Examples.Basics.BasicTests:f3 from sbv-4.4

Average Error: 0.0 → 0.0

Time: 3.3s

Precision: binary64

Cost: 7104

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Julia

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

↓

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

Wolfram

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

↓

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

Target

Original	0.0
Target	0.0
Herbie	0.0

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

Derivation

1. Initial program 0.0

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

2. Taylor expanded in x around 0 0.0

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

3. Simplified 0.0

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

   [Start] 0.0	\[ 2 \cdot \left(y \cdot x\right) + \left({y}^{2} + {x}^{2}\right) \]
   associate-+r+ [=>] 0.0	\[ \color{blue}{\left(2 \cdot \left(y \cdot x\right) + {y}^{2}\right) + {x}^{2}} \]
   unpow2 [=>] 0.0	\[ \left(2 \cdot \left(y \cdot x\right) + \color{blue}{y \cdot y}\right) + {x}^{2} \]
   fma-def [=>] 0.0	\[ \color{blue}{\mathsf{fma}\left(2, y \cdot x, y \cdot y\right)} + {x}^{2} \]
   unpow2 [=>] 0.0	\[ \mathsf{fma}\left(2, y \cdot x, y \cdot y\right) + \color{blue}{x \cdot x} \]

4. Final simplification 0.0

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

Alternatives

Alternative 1
Error	0.0
Cost	704

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

Alternative 2
Error	0.0
Cost	448

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

Alternative 3
Error	20.5
Cost	324

\[\begin{array}{l} \mathbf{if}\;y \leq 3.3 \cdot 10^{-110}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]

Alternative 4
Error	27.9
Cost	192

\[x \cdot x \]

Reproduce

herbie shell --seed 2023018 
(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)))