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

Percentage Accurate: 100.0% → 100.0%

Time: 3.9s

Alternatives: 7

Speedup: 0.1×

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, 0.1× speedup

Math

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

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (fma x (- 1.0 y) y))

C

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

Julia

x, y = sort([x, y])
function code(x, y)
	return fma(x, Float64(1.0 - y), y)
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[(x * N[(1.0 - y), $MachinePrecision] + y), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(x + y\right) - x \cdot y \]
2. Step-by-step derivation

   1. +-commutative 100.0%

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

   2. associate-+r- 100.0%

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

   3. *-commutative 100.0%

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

   4. +-commutative 100.0%

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

   5. *-lft-identity 100.0%

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

   6. metadata-eval 100.0%

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

   7. distribute-rgt-out-- 100.0%

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

   8. fma-define 100.0%

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

   9. metadata-eval 100.0%

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

3. Simplified 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 80.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := y \cdot \left(-x\right)\\ \mathbf{if}\;y \leq -61:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{+126}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 2.95 \cdot 10^{+153} \lor \neg \left(y \leq 4 \cdot 10^{+272}\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (- x))))
   (if (<= y -61.0)
     t_0
     (if (<= y 7.5e+126)
       (+ x y)
       (if (or (<= y 2.95e+153) (not (<= y 4e+272))) t_0 y)))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y * -x;
	double tmp;
	if (y <= -61.0) {
		tmp = t_0;
	} else if (y <= 7.5e+126) {
		tmp = x + y;
	} else if ((y <= 2.95e+153) || !(y <= 4e+272)) {
		tmp = t_0;
	} else {
		tmp = y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y * -x
    if (y <= (-61.0d0)) then
        tmp = t_0
    else if (y <= 7.5d+126) then
        tmp = x + y
    else if ((y <= 2.95d+153) .or. (.not. (y <= 4d+272))) then
        tmp = t_0
    else
        tmp = y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = y * -x;
	double tmp;
	if (y <= -61.0) {
		tmp = t_0;
	} else if (y <= 7.5e+126) {
		tmp = x + y;
	} else if ((y <= 2.95e+153) || !(y <= 4e+272)) {
		tmp = t_0;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y * -x
	tmp = 0
	if y <= -61.0:
		tmp = t_0
	elif y <= 7.5e+126:
		tmp = x + y
	elif (y <= 2.95e+153) or not (y <= 4e+272):
		tmp = t_0
	else:
		tmp = y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y * Float64(-x))
	tmp = 0.0
	if (y <= -61.0)
		tmp = t_0;
	elseif (y <= 7.5e+126)
		tmp = Float64(x + y);
	elseif ((y <= 2.95e+153) || !(y <= 4e+272))
		tmp = t_0;
	else
		tmp = y;
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = y * -x;
	tmp = 0.0;
	if (y <= -61.0)
		tmp = t_0;
	elseif (y <= 7.5e+126)
		tmp = x + y;
	elseif ((y <= 2.95e+153) || ~((y <= 4e+272)))
		tmp = t_0;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(y * (-x)), $MachinePrecision]}, If[LessEqual[y, -61.0], t$95$0, If[LessEqual[y, 7.5e+126], N[(x + y), $MachinePrecision], If[Or[LessEqual[y, 2.95e+153], N[Not[LessEqual[y, 4e+272]], $MachinePrecision]], t$95$0, y]]]]

Derivation

1. Split input into 3 regimes

2. if y < -61 or 7.5000000000000006e126 < y < 2.9500000000000001e153 or 4.0000000000000003e272 < y

