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

Percentage Accurate: 100.0% → 100.0%

Time: 6.3s

Alternatives: 8

Speedup: 1.0×

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

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

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. associate--l+ 100.0%

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

   2. +-commutative 100.0%

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

   3. remove-double-neg 100.0%

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

   4. unsub-neg 100.0%

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

   5. cancel-sign-sub-inv 100.0%

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

   6. associate--l+ 100.0%

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

   7. neg-mul-1 100.0%

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

   8. cancel-sign-sub-inv 100.0%

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

   9. distribute-lft-neg-out 100.0%

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

   10. distribute-rgt-neg-out 100.0%

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

   11. *-commutative 100.0%

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

   12. distribute-rgt-out 100.0%

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

   13. +-commutative 100.0%

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

   14. sub-neg 100.0%

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

   15. 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. Add Preprocessing

Alternative 2: 77.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{+183} \lor \neg \left(x \leq -1.6 \cdot 10^{+66}\right) \land x \leq 150000:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (or (<= x -9e+183) (and (not (<= x -1.6e+66)) (<= x 150000.0)))
   (+ y x)
   (* y (- x))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if ((x <= -9e+183) || (!(x <= -1.6e+66) && (x <= 150000.0))) {
		tmp = y + x;
	} else {
		tmp = y * -x;
	}
	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 ((x <= (-9d+183)) .or. (.not. (x <= (-1.6d+66))) .and. (x <= 150000.0d0)) then
        tmp = y + x
    else
        tmp = y * -x
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if ((x <= -9e+183) || (!(x <= -1.6e+66) && (x <= 150000.0))) {
		tmp = y + x;
	} else {
		tmp = y * -x;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if (x <= -9e+183) or (not (x <= -1.6e+66) and (x <= 150000.0)):
		tmp = y + x
	else:
		tmp = y * -x
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if ((x <= -9e+183) || (!(x <= -1.6e+66) && (x <= 150000.0)))
		tmp = Float64(y + x);
	else
		tmp = Float64(y * Float64(-x));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -9e+183) || (~((x <= -1.6e+66)) && (x <= 150000.0)))
		tmp = y + x;
	else
		tmp = y * -x;
	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[x, -9e+183], And[N[Not[LessEqual[x, -1.6e+66]], $MachinePrecision], LessEqual[x, 150000.0]]], N[(y + x), $MachinePrecision], N[(y * (-x)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -9.00000000000000034e183 or -1.6e66 < x < 1.5e5

