Statistics.Sample:robustSumVarWeighted from math-functions-0.1.5.2

Average Error: 0.1 → 0.1

Time: 6.7s

Precision: binary64

Cost: 6720

Math

\[x + \left(y \cdot z\right) \cdot z \]

↓

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

FPCore

(FPCore (x y z) :precision binary64 (+ x (* (* y z) z)))

↓

(FPCore (x y z) :precision binary64 (fma (* y z) z x))

C

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

↓

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

Julia

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

↓

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

Wolfram

code[x_, y_, z_] := N[(x + N[(N[(y * z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_] := N[(N[(y * z), $MachinePrecision] * z + x), $MachinePrecision]

Derivation

1. Initial program 0.1

   \[x + \left(y \cdot z\right) \cdot z \]

2. Simplified 6.1

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

   [Start] 0.1	\[ x + \left(y \cdot z\right) \cdot z \]
   associate-*l* [=>] 6.1	\[ x + \color{blue}{y \cdot \left(z \cdot z\right)} \]

3. Applied egg-rr 0.1

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

4. Final simplification 0.1

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

Alternatives

Alternative 1
Error	19.5
Cost	850

\[\begin{array}{l} \mathbf{if}\;z \leq -2.4 \cdot 10^{+63} \lor \neg \left(z \leq -420000000000 \lor \neg \left(z \leq -8.8 \cdot 10^{-38}\right) \land z \leq 2.65 \cdot 10^{+73}\right):\\ \;\;\;\;y \cdot \left(z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 2
Error	15.2
Cost	850

\[\begin{array}{l} \mathbf{if}\;z \leq -7.2 \cdot 10^{+62} \lor \neg \left(z \leq -1300000000000 \lor \neg \left(z \leq -8.8 \cdot 10^{-38}\right) \land z \leq 1.9 \cdot 10^{+74}\right):\\ \;\;\;\;z \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3
Error	1.8
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -1.35 \cdot 10^{+154} \lor \neg \left(z \leq 1.35 \cdot 10^{+154}\right):\\ \;\;\;\;z \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(z \cdot z\right)\\ \end{array} \]

Alternative 4
Error	0.1
Cost	448

\[x + z \cdot \left(y \cdot z\right) \]

Alternative 5
Error	21.0
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023055 
(FPCore (x y z)
  :name "Statistics.Sample:robustSumVarWeighted from math-functions-0.1.5.2"
  :precision binary64
  (+ x (* (* y z) z)))