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

Average Error: 0.15% → 0.15%

Time: 8.1s

Precision: binary64

Cost: 713

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -1.9 \cdot 10^{+53} \lor \neg \left(y \leq 5 \cdot 10^{+18}\right):\\ \;\;\;\;y \cdot \left(y \cdot \left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y - y \cdot y\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.9e+53) (not (<= y 5e+18)))
   (* y (* y (- x)))
   (* x (- y (* y y)))))

C

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

↓

double code(double x, double y) {
	double tmp;
	if ((y <= -1.9e+53) || !(y <= 5e+18)) {
		tmp = y * (y * -x);
	} else {
		tmp = x * (y - (y * y));
	}
	return tmp;
}

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

↓

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-1.9d+53)) .or. (.not. (y <= 5d+18))) then
        tmp = y * (y * -x)
    else
        tmp = x * (y - (y * y))
    end if
    code = tmp
end function

Java

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

↓

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.9e+53) || !(y <= 5e+18)) {
		tmp = y * (y * -x);
	} else {
		tmp = x * (y - (y * y));
	}
	return tmp;
}

Python

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

↓

def code(x, y):
	tmp = 0
	if (y <= -1.9e+53) or not (y <= 5e+18):
		tmp = y * (y * -x)
	else:
		tmp = x * (y - (y * y))
	return tmp

Julia

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

↓

function code(x, y)
	tmp = 0.0
	if ((y <= -1.9e+53) || !(y <= 5e+18))
		tmp = Float64(y * Float64(y * Float64(-x)));
	else
		tmp = Float64(x * Float64(y - Float64(y * y)));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.9e+53) || ~((y <= 5e+18)))
		tmp = y * (y * -x);
	else
		tmp = x * (y - (y * y));
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[x_, y_] := If[Or[LessEqual[y, -1.9e+53], N[Not[LessEqual[y, 5e+18]], $MachinePrecision]], N[(y * N[(y * (-x)), $MachinePrecision]), $MachinePrecision], N[(x * N[(y - N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.89999999999999999e53 or 5e18 < y

   1. Initial program 0.4

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

   2. Simplified 33.92

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

      [Start] 0.4	\[ \left(x \cdot y\right) \cdot \left(1 - y\right) \]
      distribute-lft-out-- [<=] 0.4	\[ \color{blue}{\left(x \cdot y\right) \cdot 1 - \left(x \cdot y\right) \cdot y} \]
      *-rgt-identity [=>] 0.4	\[ \color{blue}{x \cdot y} - \left(x \cdot y\right) \cdot y \]
      associate-*l* [=>] 33.92	\[ x \cdot y - \color{blue}{x \cdot \left(y \cdot y\right)} \]
      distribute-lft-out-- [=>] 33.92	\[ \color{blue}{x \cdot \left(y - y \cdot y\right)} \]

   3. Taylor expanded in y around inf 33.92

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

   4. Simplified 0.4

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

      [Start] 33.92	\[ -1 \cdot \left({y}^{2} \cdot x\right) \]
      mul-1-neg [=>] 33.92	\[ \color{blue}{-{y}^{2} \cdot x} \]
      unpow2 [=>] 33.92	\[ -\color{blue}{\left(y \cdot y\right)} \cdot x \]
      associate-*l* [=>] 0.4	\[ -\color{blue}{y \cdot \left(y \cdot x\right)} \]
      distribute-lft-neg-in [=>] 0.4	\[ \color{blue}{\left(-y\right) \cdot \left(y \cdot x\right)} \]
      *-commutative [<=] 0.4	\[ \color{blue}{\left(y \cdot x\right) \cdot \left(-y\right)} \]

   if -1.89999999999999999e53 < y < 5e18

   1. Initial program 0.07

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

   2. Simplified 0.07

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

      [Start] 0.07	\[ \left(x \cdot y\right) \cdot \left(1 - y\right) \]
      distribute-lft-out-- [<=] 0.07	\[ \color{blue}{\left(x \cdot y\right) \cdot 1 - \left(x \cdot y\right) \cdot y} \]
      *-rgt-identity [=>] 0.07	\[ \color{blue}{x \cdot y} - \left(x \cdot y\right) \cdot y \]
      associate-*l* [=>] 0.07	\[ x \cdot y - \color{blue}{x \cdot \left(y \cdot y\right)} \]
      distribute-lft-out-- [=>] 0.07	\[ \color{blue}{x \cdot \left(y - y \cdot y\right)} \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.15

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.9 \cdot 10^{+53} \lor \neg \left(y \leq 5 \cdot 10^{+18}\right):\\ \;\;\;\;y \cdot \left(y \cdot \left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y - y \cdot y\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	11.52%
Cost	649

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

Alternative 2
Error	3.05%
Cost	649

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

Alternative 3
Error	0.15%
Cost	448

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

Alternative 4
Error	33.06%
Cost	192

\[x \cdot y \]

Alternative 5
Error	96.41%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023088 
(FPCore (x y)
  :name "Statistics.Distribution.Binomial:$cvariance from math-functions-0.1.5.2"
  :precision binary64
  (* (* x y) (- 1.0 y)))