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

Percentage Accurate: 99.9% → 99.9%

Time: 5.6s

Alternatives: 7

Speedup: 0.1×

• 25.0% 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, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. associate-*l* 96.2%

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

3. Simplified 96.2%

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

   1. associate-*r* 99.9%

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

   2. sub-neg 99.9%

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

   3. distribute-lft-in 95.2%

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

   4. *-commutative 95.2%

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

   5. *-un-lft-identity 95.2%

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

   6. *-commutative 95.2%

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

   7. fma-def 99.9%

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

5. Applied egg-rr 99.9%

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

6. Final simplification 99.9%

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

Alternative 2: 97.7% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 1.0))) (* y (* y (- x))) (* y x)))

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-1.0d0)) .or. (.not. (y <= 1.0d0))) then
        tmp = y * (y * -x)
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

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

Python

def code(x, y):
	tmp = 0
	if (y <= -1.0) or not (y <= 1.0):
		tmp = y * (y * -x)
	else:
		tmp = y * x
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.0], N[Not[LessEqual[y, 1.0]], $MachinePrecision]], N[(y * N[(y * (-x)), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]

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. Step-by-step derivation

      1. associate-*l* 92.8%

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

   3. Simplified 92.8%

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

      1. associate-*r* 99.8%

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

      2. sub-neg 99.8%

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

      3. distribute-lft-in 90.7%

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

      4. *-commutative 90.7%

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

      5. *-un-lft-identity 90.7%

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

      6. *-commutative 90.7%

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

      7. fma-def 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. fma-udef 90.7%

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

      2. *-commutative 90.7%

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

      3. distribute-rgt-neg-out 90.7%

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

      4. distribute-lft-neg-in 90.7%

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

      5. add-sqr-sqrt 41.5%

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

      6. sqrt-unprod 36.8%

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

      7. sqr-neg 36.8%

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

      8. sqrt-unprod 0.3%

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

      9. add-sqr-sqrt 0.6%

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

      10. cancel-sign-sub-inv 0.6%

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

      11. add-sqr-sqrt 0.3%

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

      12. cancel-sign-sub-inv 0.3%

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

   7. Applied egg-rr 45.2%

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

      1. add-sqr-sqrt 26.4%

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

      2. sqrt-unprod 26.5%

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

      3. sqr-neg 26.5%

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

      4. sqrt-unprod 0.1%

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

      5. add-sqr-sqrt 0.3%

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

      6. cancel-sign-sub 0.3%

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

      7. associate-*r* 0.3%

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

      8. distribute-rgt-out-- 0.3%

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

      9. *-commutative 0.3%

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

      10. add-sqr-sqrt 0.1%

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

      11. sqrt-unprod 22.7%

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

      12. sqr-neg 22.7%

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

      13. sqrt-unprod 22.6%

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

   9. Applied egg-rr 99.8%

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

   10. Taylor expanded in y around inf 97.7%

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

       1. associate-*r* 97.7%

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

       2. neg-mul-1 97.7%

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

       3. *-commutative 97.7%

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

   12. Simplified 97.7%

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

   if -1 < y < 1

   1. Initial program 99.9%

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

      1. associate-*l* 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around 0 96.2%

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

4. Final simplification 97.0%

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

Alternative 3: 97.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.5 \cdot 10^{+126}:\\ \;\;\;\;x \cdot \left(y \cdot \left(1 - y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot \left(-x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.5e+126) (* x (* y (- 1.0 y))) (* y (* y (- x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.5e+126) {
		tmp = x * (y * (1.0 - y));
	} else {
		tmp = y * (y * -x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.5d+126) then
        tmp = x * (y * (1.0d0 - y))
    else
        tmp = y * (y * -x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.5e+126) {
		tmp = x * (y * (1.0 - y));
	} else {
		tmp = y * (y * -x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.5e+126:
		tmp = x * (y * (1.0 - y))
	else:
		tmp = y * (y * -x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.5e+126)
		tmp = Float64(x * Float64(y * Float64(1.0 - y)));
	else
		tmp = Float64(y * Float64(y * Float64(-x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.5e+126)
		tmp = x * (y * (1.0 - y));
	else
		tmp = y * (y * -x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.5e+126], N[(x * N[(y * N[(1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(y * (-x)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.5000000000000001e126

