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

Percentage Accurate: 99.9% → 99.8%

Time: 6.8s

Alternatives: 5

Speedup: 1.0×

• 25.2% 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: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.1 \cdot 10^{+155} \lor \neg \left(y \leq 4 \cdot 10^{+126}\right):\\ \;\;\;\;y \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \left(1 - y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -7.1e+155) (not (<= y 4e+126)))
   (* y (* x (- y)))
   (* x (* y (- 1.0 y)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -7.1e+155) || !(y <= 4e+126)) {
		tmp = y * (x * -y);
	} else {
		tmp = x * (y * (1.0 - 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 <= (-7.1d+155)) .or. (.not. (y <= 4d+126))) then
        tmp = y * (x * -y)
    else
        tmp = x * (y * (1.0d0 - y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -7.1e+155) || !(y <= 4e+126)) {
		tmp = y * (x * -y);
	} else {
		tmp = x * (y * (1.0 - y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -7.1e+155) or not (y <= 4e+126):
		tmp = y * (x * -y)
	else:
		tmp = x * (y * (1.0 - y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -7.1e+155) || !(y <= 4e+126))
		tmp = Float64(y * Float64(x * Float64(-y)));
	else
		tmp = Float64(x * Float64(y * Float64(1.0 - y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -7.1e+155) || ~((y <= 4e+126)))
		tmp = y * (x * -y);
	else
		tmp = x * (y * (1.0 - y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -7.1e+155], N[Not[LessEqual[y, 4e+126]], $MachinePrecision]], N[(y * N[(x * (-y)), $MachinePrecision]), $MachinePrecision], N[(x * N[(y * N[(1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -7.09999999999999992e155 or 3.9999999999999997e126 < y

   1. Initial program 100.0%

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

      1. associate-*l* 82.0%

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

   3. Simplified 82.0%

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

      1. add0 82.0%

         \[\leadsto \color{blue}{x \cdot \left(y \cdot \left(1 - y\right)\right) + 0} \]

      2. fma-define 82.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, y \cdot \left(1 - y\right), 0\right)} \]

      3. metadata-eval 82.0%

         \[\leadsto \mathsf{fma}\left(x, y \cdot \left(1 - y\right), \color{blue}{-0}\right) \]

      4. fma-neg 82.0%

         \[\leadsto \color{blue}{x \cdot \left(y \cdot \left(1 - y\right)\right) - 0} \]

      5. associate-*r* 100.0%

         \[\leadsto \color{blue}{\left(x \cdot y\right) \cdot \left(1 - y\right)} - 0 \]

      6. *-commutative 100.0%

         \[\leadsto \color{blue}{\left(1 - y\right) \cdot \left(x \cdot y\right)} - 0 \]

      7. add-sqr-sqrt 44.0%

         \[\leadsto \left(1 - y\right) \cdot \color{blue}{\left(\sqrt{x \cdot y} \cdot \sqrt{x \cdot y}\right)} - 0 \]

      8. associate-*r* 44.0%

         \[\leadsto \color{blue}{\left(\left(1 - y\right) \cdot \sqrt{x \cdot y}\right) \cdot \sqrt{x \cdot y}} - 0 \]

      9. fma-neg 44.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(1 - y\right) \cdot \sqrt{x \cdot y}, \sqrt{x \cdot y}, -0\right)} \]

      10. metadata-eval 44.0%

          \[\leadsto \mathsf{fma}\left(\left(1 - y\right) \cdot \sqrt{x \cdot y}, \sqrt{x \cdot y}, \color{blue}{0}\right) \]

   6. Applied egg-rr 44.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(1 - y\right) \cdot \sqrt{x \cdot y}, \sqrt{x \cdot y}, 0\right)} \]
   7. Step-by-step derivation

      1. add0 44.0%

         \[\leadsto \mathsf{fma}\left(\left(1 - y\right) \cdot \color{blue}{\left(\sqrt{x \cdot y} + 0\right)}, \sqrt{x \cdot y}, 0\right) \]

