64 x y)) < -1.9999999999999999e-234

   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. --lowering--.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 67.7

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

   5. Simplified 67.7%

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

   if -1.9999999999999999e-234 < (-.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. --lowering--.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 58.6

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

   5. Simplified 58.6%

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

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x + y\right) - x \cdot y \leq -2 \cdot 10^{-234}:\\ \;\;\;\;x - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;y - x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -19500000:\\ \;\;\;\;x - x \cdot y\\ \mathbf{elif}\;x \leq 52:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -19500000.0) (- x (* x y)) (if (<= x 52.0) (+ x y) (- (* x y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -19500000.0) {
		tmp = x - (x * y);
	} else if (x <= 52.0) {
		tmp = x + y;
	} else {
		tmp = -(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 (x <= (-19500000.0d0)) then
        tmp = x - (x * y)
    else if (x <= 52.0d0) then
        tmp = x + y
    else
        tmp = -(x * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -19500000.0) {
		tmp = x - (x * y);
	} else if (x <= 52.0) {
		tmp = x + y;
	} else {
		tmp = -(x * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -19500000.0:
		tmp = x - (x * y)
	elif x <= 52.0:
		tmp = x + y
	else:
		tmp = -(x * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -19500000.0)
		tmp = Float64(x - Float64(x * y));
	elseif (x <= 52.0)
		tmp = Float64(x + y);
	else
		tmp = Float64(-Float64(x * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -19500000.0)
		tmp = x - (x * y);
	elseif (x <= 52.0)
		tmp = x + y;
	else
		tmp = -(x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -19500000.0], N[(x - N[(x * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 52.0], N[(x + y), $MachinePrecision], (-N[(x * y), $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if x < -1.95e7

   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. --lowering--.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 100.0

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

   5. Simplified 100.0%

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

   if -1.95e7 < x < 52

   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. accelerator-lowering-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. --lowering--.f64 100.0

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

   5. Simplified 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. Simplified 96.1%

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

      1. *-rgt-identity N/A

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

      2. +-lowering-+.f64 96.1

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

   10. Applied egg-rr 96.1%

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

   if 52 < 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. --lowering--.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 98.0

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

   5. Simplified 98.0%

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

   6. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

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

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

      3. mul-1-neg N/A

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

      4. *-lowering-*.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. neg-lowering-neg.f64 52.2

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

   8. Simplified 52.2%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -19500000:\\ \;\;\;\;x - x \cdot y\\ \mathbf{elif}\;x \leq 52:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 39.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(x + y\right) - x \cdot y \leq -2 \cdot 10^{-234}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= (- (+ x y) (* x y)) -2e-234) x y))

C

double code(double x, double y) {
	double tmp;
	if (((x + y) - (x * y)) <= -2e-234) {
		tmp = x;
	} else {
		tmp = y;
	}
	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-234)) then
        tmp = x
    else
        tmp = y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((x + y) - (x * y)) <= -2e-234) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x + y) - (x * y)) <= -2e-234:
		tmp = x
	else:
		tmp = y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x + y) - Float64(x * y)) <= -2e-234)
		tmp = x;
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((x + y) - (x * y)) <= -2e-234)
		tmp = x;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x + y), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision], -2e-234], x, y]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 x y) (*.f64 x y)) < -1.9999999999999999e-234

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x} \]
   4. Step-by-step derivation

   5. Simplified 42.0%

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

   if -1.9999999999999999e-234 < (-.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 0

      \[\leadsto \color{blue}{y} \]
   4. Step-by-step derivation

   5. Simplified 37.5%

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

Alternative 8: 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. accelerator-lowering-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. --lowering--.f64 100.0

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

5. Simplified 100.0%

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

Alternative 9: 75.0% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(x + y), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

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

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

   2. *-lft-identity N/A

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

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

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

   4. +-commutative N/A

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

   5. associate-+r+ N/A

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

   6. *-rgt-identity N/A

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

   7. distribute-lft-neg-out N/A

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

   8. distribute-rgt-neg-out N/A

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

   9. distribute-lft-in N/A

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

   10. mul-1-neg N/A

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

   11. accelerator-lowering-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. --lowering--.f64 100.0

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

5. Simplified 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. Simplified 75.7%

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

   1. *-rgt-identity N/A

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

   2. +-lowering-+.f64 75.7

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

10. Applied egg-rr 75.7%

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

11. Final simplification 75.7%

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

Alternative 10: 38.8% accurate, 12.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 x)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return x

Julia

function code(x, y)
	return x
end

MATLAB

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

Wolfram

code[x_, y_] := x

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{x} \]
4. Step-by-step derivation

5. Simplified 43.0%

   \[\leadsto \color{blue}{x} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024198 
(FPCore (x y)
  :name "Data.Colour.RGBSpace.HSL:hsl from colour-2.3.3, A"
  :precision binary64
  (- (+ x y) (* x y)))