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

Average Error: 0.1 → 0.1

Time: 8.4s

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 0.1

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

2. Simplified 0.1

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

   [Start] 0.1	\[ \left(x \cdot \log y - z\right) - y \]
   associate--l- [=>] 0.1	\[ \color{blue}{x \cdot \log y - \left(z + y\right)} \]
   fma-neg [=>] 0.1	\[ \color{blue}{\mathsf{fma}\left(x, \log y, -\left(z + y\right)\right)} \]
   distribute-neg-in [=>] 0.1	\[ \mathsf{fma}\left(x, \log y, \color{blue}{\left(-z\right) + \left(-y\right)}\right) \]
   sub-neg [<=] 0.1	\[ \mathsf{fma}\left(x, \log y, \color{blue}{\left(-z\right) - y}\right) \]

3. Final simplification 0.1

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

Alternatives

Alternative 1
Error	11.6
Cost	7116

\[\begin{array}{l} t_0 := \left(-z\right) - y\\ t_1 := x \cdot \log y\\ \mathbf{if}\;z \leq -1.2 \cdot 10^{-53}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 4.3 \cdot 10^{-104}:\\ \;\;\;\;t_1 - y\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{+63}:\\ \;\;\;\;t_1 - z\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	10.3
Cost	6985

\[\begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{-51} \lor \neg \left(z \leq 2 \cdot 10^{+53}\right):\\ \;\;\;\;\left(-z\right) - y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - y\\ \end{array} \]

Alternative 3
Error	13.3
Cost	6857

\[\begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{+178} \lor \neg \left(x \leq 10^{+130}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) - y\\ \end{array} \]

Alternative 4
Error	0.1
Cost	6848

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

Alternative 5
Error	32.1
Cost	524

\[\begin{array}{l} \mathbf{if}\;y \leq 2.1 \cdot 10^{-26}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 1.22 \cdot 10^{+94}:\\ \;\;\;\;-y\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{+145}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \]

Alternative 6
Error	21.8
Cost	256

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

Alternative 7
Error	42.6
Cost	128

\[-y \]

Alternative 8
Error	62.5
Cost	64

\[z \]

Reproduce

herbie shell --seed 2022364 
(FPCore (x y z)
  :name "Statistics.Distribution.Poisson:$clogProbability from math-functions-0.1.5.2"
  :precision binary64
  (- (- (* x (log y)) z) y))