   1. Initial program 100.0%

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

      1. associate--l+ 100.0%

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

      2. +-commutative 100.0%

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

      3. remove-double-neg 100.0%

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

      4. unsub-neg 100.0%

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

      5. cancel-sign-sub-inv 100.0%

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

      6. associate--l+ 100.0%

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

      7. neg-mul-1 100.0%

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

      8. cancel-sign-sub-inv 100.0%

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

      9. distribute-lft-neg-out 100.0%

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

      10. distribute-rgt-neg-out 100.0%

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

      11. *-commutative 100.0%

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

      12. distribute-rgt-out 100.0%

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

      13. +-commutative 100.0%

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

      14. sub-neg 100.0%

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

      15. 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 89.9%

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

   if -9.00000000000000034e183 < x < -1.6e66 or 1.5e5 < x

   1. Initial program 100.0%

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

      1. associate--l+ 100.0%

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

      2. +-commutative 100.0%

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

      3. remove-double-neg 100.0%

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

      4. unsub-neg 100.0%

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

      5. cancel-sign-sub-inv 100.0%

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

      6. associate--l+ 100.0%

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

      7. neg-mul-1 100.0%

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

      8. cancel-sign-sub-inv 100.0%

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

      9. distribute-lft-neg-out 100.0%

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

      10. distribute-rgt-neg-out 100.0%

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

      11. *-commutative 100.0%

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

      12. distribute-rgt-out 100.0%

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

      13. +-commutative 100.0%

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

      14. sub-neg 100.0%

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

      15. 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 54.3%

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

      1. mul-1-neg 54.3%

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

      2. unsub-neg 54.3%

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

   7. Simplified 54.3%

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

   8. Taylor expanded in x around inf 53.5%

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

      1. mul-1-neg 53.5%

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

   10. Simplified 53.5%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{+183} \lor \neg \left(x \leq -1.6 \cdot 10^{+66}\right) \land x \leq 150000:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{+16}:\\ \;\;\;\;x \cdot \left(1 - y\right)\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-27}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;y - y \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -6.2e+16) (* x (- 1.0 y)) (if (<= x 3e-27) (+ y x) (- y (* y x)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -6.2e+16) {
		tmp = x * (1.0 - y);
	} else if (x <= 3e-27) {
		tmp = y + x;
	} else {
		tmp = y - (y * x);
	}
	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 (x <= (-6.2d+16)) then
        tmp = x * (1.0d0 - y)
    else if (x <= 3d-27) then
        tmp = y + x
    else
        tmp = y - (y * x)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -6.2e+16) {
		tmp = x * (1.0 - y);
	} else if (x <= 3e-27) {
		tmp = y + x;
	} else {
		tmp = y - (y * x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -6.2e+16:
		tmp = x * (1.0 - y)
	elif x <= 3e-27:
		tmp = y + x
	else:
		tmp = y - (y * x)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -6.2e+16)
		tmp = Float64(x * Float64(1.0 - y));
	elseif (x <= 3e-27)
		tmp = Float64(y + x);
	else
		tmp = Float64(y - Float64(y * x));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -6.2e+16)
		tmp = x * (1.0 - y);
	elseif (x <= 3e-27)
		tmp = y + x;
	else
		tmp = y - (y * x);
	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[x, -6.2e+16], N[(x * N[(1.0 - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3e-27], N[(y + x), $MachinePrecision], N[(y - N[(y * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.2e16

