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

Percentage Accurate: 99.9% → 99.9%

Time: 4.5s

Alternatives: 5

Speedup: 1.0×

• 24.7% 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 99.8%

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

   1. associate-*l* 92.7%

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

3. Simplified 92.7%

   \[\leadsto \color{blue}{x \cdot \left(y \cdot \left(1 - y\right)\right)} \]
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 91.6%

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

   4. *-commutative 91.6%

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

   5. *-un-lft-identity 91.6%

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

   6. *-commutative 91.6%

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

   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: 93.8% accurate, 1.0× speedup

Math

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

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(x * Float64(y * Float64(1.0 - y)))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate-*l* 92.7%

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

3. Simplified 92.7%

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

4. Final simplification 92.7%

   \[\leadsto x \cdot \left(y \cdot \left(1 - y\right)\right) \]

Alternative 3: 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 99.8%

   \[\left(x \cdot y\right) \cdot \left(1 - y\right) \]

2. Final simplification 99.8%

   \[\leadsto \left(y \cdot x\right) \cdot \left(1 - y\right) \]

Alternative 4: 63.4% 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* 96.9%

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

   3. Simplified 96.9%

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

   4. Taylor expanded in y around 0 72.5%

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

   if 1 < y

   1. Initial program 99.7%

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

      1. associate-*l* 81.7%

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

   3. Simplified 81.7%

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

      1. associate-*r* 99.7%

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

      2. flip-- 81.7%

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

      3. associate-*r/ 77.7%

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

      4. metadata-eval 77.7%

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

      5. pow2 77.7%

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

      6. +-commutative 77.7%

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

   5. Applied egg-rr 77.7%

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

      1. associate-*l* 75.0%

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

      2. associate-/l* 76.3%

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

      3. sub-neg 76.3%

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

      4. distribute-lft-in 76.3%

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

      5. *-rgt-identity 76.3%

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

      6. distribute-rgt-neg-in 76.3%

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

      7. unpow2 76.3%

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

      8. cube-mult 76.3%

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

      9. unsub-neg 76.3%

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

   7. Simplified 76.3%

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

   8. Taylor expanded in y around 0 0.6%

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

      1. frac-2neg 0.6%

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

      2. neg-sub0 0.6%

         \[\leadsto \frac{\color{blue}{0 - x}}{-\frac{1}{y}} \]

      3. div-sub 0.6%

         \[\leadsto \color{blue}{\frac{0}{-\frac{1}{y}} - \frac{x}{-\frac{1}{y}}} \]

      4. distribute-neg-frac 0.6%

         \[\leadsto \frac{0}{\color{blue}{\frac{-1}{y}}} - \frac{x}{-\frac{1}{y}} \]

      5. add-sqr-sqrt 0.6%

         \[\leadsto \frac{0}{\frac{-1}{\color{blue}{\sqrt{y} \cdot \sqrt{y}}}} - \frac{x}{-\frac{1}{y}} \]

      6. sqrt-unprod 0.5%

         \[\leadsto \frac{0}{\frac{-1}{\color{blue}{\sqrt{y \cdot y}}}} - \frac{x}{-\frac{1}{y}} \]

      7. sqr-neg 0.5%

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

      8. sqrt-unprod 0.0%

         \[\leadsto \frac{0}{\frac{-1}{\color{blue}{\sqrt{-y} \cdot \sqrt{-y}}}} - \frac{x}{-\frac{1}{y}} \]

      9. add-sqr-sqrt 0.6%

         \[\leadsto \frac{0}{\frac{-1}{\color{blue}{-y}}} - \frac{x}{-\frac{1}{y}} \]

      10. frac-2neg 0.6%

          \[\leadsto \frac{0}{\color{blue}{\frac{1}{y}}} - \frac{x}{-\frac{1}{y}} \]

      11. add-sqr-sqrt 0.3%

          \[\leadsto \frac{0}{\frac{1}{y}} - \frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{-\frac{1}{y}} \]

      12. sqrt-unprod 17.9%

          \[\leadsto \frac{0}{\frac{1}{y}} - \frac{\color{blue}{\sqrt{x \cdot x}}}{-\frac{1}{y}} \]

      13. sqr-neg 17.9%

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

      14. sqrt-unprod 15.5%

          \[\leadsto \frac{0}{\frac{1}{y}} - \frac{\color{blue}{\sqrt{-x} \cdot \sqrt{-x}}}{-\frac{1}{y}} \]

      15. add-sqr-sqrt 34.5%

          \[\leadsto \frac{0}{\frac{1}{y}} - \frac{\color{blue}{-x}}{-\frac{1}{y}} \]

      16. frac-2neg 34.5%

          \[\leadsto \frac{0}{\frac{1}{y}} - \color{blue}{\frac{x}{\frac{1}{y}}} \]

      17. associate-/r/ 34.5%

          \[\leadsto \frac{0}{\frac{1}{y}} - \color{blue}{\frac{x}{1} \cdot y} \]

      18. /-rgt-identity 34.5%

          \[\leadsto \frac{0}{\frac{1}{y}} - \color{blue}{x} \cdot y \]

   10. Applied egg-rr 34.5%

       \[\leadsto \color{blue}{\frac{0}{\frac{1}{y}} - x \cdot y} \]
   11. Step-by-step derivation

       1. div0 34.5%

          \[\leadsto \color{blue}{0} - x \cdot y \]

       2. neg-sub0 34.5%

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

       3. distribute-rgt-neg-in 34.5%

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

   12. Simplified 34.5%

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

4. Final simplification 61.9%

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

Alternative 5: 56.4% 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 99.8%

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

   1. associate-*l* 92.7%

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

3. Simplified 92.7%

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

4. Taylor expanded in y around 0 52.5%

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

5. Final simplification 52.5%

   \[\leadsto y \cdot x \]

Reproduce

herbie shell --seed 2023320 
(FPCore (x y)
  :name "Statistics.Distribution.Binomial:$cvariance from math-functions-0.1.5.2"
  :precision binary64
  (* (* x y) (- 1.0 y)))