   1. Initial program 99.9%

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

      1. associate-*l* 98.6%

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

   3. Simplified 98.6%

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

   if 1.5000000000000001e126 < y

   1. Initial program 99.9%

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

      1. associate-*l* 82.1%

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

   3. Simplified 82.1%

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

      1. associate-*r* 99.9%

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

      2. sub-neg 99.9%

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

      3. distribute-lft-in 67.5%

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

      4. *-commutative 67.5%

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

      5. *-un-lft-identity 67.5%

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

      6. *-commutative 67.5%

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

      7. fma-def 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. fma-udef 67.5%

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

      2. *-commutative 67.5%

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

      3. distribute-rgt-neg-out 67.5%

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

      4. distribute-lft-neg-in 67.5%

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

      5. add-sqr-sqrt 67.4%

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

      6. sqrt-unprod 49.7%

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

      7. sqr-neg 49.7%

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

      8. sqrt-unprod 0.0%

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

      9. add-sqr-sqrt 0.2%

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

      10. cancel-sign-sub-inv 0.2%

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

      11. add-sqr-sqrt 0.1%

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

      12. cancel-sign-sub-inv 0.1%

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

   7. Applied egg-rr 37.6%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 0.1%

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

      3. sqr-neg 0.1%

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

      4. sqrt-unprod 0.1%

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

      5. add-sqr-sqrt 0.1%

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

      6. cancel-sign-sub 0.1%

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

      7. associate-*r* 0.1%

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

      8. distribute-rgt-out-- 0.1%

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

      9. *-commutative 0.1%

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

      10. add-sqr-sqrt 0.0%

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

      11. sqrt-unprod 51.2%

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

      12. sqr-neg 51.2%

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

      13. sqrt-unprod 51.2%

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

   9. Applied egg-rr 99.9%

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

   10. Taylor expanded in y around inf 99.9%

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

       1. associate-*r* 99.9%

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

       2. neg-mul-1 99.9%

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

       3. *-commutative 99.9%

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

   12. Simplified 99.9%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.5 \cdot 10^{+126}:\\ \;\;\;\;x \cdot \left(y \cdot \left(1 - y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot \left(-x\right)\right)\\ \end{array} \]

Alternative 4: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. associate-*l* 96.2%

