Average Error: 0.1 → 0.1
Time: 5.6s
Precision: binary64
Cost: 13376
\[\left(\left(x \cdot x + y \cdot y\right) + y \cdot y\right) + y \cdot y \]
↓
\[\mathsf{fma}\left(y, y, \mathsf{fma}\left(x, x, y \cdot \left(y + y\right)\right)\right) \]
(FPCore (x y) :precision binary64 (+ (+ (+ (* x x) (* y y)) (* y y)) (* y y)))
↓
(FPCore (x y) :precision binary64 (fma y y (fma x x (* y (+ y y)))))
double code(double x, double y) {
	return (((x * x) + (y * y)) + (y * y)) + (y * y);
}
↓
double code(double x, double y) {
	return fma(y, y, fma(x, x, (y * (y + y))));
}

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

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

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

Error

Target

Original0.1
Target0.1
Herbie0.1
\[x \cdot x + y \cdot \left(y + \left(y + y\right)\right) \]

Derivation

  1. Initial program 0.1

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

  2. Simplified0.1

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

    [Start]0.1

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

    +-commutative [=>]0.1

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

    fma-def [=>]0.1

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

    associate-+l+ [=>]0.1

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

    fma-def [=>]0.1

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

    distribute-lft-out [=>]0.1

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

  3. Final simplification0.1

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

Alternatives

Alternative 1
Error12.7
Cost1101
\[\begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{-164} \lor \neg \left(x \cdot x \leq 1.95 \cdot 10^{-73}\right) \land x \cdot x \leq 4.3 \cdot 10^{-45}:\\ \;\;\;\;3 \cdot \left(y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]
Alternative 2
Error11.2
Cost713
\[\begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{-80} \lor \neg \left(x \leq 4.9 \cdot 10^{-88}\right):\\ \;\;\;\;y \cdot y + x \cdot x\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(y \cdot y\right)\\ \end{array} \]
Alternative 3
Error0.1
Cost576
\[3 \cdot \left(y \cdot y\right) + x \cdot x \]
Alternative 4
Error26.2
Cost456
\[\begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{-143}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-153}:\\ \;\;\;\;y \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]
Alternative 5
Error27.9
Cost192
\[x \cdot x \]

Error

Reproduce

herbie shell --seed 2022364 
(FPCore (x y)
  :name "Linear.Quaternion:$c/ from linear-1.19.1.3, E"
  :precision binary64

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

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