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

Percentage Accurate: 100.0% → 100.0%

Time: 5.9s

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

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

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

   2. *-rgt-identity N/A

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

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

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

   4. mul-1-neg N/A

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

   5. +-commutative N/A

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

   6. associate-+r+ N/A

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

   7. +-commutative N/A

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

   8. distribute-lft1-in N/A

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

   9. +-commutative N/A

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

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

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

   11. metadata-eval N/A

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

   12. *-lft-identity N/A

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

   13. lower-fma.f64 N/A

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

   14. lower--.f64 100.0

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

5. Applied rewrites 100.0%

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

Alternative 2: 86.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x + y\right) - x \cdot y\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+302} \lor \neg \left(t\_0 \leq 5 \cdot 10^{+299}\right):\\ \;\;\;\;\left(-y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (+ x y) (* x y))))
   (if (or (<= t_0 -5e+302) (not (<= t_0 5e+299))) (* (- y) x) (fma 1.0 y x))))

C

double code(double x, double y) {
	double t_0 = (x + y) - (x * y);
	double tmp;
	if ((t_0 <= -5e+302) || !(t_0 <= 5e+299)) {
		tmp = -y * x;
	} else {
		tmp = fma(1.0, y, x);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(x + y) - Float64(x * y))
	tmp = 0.0
	if ((t_0 <= -5e+302) || !(t_0 <= 5e+299))
		tmp = Float64(Float64(-y) * x);
	else
		tmp = fma(1.0, y, x);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x + y), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -5e+302], N[Not[LessEqual[t$95$0, 5e+299]], $MachinePrecision]], N[((-y) * x), $MachinePrecision], N[(1.0 * y + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 x y) (*.f64 x y)) < -5e302 or 5.0000000000000003e299 < (-.f64 (+.f64 x y) (*.f64 x y))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 97.8%

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

   if -5e302 < (-.f64 (+.f64 x y) (*.f64 x y)) < 5.0000000000000003e299

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

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

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

      2. *-rgt-identity N/A

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

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

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. associate-+r+ N/A

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

      7. +-commutative N/A

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

      8. distribute-lft1-in N/A

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

      9. +-commutative N/A

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

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

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

      11. metadata-eval N/A

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

      12. *-lft-identity N/A

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

      13. lower-fma.f64 N/A

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

      14. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 79.3%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x + y\right) - x \cdot y \leq -5 \cdot 10^{+302} \lor \neg \left(\left(x + y\right) - x \cdot y \leq 5 \cdot 10^{+299}\right):\\ \;\;\;\;\left(-y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 62.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(x + y\right) - x \cdot y \leq -2 \cdot 10^{-268}:\\ \;\;\;\;\left(1 - y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;y - y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (- (+ x y) (* x y)) -2e-268) (* (- 1.0 y) x) (- y (* y x))))

C

double code(double x, double y) {
	double tmp;
	if (((x + y) - (x * y)) <= -2e-268) {
		tmp = (1.0 - y) * x;
	} else {
		tmp = y - (y * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (((x + y) - (x * y)) <= (-2d-268)) then
        tmp = (1.0d0 - y) * x
    else
        tmp = y - (y * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((x + y) - (x * y)) <= -2e-268) {
		tmp = (1.0 - y) * x;
	} else {
		tmp = y - (y * x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x + y) - (x * y)) <= -2e-268:
		tmp = (1.0 - y) * x
	else:
		tmp = y - (y * x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x + y) - Float64(x * y)) <= -2e-268)
		tmp = Float64(Float64(1.0 - y) * x);
	else
		tmp = Float64(y - Float64(y * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((x + y) - (x * y)) <= -2e-268)
		tmp = (1.0 - y) * x;
	else
		tmp = y - (y * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x + y), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision], -2e-268], N[(N[(1.0 - y), $MachinePrecision] * x), $MachinePrecision], N[(y - N[(y * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 x y) (*.f64 x y)) < -1.99999999999999992e-268

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 76.4

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

   5. Applied rewrites 76.4%

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

   if -1.99999999999999992e-268 < (-.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. lower-*.f64 N/A

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

      3. lower--.f64 68.7

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

   5. Applied rewrites 68.7%

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

   7. Applied rewrites 68.7%

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

Alternative 4: 62.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(x + y\right) - x \cdot y \leq -2 \cdot 10^{-268}:\\ \;\;\;\;\left(1 - y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(1 - x\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (- (+ x y) (* x y)) -2e-268) (* (- 1.0 y) x) (* (- 1.0 x) y)))

C

double code(double x, double y) {
	double tmp;
	if (((x + y) - (x * y)) <= -2e-268) {
		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 (((x + y) - (x * y)) <= (-2d-268)) 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 (((x + y) - (x * y)) <= -2e-268) {
		tmp = (1.0 - y) * x;
	} else {
		tmp = (1.0 - x) * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x + y) - (x * y)) <= -2e-268:
		tmp = (1.0 - y) * x
	else:
		tmp = (1.0 - x) * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x + y) - Float64(x * y)) <= -2e-268)
		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 (((x + y) - (x * y)) <= -2e-268)
		tmp = (1.0 - y) * x;
	else
		tmp = (1.0 - x) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x + y), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision], -2e-268], 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)) < -1.99999999999999992e-268

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 76.4

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

   5. Applied rewrites 76.4%

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

   if -1.99999999999999992e-268 < (-.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. lower-*.f64 N/A

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

      3. lower--.f64 68.7

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

   5. Applied rewrites 68.7%

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

Alternative 5: 86.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -75:\\ \;\;\;\;\left(-y\right) \cdot x\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\mathsf{fma}\left(1, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - x\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -75.0) (* (- 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 <= -75.0) {
		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 <= -75.0)
		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, -75.0], 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 < -75

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 67.9

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

   5. Applied rewrites 67.9%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 65.8%

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

   if -75 < y < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

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

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

      2. *-rgt-identity N/A

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

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

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. associate-+r+ N/A

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

      7. +-commutative N/A

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

      8. distribute-lft1-in N/A

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

      9. +-commutative N/A

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

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

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

      11. metadata-eval N/A

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

      12. *-lft-identity N/A

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

      13. lower-fma.f64 N/A

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

      14. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 98.5%

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

   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. lower-*.f64 N/A

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

      3. lower--.f64 99.5

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

   5. Applied rewrites 99.5%

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

Alternative 6: 74.5% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

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

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

   2. *-rgt-identity N/A

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

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

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

   4. mul-1-neg N/A

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

   5. +-commutative N/A

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

   6. associate-+r+ N/A

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

   7. +-commutative N/A

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

   8. distribute-lft1-in N/A

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

   9. +-commutative N/A

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

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

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

   11. metadata-eval N/A

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

   12. *-lft-identity N/A

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

   13. lower-fma.f64 N/A

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

   14. lower--.f64 100.0

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

5. Applied rewrites 100.0%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 66.8%

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

Alternative 7: 37.7% 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. lower-*.f64 N/A

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

   3. lower--.f64 65.2

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

5. Applied rewrites 65.2%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 32.8%

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

Reproduce

herbie shell --seed 2024338 
(FPCore (x y)
  :name "Data.Colour.RGBSpace.HSL:hsl from colour-2.3.3, A"
  :precision binary64
  (- (+ x y) (* x y)))