   1. Initial program 100.0%

      \[\left(x + y\right) - x \cdot y \]
   2. Step-by-step derivation

      1. +-commutative 100.0%

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

      2. associate-+r- 100.0%

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

      3. cancel-sign-sub-inv 100.0%

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

      4. *-commutative 100.0%

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

      5. cancel-sign-sub 100.0%

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

      6. remove-double-neg 100.0%

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

      7. neg-mul-1 100.0%

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

      8. distribute-rgt-out-- 100.0%

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

      9. distribute-lft-out-- 100.0%

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

      10. distribute-lft-neg-in 100.0%

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

      11. distribute-rgt-neg-out 100.0%

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

      12. remove-double-neg 100.0%

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

      13. distribute-lft-neg-in 100.0%

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

      14. distribute-rgt-neg-in 100.0%

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

      15. distribute-lft-out-- 100.0%

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

      16. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around inf 97.2%

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

      1. neg-mul-1 97.2%

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

      2. sub-neg 97.2%

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

   7. Simplified 97.2%

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

   8. Taylor expanded in x around inf 59.0%

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

      1. mul-1-neg 59.0%

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

      2. *-commutative 59.0%

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

      3. distribute-rgt-neg-in 59.0%

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

   10. Simplified 59.0%

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

   if -61 < y < 7.5000000000000006e126

   1. Initial program 100.0%

      \[\left(x + y\right) - x \cdot y \]
   2. Step-by-step derivation

      1. +-commutative 100.0%

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

      2. associate-+r- 100.0%

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

      3. cancel-sign-sub-inv 100.0%

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

      4. *-commutative 100.0%

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

      5. cancel-sign-sub 100.0%

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

      6. remove-double-neg 100.0%

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

      7. neg-mul-1 100.0%

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

      8. distribute-rgt-out-- 100.0%

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

      9. distribute-lft-out-- 100.0%

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

      10. distribute-lft-neg-in 100.0%

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

      11. distribute-rgt-neg-out 100.0%

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

      12. remove-double-neg 100.0%

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

      13. distribute-lft-neg-in 100.0%

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

      14. distribute-rgt-neg-in 100.0%

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

      15. distribute-lft-out-- 100.0%

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

      16. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 90.8%

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

   if 2.9500000000000001e153 < y < 4.0000000000000003e272

   1. Initial program 100.0%

      \[\left(x + y\right) - x \cdot y \]
   2. Step-by-step derivation

      1. +-commutative 100.0%

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

      2. associate-+r- 100.0%

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

      3. cancel-sign-sub-inv 100.0%

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

      4. *-commutative 100.0%

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

      5. cancel-sign-sub 100.0%

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

      6. remove-double-neg 100.0%

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

      7. neg-mul-1 100.0%

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

      8. distribute-rgt-out-- 100.0%

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

      9. distribute-lft-out-- 100.0%

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

      10. distribute-lft-neg-in 100.0%

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

      11. distribute-rgt-neg-out 100.0%

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

      12. remove-double-neg 100.0%

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

      13. distribute-lft-neg-in 100.0%

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

      14. distribute-rgt-neg-in 100.0%

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

      15. distribute-lft-out-- 100.0%

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

      16. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 54.8%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -61:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{+126}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 2.95 \cdot 10^{+153} \lor \neg \left(y \leq 4 \cdot 10^{+272}\right):\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 6.5 \cdot 10^{-7}\right):\\ \;\;\;\;y \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 6.5e-7))) (* y (- 1.0 x)) (+ x y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 6.5e-7)) {
		tmp = y * (1.0 - x);
	} else {
		tmp = x + y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-1.0d0)) .or. (.not. (y <= 6.5d-7))) then
        tmp = y * (1.0d0 - x)
    else
        tmp = x + y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 6.5e-7)) {
		tmp = y * (1.0 - x);
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if (y <= -1.0) or not (y <= 6.5e-7):
		tmp = y * (1.0 - x)
	else:
		tmp = x + y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if ((y <= -1.0) || !(y <= 6.5e-7))
		tmp = Float64(y * Float64(1.0 - x));
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.0) || ~((y <= 6.5e-7)))
		tmp = y * (1.0 - x);
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[Or[LessEqual[y, -1.0], N[Not[LessEqual[y, 6.5e-7]], $MachinePrecision]], N[(y * N[(1.0 - x), $MachinePrecision]), $MachinePrecision], N[(x + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 6.50000000000000024e-7 < y

