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

Average Error: 0.1 → 0.1

Time: 7.0s

Precision: binary64

Cost: 712

Math

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

↓

\[\begin{array}{l} t_0 := y \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{if}\;y \leq -50554366478291380:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 10^{+63}:\\ \;\;\;\;x \cdot \left(y \cdot \left(1 - y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* x (- y)))))
   (if (<= y -50554366478291380.0)
     t_0
     (if (<= y 1e+63) (* x (* y (- 1.0 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 * (x * -y);
	double tmp;
	if (y <= -50554366478291380.0) {
		tmp = t_0;
	} else if (y <= 1e+63) {
		tmp = x * (y * (1.0 - 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 * (x * -y)
    if (y <= (-50554366478291380.0d0)) then
        tmp = t_0
    else if (y <= 1d+63) then
        tmp = x * (y * (1.0d0 - 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 * (x * -y);
	double tmp;
	if (y <= -50554366478291380.0) {
		tmp = t_0;
	} else if (y <= 1e+63) {
		tmp = x * (y * (1.0 - 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 * (x * -y)
	tmp = 0
	if y <= -50554366478291380.0:
		tmp = t_0
	elif y <= 1e+63:
		tmp = x * (y * (1.0 - 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(x * Float64(-y)))
	tmp = 0.0
	if (y <= -50554366478291380.0)
		tmp = t_0;
	elseif (y <= 1e+63)
		tmp = Float64(x * Float64(y * Float64(1.0 - 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 * (x * -y);
	tmp = 0.0;
	if (y <= -50554366478291380.0)
		tmp = t_0;
	elseif (y <= 1e+63)
		tmp = x * (y * (1.0 - 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[(x * (-y)), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -50554366478291380.0], t$95$0, If[LessEqual[y, 1e+63], N[(x * N[(y * N[(1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -50554366478291376 or 1.00000000000000006e63 < y

   1. Initial program 0.3

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

   2. Simplified 21.6

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(y, -y, y\right)} \]
      Proof
      (*.f64 x (fma.f64 y (neg.f64 y) y)): 0 points increase in error, 0 points decrease in error
      (*.f64 x (Rewrite<= fma-def_binary64 (+.f64 (*.f64 y (neg.f64 y)) y))): 3 points increase in error, 0 points decrease in error
      (*.f64 x (+.f64 (*.f64 y (neg.f64 y)) (Rewrite<= *-rgt-identity_binary64 (*.f64 y 1)))): 0 points increase in error, 0 points decrease in error
      (*.f64 x (Rewrite<= distribute-lft-in_binary64 (*.f64 y (+.f64 (neg.f64 y) 1)))): 1 points increase in error, 5 points decrease in error
      (*.f64 x (*.f64 y (Rewrite<= +-commutative_binary64 (+.f64 1 (neg.f64 y))))): 0 points increase in error, 0 points decrease in error
      (*.f64 x (*.f64 y (Rewrite<= sub-neg_binary64 (-.f64 1 y)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 x y) (-.f64 1 y))): 12 points increase in error, 39 points decrease in error

   3. Taylor expanded in y around inf 21.6

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

   4. Simplified 0.3

      \[\leadsto \color{blue}{y \cdot \left(y \cdot \left(-x\right)\right)} \]
      Proof
      (*.f64 y (*.f64 y (neg.f64 x))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 y y) (neg.f64 x))): 61 points increase in error, 39 points decrease in error
      (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 y 2)) (neg.f64 x)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (pow.f64 y 2) x))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (*.f64 (pow.f64 y 2) x))): 0 points increase in error, 0 points decrease in error

   if -50554366478291376 < y < 1.00000000000000006e63

   1. Initial program 0.0

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

   2. Simplified 0.0

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(y, -y, y\right)} \]
      Proof
      (*.f64 x (fma.f64 y (neg.f64 y) y)): 0 points increase in error, 0 points decrease in error
      (*.f64 x (Rewrite<= fma-def_binary64 (+.f64 (*.f64 y (neg.f64 y)) y))): 3 points increase in error, 0 points decrease in error
      (*.f64 x (+.f64 (*.f64 y (neg.f64 y)) (Rewrite<= *-rgt-identity_binary64 (*.f64 y 1)))): 0 points increase in error, 0 points decrease in error
      (*.f64 x (Rewrite<= distribute-lft-in_binary64 (*.f64 y (+.f64 (neg.f64 y) 1)))): 1 points increase in error, 5 points decrease in error
      (*.f64 x (*.f64 y (Rewrite<= +-commutative_binary64 (+.f64 1 (neg.f64 y))))): 0 points increase in error, 0 points decrease in error
      (*.f64 x (*.f64 y (Rewrite<= sub-neg_binary64 (-.f64 1 y)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 x y) (-.f64 1 y))): 12 points increase in error, 39 points decrease in error

   3. Taylor expanded in x around 0 0.0

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

   4. Simplified 0.0

      \[\leadsto \color{blue}{\left(y \cdot \left(1 - y\right)\right) \cdot x} \]
      Proof
      (*.f64 (*.f64 y (-.f64 1 y)) x): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 y (-.f64 (Rewrite<= metadata-eval (/.f64 2 2)) y)) x): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= distribute-lft-out--_binary64 (-.f64 (*.f64 y (/.f64 2 2)) (*.f64 y y))) x): 5 points increase in error, 1 points decrease in error
      (*.f64 (-.f64 (*.f64 y (/.f64 2 2)) (Rewrite<= unpow2_binary64 (pow.f64 y 2))) x): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= unsub-neg_binary64 (+.f64 (*.f64 y (/.f64 2 2)) (neg.f64 (pow.f64 y 2)))) x): 0 points increase in error, 0 points decrease in error
      (*.f64 (+.f64 (*.f64 y (Rewrite=> metadata-eval 1)) (neg.f64 (pow.f64 y 2))) x): 0 points increase in error, 0 points decrease in error
      (*.f64 (+.f64 (Rewrite=> *-rgt-identity_binary64 y) (neg.f64 (pow.f64 y 2))) x): 0 points increase in error, 0 points decrease in error
      (*.f64 (+.f64 y (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (pow.f64 y 2)))) x): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 -1 (pow.f64 y 2)) y)) x): 0 points increase in error, 0 points decrease in error
3. Recombined 2 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -50554366478291380:\\ \;\;\;\;y \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{elif}\;y \leq 10^{+63}:\\ \;\;\;\;x \cdot \left(y \cdot \left(1 - y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot \left(-y\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	7.3
Cost	648

\[\begin{array}{l} t_0 := x \cdot \left(y \cdot \left(-y\right)\right)\\ \mathbf{if}\;y \leq -326215.46631252294:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 0.014842649333539487:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	1.9
Cost	648

\[\begin{array}{l} t_0 := y \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{if}\;y \leq -326215.46631252294:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 0.014842649333539487:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	0.1
Cost	448

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

Alternative 4
Error	21.3
Cost	192

\[x \cdot y \]

Reproduce

herbie shell --seed 2022291 
(FPCore (x y)
  :name "Statistics.Distribution.Binomial:$cvariance from math-functions-0.1.5.2"
  :precision binary64
  (* (* x y) (- 1.0 y)))