?

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

↓

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

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

↓

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

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

↓

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

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

↓

function code(x, y)
	return fma(x, Float64(x + 2.0), Float64(y * y))
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[(x * N[(x + 2.0), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]

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

↓

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

Original	0.02%
Target	0.02%
Herbie	0.02%

Derivation?

Initial program 0.02
\[\left(x \cdot 2 + x \cdot x\right) + y \cdot y \]

Simplified0.02

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

Proof

[Start]0.02	\[ \left(x \cdot 2 + x \cdot x\right) + y \cdot y \]
distribute-lft-out [=>]0.02	\[ \color{blue}{x \cdot \left(2 + x\right)} + y \cdot y \]
fma-def [=>]0.02	\[ \color{blue}{\mathsf{fma}\left(x, 2 + x, y \cdot y\right)} \]
+-commutative [=>]0.02	\[ \mathsf{fma}\left(x, \color{blue}{x + 2}, y \cdot y\right) \]

Final simplification0.02
\[\leadsto \mathsf{fma}\left(x, x + 2, y \cdot y\right) \]

Alternatives

Alternative 1
Error	35.29%
Cost	1248

\[\begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{-11}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq -5.7 \cdot 10^{-86}:\\ \;\;\;\;x + x\\ \mathbf{elif}\;x \leq -9.6 \cdot 10^{-144}:\\ \;\;\;\;y \cdot y\\ \mathbf{elif}\;x \leq -3.6 \cdot 10^{-187}:\\ \;\;\;\;x + x\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-145}:\\ \;\;\;\;y \cdot y\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{-107}:\\ \;\;\;\;x + x\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{-49}:\\ \;\;\;\;y \cdot y\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;x + x\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 2
Error	5.89%
Cost	713

\[\begin{array}{l} \mathbf{if}\;x \leq -140 \lor \neg \left(x \leq 9.6 \cdot 10^{-8}\right):\\ \;\;\;\;x \cdot \left(x + 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot y + x \cdot 2\\ \end{array} \]

Alternative 3
Error	1.84%
Cost	713

\[\begin{array}{l} \mathbf{if}\;x \leq -140 \lor \neg \left(x \leq 2\right):\\ \;\;\;\;y \cdot y + x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y + x \cdot 2\\ \end{array} \]

Alternative 4
Error	15.26%
Cost	580

\[\begin{array}{l} \mathbf{if}\;y \cdot y \leq 10^{-95}:\\ \;\;\;\;x \cdot \left(x + 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]

Alternative 5
Error	0.02%
Cost	576

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

Alternative 6
Error	37.56%
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -86000000:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{-7}:\\ \;\;\;\;y \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 7
Error	69.24%
Cost	192

\[x \cdot x \]

Reproduce?

herbie shell --seed 2023101 
(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?