?

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

↓

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

(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)

Original	0.0
Target	0.0
Herbie	0.0

Derivation?

Initial program 0.0
\[\left(x \cdot 2 + x \cdot x\right) + y \cdot y \]
Simplified0.0
\[\leadsto \color{blue}{x \cdot \left(2 + x\right) + y \cdot y} \]
Proof
[Start]0.0
\[ \left(x \cdot 2 + x \cdot x\right) + y \cdot y \]
distribute-lft-out [=>]0.0
\[ \color{blue}{x \cdot \left(2 + x\right)} + y \cdot y \]
Applied egg-rr0.0
\[\leadsto \color{blue}{\mathsf{fma}\left(y, y, x \cdot \left(x + 2\right)\right)} \]
Final simplification0.0
\[\leadsto \mathsf{fma}\left(y, y, x \cdot \left(x + 2\right)\right) \]

[Start]0.0	\[ \left(x \cdot 2 + x \cdot x\right) + y \cdot y \]
distribute-lft-out [=>]0.0	\[ \color{blue}{x \cdot \left(2 + x\right)} + y \cdot y \]

Alternatives

Alternative 1
Error	1.7
Cost	844

\[\begin{array}{l} t_0 := y \cdot y + x \cdot x\\ \mathbf{if}\;x \leq -480000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{-33}:\\ \;\;\;\;y \cdot y + x \cdot 2\\ \mathbf{elif}\;x \leq 70000000:\\ \;\;\;\;x \cdot x + x \cdot 2\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	21.9
Cost	720

\[\begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq -1.5 \cdot 10^{-143}:\\ \;\;\;\;x + x\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-99}:\\ \;\;\;\;y \cdot y\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;x + x\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 3
Error	5.0
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{+25}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{-33}:\\ \;\;\;\;y \cdot y + x \cdot 2\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x + 2\right)\\ \end{array} \]

Alternative 4
Error	9.9
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{-32}:\\ \;\;\;\;y \cdot y\\ \mathbf{elif}\;y \leq 6.8 \cdot 10^{-66}:\\ \;\;\;\;x \cdot \left(x + 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]

Alternative 5
Error	0.0
Cost	576

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

Alternative 6
Error	25.8
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -8.2 \cdot 10^{+23}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{-33}:\\ \;\;\;\;y \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 7
Error	45.0
Cost	192

\[x \cdot x \]

Reproduce?

herbie shell --seed 2023054 
(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)))

?

?

Error?

Target

Derivation?

Alternatives

Error

Reproduce?