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. +-commutative 99.9%

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

      3. remove-double-neg 99.9%

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

      4. unsub-neg 99.9%

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

      5. cancel-sign-sub-inv 99.9%

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

      6. associate--l+ 99.9%

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

      7. neg-mul-1 99.9%

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

      8. cancel-sign-sub-inv 99.9%

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

      9. distribute-lft-neg-out 99.9%

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

      10. distribute-rgt-neg-out 99.9%

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

      11. *-commutative 99.9%

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

      12. distribute-rgt-out 100.0%

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

      13. +-commutative 100.0%

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

      14. sub-neg 100.0%

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

      15. 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 99.9%

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

      1. +-commutative 99.9%

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

      2. fma-define 100.0%

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

      3. mul-1-neg 100.0%

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

      4. unsub-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around inf 100.0%

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

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

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

   10. Simplified 100.0%

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

   if -6.2e16 < x < 3.0000000000000001e-27

   1. Initial program 100.0%

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

      1. associate--l+ 100.0%

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

      2. +-commutative 100.0%

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

      3. remove-double-neg 100.0%

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

      4. unsub-neg 100.0%

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

      5. cancel-sign-sub-inv 100.0%

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

      6. associate--l+ 100.0%

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

      7. neg-mul-1 100.0%

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

      8. cancel-sign-sub-inv 100.0%

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

      9. distribute-lft-neg-out 100.0%

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

      10. distribute-rgt-neg-out 100.0%

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

      11. *-commutative 100.0%

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

      12. distribute-rgt-out 100.0%

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

      13. +-commutative 100.0%

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

      14. sub-neg 100.0%

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

      15. 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 98.7%

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

   if 3.0000000000000001e-27 < x

   1. Initial program 100.0%

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

      1. associate--l+ 100.0%

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

      2. +-commutative 100.0%

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

      3. remove-double-neg 100.0%

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

      4. unsub-neg 100.0%

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

      5. cancel-sign-sub-inv 100.0%

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

      6. associate--l+ 100.0%

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

      7. neg-mul-1 100.0%

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

      8. cancel-sign-sub-inv 100.0%

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

      9. distribute-lft-neg-out 100.0%

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

      10. distribute-rgt-neg-out 100.0%

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

      11. *-commutative 100.0%

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

      12. distribute-rgt-out 100.0%

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

      13. +-commutative 100.0%

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

      14. sub-neg 100.0%

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

      15. 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 49.5%

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

      1. mul-1-neg 49.5%

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

      2. unsub-neg 49.5%

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

   7. Simplified 49.5%

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

      1. sub-neg 49.5%

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

      2. distribute-rgt-in 49.5%

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

      3. *-un-lft-identity 49.5%

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

      4. *-commutative 49.5%

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

      5. distribute-rgt-neg-out 49.5%

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

      6. unsub-neg 49.5%

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

   9. Applied egg-rr 49.5%

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

Alternative 4: 99.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{+16}:\\ \;\;\;\;x \cdot \left(1 - y\right)\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-27}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - x\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -6.2e+16)
   (* x (- 1.0 y))
   (if (<= x 3e-27) (+ y x) (* y (- 1.0 x)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -6.2e+16) {
		tmp = x * (1.0 - y);
	} else if (x <= 3e-27) {
		tmp = y + x;
	} else {
		tmp = y * (1.0 - x);
	}
	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 (x <= (-6.2d+16)) then
        tmp = x * (1.0d0 - y)
    else if (x <= 3d-27) then
        tmp = y + x
    else
        tmp = y * (1.0d0 - x)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -6.2e+16) {
		tmp = x * (1.0 - y);
	} else if (x <= 3e-27) {
		tmp = y + x;
	} else {
		tmp = y * (1.0 - x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -6.2e+16:
		tmp = x * (1.0 - y)
	elif x <= 3e-27:
		tmp = y + x
	else:
		tmp = y * (1.0 - x)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -6.2e+16)
		tmp = Float64(x * Float64(1.0 - y));
	elseif (x <= 3e-27)
		tmp = Float64(y + x);
	else
		tmp = Float64(y * Float64(1.0 - x));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -6.2e+16)
		tmp = x * (1.0 - y);
	elseif (x <= 3e-27)
		tmp = y + x;
	else
		tmp = y * (1.0 - x);
	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[x, -6.2e+16], N[(x * N[(1.0 - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3e-27], N[(y + x), $MachinePrecision], N[(y * N[(1.0 - x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.2e16

