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

Average Error: 0.1 → 0.1

Time: 8.2s

Precision: binary64

Cost: 13184

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Julia

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

↓

function code(x, y, z)
	return Float64(fma(log(y), x, Float64(-z)) - y)
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[Log[y], $MachinePrecision] * x + (-z)), $MachinePrecision] - y), $MachinePrecision]

Derivation

1. Initial program 0.1

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

2. Taylor expanded in y around 0 0.1

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

3. Simplified 0.1

   \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, x, -z\right)} - y \]
   Proof
   (fma.f64 (log.f64 y) x (neg.f64 z)): 0 points increase in error, 0 points decrease in error
   (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 (log.f64 y) x) z)): 4 points increase in error, 0 points decrease in error

4. Final simplification 0.1

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

Alternatives

Alternative 1
Error	11.8
Cost	7248

\[\begin{array}{l} t_0 := \log y \cdot x - y\\ t_1 := \left(-z\right) - y\\ \mathbf{if}\;x \leq -3.032427623150505 \cdot 10^{+80}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -313394540240688500:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -7.063372647782073 \cdot 10^{-163}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 9.470811504927631 \cdot 10^{+83}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	0.1
Cost	6848

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

Alternative 3
Error	21.1
Cost	256

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

Alternative 4
Error	42.6
Cost	192

\[z - y \]

Reproduce

herbie shell --seed 2022310 
(FPCore (x y z)
  :name "Statistics.Distribution.Poisson:$clogProbability from math-functions-0.1.5.2"
  :precision binary64
  (- (- (* x (log y)) z) y))