Average Error: 0.1 → 0.1
Time: 1.7s
Precision: binary64
\[\left(\left(x \cdot x + y \cdot y\right) + y \cdot y\right) + y \cdot y \]
↓
\[\mathsf{fma}\left(x, x, y \cdot \left(3 \cdot y\right)\right) \]
(FPCore (x y) :precision binary64 (+ (+ (+ (* x x) (* y y)) (* y y)) (* y y)))
↓
(FPCore (x y) :precision binary64 (fma x x (* y (* 3.0 y))))
double code(double x, double y) {
	return (((x * x) + (y * y)) + (y * y)) + (y * y);
}
↓
double code(double x, double y) {
	return fma(x, x, (y * (3.0 * 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(x, x, Float64(y * Float64(3.0 * 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[(x * x + N[(y * N[(3.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\left(\left(x \cdot x + y \cdot y\right) + y \cdot y\right) + y \cdot y
↓
\mathsf{fma}\left(x, x, y \cdot \left(3 \cdot y\right)\right)

Error

Bits error versus x

Bits error versus y

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

    \[\leadsto \color{blue}{\mathsf{fma}\left(y, y \cdot 3, x \cdot x\right)} \]

  3. Taylor expanded in y around 0 0.1

    \[\leadsto \color{blue}{3 \cdot {y}^{2} + {x}^{2}} \]

  4. Simplified0.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, 3 \cdot \left(y \cdot y\right)\right)} \]

  5. Applied associate-*r*_binary640.1

    \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{\left(3 \cdot y\right) \cdot y}\right) \]

  6. Final simplification0.1

    \[\leadsto \mathsf{fma}\left(x, x, y \cdot \left(3 \cdot y\right)\right) \]

Reproduce

herbie shell --seed 2022131 
(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)))