Examples.Basics.ProofTests:f4 from sbv-4.4

Average Accuracy: 100.0% → 100.0%

Time: 3.8s

Precision: binary64

Cost: 6976

\[ \begin{array}{c}[x, y] = \mathsf{sort}([x, y])\\ \end{array} \]

Math

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

↓

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

FPCore

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

↓

(FPCore (x y) :precision binary64 (fma x x (* y (+ y (* x 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(x, x, (y * (y + (x * 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(x, x, Float64(y * Float64(y + Float64(x * 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[(x * x + N[(y * N[(y + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	100.0%
Target	100.0%
Herbie	100.0%

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

Derivation

1. Initial program 100.0%

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

2. Simplified 100.0%

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

   [Start] 100.0	\[ \left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y \]
   associate-+l+ [=>] 100.0	\[ \color{blue}{x \cdot x + \left(\left(x \cdot 2\right) \cdot y + y \cdot y\right)} \]
   fma-def [=>] 100.0	\[ \color{blue}{\mathsf{fma}\left(x, x, \left(x \cdot 2\right) \cdot y + y \cdot y\right)} \]
   distribute-rgt-out [=>] 100.0	\[ \mathsf{fma}\left(x, x, \color{blue}{y \cdot \left(x \cdot 2 + y\right)}\right) \]

3. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	87.7%
Cost	844

\[\begin{array}{l} \mathbf{if}\;y \leq 3.6 \cdot 10^{-137}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{-119}:\\ \;\;\;\;y \cdot y\\ \mathbf{elif}\;y \leq 5.4 \cdot 10^{-105}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y + \left(x + x\right)\right)\\ \end{array} \]

Alternative 2
Accuracy	100.0%
Cost	704

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

Alternative 3
Accuracy	87.7%
Cost	589

\[\begin{array}{l} \mathbf{if}\;y \leq 2.7 \cdot 10^{-137} \lor \neg \left(y \leq 2 \cdot 10^{-117}\right) \land y \leq 1.65 \cdot 10^{-103}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]

Alternative 4
Accuracy	98.3%
Cost	448

\[y \cdot y + x \cdot x \]

Alternative 5
Accuracy	56.8%
Cost	192

\[x \cdot x \]

Reproduce

herbie shell --seed 2023151 
(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)))