Statistics.Distribution.Poisson:$clogProbability from math-functions-0.1.5.2

Average Error: 0.1 → 0.1

Time: 7.2s

Precision: binary64

Cost: 6848

Math

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

↓

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

FPCore

(FPCore (x y z) :precision binary64 (- (- (* x (log y)) z) y))

↓

(FPCore (x y z) :precision binary64 (- (- (* (log y) x) y) z))

C

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

↓

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = ((x * log(y)) - z) - y
end function

↓

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = ((log(y) * x) - y) - z
end function

Java

public static double code(double x, double y, double z) {
	return ((x * Math.log(y)) - z) - y;
}

↓

public static double code(double x, double y, double z) {
	return ((Math.log(y) * x) - y) - z;
}

Python

def code(x, y, z):
	return ((x * math.log(y)) - z) - y

↓

def code(x, y, z):
	return ((math.log(y) * x) - y) - z

Julia

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

↓

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

MATLAB

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

↓

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

Wolfram

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

↓

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

Derivation

1. Initial program 0.1

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

2. Simplified 0.1

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

   [Start] 0.1	\[ \left(x \cdot \log y - z\right) - y \]
   associate--l- [=>] 0.1	\[ \color{blue}{x \cdot \log y - \left(z + y\right)} \]

3. Taylor expanded in y around inf 0.1

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

4. Simplified 0.1

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

   [Start] 0.1	\[ \left(-1 \cdot y + -1 \cdot \left(\log \left(\frac{1}{y}\right) \cdot x\right)\right) - z \]
   *-commutative [<=] 0.1	\[ \left(\color{blue}{y \cdot -1} + -1 \cdot \left(\log \left(\frac{1}{y}\right) \cdot x\right)\right) - z \]
   log-rec [=>] 0.1	\[ \left(y \cdot -1 + -1 \cdot \left(\color{blue}{\left(-\log y\right)} \cdot x\right)\right) - z \]
   distribute-lft-neg-in [<=] 0.1	\[ \left(y \cdot -1 + -1 \cdot \color{blue}{\left(-\log y \cdot x\right)}\right) - z \]
   *-commutative [<=] 0.1	\[ \left(y \cdot -1 + -1 \cdot \left(-\color{blue}{x \cdot \log y}\right)\right) - z \]
   mul-1-neg [=>] 0.1	\[ \left(y \cdot -1 + \color{blue}{\left(-\left(-x \cdot \log y\right)\right)}\right) - z \]
   unsub-neg [=>] 0.1	\[ \color{blue}{\left(y \cdot -1 - \left(-x \cdot \log y\right)\right)} - z \]
   *-commutative [=>] 0.1	\[ \left(\color{blue}{-1 \cdot y} - \left(-x \cdot \log y\right)\right) - z \]
   mul-1-neg [=>] 0.1	\[ \left(\color{blue}{\left(-y\right)} - \left(-x \cdot \log y\right)\right) - z \]
   *-commutative [=>] 0.1	\[ \left(\left(-y\right) - \left(-\color{blue}{\log y \cdot x}\right)\right) - z \]
   distribute-rgt-neg-in [=>] 0.1	\[ \left(\left(-y\right) - \color{blue}{\log y \cdot \left(-x\right)}\right) - z \]

5. Final simplification 0.1

   \[\leadsto \left(\log y \cdot x - y\right) - z \]

Alternatives

Alternative 1
Error	9.6
Cost	6985

\[\begin{array}{l} \mathbf{if}\;x \leq -250 \lor \neg \left(x \leq 3.1 \cdot 10^{-47}\right):\\ \;\;\;\;\log y \cdot x - y\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) - z\\ \end{array} \]

Alternative 2
Error	13.4
Cost	6857

\[\begin{array}{l} \mathbf{if}\;x \leq -3.3 \cdot 10^{+62} \lor \neg \left(x \leq 1.62 \cdot 10^{+94}\right):\\ \;\;\;\;\log y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) - z\\ \end{array} \]

Alternative 3
Error	10.3
Cost	6852

\[\begin{array}{l} \mathbf{if}\;y \leq 6.2 \cdot 10^{-31}:\\ \;\;\;\;\log y \cdot x - z\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) - z\\ \end{array} \]

Alternative 4
Error	0.1
Cost	6848

\[\log y \cdot x - \left(y + z\right) \]

Alternative 5
Error	30.4
Cost	260

\[\begin{array}{l} \mathbf{if}\;y \leq 0.00064:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \]

Alternative 6
Error	21.6
Cost	256

\[\left(-y\right) - z \]

Alternative 7
Error	41.8
Cost	128

\[-y \]

Reproduce

herbie shell --seed 2023016 
(FPCore (x y z)
  :name "Statistics.Distribution.Poisson:$clogProbability from math-functions-0.1.5.2"
  :precision binary64
  (- (- (* x (log y)) z) y))