Average Error: 0.0 → 0.0

Time: 5.2s

Precision: binary64

Cost: 7104

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

↓

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

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

↓

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

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 * x) * 2.0)) + (y * y);
}

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 Float64(fma(x, x, Float64(Float64(y * x) * 2.0)) + Float64(y * y))
end

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[(N[(x * x + N[(N[(y * x), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]

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

↓

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

Target

Original	0.0
Target	0.0
Herbie	0.0

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

Derivation

Initial program 0.0
\[\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y \]
Applied egg-rr0.0
\[\leadsto \color{blue}{\mathsf{fma}\left(x, x, \left(y \cdot x\right) \cdot 2\right)} + y \cdot y \]

Alternatives

Alternative 1
Error	0.0
Cost	704

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

Alternative 2
Error	1.2
Cost	448

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

Alternative 3
Error	27.9
Cost	192

\[y \cdot y \]

Reproduce

herbie shell --seed 2023010 
(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)))