Average Error: 0.0 → 0.0
Time: 7.0s
Precision: binary64
\[\left(x \cdot x + y\right) + y \]
↓
\[\mathsf{fma}\left(2, y, x \cdot x\right) \]
(FPCore (x y) :precision binary64 (+ (+ (* x x) y) y))
↓
(FPCore (x y) :precision binary64 (fma 2.0 y (* x x)))
double code(double x, double y) {
	return ((x * x) + y) + y;
}
↓
double code(double x, double y) {
	return fma(2.0, y, (x * x));
}

function code(x, y)
	return Float64(Float64(Float64(x * x) + y) + y)
end
↓
function code(x, y)
	return fma(2.0, y, Float64(x * x))
end

code[x_, y_] := N[(N[(N[(x * x), $MachinePrecision] + y), $MachinePrecision] + y), $MachinePrecision]
↓
code[x_, y_] := N[(2.0 * y + N[(x * x), $MachinePrecision]), $MachinePrecision]

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

Error

Bits error versus x

Bits error versus y

Target

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

Derivation

  1. Initial program 0.0

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

  2. Simplified0.0

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

  3. Taylor expanded in x around 0 0.0

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

  4. Simplified0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2022152 
(FPCore (x y)
  :name "Data.Random.Distribution.Normal:normalTail from random-fu-0.2.6.2"
  :precision binary64

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

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