Average Error: 0.0 → 0.0
Time: 2.0s
Precision: binary64
\[\left(x \cdot 2 + x \cdot x\right) + y \cdot y \]
↓
\[\mathsf{fma}\left(\mathsf{hypot}\left(x, y\right), \mathsf{hypot}\left(x, y\right), x + x\right) \]
(FPCore (x y) :precision binary64 (+ (+ (* x 2.0) (* x x)) (* y y)))
↓
(FPCore (x y) :precision binary64 (fma (hypot x y) (hypot x y) (+ x x)))
double code(double x, double y) {
	return ((x * 2.0) + (x * x)) + (y * y);
}
↓
double code(double x, double y) {
	return fma(hypot(x, y), hypot(x, y), (x + 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(hypot(x, y), hypot(x, y), Float64(x + 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[(N[Sqrt[x ^ 2 + y ^ 2], $MachinePrecision] * N[Sqrt[x ^ 2 + y ^ 2], $MachinePrecision] + N[(x + x), $MachinePrecision]), $MachinePrecision]

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

Error

Bits error versus x

Bits error versus y

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. Applied egg-rr0.0

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2022162 
(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)))