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

3. Simplified 96.2%

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

   1. associate-*r* 99.9%

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

   2. sub-neg 99.9%

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

   3. distribute-lft-in 95.2%

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

   4. *-commutative 95.2%

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

   5. *-un-lft-identity 95.2%

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

   6. *-commutative 95.2%

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

   7. fma-def 99.9%

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

5. Applied egg-rr 99.9%

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

   1. fma-udef 95.2%

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

   2. *-commutative 95.2%

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

   3. distribute-rgt-neg-out 95.2%

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

   4. distribute-lft-neg-in 95.2%

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

   5. add-sqr-sqrt 43.6%

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

   6. sqrt-unprod 66.3%

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

   7. sqr-neg 66.3%

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

   8. sqrt-unprod 25.2%

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

   9. add-sqr-sqrt 46.9%

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

   10. cancel-sign-sub-inv 46.9%

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

   11. add-sqr-sqrt 35.3%

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

   12. cancel-sign-sub-inv 35.3%

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

7. Applied egg-rr 47.5%

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

   1. add-sqr-sqrt 29.6%

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

   2. sqrt-unprod 37.9%

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

   3. sqr-neg 37.9%

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

   4. sqrt-unprod 12.2%

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

   5. add-sqr-sqrt 23.3%

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

   6. cancel-sign-sub 23.3%

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

   7. associate-*r* 23.3%

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

   8. distribute-rgt-out-- 23.3%

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

   9. *-commutative 23.3%

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

   10. add-sqr-sqrt 15.0%

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

   11. sqrt-unprod 34.9%

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

   12. sqr-neg 34.9%

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

   13. sqrt-unprod 23.8%

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

9. Applied egg-rr 99.9%

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

10. Final simplification 99.9%

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

Alternative 5: 63.1% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

      1. associate-*l* 98.4%

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

   3. Simplified 98.4%

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

   4. Taylor expanded in y around 0 74.3%

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

   if 1 < y

   1. Initial program 99.8%

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

      1. associate-*l* 90.1%

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

   3. Simplified 90.1%

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

      1. associate-*r* 99.8%

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

      2. flip-- 89.9%

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

      3. associate-*r/ 83.1%

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

      4. metadata-eval 83.1%

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

      5. pow2 83.1%

         \[\leadsto \frac{\left(x \cdot y\right) \cdot \left(1 - \color{blue}{{y}^{2}}\right)}{1 + y} \]

      6. +-commutative 83.1%

         \[\leadsto \frac{\left(x \cdot y\right) \cdot \left(1 - {y}^{2}\right)}{\color{blue}{y + 1}} \]

   5. Applied egg-rr 83.1%

      \[\leadsto \color{blue}{\frac{\left(x \cdot y\right) \cdot \left(1 - {y}^{2}\right)}{y + 1}} \]
   6. Step-by-step derivation

      1. associate-*l* 78.8%

         \[\leadsto \frac{\color{blue}{x \cdot \left(y \cdot \left(1 - {y}^{2}\right)\right)}}{y + 1} \]

      2. associate-/l* 80.0%

         \[\leadsto \color{blue}{\frac{x}{\frac{y + 1}{y \cdot \left(1 - {y}^{2}\right)}}} \]

      3. sub-neg 80.0%

         \[\leadsto \frac{x}{\frac{y + 1}{y \cdot \color{blue}{\left(1 + \left(-{y}^{2}\right)\right)}}} \]

      4. distribute-lft-in 80.0%

         \[\leadsto \frac{x}{\frac{y + 1}{\color{blue}{y \cdot 1 + y \cdot \left(-{y}^{2}\right)}}} \]

      5. *-rgt-identity 80.0%

         \[\leadsto \frac{x}{\frac{y + 1}{\color{blue}{y} + y \cdot \left(-{y}^{2}\right)}} \]

      6. distribute-rgt-neg-in 80.0%

         \[\leadsto \frac{x}{\frac{y + 1}{y + \color{blue}{\left(-y \cdot {y}^{2}\right)}}} \]

      7. unpow2 80.0%

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

      8. cube-mult 80.1%

         \[\leadsto \frac{x}{\frac{y + 1}{y + \left(-\color{blue}{{y}^{3}}\right)}} \]

      9. unsub-neg 80.1%

         \[\leadsto \frac{x}{\frac{y + 1}{\color{blue}{y - {y}^{3}}}} \]

   7. Simplified 80.1%

      \[\leadsto \color{blue}{\frac{x}{\frac{y + 1}{y - {y}^{3}}}} \]

   8. Taylor expanded in y around 0 0.9%

      \[\leadsto \frac{x}{\color{blue}{\frac{1}{y}}} \]
   9. Step-by-step derivation

      1. associate-/r/ 0.9%

         \[\leadsto \color{blue}{\frac{x}{1} \cdot y} \]

      2. /-rgt-identity 0.9%

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

      3. add-sqr-sqrt 0.3%

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

      4. associate-*l* 0.3%

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

      5. add-sqr-sqrt 0.3%

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

      6. sqrt-unprod 0.3%

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

      7. sqr-neg 0.3%

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

      8. sqrt-unprod 0.0%

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

      9. add-sqr-sqrt 9.8%

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

      10. *-commutative 9.8%

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

      11. neg-mul-1 9.8%

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

      12. associate-*l* 9.8%

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

      13. *-commutative 9.8%

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

      14. associate-*r* 9.8%

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

      15. add-sqr-sqrt 23.5%

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

      16. *-commutative 23.5%

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

   10. Applied egg-rr 23.5%

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

       1. neg-mul-1 23.5%

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

       2. distribute-rgt-neg-in 23.5%

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

   12. Simplified 23.5%

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

4. Final simplification 61.0%

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

Alternative 6: 55.3% 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. Step-by-step derivation

