?

Average Error: 0.02% → 0.01%
Time: 3.7s
Precision: binary64
Cost: 6976

?

\[\left(x \cdot 2 + x \cdot x\right) + y \cdot y \]
↓
\[\mathsf{fma}\left(x, x, x \cdot 2\right) + y \cdot y \]
(FPCore (x y) :precision binary64 (+ (+ (* x 2.0) (* x x)) (* y y)))
↓
(FPCore (x y) :precision binary64 (+ (fma x 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, (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 Float64(fma(x, 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[(N[(x * x + N[(x * 2.0), $MachinePrecision]), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]

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

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. Applied egg-rr0.01

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

  3. Final simplification0.01

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

Alternatives

Alternative 1
Error34.96%
Cost984
\[\begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq -1.25 \cdot 10^{-86}:\\ \;\;\;\;x + x\\ \mathbf{elif}\;x \leq -1.6 \cdot 10^{-203}:\\ \;\;\;\;y \cdot y\\ \mathbf{elif}\;x \leq -4.8 \cdot 10^{-227}:\\ \;\;\;\;x + x\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-137}:\\ \;\;\;\;y \cdot y\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;x + x\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]
Alternative 2
Error6.75%
Cost713
\[\begin{array}{l} \mathbf{if}\;x \leq -0.0014 \lor \neg \left(x \leq 20500000000\right):\\ \;\;\;\;x \cdot \left(x + 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot y + x \cdot 2\\ \end{array} \]
Alternative 3
Error1.77%
Cost713
\[\begin{array}{l} \mathbf{if}\;x \leq -2 \lor \neg \left(x \leq 0.00024\right):\\ \;\;\;\;y \cdot y + x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y + x \cdot 2\\ \end{array} \]
Alternative 4
Error15.15%
Cost584
\[\begin{array}{l} \mathbf{if}\;y \leq -1.15 \cdot 10^{-46}:\\ \;\;\;\;y \cdot y\\ \mathbf{elif}\;y \leq 8 \cdot 10^{-36}:\\ \;\;\;\;x \cdot \left(x + 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]
Alternative 5
Error0.02%
Cost576
\[y \cdot y + x \cdot \left(x + 2\right) \]
Alternative 6
Error38.44%
Cost456
\[\begin{array}{l} \mathbf{if}\;x \leq -400:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq 135000000000:\\ \;\;\;\;y \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]
Alternative 7
Error68.91%
Cost192
\[x \cdot x \]

Error

Reproduce?

herbie shell --seed 2023090 
(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)))