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

Average Error: 0.15% → 0.14%

Time: 10.2s

Precision: binary64

Cost: 13184

Math

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

↓

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

FPCore

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

↓

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

Julia

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

↓

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

Derivation

1. Initial program 0.15

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

2. Simplified 0.14

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

   [Start] 0.15	\[ \left(x \cdot \log y - z\right) - y \]
   associate--l- [=>] 0.15	\[ \color{blue}{x \cdot \log y - \left(z + y\right)} \]
   fma-neg [=>] 0.14	\[ \color{blue}{\mathsf{fma}\left(x, \log y, -\left(z + y\right)\right)} \]
   distribute-neg-in [=>] 0.14	\[ \mathsf{fma}\left(x, \log y, \color{blue}{\left(-z\right) + \left(-y\right)}\right) \]
   sub-neg [<=] 0.14	\[ \mathsf{fma}\left(x, \log y, \color{blue}{\left(-z\right) - y}\right) \]

3. Final simplification 0.14

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

Alternatives

Alternative 1
Error	15.34%
Cost	6985

\[\begin{array}{l} \mathbf{if}\;z \leq -3.1 \cdot 10^{+62} \lor \neg \left(z \leq 2.6 \cdot 10^{+128}\right):\\ \;\;\;\;\left(-z\right) - y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - y\\ \end{array} \]

Alternative 2
Error	21.22%
Cost	6857

\[\begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{+192} \lor \neg \left(x \leq 6.8 \cdot 10^{+174}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) - y\\ \end{array} \]

Alternative 3
Error	14.53%
Cost	6852

\[\begin{array}{l} \mathbf{if}\;y \leq 3.6 \cdot 10^{+50}:\\ \;\;\;\;x \cdot \log y - z\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) - y\\ \end{array} \]

Alternative 4
Error	0.15%
Cost	6848

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

Alternative 5
Error	48.34%
Cost	260

\[\begin{array}{l} \mathbf{if}\;y \leq 3.6 \cdot 10^{+113}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \]

Alternative 6
Error	33.93%
Cost	256

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

Alternative 7
Error	66.95%
Cost	128

\[-y \]

Reproduce

herbie shell --seed 2023088 
(FPCore (x y z)
  :name "Statistics.Distribution.Poisson:$clogProbability from math-functions-0.1.5.2"
  :precision binary64
  (- (- (* x (log y)) z) y))