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

Average Error: 0.1 → 0.1

Time: 3.0s

Precision: binary64

Math

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

↓

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

FPCore

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

↓

(FPCore (x y z) :precision binary64 (- (- (* x (- (log (/ 1.0 y)))) 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 ((x * -log((1.0 / y))) - 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 = ((x * -log((1.0d0 / y))) - 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 ((x * -Math.log((1.0 / y))) - y) - z;
}

Python

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

↓

def code(x, y, z):
	return ((x * -math.log((1.0 / y))) - 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(x * Float64(-log(Float64(1.0 / y)))) - y) - z)
end

MATLAB

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

↓

function tmp = code(x, y, z)
	tmp = ((x * -log((1.0 / y))) - 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[(x * (-N[Log[N[(1.0 / y), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] - y), $MachinePrecision] - z), $MachinePrecision]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

1. Initial program 0.1

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

2. Taylor expanded in x around 0 0.1

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

3. Simplified 0.1

   \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, x, -y\right) - z} \]

4. Taylor expanded in y around inf 0.1

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

5. Final simplification 0.1

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

Reproduce

herbie shell --seed 2022166 
(FPCore (x y z)
  :name "Statistics.Distribution.Poisson:$clogProbability from math-functions-0.1.5.2"
  :precision binary64
  (- (- (* x (log y)) z) y))