Statistics.Distribution.Binomial:$cvariance from math-functions-0.1.5.2

Percentage Accurate: 99.9% → 99.9%

Time: 7.4s

Alternatives: 7

Speedup: 1.0×

• 24.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

   \[\left(x \cdot y\right) \cdot \left(1 - y\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. sub-neg N/A

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

   2. distribute-lft-in N/A

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

   3. *-rgt-identity N/A

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

   4. *-commutative N/A

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

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

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

   6. *-commutative N/A

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

   7. associate-*l* N/A

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

   8. *-lowering-*.f64 N/A

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

   9. *-lowering-*.f64 N/A

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

   10. neg-lowering-neg.f64 99.9

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

4. Applied egg-rr 99.9%

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

Alternative 2: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{if}\;y \leq -7 \cdot 10^{+164}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 10^{+126}:\\ \;\;\;\;x \cdot \left(y - y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* x (- y)))))
   (if (<= y -7e+164) t_0 (if (<= y 1e+126) (* x (- y (* y y))) t_0))))

C

double code(double x, double y) {
	double t_0 = y * (x * -y);
	double tmp;
	if (y <= -7e+164) {
		tmp = t_0;
	} else if (y <= 1e+126) {
		tmp = x * (y - (y * y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

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 * -y)
    if (y <= (-7d+164)) then
        tmp = t_0
    else if (y <= 1d+126) then
        tmp = x * (y - (y * y))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y * (x * -y);
	double tmp;
	if (y <= -7e+164) {
		tmp = t_0;
	} else if (y <= 1e+126) {
		tmp = x * (y - (y * y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (x * -y)
	tmp = 0
	if y <= -7e+164:
		tmp = t_0
	elif y <= 1e+126:
		tmp = x * (y - (y * y))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(x * Float64(-y)))
	tmp = 0.0
	if (y <= -7e+164)
		tmp = t_0;
	elseif (y <= 1e+126)
		tmp = Float64(x * Float64(y - Float64(y * y)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * (x * -y);
	tmp = 0.0;
	if (y <= -7e+164)
		tmp = t_0;
	elseif (y <= 1e+126)
		tmp = x * (y - (y * y));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(x * (-y)), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -7e+164], t$95$0, If[LessEqual[y, 1e+126], N[(x * N[(y - N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -6.9999999999999995e164 or 9.99999999999999925e125 < y

   1. Initial program 99.9%

      \[\left(x \cdot y\right) \cdot \left(1 - y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 99.9

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

   5. Simplified 99.9%

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

   if -6.9999999999999995e164 < y < 9.99999999999999925e125

   1. Initial program 99.9%

      \[\left(x \cdot y\right) \cdot \left(1 - y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 N/A

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

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

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

      3. *-rgt-identity N/A

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

      4. unpow2 N/A

         \[\leadsto x \cdot \left(y - \color{blue}{{y}^{2}}\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(y - {y}^{2}\right)} \]

      6. unpow2 N/A

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

      7. *-lowering-*.f64 99.9

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

   5. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{+164}:\\ \;\;\;\;y \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{elif}\;y \leq 10^{+126}:\\ \;\;\;\;x \cdot \left(y - y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot \left(-y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{if}\;y \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* x (- y)))))
   (if (<= y -1.0) t_0 (if (<= y 1.0) (* y x) t_0))))

C

double code(double x, double y) {
	double t_0 = y * (x * -y);
	double tmp;
	if (y <= -1.0) {
		tmp = t_0;
	} else if (y <= 1.0) {
		tmp = y * x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

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 * -y)
    if (y <= (-1.0d0)) then
        tmp = t_0
    else if (y <= 1.0d0) then
        tmp = y * x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y * (x * -y);
	double tmp;
	if (y <= -1.0) {
		tmp = t_0;
	} else if (y <= 1.0) {
		tmp = y * x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (x * -y)
	tmp = 0
	if y <= -1.0:
		tmp = t_0
	elif y <= 1.0:
		tmp = y * x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(x * Float64(-y)))
	tmp = 0.0
	if (y <= -1.0)
		tmp = t_0;
	elseif (y <= 1.0)
		tmp = Float64(y * x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * (x * -y);
	tmp = 0.0;
	if (y <= -1.0)
		tmp = t_0;
	elseif (y <= 1.0)
		tmp = y * x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(x * (-y)), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.0], t$95$0, If[LessEqual[y, 1.0], N[(y * x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 99.8%

      \[\left(x \cdot y\right) \cdot \left(1 - y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 97.2

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

   5. Simplified 97.2%

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

   if -1 < y < 1

   1. Initial program 100.0%

      \[\left(x \cdot y\right) \cdot \left(1 - y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. *-lowering-*.f64 98.5

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

   5. Simplified 98.5%

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

4. Final simplification 97.8%

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

Alternative 4: 63.6% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 (if (<= y 1.0) (* y x) (* x (- y))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < 1