      2. flip-+ 27.8%

         \[\leadsto \mathsf{fma}\left(\left(1 - y\right) \cdot \color{blue}{\frac{\sqrt{x \cdot y} \cdot \sqrt{x \cdot y} - 0 \cdot 0}{\sqrt{x \cdot y} - 0}}, \sqrt{x \cdot y}, 0\right) \]

      3. add-sqr-sqrt 27.9%

         \[\leadsto \mathsf{fma}\left(\left(1 - y\right) \cdot \frac{\color{blue}{x \cdot y} - 0 \cdot 0}{\sqrt{x \cdot y} - 0}, \sqrt{x \cdot y}, 0\right) \]

      4. metadata-eval 27.9%

         \[\leadsto \mathsf{fma}\left(\left(1 - y\right) \cdot \frac{x \cdot y - \color{blue}{0}}{\sqrt{x \cdot y} - 0}, \sqrt{x \cdot y}, 0\right) \]

   8. Applied egg-rr 27.9%

      \[\leadsto \mathsf{fma}\left(\left(1 - y\right) \cdot \color{blue}{\frac{x \cdot y - 0}{\sqrt{x \cdot y} - 0}}, \sqrt{x \cdot y}, 0\right) \]
   9. Step-by-step derivation

      1. --rgt-identity 27.9%

         \[\leadsto \mathsf{fma}\left(\left(1 - y\right) \cdot \frac{\color{blue}{x \cdot y}}{\sqrt{x \cdot y} - 0}, \sqrt{x \cdot y}, 0\right) \]

      2. --rgt-identity 27.9%

         \[\leadsto \mathsf{fma}\left(\left(1 - y\right) \cdot \frac{x \cdot y}{\color{blue}{\sqrt{x \cdot y}}}, \sqrt{x \cdot y}, 0\right) \]

   10. Simplified 27.9%

       \[\leadsto \mathsf{fma}\left(\left(1 - y\right) \cdot \color{blue}{\frac{x \cdot y}{\sqrt{x \cdot y}}}, \sqrt{x \cdot y}, 0\right) \]
   11. Step-by-step derivation

       1. fma-undefine 27.9%

          \[\leadsto \color{blue}{\left(\left(1 - y\right) \cdot \frac{x \cdot y}{\sqrt{x \cdot y}}\right) \cdot \sqrt{x \cdot y} + 0} \]

       2. add0 27.9%

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

       3. pow1 27.9%

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

       4. pow1/2 27.9%

          \[\leadsto \left(\left(1 - y\right) \cdot \frac{{\left(x \cdot y\right)}^{1}}{\color{blue}{{\left(x \cdot y\right)}^{0.5}}}\right) \cdot \sqrt{x \cdot y} \]

       5. pow-div 44.0%

          \[\leadsto \left(\left(1 - y\right) \cdot \color{blue}{{\left(x \cdot y\right)}^{\left(1 - 0.5\right)}}\right) \cdot \sqrt{x \cdot y} \]

       6. metadata-eval 44.0%

          \[\leadsto \left(\left(1 - y\right) \cdot {\left(x \cdot y\right)}^{\color{blue}{0.5}}\right) \cdot \sqrt{x \cdot y} \]

       7. pow1/2 44.0%

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

       8. associate-*r* 44.0%

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

       9. add-sqr-sqrt 100.0%

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

       10. associate-*r* 100.0%

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

   12. Applied egg-rr 100.0%

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

   13. Taylor expanded in y around inf 100.0%

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

       1. mul-1-neg 100.0%

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

       2. distribute-rgt-neg-out 100.0%

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

   15. Simplified 100.0%

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

   if -7.09999999999999992e155 < y < 3.9999999999999997e126

   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. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.1 \cdot 10^{+155} \lor \neg \left(y \leq 4 \cdot 10^{+126}\right):\\ \;\;\;\;y \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \left(1 - y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.9% accurate, 0.4× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 1.0)) {
		tmp = y * (x * -y);
	} 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)) .or. (.not. (y <= 1.0d0))) then
        tmp = y * (x * -y)
    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) || !(y <= 1.0)) {
		tmp = y * (x * -y);
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.0) || ~((y <= 1.0)))
		tmp = y * (x * -y);
	else
		tmp = x * y;
	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[(x * (-y)), $MachinePrecision]), $MachinePrecision], N[(x * y), $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* 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. Add Preprocessing
   5. Step-by-step derivation

