Examples.Basics.ProofTests:f4 from sbv-4.4

Average Error: 0.03% → 0.01%

Time: 3.8s

Precision: binary64

Cost: 6976

Math

\[\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y \]

↓

\[\mathsf{fma}\left(y, y, x \cdot \left(x + y \cdot 2\right)\right) \]

FPCore

(FPCore (x y) :precision binary64 (+ (+ (* x x) (* (* x 2.0) y)) (* y y)))

↓

(FPCore (x y) :precision binary64 (fma y y (* x (+ x (* y 2.0)))))

C

double code(double x, double y) {
	return ((x * x) + ((x * 2.0) * y)) + (y * y);
}

↓

double code(double x, double y) {
	return fma(y, y, (x * (x + (y * 2.0))));
}

Julia

function code(x, y)
	return Float64(Float64(Float64(x * x) + Float64(Float64(x * 2.0) * y)) + Float64(y * y))
end

↓

function code(x, y)
	return fma(y, y, Float64(x * Float64(x + Float64(y * 2.0))))
end

Wolfram

code[x_, y_] := N[(N[(N[(x * x), $MachinePrecision] + N[(N[(x * 2.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_] := N[(y * y + N[(x * N[(x + N[(y * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	0.03%
Target	0.02%
Herbie	0.01%

\[x \cdot x + \left(y \cdot y + \left(x \cdot y\right) \cdot 2\right) \]

Derivation

1. Initial program 0.03

   \[\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y \]

2. Simplified 0.01

   \[\leadsto \color{blue}{\mathsf{fma}\left(y, y, x \cdot \left(2 \cdot y + x\right)\right)} \]
   Proof

   [Start] 0.03	\[ \left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y \]
   +-commutative [=>] 0.03	\[ \color{blue}{y \cdot y + \left(x \cdot x + \left(x \cdot 2\right) \cdot y\right)} \]
   fma-def [=>] 0.02	\[ \color{blue}{\mathsf{fma}\left(y, y, x \cdot x + \left(x \cdot 2\right) \cdot y\right)} \]
   +-commutative [=>] 0.02	\[ \mathsf{fma}\left(y, y, \color{blue}{\left(x \cdot 2\right) \cdot y + x \cdot x}\right) \]
   associate-*l* [=>] 0.02	\[ \mathsf{fma}\left(y, y, \color{blue}{x \cdot \left(2 \cdot y\right)} + x \cdot x\right) \]
   distribute-lft-out [=>] 0.01	\[ \mathsf{fma}\left(y, y, \color{blue}{x \cdot \left(2 \cdot y + x\right)}\right) \]

3. Final simplification 0.01

   \[\leadsto \mathsf{fma}\left(y, y, x \cdot \left(x + y \cdot 2\right)\right) \]

Alternatives

Alternative 1
Error	32.75%
Cost	845

\[\begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-91} \lor \neg \left(x \leq -3.7 \cdot 10^{-103}\right) \land x \leq -5.5 \cdot 10^{-136}:\\ \;\;\;\;x \cdot \left(x + y \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]

Alternative 2
Error	32.46%
Cost	844

\[\begin{array}{l} t_0 := x \cdot \left(x + y \cdot 2\right)\\ \mathbf{if}\;x \leq -6.5 \cdot 10^{-93}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -2.5 \cdot 10^{-103}:\\ \;\;\;\;y \cdot y\\ \mathbf{elif}\;x \leq -1.2 \cdot 10^{-134}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y + x \cdot 2\right)\\ \end{array} \]

Alternative 3
Error	32.46%
Cost	844

\[\begin{array}{l} \mathbf{if}\;x \leq -1.85 \cdot 10^{-91}:\\ \;\;\;\;x \cdot x + x \cdot \left(y \cdot 2\right)\\ \mathbf{elif}\;x \leq -5.6 \cdot 10^{-101}:\\ \;\;\;\;y \cdot y\\ \mathbf{elif}\;x \leq -1.22 \cdot 10^{-133}:\\ \;\;\;\;x \cdot \left(x + y \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y + x \cdot 2\right)\\ \end{array} \]

Alternative 4
Error	0.03%
Cost	832

\[\left(x \cdot x + y \cdot \left(x \cdot 2\right)\right) + y \cdot y \]

Alternative 5
Error	0.03%
Cost	704

\[y \cdot y + x \cdot \left(x + y \cdot 2\right) \]

Alternative 6
Error	32.83%
Cost	589

\[\begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-92} \lor \neg \left(x \leq -2.5 \cdot 10^{-101}\right) \land x \leq -1.22 \cdot 10^{-133}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]

Alternative 7
Error	43.6%
Cost	192

\[x \cdot x \]

Reproduce

herbie shell --seed 2023121 
(FPCore (x y)
  :name "Examples.Basics.ProofTests:f4 from sbv-4.4"
  :precision binary64

  :herbie-target
  (+ (* x x) (+ (* y y) (* (* x y) 2.0)))

  (+ (+ (* x x) (* (* x 2.0) y)) (* y y)))