   1. Initial program 99.9%

      \[\left(x \cdot y\right) \cdot \left(1 - y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. *-lowering-*.f64 76.7

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

   5. Simplified 76.7%

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

   if 1 < y

   1. Initial program 99.9%

      \[\left(x \cdot y\right) \cdot \left(1 - y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. *-lowering-*.f64 0.6

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

   5. Simplified 0.6%

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

      1. remove-double-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto x \cdot \left(\mathsf{neg}\left(\color{blue}{\left(0 - y\right)}\right)\right) \]

      3. flip3-- N/A

         \[\leadsto x \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{{0}^{3} - {y}^{3}}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)}}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto x \cdot \left(\mathsf{neg}\left(\frac{\color{blue}{0} - {y}^{3}}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)}\right)\right) \]

      5. sub0-neg N/A

         \[\leadsto x \cdot \left(\mathsf{neg}\left(\frac{\color{blue}{\mathsf{neg}\left({y}^{3}\right)}}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)}\right)\right) \]

      6. cube-neg N/A

         \[\leadsto x \cdot \left(\mathsf{neg}\left(\frac{\color{blue}{{\left(\mathsf{neg}\left(y\right)\right)}^{3}}}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)}\right)\right) \]

      7. distribute-neg-frac N/A

         \[\leadsto x \cdot \color{blue}{\frac{\mathsf{neg}\left({\left(\mathsf{neg}\left(y\right)\right)}^{3}\right)}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)}} \]

      8. metadata-eval N/A

         \[\leadsto x \cdot \frac{\mathsf{neg}\left({\left(\mathsf{neg}\left(y\right)\right)}^{\color{blue}{\left(2 \cdot \frac{3}{2}\right)}}\right)}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)} \]

      9. metadata-eval N/A

         \[\leadsto x \cdot \frac{\mathsf{neg}\left({\left(\mathsf{neg}\left(y\right)\right)}^{\left(2 \cdot \color{blue}{\frac{3}{2}}\right)}\right)}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)} \]

      10. pow-pow N/A

          \[\leadsto x \cdot \frac{\mathsf{neg}\left(\color{blue}{{\left({\left(\mathsf{neg}\left(y\right)\right)}^{2}\right)}^{\left(\frac{3}{2}\right)}}\right)}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)} \]

      11. pow2 N/A

          \[\leadsto x \cdot \frac{\mathsf{neg}\left({\color{blue}{\left(\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}}^{\left(\frac{3}{2}\right)}\right)}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)} \]

      12. sqr-neg N/A

          \[\leadsto x \cdot \frac{\mathsf{neg}\left({\color{blue}{\left(y \cdot y\right)}}^{\left(\frac{3}{2}\right)}\right)}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)} \]

      13. unpow-prod-down N/A

          \[\leadsto x \cdot \frac{\mathsf{neg}\left(\color{blue}{{y}^{\left(\frac{3}{2}\right)} \cdot {y}^{\left(\frac{3}{2}\right)}}\right)}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)} \]

      14. sqr-pow N/A

          \[\leadsto x \cdot \frac{\mathsf{neg}\left(\color{blue}{{y}^{3}}\right)}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)} \]

      15. sub0-neg N/A

          \[\leadsto x \cdot \frac{\color{blue}{0 - {y}^{3}}}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)} \]

      16. metadata-eval N/A

          \[\leadsto x \cdot \frac{\color{blue}{{0}^{3}} - {y}^{3}}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)} \]

      17. flip3-- N/A

          \[\leadsto x \cdot \color{blue}{\left(0 - y\right)} \]

      18. neg-sub0 N/A

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

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

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

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

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

      21. *-lowering-*.f64 N/A

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

   7. Applied egg-rr 32.7%

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

4. Final simplification 65.7%

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

Alternative 5: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

   \[\left(x \cdot y\right) \cdot \left(1 - y\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. associate-*l* N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-lowering-*.f64 N/A

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

   5. sub-neg N/A

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

   6. +-commutative N/A

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

   7. distribute-lft-in N/A

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

   8. *-rgt-identity N/A

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

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

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

   10. neg-lowering-neg.f64 99.9

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

4. Applied egg-rr 99.9%

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

5. Final simplification 99.9%

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

Alternative 6: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

   \[\left(x \cdot y\right) \cdot \left(1 - y\right) \]
2. Add Preprocessing

3. Final simplification 99.9%

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

Alternative 7: 56.2% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

   \[\left(x \cdot y\right) \cdot \left(1 - y\right) \]
2. Add Preprocessing

3. Taylor expanded in y around 0

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

   1. *-lowering-*.f64 57.7

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

5. Simplified 57.7%

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

6. Final simplification 57.7%

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

Reproduce

herbie shell --seed 2024198 
(FPCore (x y)
  :name "Statistics.Distribution.Binomial:$cvariance from math-functions-0.1.5.2"
  :precision binary64
  (* (* x y) (- 1.0 y)))