   1. Initial program 100.0%

      \[\left(x + y\right) - x \cdot y \]
   2. Step-by-step derivation

      1. +-commutative 100.0%

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

      2. associate-+r- 100.0%

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

      3. cancel-sign-sub-inv 100.0%

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

      4. *-commutative 100.0%

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

      5. cancel-sign-sub 100.0%

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

      6. remove-double-neg 100.0%

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

      7. neg-mul-1 100.0%

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

      8. distribute-rgt-out-- 100.0%

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

      9. distribute-lft-out-- 100.0%

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

      10. distribute-lft-neg-in 100.0%

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

      11. distribute-rgt-neg-out 100.0%

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

      12. remove-double-neg 100.0%

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

      13. distribute-lft-neg-in 100.0%

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

      14. distribute-rgt-neg-in 100.0%

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

      15. distribute-lft-out-- 100.0%

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

      16. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around inf 95.9%

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

      1. neg-mul-1 95.9%

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

      2. sub-neg 95.9%

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

   7. Simplified 95.9%

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

   if -1 < y < 6.50000000000000024e-7

   1. Initial program 100.0%

      \[\left(x + y\right) - x \cdot y \]
   2. Step-by-step derivation

      1. +-commutative 100.0%

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

      2. associate-+r- 100.0%

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

      3. cancel-sign-sub-inv 100.0%

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

      4. *-commutative 100.0%

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

      5. cancel-sign-sub 100.0%

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

      6. remove-double-neg 100.0%

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

      7. neg-mul-1 100.0%

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

      8. distribute-rgt-out-- 100.0%

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

      9. distribute-lft-out-- 100.0%

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

      10. distribute-lft-neg-in 100.0%

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

      11. distribute-rgt-neg-out 100.0%

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

      12. remove-double-neg 100.0%

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

      13. distribute-lft-neg-in 100.0%

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

      14. distribute-rgt-neg-in 100.0%

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

      15. distribute-lft-out-- 100.0%

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

      16. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 98.4%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 6.5 \cdot 10^{-7}\right):\\ \;\;\;\;y \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;y \cdot \left(1 - x\right)\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{-7}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;y - x \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y -1.0) (* y (- 1.0 x)) (if (<= y 6.5e-7) (+ x y) (- y (* x y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = y * (1.0 - x);
	} else if (y <= 6.5e-7) {
		tmp = x + y;
	} else {
		tmp = y - (x * y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.0d0)) then
        tmp = y * (1.0d0 - x)
    else if (y <= 6.5d-7) then
        tmp = x + y
    else
        tmp = y - (x * y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = y * (1.0 - x);
	} else if (y <= 6.5e-7) {
		tmp = x + y;
	} else {
		tmp = y - (x * y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = y * (1.0 - x)
	elif y <= 6.5e-7:
		tmp = x + y
	else:
		tmp = y - (x * y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = Float64(y * Float64(1.0 - x));
	elseif (y <= 6.5e-7)
		tmp = Float64(x + y);
	else
		tmp = Float64(y - Float64(x * y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.0)
		tmp = y * (1.0 - x);
	elseif (y <= 6.5e-7)
		tmp = x + y;
	else
		tmp = y - (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, -1.0], N[(y * N[(1.0 - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6.5e-7], N[(x + y), $MachinePrecision], N[(y - N[(x * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1

   1. Initial program 100.0%

      \[\left(x + y\right) - x \cdot y \]
   2. Step-by-step derivation

      1. +-commutative 100.0%

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

      2. associate-+r- 100.0%

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

      3. cancel-sign-sub-inv 100.0%

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

      4. *-commutative 100.0%

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

      5. cancel-sign-sub 100.0%

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

      6. remove-double-neg 100.0%

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

      7. neg-mul-1 100.0%

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

      8. distribute-rgt-out-- 100.0%

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

      9. distribute-lft-out-- 100.0%

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

      10. distribute-lft-neg-in 100.0%

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

      11. distribute-rgt-neg-out 100.0%

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

      12. remove-double-neg 100.0%

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

      13. distribute-lft-neg-in 100.0%

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

      14. distribute-rgt-neg-in 100.0%

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

      15. distribute-lft-out-- 100.0%

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

      16. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around inf 95.4%

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

      1. neg-mul-1 95.4%

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

      2. sub-neg 95.4%

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

   7. Simplified 95.4%

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

   if -1 < y < 6.50000000000000024e-7

   1. Initial program 100.0%

      \[\left(x + y\right) - x \cdot y \]
   2. Step-by-step derivation

      1. +-commutative 100.0%

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

      2. associate-+r- 100.0%

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

      3. cancel-sign-sub-inv 100.0%

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

      4. *-commutative 100.0%

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

      5. cancel-sign-sub 100.0%

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

      6. remove-double-neg 100.0%

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

      7. neg-mul-1 100.0%

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

      8. distribute-rgt-out-- 100.0%

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

      9. distribute-lft-out-- 100.0%

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

      10. distribute-lft-neg-in 100.0%

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

      11. distribute-rgt-neg-out 100.0%

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

      12. remove-double-neg 100.0%

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

      13. distribute-lft-neg-in 100.0%

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

      14. distribute-rgt-neg-in 100.0%

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

      15. distribute-lft-out-- 100.0%

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

      16. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 98.4%

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

   if 6.50000000000000024e-7 < y

   1. Initial program 100.0%

      \[\left(x + y\right) - x \cdot y \]
   2. Step-by-step derivation

      1. +-commutative 100.0%

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

      2. associate-+r- 100.0%

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

      3. cancel-sign-sub-inv 100.0%

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

      4. *-commutative 100.0%

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

      5. cancel-sign-sub 100.0%

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

      6. remove-double-neg 100.0%

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

      7. neg-mul-1 100.0%

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

      8. distribute-rgt-out-- 100.0%

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

      9. distribute-lft-out-- 100.0%

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

      10. distribute-lft-neg-in 100.0%

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

      11. distribute-rgt-neg-out 100.0%

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

      12. remove-double-neg 100.0%

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

      13. distribute-lft-neg-in 100.0%

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

      14. distribute-rgt-neg-in 100.0%

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

      15. distribute-lft-out-- 100.0%

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

      16. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around inf 96.5%

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

      1. neg-mul-1 96.5%

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

      2. sub-neg 96.5%

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

   7. Simplified 96.5%

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

   8. Taylor expanded in x around 0 96.5%

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

      1. mul-1-neg 96.5%

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

      2. *-commutative 96.5%

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

      3. sub-neg 96.5%

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

   10. Simplified 96.5%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;y \cdot \left(1 - x\right)\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{-7}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;y - x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ y + x \cdot \left(1 - y\right) \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (+ y (* x (- 1.0 y))))