      1. add0 90.1%

         \[\leadsto \color{blue}{x \cdot \left(y \cdot \left(1 - y\right)\right) + 0} \]

      2. fma-define 90.1%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, y \cdot \left(1 - y\right), 0\right)} \]

      3. metadata-eval 90.1%

         \[\leadsto \mathsf{fma}\left(x, y \cdot \left(1 - y\right), \color{blue}{-0}\right) \]

      4. fma-neg 90.1%

         \[\leadsto \color{blue}{x \cdot \left(y \cdot \left(1 - y\right)\right) - 0} \]

      5. associate-*r* 99.8%

         \[\leadsto \color{blue}{\left(x \cdot y\right) \cdot \left(1 - y\right)} - 0 \]

      6. *-commutative 99.8%

         \[\leadsto \color{blue}{\left(1 - y\right) \cdot \left(x \cdot y\right)} - 0 \]

      7. add-sqr-sqrt 45.4%

         \[\leadsto \left(1 - y\right) \cdot \color{blue}{\left(\sqrt{x \cdot y} \cdot \sqrt{x \cdot y}\right)} - 0 \]

      8. associate-*r* 45.4%

         \[\leadsto \color{blue}{\left(\left(1 - y\right) \cdot \sqrt{x \cdot y}\right) \cdot \sqrt{x \cdot y}} - 0 \]

      9. fma-neg 45.4%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(1 - y\right) \cdot \sqrt{x \cdot y}, \sqrt{x \cdot y}, -0\right)} \]

      10. metadata-eval 45.4%

          \[\leadsto \mathsf{fma}\left(\left(1 - y\right) \cdot \sqrt{x \cdot y}, \sqrt{x \cdot y}, \color{blue}{0}\right) \]

   6. Applied egg-rr 45.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(1 - y\right) \cdot \sqrt{x \cdot y}, \sqrt{x \cdot y}, 0\right)} \]
   7. Step-by-step derivation

      1. add0 45.4%

         \[\leadsto \mathsf{fma}\left(\left(1 - y\right) \cdot \color{blue}{\left(\sqrt{x \cdot y} + 0\right)}, \sqrt{x \cdot y}, 0\right) \]

      2. flip-+ 33.4%

         \[\leadsto \mathsf{fma}\left(\left(1 - y\right) \cdot \color{blue}{\frac{\sqrt{x \cdot y} \cdot \sqrt{x \cdot y} - 0 \cdot 0}{\sqrt{x \cdot y} - 0}}, \sqrt{x \cdot y}, 0\right) \]

      3. add-sqr-sqrt 33.5%

         \[\leadsto \mathsf{fma}\left(\left(1 - y\right) \cdot \frac{\color{blue}{x \cdot y} - 0 \cdot 0}{\sqrt{x \cdot y} - 0}, \sqrt{x \cdot y}, 0\right) \]

      4. metadata-eval 33.5%

         \[\leadsto \mathsf{fma}\left(\left(1 - y\right) \cdot \frac{x \cdot y - \color{blue}{0}}{\sqrt{x \cdot y} - 0}, \sqrt{x \cdot y}, 0\right) \]

   8. Applied egg-rr 33.5%

      \[\leadsto \mathsf{fma}\left(\left(1 - y\right) \cdot \color{blue}{\frac{x \cdot y - 0}{\sqrt{x \cdot y} - 0}}, \sqrt{x \cdot y}, 0\right) \]
   9. Step-by-step derivation

