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

Percentage Accurate: 99.9% → 99.7%

Time: 6.9s

Alternatives: 5

Speedup: 1.0×

• 25.3% 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: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-y\right) \cdot \left(y \cdot x\right)\\ \mathbf{if}\;y \leq -3.15 \cdot 10^{+159}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2.75 \cdot 10^{+16}:\\ \;\;\;\;\left(\left(1 - y\right) \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (- y) (* y x))))
   (if (<= y -3.15e+159) t_0 (if (<= y 2.75e+16) (* (* (- 1.0 y) y) x) t_0))))

C

double code(double x, double y) {
	double t_0 = -y * (y * x);
	double tmp;
	if (y <= -3.15e+159) {
		tmp = t_0;
	} else if (y <= 2.75e+16) {
		tmp = ((1.0 - y) * 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 * (y * x)
    if (y <= (-3.15d+159)) then
        tmp = t_0
    else if (y <= 2.75d+16) then
        tmp = ((1.0d0 - y) * 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 * (y * x);
	double tmp;
	if (y <= -3.15e+159) {
		tmp = t_0;
	} else if (y <= 2.75e+16) {
		tmp = ((1.0 - y) * y) * x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = -y * (y * x)
	tmp = 0
	if y <= -3.15e+159:
		tmp = t_0
	elif y <= 2.75e+16:
		tmp = ((1.0 - y) * y) * x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(-y) * Float64(y * x))
	tmp = 0.0
	if (y <= -3.15e+159)
		tmp = t_0;
	elseif (y <= 2.75e+16)
		tmp = Float64(Float64(Float64(1.0 - y) * y) * x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = -y * (y * x);
	tmp = 0.0;
	if (y <= -3.15e+159)
		tmp = t_0;
	elseif (y <= 2.75e+16)
		tmp = ((1.0 - y) * y) * x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[((-y) * N[(y * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3.15e+159], t$95$0, If[LessEqual[y, 2.75e+16], N[(N[(N[(1.0 - y), $MachinePrecision] * y), $MachinePrecision] * x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.15000000000000003e159 or 2.75e16 < 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. lower-neg.f64 99.9

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

   5. Applied rewrites 99.9%

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

   if -3.15000000000000003e159 < y < 2.75e16

   1. Initial program 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

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

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

      10. *-lft-identity N/A

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

      11. lower-fma.f64 N/A

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

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

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 100.0

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. unpow2 N/A

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

      6. *-rgt-identity N/A

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

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

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 100.0

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

   7. Applied rewrites 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.15 \cdot 10^{+159}:\\ \;\;\;\;\left(-y\right) \cdot \left(y \cdot x\right)\\ \mathbf{elif}\;y \leq 2.75 \cdot 10^{+16}:\\ \;\;\;\;\left(\left(1 - y\right) \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot \left(y \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.7% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (- y) (* y x))))
   (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 * (y * x);
	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 * (y * x)
    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 * (y * x);
	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 * (y * x)
	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(Float64(-y) * Float64(y * x))
	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 * (y * x);
	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[(y * x), $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 99.7%

      \[\left(x \cdot y\right) \cdot \left(1 - y\right) \]
   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. lower-neg.f64 97.3

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

   5. Applied rewrites 97.3%

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

   if -1 < y < 1

   1. Initial program 100.0%

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

   4. Applied rewrites 100.0%

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

   6. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. clear-num N/A

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

      8. lift-/.f64 N/A

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

      9. inv-pow N/A

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

      10. lower-pow.f64 99.6

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

      11. lift-/.f64 N/A

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

      12. clear-num N/A

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

      13. associate-/r/ N/A

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

      14. lift-pow.f64 N/A

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

      15. unpow-1 N/A

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

      16. remove-double-div N/A

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

      17. lower-*.f64 99.6

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

   8. Applied rewrites 99.6%

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

   9. Taylor expanded in y around 0

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

       1. lower-*.f64 98.4

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

   11. Applied rewrites 98.4%

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

4. Final simplification 97.9%

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

Alternative 3: 91.9% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* (- y) y) x)))
   (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 * y) * x;
	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 * y) * x
    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 * y) * x;
	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 * y) * x
	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(Float64(Float64(-y) * y) * x)
	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 * y) * x;
	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[(N[((-y) * y), $MachinePrecision] * x), $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 99.7%

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

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 85.2

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

   5. Applied rewrites 85.2%

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

   if -1 < y < 1

   1. Initial program 100.0%

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

   4. Applied rewrites 100.0%

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

   6. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. clear-num N/A

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

      8. lift-/.f64 N/A

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

      9. inv-pow N/A

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

      10. lower-pow.f64 99.6

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

      11. lift-/.f64 N/A

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

      12. clear-num N/A

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

      13. associate-/r/ N/A

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

      14. lift-pow.f64 N/A

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

      15. unpow-1 N/A

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

      16. remove-double-div N/A

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

      17. lower-*.f64 99.6

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

   8. Applied rewrites 99.6%

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

   9. Taylor expanded in y around 0

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

       1. lower-*.f64 98.4

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

   11. Applied rewrites 98.4%

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

4. Final simplification 91.7%

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

Alternative 4: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

Alternative 5: 56.0% 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

4. Applied rewrites 99.8%

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

6. Applied rewrites 99.8%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. clear-num N/A

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

   5. un-div-inv N/A

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

   6. lower-/.f64 N/A

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

   7. clear-num N/A

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

   8. lift-/.f64 N/A

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

   9. inv-pow N/A

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

   10. lower-pow.f64 99.6

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

   11. lift-/.f64 N/A

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

   12. clear-num N/A

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

   13. associate-/r/ N/A

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

   14. lift-pow.f64 N/A

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

   15. unpow-1 N/A

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

   16. remove-double-div N/A

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

   17. lower-*.f64 99.7

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

8. Applied rewrites 99.7%

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

9. Taylor expanded in y around 0

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

    1. lower-*.f64 56.9

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

11. Applied rewrites 56.9%

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

12. Final simplification 56.9%

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

Reproduce

herbie shell --seed 2024332 
(FPCore (x y)
  :name "Statistics.Distribution.Binomial:$cvariance from math-functions-0.1.5.2"
  :precision binary64
  (* (* x y) (- 1.0 y)))