C

assert(x < y);
double code(double x, double y) {
	return y + (x * (1.0 - y));
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = y + (x * (1.0d0 - y))
end function

Java

assert x < y;
public static double code(double x, double y) {
	return y + (x * (1.0 - y));
}

Python

[x, y] = sort([x, y])
def code(x, y):
	return y + (x * (1.0 - y))

Julia

x, y = sort([x, y])
function code(x, y)
	return Float64(y + Float64(x * Float64(1.0 - y)))
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = y + (x * (1.0 - y));
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[(y + N[(x * N[(1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(x + y\right) - x \cdot y \]
2. Step-by-step derivation

   1. +-commutative 100.0%

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

   2. associate-+r- 100.0%

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

   3. cancel-sign-sub-inv 100.0%

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

   4. *-commutative 100.0%

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

   5. cancel-sign-sub 100.0%

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

   6. remove-double-neg 100.0%

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

   7. neg-mul-1 100.0%

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

   8. distribute-rgt-out-- 100.0%

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

   9. distribute-lft-out-- 100.0%

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

   10. distribute-lft-neg-in 100.0%

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

   11. distribute-rgt-neg-out 100.0%

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

   12. remove-double-neg 100.0%

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

   13. distribute-lft-neg-in 100.0%

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

   14. distribute-rgt-neg-in 100.0%

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

   15. distribute-lft-out-- 100.0%

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

   16. metadata-eval 100.0%

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

3. Simplified 100.0%

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

5. Final simplification 100.0%

   \[\leadsto y + x \cdot \left(1 - y\right) \]
6. Add Preprocessing

Alternative 6: 74.8% accurate, 2.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ x + y \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (+ x y))

C

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

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x + y
end function

Java

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

Python

[x, y] = sort([x, y])
def code(x, y):
	return x + y

Julia

x, y = sort([x, y])
function code(x, y)
	return Float64(x + y)
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = x + y;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[(x + y), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(x + y\right) - x \cdot y \]
2. Step-by-step derivation

   1. +-commutative 100.0%

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

   2. associate-+r- 100.0%

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

   3. cancel-sign-sub-inv 100.0%

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

   4. *-commutative 100.0%

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

   5. cancel-sign-sub 100.0%

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

   6. remove-double-neg 100.0%

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

   7. neg-mul-1 100.0%

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

   8. distribute-rgt-out-- 100.0%

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

   9. distribute-lft-out-- 100.0%

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

   10. distribute-lft-neg-in 100.0%

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

   11. distribute-rgt-neg-out 100.0%

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

   12. remove-double-neg 100.0%

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

   13. distribute-lft-neg-in 100.0%

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

   14. distribute-rgt-neg-in 100.0%

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

   15. distribute-lft-out-- 100.0%

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

   16. metadata-eval 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in y around 0 73.0%

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

6. Final simplification 73.0%

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

Alternative 7: 37.9% accurate, 7.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ y \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 y)

C

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

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = y
end function

Java

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

Python

[x, y] = sort([x, y])
def code(x, y):
	return y

Julia

x, y = sort([x, y])
function code(x, y)
	return y
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = y;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := y

Derivation

1. Initial program 100.0%

   \[\left(x + y\right) - x \cdot y \]
2. Step-by-step derivation

   1. +-commutative 100.0%

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

   2. associate-+r- 100.0%

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

   3. cancel-sign-sub-inv 100.0%

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

   4. *-commutative 100.0%

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

   5. cancel-sign-sub 100.0%

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

   6. remove-double-neg 100.0%

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

   7. neg-mul-1 100.0%

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

   8. distribute-rgt-out-- 100.0%

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

   9. distribute-lft-out-- 100.0%

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

   10. distribute-lft-neg-in 100.0%

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

   11. distribute-rgt-neg-out 100.0%

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

   12. remove-double-neg 100.0%

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

   13. distribute-lft-neg-in 100.0%

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

   14. distribute-rgt-neg-in 100.0%

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

   15. distribute-lft-out-- 100.0%

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

   16. metadata-eval 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0 33.7%

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

6. Final simplification 33.7%

   \[\leadsto y \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024034 
(FPCore (x y)
  :name "Data.Colour.RGBSpace.HSL:hsl from colour-2.3.3, A"
  :precision binary64
  (- (+ x y) (* x y)))