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

Average Error: 0.16% → 0.16%

Time: 3.6s

Precision: binary64

Cost: 713

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -8.6 \cdot 10^{+62} \lor \neg \left(y \leq 9 \cdot 10^{+15}\right):\\ \;\;\;\;y \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \left(1 - y\right)\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (if (or (<= y -8.6e+62) (not (<= y 9e+15)))
   (* y (* x (- y)))
   (* x (* y (- 1.0 y)))))

C

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

↓

double code(double x, double y) {
	double tmp;
	if ((y <= -8.6e+62) || !(y <= 9e+15)) {
		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
    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 <= (-8.6d+62)) .or. (.not. (y <= 9d+15))) 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) {
	return (x * y) * (1.0 - y);
}

↓

public static double code(double x, double y) {
	double tmp;
	if ((y <= -8.6e+62) || !(y <= 9e+15)) {
		tmp = y * (x * -y);
	} else {
		tmp = x * (y * (1.0 - y));
	}
	return tmp;
}

Python

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

↓

def code(x, y):
	tmp = 0
	if (y <= -8.6e+62) or not (y <= 9e+15):
		tmp = y * (x * -y)
	else:
		tmp = x * (y * (1.0 - 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 <= -8.6e+62) || !(y <= 9e+15))
		tmp = Float64(y * Float64(x * Float64(-y)));
	else
		tmp = Float64(x * Float64(y * Float64(1.0 - 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 <= -8.6e+62) || ~((y <= 9e+15)))
		tmp = y * (x * -y);
	else
		tmp = x * (y * (1.0 - 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, -8.6e+62], N[Not[LessEqual[y, 9e+15]], $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 < -8.5999999999999994e62 or 9e15 < y

   1. Initial program 0.41

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

   2. Simplified 33.7

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

      [Start] 0.41	\[ \left(x \cdot y\right) \cdot \left(1 - y\right) \]
      associate-*l* [=>] 33.7	\[ \color{blue}{x \cdot \left(y \cdot \left(1 - y\right)\right)} \]

   3. Taylor expanded in y around inf 33.7

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

   4. Simplified 0.41

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

      [Start] 33.7	\[ -1 \cdot \left({y}^{2} \cdot x\right) \]
      mul-1-neg [=>] 33.7	\[ \color{blue}{-{y}^{2} \cdot x} \]
      distribute-lft-neg-in [=>] 33.7	\[ \color{blue}{\left(-{y}^{2}\right) \cdot x} \]
      unpow2 [=>] 33.7	\[ \left(-\color{blue}{y \cdot y}\right) \cdot x \]
      distribute-rgt-neg-out [<=] 33.7	\[ \color{blue}{\left(y \cdot \left(-y\right)\right)} \cdot x \]
      associate-*l* [=>] 0.41	\[ \color{blue}{y \cdot \left(\left(-y\right) \cdot x\right)} \]
      *-commutative [=>] 0.41	\[ y \cdot \color{blue}{\left(x \cdot \left(-y\right)\right)} \]

   if -8.5999999999999994e62 < y < 9e15

   1. Initial program 0.07

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

   2. Simplified 0.08

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

      [Start] 0.07	\[ \left(x \cdot y\right) \cdot \left(1 - y\right) \]
      associate-*l* [=>] 0.08	\[ \color{blue}{x \cdot \left(y \cdot \left(1 - y\right)\right)} \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.16

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.6 \cdot 10^{+62} \lor \neg \left(y \leq 9 \cdot 10^{+15}\right):\\ \;\;\;\;y \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \left(1 - y\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	3.05%
Cost	649

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

Alternative 2
Error	0.16%
Cost	448

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

Alternative 3
Error	33.89%
Cost	192

\[x \cdot y \]

Reproduce

herbie shell --seed 2023090 
(FPCore (x y)
  :name "Statistics.Distribution.Binomial:$cvariance from math-functions-0.1.5.2"
  :precision binary64
  (* (* x y) (- 1.0 y)))