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

Average Error: 0.1 → 0.1

Time: 5.5s

Precision: binary64

Cost: 712

Math

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

↓

\[\begin{array}{l} t_0 := \left(y \cdot x\right) \cdot \left(-y\right)\\ \mathbf{if}\;y \leq -8.6 \cdot 10^{+62}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{+15}:\\ \;\;\;\;x \cdot \left(y - y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y x) (- y))))
   (if (<= y -8.6e+62) t_0 (if (<= y 9.5e+15) (* 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 * x) * -y;
	double tmp;
	if (y <= -8.6e+62) {
		tmp = t_0;
	} else if (y <= 9.5e+15) {
		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 * x) * -y
    if (y <= (-8.6d+62)) then
        tmp = t_0
    else if (y <= 9.5d+15) 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 * x) * -y;
	double tmp;
	if (y <= -8.6e+62) {
		tmp = t_0;
	} else if (y <= 9.5e+15) {
		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 * x) * -y
	tmp = 0
	if y <= -8.6e+62:
		tmp = t_0
	elif y <= 9.5e+15:
		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(Float64(y * x) * Float64(-y))
	tmp = 0.0
	if (y <= -8.6e+62)
		tmp = t_0;
	elseif (y <= 9.5e+15)
		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 * x) * -y;
	tmp = 0.0;
	if (y <= -8.6e+62)
		tmp = t_0;
	elseif (y <= 9.5e+15)
		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[(N[(y * x), $MachinePrecision] * (-y)), $MachinePrecision]}, If[LessEqual[y, -8.6e+62], t$95$0, If[LessEqual[y, 9.5e+15], N[(x * N[(y - N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -8.5999999999999994e62 or 9.5e15 < y

   1. Initial program 0.3

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

   2. Applied egg-rr 0.3

      \[\leadsto \color{blue}{0 \cdot \left(\left(x \cdot y\right) \cdot \frac{1 - y}{x \cdot y}\right) - \left(\left(x \cdot y\right) \cdot \frac{1 - y}{x \cdot y}\right) \cdot \left(y \cdot \left(-x\right)\right)} \]