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. +-commutative 99.9%

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

      3. remove-double-neg 99.9%

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

      4. unsub-neg 99.9%

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

      5. cancel-sign-sub-inv 99.9%

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

      6. associate--l+ 99.9%

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

      7. neg-mul-1 99.9%

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

      8. cancel-sign-sub-inv 99.9%

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

      9. distribute-lft-neg-out 99.9%

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

      10. distribute-rgt-neg-out 99.9%

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

      11. *-commutative 99.9%

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

      12. distribute-rgt-out 100.0%

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

      13. +-commutative 100.0%

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

      14. sub-neg 100.0%

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

      15. 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 99.9%

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

      1. +-commutative 99.9%

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

      2. fma-define 100.0%

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

      3. mul-1-neg 100.0%

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

      4. unsub-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around inf 100.0%

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

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

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

   10. Simplified 100.0%

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

   if -6.2e16 < x < 3.0000000000000001e-27

   1. Initial program 100.0%

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

      1. associate--l+ 100.0%

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

      2. +-commutative 100.0%

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

      3. remove-double-neg 100.0%

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

      4. unsub-neg 100.0%

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

      5. cancel-sign-sub-inv 100.0%

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

      6. associate--l+ 100.0%

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

      7. neg-mul-1 100.0%

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

      8. cancel-sign-sub-inv 100.0%

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

      9. distribute-lft-neg-out 100.0%

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

      10. distribute-rgt-neg-out 100.0%

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

      11. *-commutative 100.0%

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

      12. distribute-rgt-out 100.0%

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

      13. +-commutative 100.0%

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

      14. sub-neg 100.0%

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

      15. 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 98.7%

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

   if 3.0000000000000001e-27 < x

   1. Initial program 100.0%

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

      1. associate--l+ 100.0%

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

      2. +-commutative 100.0%

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

      3. remove-double-neg 100.0%

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

      4. unsub-neg 100.0%

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

      5. cancel-sign-sub-inv 100.0%

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

      6. associate--l+ 100.0%

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

      7. neg-mul-1 100.0%

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

      8. cancel-sign-sub-inv 100.0%

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

      9. distribute-lft-neg-out 100.0%

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

      10. distribute-rgt-neg-out 100.0%

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

      11. *-commutative 100.0%

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

      12. distribute-rgt-out 100.0%

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

      13. +-commutative 100.0%

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

      14. sub-neg 100.0%

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

      15. 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 49.5%

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

      1. mul-1-neg 49.5%

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

      2. unsub-neg 49.5%

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

   7. Simplified 49.5%

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

Alternative 5: 98.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{+16}:\\ \;\;\;\;x \cdot \left(1 - y\right)\\ \mathbf{elif}\;x \leq 240000:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -6.2e+16) (* x (- 1.0 y)) (if (<= x 240000.0) (+ y x) (* y (- x)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -6.2e+16) {
		tmp = x * (1.0 - y);
	} else if (x <= 240000.0) {
		tmp = y + x;
	} else {
		tmp = y * -x;
	}
	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 (x <= (-6.2d+16)) then
        tmp = x * (1.0d0 - y)
    else if (x <= 240000.0d0) then
        tmp = y + x
    else
        tmp = y * -x
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -6.2e+16) {
		tmp = x * (1.0 - y);
	} else if (x <= 240000.0) {
		tmp = y + x;
	} else {
		tmp = y * -x;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -6.2e+16:
		tmp = x * (1.0 - y)
	elif x <= 240000.0:
		tmp = y + x
	else:
		tmp = y * -x
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -6.2e+16)
		tmp = Float64(x * Float64(1.0 - y));
	elseif (x <= 240000.0)
		tmp = Float64(y + x);
	else
		tmp = Float64(y * Float64(-x));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -6.2e+16)
		tmp = x * (1.0 - y);
	elseif (x <= 240000.0)
		tmp = y + x;
	else
		tmp = y * -x;
	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[x, -6.2e+16], N[(x * N[(1.0 - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 240000.0], N[(y + x), $MachinePrecision], N[(y * (-x)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.2e16

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. +-commutative 99.9%

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

      3. remove-double-neg 99.9%

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

      4. unsub-neg 99.9%

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

      5. cancel-sign-sub-inv 99.9%

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

      6. associate--l+ 99.9%

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

      7. neg-mul-1 99.9%

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

      8. cancel-sign-sub-inv 99.9%

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

      9. distribute-lft-neg-out 99.9%

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

      10. distribute-rgt-neg-out 99.9%

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

      11. *-commutative 99.9%

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

      12. distribute-rgt-out 100.0%

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

      13. +-commutative 100.0%

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

      14. sub-neg 100.0%

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

      15. 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 99.9%

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

      1. +-commutative 99.9%

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

      2. fma-define 100.0%

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

      3. mul-1-neg 100.0%

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

      4. unsub-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around inf 100.0%

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

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

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

   10. Simplified 100.0%

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

   if -6.2e16 < x < 2.4e5

   1. Initial program 100.0%

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

      1. associate--l+ 100.0%

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

      2. +-commutative 100.0%

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

      3. remove-double-neg 100.0%

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

      4. unsub-neg 100.0%

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

      5. cancel-sign-sub-inv 100.0%

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

      6. associate--l+ 100.0%

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

      7. neg-mul-1 100.0%

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

      8. cancel-sign-sub-inv 100.0%

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

      9. distribute-lft-neg-out 100.0%

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

      10. distribute-rgt-neg-out 100.0%

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

      11. *-commutative 100.0%

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

      12. distribute-rgt-out 100.0%

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

      13. +-commutative 100.0%

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

      14. sub-neg 100.0%

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

      15. 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 96.6%

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

   if 2.4e5 < x

   1. Initial program 100.0%

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

      1. associate--l+ 100.0%

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

      2. +-commutative 100.0%

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

      3. remove-double-neg 100.0%

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

      4. unsub-neg 100.0%

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

      5. cancel-sign-sub-inv 100.0%

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

      6. associate--l+ 100.0%

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

      7. neg-mul-1 100.0%

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

      8. cancel-sign-sub-inv 100.0%

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

      9. distribute-lft-neg-out 100.0%

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

      10. distribute-rgt-neg-out 100.0%

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

      11. *-commutative 100.0%

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

      12. distribute-rgt-out 100.0%

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

      13. +-commutative 100.0%

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

      14. sub-neg 100.0%

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

      15. 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 47.6%

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

      1. mul-1-neg 47.6%

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

      2. unsub-neg 47.6%

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

   7. Simplified 47.6%

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

   8. Taylor expanded in x around inf 46.6%

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

      1. mul-1-neg 46.6%

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

   10. Simplified 46.6%

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

Alternative 6: 63.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.35 \cdot 10^{-27}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (if (<= y 2.35e-27) x y))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 2.35e-27) {
		tmp = x;
	} 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) :: tmp
    if (y <= 2.35d-27) then
        tmp = x
    else
        tmp = y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 2.35e-27) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 2.35e-27:
		tmp = x
	else:
		tmp = y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 2.35e-27)
		tmp = x;
	else
		tmp = y;
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.35e-27)
		tmp = x;
	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_] := If[LessEqual[y, 2.35e-27], x, y]