      1. --rgt-identity 33.5%

         \[\leadsto \mathsf{fma}\left(\left(1 - y\right) \cdot \frac{\color{blue}{x \cdot y}}{\sqrt{x \cdot y} - 0}, \sqrt{x \cdot y}, 0\right) \]

      2. --rgt-identity 33.5%

         \[\leadsto \mathsf{fma}\left(\left(1 - y\right) \cdot \frac{x \cdot y}{\color{blue}{\sqrt{x \cdot y}}}, \sqrt{x \cdot y}, 0\right) \]

   10. Simplified 33.5%

       \[\leadsto \mathsf{fma}\left(\left(1 - y\right) \cdot \color{blue}{\frac{x \cdot y}{\sqrt{x \cdot y}}}, \sqrt{x \cdot y}, 0\right) \]
   11. Step-by-step derivation

       1. fma-undefine 33.5%

          \[\leadsto \color{blue}{\left(\left(1 - y\right) \cdot \frac{x \cdot y}{\sqrt{x \cdot y}}\right) \cdot \sqrt{x \cdot y} + 0} \]

       2. add0 33.5%

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

       3. pow1 33.5%

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

       4. pow1/2 33.5%

          \[\leadsto \left(\left(1 - y\right) \cdot \frac{{\left(x \cdot y\right)}^{1}}{\color{blue}{{\left(x \cdot y\right)}^{0.5}}}\right) \cdot \sqrt{x \cdot y} \]

       5. pow-div 45.4%

          \[\leadsto \left(\left(1 - y\right) \cdot \color{blue}{{\left(x \cdot y\right)}^{\left(1 - 0.5\right)}}\right) \cdot \sqrt{x \cdot y} \]

       6. metadata-eval 45.4%

          \[\leadsto \left(\left(1 - y\right) \cdot {\left(x \cdot y\right)}^{\color{blue}{0.5}}\right) \cdot \sqrt{x \cdot y} \]

       7. pow1/2 45.4%

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

       8. associate-*r* 45.4%

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

       9. add-sqr-sqrt 99.8%

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

       10. associate-*r* 99.8%

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

   12. Applied egg-rr 99.8%

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

   13. Taylor expanded in y around inf 94.3%

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

       1. mul-1-neg 94.3%

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

       2. distribute-rgt-neg-out 94.3%

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

   15. Simplified 94.3%

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

   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* 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 97.1%

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

4. Final simplification 95.7%

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

Alternative 3: 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]

Derivation

1. Initial program 99.9%

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

3. Final simplification 99.9%

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

Alternative 4: 56.3% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(x * y), $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* 95.2%

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

3. Simplified 95.2%

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

5. Taylor expanded in y around 0 58.5%

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

6. Final simplification 58.5%

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

Alternative 5: 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* 95.2%

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

3. Simplified 95.2%

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

   1. associate-*r* 99.9%

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

   2. flip-- 95.1%

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

   3. associate-*r/ 93.6%

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

   4. metadata-eval 93.6%

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

   5. pow2 93.6%

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

   6. +-commutative 93.6%

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

6. Applied egg-rr 93.6%

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

   1. associate-*l* 91.8%

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

   2. associate-/l* 92.4%

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

   3. associate-/r/ 88.0%

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

   4. sub-neg 88.0%

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

   5. distribute-lft-in 88.0%

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

   6. distribute-rgt-neg-in 88.0%

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

   7. unpow2 88.0%

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

   8. cube-mult 88.0%

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

   9. unsub-neg 88.0%

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

   10. *-rgt-identity 88.0%

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

8. Simplified 88.0%

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

9. Taylor expanded in y around inf 36.2%

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

10. Taylor expanded in y around 0 2.6%

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

11. Final simplification 2.6%

    \[\leadsto x \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024034 
(FPCore (x y)
  :name "Statistics.Distribution.Binomial:$cvariance from math-functions-0.1.5.2"
  :precision binary64
  (* (* x y) (- 1.0 y)))