Examples.Basics.BasicTests:f3 from sbv-4.4

Average Accuracy: 100.0% → 100.0%

Time: 4.7s

Precision: binary64

Cost: 13248

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

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Julia

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

↓

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

Wolfram

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

↓

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

Target

Original	100.0%
Target	100.0%
Herbie	100.0%

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

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in x around 0 100.0%

   \[\leadsto \color{blue}{2 \cdot \left(y \cdot x\right) + \left({y}^{2} + {x}^{2}\right)} \]

3. Simplified 100.0%

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

   [Start] 100.0	\[ 2 \cdot \left(y \cdot x\right) + \left({y}^{2} + {x}^{2}\right) \]
   associate-+r+ [=>] 100.0	\[ \color{blue}{\left(2 \cdot \left(y \cdot x\right) + {y}^{2}\right) + {x}^{2}} \]
   +-commutative [=>] 100.0	\[ \color{blue}{\left({y}^{2} + 2 \cdot \left(y \cdot x\right)\right)} + {x}^{2} \]
   associate-+l+ [=>] 100.0	\[ \color{blue}{{y}^{2} + \left(2 \cdot \left(y \cdot x\right) + {x}^{2}\right)} \]
   unpow2 [=>] 100.0	\[ \color{blue}{y \cdot y} + \left(2 \cdot \left(y \cdot x\right) + {x}^{2}\right) \]
   fma-def [=>] 100.0	\[ \color{blue}{\mathsf{fma}\left(y, y, 2 \cdot \left(y \cdot x\right) + {x}^{2}\right)} \]
   associate-*r* [=>] 100.0	\[ \mathsf{fma}\left(y, y, \color{blue}{\left(2 \cdot y\right) \cdot x} + {x}^{2}\right) \]
   unpow2 [=>] 100.0	\[ \mathsf{fma}\left(y, y, \left(2 \cdot y\right) \cdot x + \color{blue}{x \cdot x}\right) \]
   distribute-rgt-out [=>] 100.0	\[ \mathsf{fma}\left(y, y, \color{blue}{x \cdot \left(2 \cdot y + x\right)}\right) \]
   *-commutative [=>] 100.0	\[ \mathsf{fma}\left(y, y, x \cdot \left(\color{blue}{y \cdot 2} + x\right)\right) \]
   fma-def [=>] 100.0	\[ \mathsf{fma}\left(y, y, x \cdot \color{blue}{\mathsf{fma}\left(y, 2, x\right)}\right) \]

4. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	100.0%
Cost	704

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

Alternative 2
Accuracy	100.0%
Cost	704

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

Alternative 3
Accuracy	87.0%
Cost	580

\[\begin{array}{l} \mathbf{if}\;x \leq -6.6 \cdot 10^{-89}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y + x \cdot 2\right)\\ \end{array} \]

Alternative 4
Accuracy	100.0%
Cost	448

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

Alternative 5
Accuracy	86.8%
Cost	324

\[\begin{array}{l} \mathbf{if}\;x \leq -1.15 \cdot 10^{-89}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]

Alternative 6
Accuracy	56.2%
Cost	192

\[x \cdot x \]

Reproduce

herbie shell --seed 2023147 
(FPCore (x y)
  :name "Examples.Basics.BasicTests:f3 from sbv-4.4"
  :precision binary64

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

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