?

Average Error: 0.02% → 0.01%
Time: 5.7s
Precision: binary64
Cost: 6848

?

\[\left(x \cdot 2 + x \cdot x\right) + y \cdot y \]
↓
\[\mathsf{fma}\left(y, y, x \cdot \left(x + 2\right)\right) \]
(FPCore (x y) :precision binary64 (+ (+ (* x 2.0) (* x x)) (* y y)))
↓
(FPCore (x y) :precision binary64 (fma y y (* x (+ x 2.0))))
double code(double x, double y) {
	return ((x * 2.0) + (x * x)) + (y * y);
}
↓
double code(double x, double y) {
	return fma(y, y, (x * (x + 2.0)));
}

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

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

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

Error?

Target

Original0.02%
Target0.02%
Herbie0.01%
\[y \cdot y + \left(2 \cdot x + x \cdot x\right) \]

Derivation?

  1. Initial program 0.02

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

  2. Simplified0.02

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

    [Start]0.02

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

    distribute-lft-out [=>]0.02

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

  3. Applied egg-rr0.01

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

  4. Final simplification0.01

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

Alternatives

Alternative 1
Error6.75%
Cost713
\[\begin{array}{l} \mathbf{if}\;x \leq -0.0001 \lor \neg \left(x \leq 3000000000\right):\\ \;\;\;\;x \cdot \left(x + 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot y + x \cdot 2\\ \end{array} \]
Alternative 2
Error1.92%
Cost713
\[\begin{array}{l} \mathbf{if}\;x \leq -2 \lor \neg \left(x \leq 2\right):\\ \;\;\;\;y \cdot y + x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y + x \cdot 2\\ \end{array} \]
Alternative 3
Error15.72%
Cost584
\[\begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{-48}:\\ \;\;\;\;y \cdot y\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{-77}:\\ \;\;\;\;x \cdot \left(x + 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]
Alternative 4
Error0.02%
Cost576
\[y \cdot y + x \cdot \left(x + 2\right) \]
Alternative 5
Error39.14%
Cost456
\[\begin{array}{l} \mathbf{if}\;x \leq -17000000000:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq 3200000000:\\ \;\;\;\;y \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]
Alternative 6
Error32.88%
Cost456
\[\begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{-97}:\\ \;\;\;\;y \cdot y\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{-81}:\\ \;\;\;\;x + x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]
Alternative 7
Error70.45%
Cost192
\[x \cdot x \]

Error

Reproduce?

herbie shell --seed 2023115 
(FPCore (x y)
  :name "Numeric.Log:$clog1p from log-domain-0.10.2.1, A"
  :precision binary64

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

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