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

Average Error: 0.1 → 0.1

Time: 2.8s

Precision: binary64

Cost: 712

Math

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

↓

\[\begin{array}{l} t_0 := y \cdot \left(y \cdot \left(-x\right)\right)\\ \mathbf{if}\;y \leq -1.96 \cdot 10^{+96}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{+138}:\\ \;\;\;\;x \cdot \left(y - y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y (- x)))))
   (if (<= y -1.96e+96) t_0 (if (<= y 6.2e+138) (* x (- y (* y y))) t_0))))

C

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

↓

double code(double x, double y) {
	double t_0 = y * (y * -x);
	double tmp;
	if (y <= -1.96e+96) {
		tmp = t_0;
	} else if (y <= 6.2e+138) {
		tmp = x * (y - (y * y));
	} else {
		tmp = t_0;
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = y * (y * -x)
    if (y <= (-1.96d+96)) then
        tmp = t_0
    else if (y <= 6.2d+138) then
        tmp = x * (y - (y * y))
    else
        tmp = t_0
    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 t_0 = y * (y * -x);
	double tmp;
	if (y <= -1.96e+96) {
		tmp = t_0;
	} else if (y <= 6.2e+138) {
		tmp = x * (y - (y * y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

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

↓

def code(x, y):
	t_0 = y * (y * -x)
	tmp = 0
	if y <= -1.96e+96:
		tmp = t_0
	elif y <= 6.2e+138:
		tmp = x * (y - (y * y))
	else:
		tmp = t_0
	return tmp

Julia

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

↓

function code(x, y)
	t_0 = Float64(y * Float64(y * Float64(-x)))
	tmp = 0.0
	if (y <= -1.96e+96)
		tmp = t_0;
	elseif (y <= 6.2e+138)
		tmp = Float64(x * Float64(y - Float64(y * y)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x, y)
	t_0 = y * (y * -x);
	tmp = 0.0;
	if (y <= -1.96e+96)
		tmp = t_0;
	elseif (y <= 6.2e+138)
		tmp = x * (y - (y * y));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * (-x)), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.96e+96], t$95$0, If[LessEqual[y, 6.2e+138], N[(x * N[(y - N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.96e96 or 6.1999999999999995e138 < y

   1. Initial program 0.2

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

   2. Taylor expanded in y around inf 41.6

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

   3. Simplified 0.2

      \[\leadsto \color{blue}{\left(y \cdot x\right) \cdot \left(-y\right)} \]
      Proof
      (*.f64 (*.f64 y x) (neg.f64 y)): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 x y)) (neg.f64 y)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 (neg.f64 y) (*.f64 x y))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> distribute-lft-neg-out_binary64 (neg.f64 (*.f64 y (*.f64 x y)))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> neg-mul-1_binary64 (*.f64 -1 (*.f64 y (*.f64 x y)))): 0 points increase in error, 0 points decrease in error
      (*.f64 -1 (*.f64 y (Rewrite=> *-commutative_binary64 (*.f64 y x)))): 0 points increase in error, 0 points decrease in error
      (*.f64 -1 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 y y) x))): 52 points increase in error, 39 points decrease in error
      (*.f64 -1 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 y 2)) x)): 0 points increase in error, 0 points decrease in error

   if -1.96e96 < y < 6.1999999999999995e138

   1. Initial program 0.1

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

   2. Simplified 0.1

      \[\leadsto \color{blue}{x \cdot \left(y - y \cdot y\right)} \]
      Proof
      (*.f64 x (-.f64 y (*.f64 y y))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= distribute-lft-out--_binary64 (-.f64 (*.f64 x y) (*.f64 x (*.f64 y y)))): 1 points increase in error, 2 points decrease in error
      (-.f64 (Rewrite<= *-rgt-identity_binary64 (*.f64 (*.f64 x y) 1)) (*.f64 x (*.f64 y y))): 0 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 (*.f64 x y) 1) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 x y) y))): 13 points increase in error, 24 points decrease in error
      (Rewrite=> distribute-lft-out--_binary64 (*.f64 (*.f64 x y) (-.f64 1 y))): 5 points increase in error, 1 points decrease in error
3. Recombined 2 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.96 \cdot 10^{+96}:\\ \;\;\;\;y \cdot \left(y \cdot \left(-x\right)\right)\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{+138}:\\ \;\;\;\;x \cdot \left(y - y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot \left(-x\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	7.0
Cost	648

\[\begin{array}{l} t_0 := x \cdot \left(-y \cdot y\right)\\ \mathbf{if}\;y \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	1.9
Cost	648

\[\begin{array}{l} t_0 := y \cdot \left(y \cdot \left(-x\right)\right)\\ \mathbf{if}\;y \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	0.1
Cost	448

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

Alternative 4
Error	21.1
Cost	192

\[x \cdot y \]

Alternative 5
Error	61.6
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022329 
(FPCore (x y)
  :name "Statistics.Distribution.Binomial:$cvariance from math-functions-0.1.5.2"
  :precision binary64
  (* (* x y) (- 1.0 y)))