Statistics.Sample:robustSumVarWeighted from math-functions-0.1.5.2

Average Accuracy: 99.9% → 99.9%

Time: 8.3s

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 99.9%

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

2. Simplified 90.2%

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

   [Start] 99.9	\[ x + \left(y \cdot z\right) \cdot z \]
   associate-*l* [=>] 90.2	\[ x + \color{blue}{y \cdot \left(z \cdot z\right)} \]

3. Applied egg-rr 99.9%

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

   [Start] 90.2	\[ x + y \cdot \left(z \cdot z\right) \]
   +-commutative [=>] 90.2	\[ \color{blue}{y \cdot \left(z \cdot z\right) + x} \]
   associate-*r* [=>] 99.9	\[ \color{blue}{\left(y \cdot z\right) \cdot z} + x \]
   fma-def [=>] 99.9	\[ \color{blue}{\mathsf{fma}\left(y \cdot z, z, x\right)} \]

4. Final simplification 99.9%

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

Alternatives

Alternative 1
Accuracy	68.0%
Cost	850

\[\begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{+18} \lor \neg \left(z \leq -4 \cdot 10^{-39}\right) \land \left(z \leq -4.9 \cdot 10^{-106} \lor \neg \left(z \leq 1.1 \cdot 10^{-17}\right)\right):\\ \;\;\;\;y \cdot \left(z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 2
Accuracy	74.9%
Cost	848

\[\begin{array}{l} t_0 := z \cdot \left(y \cdot z\right)\\ \mathbf{if}\;z \leq -3.6 \cdot 10^{+16}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -9.6 \cdot 10^{-39}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -4.5 \cdot 10^{-106}:\\ \;\;\;\;y \cdot \left(z \cdot z\right)\\ \mathbf{elif}\;z \leq 2.25 \cdot 10^{-17}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Accuracy	96.5%
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{+147} \lor \neg \left(z \leq 3.7 \cdot 10^{+123}\right):\\ \;\;\;\;z \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(z \cdot z\right)\\ \end{array} \]

Alternative 4
Accuracy	99.9%
Cost	448

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

Alternative 5
Accuracy	66.8%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023137 
(FPCore (x y z)
  :name "Statistics.Sample:robustSumVarWeighted from math-functions-0.1.5.2"
  :precision binary64
  (+ x (* (* y z) z)))