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

Average Accuracy: 99.9% → 99.9%

Time: 10.5s

Precision: binary64

Cost: 13184

Math

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

↓

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

FPCore

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

↓

(FPCore (x y z) :precision binary64 (- (fma x (log 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 fma(x, log(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(fma(x, log(y), Float64(-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[(x * N[Log[y], $MachinePrecision] + (-y)), $MachinePrecision] - z), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

2. Applied egg-rr 32.9%

   \[\leadsto \color{blue}{{\left(\sqrt{\mathsf{fma}\left(x, \log y, -\left(y + z\right)\right)}\right)}^{2}} \]
   Proof

   [Start] 99.9	\[ \left(x \cdot \log y - z\right) - y \]
   add-sqr-sqrt [=>] 32.9	\[ \color{blue}{\sqrt{\left(x \cdot \log y - z\right) - y} \cdot \sqrt{\left(x \cdot \log y - z\right) - y}} \]
   pow2 [=>] 32.9	\[ \color{blue}{{\left(\sqrt{\left(x \cdot \log y - z\right) - y}\right)}^{2}} \]
   associate--l- [=>] 32.9	\[ {\left(\sqrt{\color{blue}{x \cdot \log y - \left(z + y\right)}}\right)}^{2} \]
   fma-neg [=>] 32.9	\[ {\left(\sqrt{\color{blue}{\mathsf{fma}\left(x, \log y, -\left(z + y\right)\right)}}\right)}^{2} \]
   +-commutative [=>] 32.9	\[ {\left(\sqrt{\mathsf{fma}\left(x, \log y, -\color{blue}{\left(y + z\right)}\right)}\right)}^{2} \]

3. Applied egg-rr 99.9%

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

   [Start] 32.9	\[ {\left(\sqrt{\mathsf{fma}\left(x, \log y, -\left(y + z\right)\right)}\right)}^{2} \]
   unpow2 [=>] 32.9	\[ \color{blue}{\sqrt{\mathsf{fma}\left(x, \log y, -\left(y + z\right)\right)} \cdot \sqrt{\mathsf{fma}\left(x, \log y, -\left(y + z\right)\right)}} \]
   add-sqr-sqrt [<=] 99.9	\[ \color{blue}{\mathsf{fma}\left(x, \log y, -\left(y + z\right)\right)} \]
   fma-udef [=>] 99.9	\[ \color{blue}{x \cdot \log y + \left(-\left(y + z\right)\right)} \]
   distribute-neg-in [=>] 99.9	\[ x \cdot \log y + \color{blue}{\left(\left(-y\right) + \left(-z\right)\right)} \]
   associate-+r+ [=>] 99.9	\[ \color{blue}{\left(x \cdot \log y + \left(-y\right)\right) + \left(-z\right)} \]
   fma-def [=>] 99.9	\[ \color{blue}{\mathsf{fma}\left(x, \log y, -y\right)} + \left(-z\right) \]

4. Final simplification 99.9%

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

Alternatives

Alternative 1
Accuracy	99.9%
Cost	13184

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

Alternative 2
Accuracy	85.0%
Cost	7249

\[\begin{array}{l} t_0 := x \cdot \log y\\ \mathbf{if}\;y \leq 2.75 \cdot 10^{+23}:\\ \;\;\;\;t_0 - z\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{+146} \lor \neg \left(y \leq 4.4 \cdot 10^{+173}\right) \land y \leq 1.15 \cdot 10^{+214}:\\ \;\;\;\;t_0 - y\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) - z\\ \end{array} \]

Alternative 3
Accuracy	84.7%
Cost	6985

\[\begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{-23} \lor \neg \left(x \leq 2.3 \cdot 10^{+29}\right):\\ \;\;\;\;x \cdot \log y - y\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) - z\\ \end{array} \]

Alternative 4
Accuracy	78.4%
Cost	6857

\[\begin{array}{l} \mathbf{if}\;x \leq -3.7 \cdot 10^{+53} \lor \neg \left(x \leq 8.5 \cdot 10^{+133}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) - z\\ \end{array} \]

Alternative 5
Accuracy	99.9%
Cost	6848

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

Alternative 6
Accuracy	51.3%
Cost	656

\[\begin{array}{l} \mathbf{if}\;z \leq -2.85 \cdot 10^{+112}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 8.8 \cdot 10^{-7}:\\ \;\;\;\;-y\\ \mathbf{elif}\;z \leq 8.2 \cdot 10^{+50}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 1.08 \cdot 10^{+95}:\\ \;\;\;\;-y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 7
Accuracy	66.4%
Cost	256

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

Alternative 8
Accuracy	34.1%
Cost	128

\[-y \]

Alternative 9
Accuracy	2.3%
Cost	64

\[y \]

Alternative 10
Accuracy	2.3%
Cost	64

\[z \]

Reproduce

herbie shell --seed 2023151 
(FPCore (x y z)
  :name "Statistics.Distribution.Poisson:$clogProbability from math-functions-0.1.5.2"
  :precision binary64
  (- (- (* x (log y)) z) y))