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

Average Accuracy: 99.9% → 99.9%

Time: 7.9s

Precision: binary64

Cost: 713

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{+32} \lor \neg \left(y \leq 4 \cdot 10^{+17}\right):\\ \;\;\;\;y \cdot \left(y \cdot \left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y - y \cdot y\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (if (or (<= y -8e+32) (not (<= y 4e+17)))
   (* 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 <= -8e+32) || !(y <= 4e+17)) {
		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 <= (-8d+32)) .or. (.not. (y <= 4d+17))) 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 <= -8e+32) || !(y <= 4e+17)) {
		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 <= -8e+32) or not (y <= 4e+17):
		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 <= -8e+32) || !(y <= 4e+17))
		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 <= -8e+32) || ~((y <= 4e+17)))
		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, -8e+32], N[Not[LessEqual[y, 4e+17]], $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 < -8.00000000000000043e32 or 4e17 < y

   1. Initial program 99.6%

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

   2. Simplified 70.6%

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

      [Start] 99.6	\[ \left(x \cdot y\right) \cdot \left(1 - y\right) \]
      distribute-lft-out-- [<=] 99.6	\[ \color{blue}{\left(x \cdot y\right) \cdot 1 - \left(x \cdot y\right) \cdot y} \]
      *-rgt-identity [=>] 99.6	\[ \color{blue}{x \cdot y} - \left(x \cdot y\right) \cdot y \]
      associate-*l* [=>] 70.6	\[ x \cdot y - \color{blue}{x \cdot \left(y \cdot y\right)} \]
      distribute-lft-out-- [=>] 70.6	\[ \color{blue}{x \cdot \left(y - y \cdot y\right)} \]

   3. Taylor expanded in y around inf 70.6%

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

   4. Simplified 99.6%

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

      [Start] 70.6	\[ -1 \cdot \left({y}^{2} \cdot x\right) \]
      associate-*r* [=>] 70.6	\[ \color{blue}{\left(-1 \cdot {y}^{2}\right) \cdot x} \]
      mul-1-neg [=>] 70.6	\[ \color{blue}{\left(-{y}^{2}\right)} \cdot x \]
      unpow2 [=>] 70.6	\[ \left(-\color{blue}{y \cdot y}\right) \cdot x \]
      distribute-rgt-neg-in [=>] 70.6	\[ \color{blue}{\left(y \cdot \left(-y\right)\right)} \cdot x \]
      associate-*l* [=>] 99.6	\[ \color{blue}{y \cdot \left(\left(-y\right) \cdot x\right)} \]

   if -8.00000000000000043e32 < y < 4e17

   1. Initial program 99.9%

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

   2. Simplified 99.9%

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

      [Start] 99.9	\[ \left(x \cdot y\right) \cdot \left(1 - y\right) \]
      distribute-lft-out-- [<=] 99.9	\[ \color{blue}{\left(x \cdot y\right) \cdot 1 - \left(x \cdot y\right) \cdot y} \]
      *-rgt-identity [=>] 99.9	\[ \color{blue}{x \cdot y} - \left(x \cdot y\right) \cdot y \]
      associate-*l* [=>] 99.9	\[ x \cdot y - \color{blue}{x \cdot \left(y \cdot y\right)} \]
      distribute-lft-out-- [=>] 99.9	\[ \color{blue}{x \cdot \left(y - y \cdot y\right)} \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{+32} \lor \neg \left(y \leq 4 \cdot 10^{+17}\right):\\ \;\;\;\;y \cdot \left(y \cdot \left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y - y \cdot y\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	89.2%
Cost	649

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

Alternative 2
Accuracy	97.2%
Cost	649

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

Alternative 3
Accuracy	99.9%
Cost	448

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

Alternative 4
Accuracy	67.2%
Cost	192

\[y \cdot x \]

Reproduce

herbie shell --seed 2023137 
(FPCore (x y)
  :name "Statistics.Distribution.Binomial:$cvariance from math-functions-0.1.5.2"
  :precision binary64
  (* (* x y) (- 1.0 y)))