Derivation

1. Split input into 2 regimes

2. if y < 2.35000000000000016e-27

   1. Initial program 100.0%

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

      1. associate--l+ 100.0%

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

      2. +-commutative 100.0%

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

      3. remove-double-neg 100.0%

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

      4. unsub-neg 100.0%

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

      5. cancel-sign-sub-inv 100.0%

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

      6. associate--l+ 100.0%

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

      7. neg-mul-1 100.0%

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

      8. cancel-sign-sub-inv 100.0%

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

      9. distribute-lft-neg-out 100.0%

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

      10. distribute-rgt-neg-out 100.0%

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

      11. *-commutative 100.0%

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

      12. distribute-rgt-out 100.0%

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

      13. +-commutative 100.0%

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

      14. sub-neg 100.0%

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

      15. 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 0 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-define 100.0%

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

      3. mul-1-neg 100.0%

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

      4. unsub-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in y around 0 50.9%

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

   if 2.35000000000000016e-27 < y

   1. Initial program 100.0%

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

      1. associate--l+ 100.0%

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

      2. +-commutative 100.0%

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

      3. remove-double-neg 100.0%

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

      4. unsub-neg 100.0%

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

      5. cancel-sign-sub-inv 100.0%

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

      6. associate--l+ 100.0%

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

      7. neg-mul-1 100.0%

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

      8. cancel-sign-sub-inv 100.0%

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

      9. distribute-lft-neg-out 100.0%

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

      10. distribute-rgt-neg-out 100.0%

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

      11. *-commutative 100.0%

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

      12. distribute-rgt-out 100.0%

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

      13. +-commutative 100.0%

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

      14. sub-neg 100.0%

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

      15. 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 44.2%

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

Alternative 7: 74.8% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. associate--l+ 100.0%

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

   2. +-commutative 100.0%

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

   3. remove-double-neg 100.0%

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

   4. unsub-neg 100.0%

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

   5. cancel-sign-sub-inv 100.0%

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

   6. associate--l+ 100.0%

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

   7. neg-mul-1 100.0%

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

   8. cancel-sign-sub-inv 100.0%

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

   9. distribute-lft-neg-out 100.0%

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

   10. distribute-rgt-neg-out 100.0%

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

   11. *-commutative 100.0%

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

   12. distribute-rgt-out 100.0%

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

   13. +-commutative 100.0%

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

   14. sub-neg 100.0%

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

   15. 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 72.0%

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

Alternative 8: 38.3% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. associate--l+ 100.0%

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

   2. +-commutative 100.0%

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

   3. remove-double-neg 100.0%

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

   4. unsub-neg 100.0%

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

   5. cancel-sign-sub-inv 100.0%

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

   6. associate--l+ 100.0%

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

   7. neg-mul-1 100.0%

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

   8. cancel-sign-sub-inv 100.0%

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

   9. distribute-lft-neg-out 100.0%

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

   10. distribute-rgt-neg-out 100.0%

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

   11. *-commutative 100.0%

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

   12. distribute-rgt-out 100.0%

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

   13. +-commutative 100.0%

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

   14. sub-neg 100.0%

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

   15. 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 0 100.0%

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

   1. +-commutative 100.0%

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

   2. fma-define 100.0%

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

   3. mul-1-neg 100.0%

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

   4. unsub-neg 100.0%

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

7. Simplified 100.0%

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

8. Taylor expanded in y around 0 38.3%

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

Reproduce

herbie shell --seed 2024123 
(FPCore (x y)
  :name "Data.Colour.RGBSpace.HSL:hsl from colour-2.3.3, A"
  :precision binary64
  (- (+ x y) (* x y)))