Average Error: 0.0 → 0.0
Time: 6.6s
Precision: binary64
Cost: 6848
\[\left(x \cdot 2 + x \cdot x\right) + y \cdot y \]
↓
\[\mathsf{fma}\left(y, y, x \cdot \left(2 + x\right)\right) \]
(FPCore (x y) :precision binary64 (+ (+ (* x 2.0) (* x x)) (* y y)))
↓
(FPCore (x y) :precision binary64 (fma y y (* x (+ 2.0 x))))
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 * (2.0 + x)));
}

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(2.0 + x)))
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[(2.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

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

Error

Target

Original0.0
Target0.0
Herbie0.0
\[y \cdot y + \left(2 \cdot x + x \cdot x\right) \]

Derivation

  1. Initial program 0.0

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

  2. Simplified0.0

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

Alternatives

Alternative 1
Error11.9
Cost1100
\[\begin{array}{l} t_0 := \left(2 + x\right) \cdot x\\ \mathbf{if}\;y \cdot y \leq 5.1 \cdot 10^{+14}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \cdot y \leq 4.4 \cdot 10^{+79}:\\ \;\;\;\;y \cdot y\\ \mathbf{elif}\;y \cdot y \leq 4.4 \cdot 10^{+97}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]
Alternative 2
Error4.9
Cost712
\[\begin{array}{l} t_0 := \left(2 + x\right) \cdot x\\ \mathbf{if}\;x \leq -1.9 \cdot 10^{-6}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{+36}:\\ \;\;\;\;2 \cdot x + y \cdot y\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 3
Error0.0
Cost576
\[\left(2 + x\right) \cdot x + y \cdot y \]
Alternative 4
Error21.5
Cost456
\[\begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{-61}:\\ \;\;\;\;y \cdot y\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{-109}:\\ \;\;\;\;2 \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]
Alternative 5
Error42.0
Cost192
\[2 \cdot x \]

Error

Reproduce

herbie shell --seed 2023010 
(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)))