   3. Simplified 0.4

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

      [Start] 0.3	\[ 0 \cdot \left(\left(x \cdot y\right) \cdot \frac{1 - y}{x \cdot y}\right) - \left(\left(x \cdot y\right) \cdot \frac{1 - y}{x \cdot y}\right) \cdot \left(y \cdot \left(-x\right)\right) \]
      rational_best_oopsla_all_46_json_45_simplify-102 [=>] 0.3	\[ \color{blue}{\left(\left(x \cdot y\right) \cdot \frac{1 - y}{x \cdot y}\right) \cdot \left(0 - y \cdot \left(-x\right)\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 0.3	\[ \color{blue}{\left(0 - y \cdot \left(-x\right)\right) \cdot \left(\left(x \cdot y\right) \cdot \frac{1 - y}{x \cdot y}\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-97 [=>] 0.3	\[ \left(0 - y \cdot \color{blue}{\left(0 - x\right)}\right) \cdot \left(\left(x \cdot y\right) \cdot \frac{1 - y}{x \cdot y}\right) \]
      rational_best_oopsla_all_46_json_45_simplify-13 [=>] 0.3	\[ \left(0 - \color{blue}{\left(0 \cdot y - y \cdot x\right)}\right) \cdot \left(\left(x \cdot y\right) \cdot \frac{1 - y}{x \cdot y}\right) \]
      rational_best_oopsla_all_46_json_45_simplify-74 [<=] 0.3	\[ \left(0 - \left(\color{blue}{y \cdot 0} - y \cdot x\right)\right) \cdot \left(\left(x \cdot y\right) \cdot \frac{1 - y}{x \cdot y}\right) \]
      rational_best_oopsla_all_46_json_45_simplify-38 [=>] 0.3	\[ \left(0 - \left(\color{blue}{0} - y \cdot x\right)\right) \cdot \left(\left(x \cdot y\right) \cdot \frac{1 - y}{x \cdot y}\right) \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 0.3	\[ \left(0 - \left(0 - \color{blue}{x \cdot y}\right)\right) \cdot \left(\left(x \cdot y\right) \cdot \frac{1 - y}{x \cdot y}\right) \]
      rational_best_oopsla_all_46_json_45_simplify-36 [=>] 0.3	\[ \color{blue}{\left(x \cdot y - \left(0 - 0\right)\right)} \cdot \left(\left(x \cdot y\right) \cdot \frac{1 - y}{x \cdot y}\right) \]
      metadata-eval [=>] 0.3	\[ \left(x \cdot y - \color{blue}{0}\right) \cdot \left(\left(x \cdot y\right) \cdot \frac{1 - y}{x \cdot y}\right) \]
      rational_best_oopsla_all_46_json_45_simplify-81 [=>] 0.3	\[ \color{blue}{\left(x \cdot y\right)} \cdot \left(\left(x \cdot y\right) \cdot \frac{1 - y}{x \cdot y}\right) \]
      rational_best_oopsla_all_46_json_45_simplify-74 [<=] 0.3	\[ \color{blue}{\left(y \cdot x\right)} \cdot \left(\left(x \cdot y\right) \cdot \frac{1 - y}{x \cdot y}\right) \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 0.3	\[ \left(y \cdot x\right) \cdot \color{blue}{\left(\frac{1 - y}{x \cdot y} \cdot \left(x \cdot y\right)\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [<=] 0.3	\[ \left(y \cdot x\right) \cdot \left(\frac{1 - y}{x \cdot y} \cdot \color{blue}{\left(y \cdot x\right)}\right) \]
      rational_best_oopsla_all_46_json_45_simplify-7 [=>] 0.4	\[ \left(y \cdot x\right) \cdot \color{blue}{\left(y \cdot \left(\frac{1 - y}{x \cdot y} \cdot x\right)\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [<=] 0.4	\[ \left(y \cdot x\right) \cdot \left(y \cdot \left(\frac{1 - y}{\color{blue}{y \cdot x}} \cdot x\right)\right) \]

   4. Taylor expanded in y around inf 0.3

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

   5. Simplified 0.3

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

      [Start] 0.3	\[ \left(y \cdot x\right) \cdot \left(-1 \cdot y\right) \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 0.3	\[ \left(y \cdot x\right) \cdot \color{blue}{\left(y \cdot -1\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-94 [<=] 0.3	\[ \left(y \cdot x\right) \cdot \color{blue}{\left(-y\right)} \]

   if -8.5999999999999994e62 < y < 9.5e15

   1. Initial program 0.0

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

   2. Simplified 0.0

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

      [Start] 0.0	\[ \left(x \cdot y\right) \cdot \left(1 - y\right) \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 0.0	\[ \color{blue}{\left(1 - y\right) \cdot \left(x \cdot y\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-7 [=>] 0.0	\[ \color{blue}{x \cdot \left(\left(1 - y\right) \cdot y\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 0.0	\[ x \cdot \color{blue}{\left(y \cdot \left(1 - y\right)\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-13 [=>] 0.0	\[ x \cdot \color{blue}{\left(1 \cdot y - y \cdot y\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 0.0	\[ x \cdot \left(\color{blue}{y \cdot 1} - y \cdot y\right) \]
      rational_best_oopsla_all_46_json_45_simplify-52 [=>] 0.0	\[ x \cdot \left(\color{blue}{y} - y \cdot y\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.6 \cdot 10^{+62}:\\ \;\;\;\;\left(y \cdot x\right) \cdot \left(-y\right)\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{+15}:\\ \;\;\;\;x \cdot \left(y - y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot x\right) \cdot \left(-y\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	7.5
Cost	648

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

Alternative 2
Error	2.0
Cost	648

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

Alternative 3
Error	0.1
Cost	448

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

Alternative 4
Error	21.7
Cost	192

\[y \cdot x \]

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)))