   1. associate-*l* 96.2%

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

3. Simplified 96.2%

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

4. Taylor expanded in y around 0 55.1%

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

5. Final simplification 55.1%

   \[\leadsto y \cdot x \]

Alternative 7: 2.8% accurate, 7.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 x)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return x

Julia

function code(x, y)
	return x
end

MATLAB

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

Wolfram

code[x_, y_] := x

Derivation

1. Initial program 99.9%

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

   1. associate-*l* 96.2%

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

3. Simplified 96.2%

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

   1. associate-*r* 99.9%

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

   2. flip-- 96.2%

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

   3. associate-*r/ 92.3%

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

   4. metadata-eval 92.3%

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

   5. pow2 92.3%

      \[\leadsto \frac{\left(x \cdot y\right) \cdot \left(1 - \color{blue}{{y}^{2}}\right)}{1 + y} \]

   6. +-commutative 92.3%

      \[\leadsto \frac{\left(x \cdot y\right) \cdot \left(1 - {y}^{2}\right)}{\color{blue}{y + 1}} \]

5. Applied egg-rr 92.3%

   \[\leadsto \color{blue}{\frac{\left(x \cdot y\right) \cdot \left(1 - {y}^{2}\right)}{y + 1}} \]
6. Step-by-step derivation

   1. associate-*l* 89.7%

      \[\leadsto \frac{\color{blue}{x \cdot \left(y \cdot \left(1 - {y}^{2}\right)\right)}}{y + 1} \]

   2. associate-/l* 91.2%

      \[\leadsto \color{blue}{\frac{x}{\frac{y + 1}{y \cdot \left(1 - {y}^{2}\right)}}} \]

   3. sub-neg 91.2%

      \[\leadsto \frac{x}{\frac{y + 1}{y \cdot \color{blue}{\left(1 + \left(-{y}^{2}\right)\right)}}} \]

   4. distribute-lft-in 91.2%

      \[\leadsto \frac{x}{\frac{y + 1}{\color{blue}{y \cdot 1 + y \cdot \left(-{y}^{2}\right)}}} \]

   5. *-rgt-identity 91.2%

      \[\leadsto \frac{x}{\frac{y + 1}{\color{blue}{y} + y \cdot \left(-{y}^{2}\right)}} \]

   6. distribute-rgt-neg-in 91.2%

      \[\leadsto \frac{x}{\frac{y + 1}{y + \color{blue}{\left(-y \cdot {y}^{2}\right)}}} \]

   7. unpow2 91.2%

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

   8. cube-mult 91.3%

      \[\leadsto \frac{x}{\frac{y + 1}{y + \left(-\color{blue}{{y}^{3}}\right)}} \]

   9. unsub-neg 91.3%

      \[\leadsto \frac{x}{\frac{y + 1}{\color{blue}{y - {y}^{3}}}} \]

7. Simplified 91.3%

   \[\leadsto \color{blue}{\frac{x}{\frac{y + 1}{y - {y}^{3}}}} \]

8. Taylor expanded in y around 0 47.5%

   \[\leadsto \frac{x}{\color{blue}{1 + \frac{1}{y}}} \]

9. Taylor expanded in y around inf 2.8%

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

10. Final simplification 2.8%

    \[\leadsto x \]

Reproduce

herbie shell --seed 2023322 
(FPCore (x y)
  :name "Statistics.Distribution.Binomial:$cvariance from math-functions-0.1.5.2"
  :precision binary64
  (* (* x y) (- 1.0 y)))