Average Error: 0.0 → 0
Time: 1.1s
Precision: binary64
\[\left(x \cdot x + y\right) + y \]
↓
\[\mathsf{fma}\left(x, x, 2 \cdot y\right) \]
(FPCore (x y) :precision binary64 (+ (+ (* x x) y) y))
↓
(FPCore (x y) :precision binary64 (fma x x (* 2.0 y)))
double code(double x, double y) {
	return ((x * x) + y) + y;
}
↓
double code(double x, double y) {
	return fma(x, x, (2.0 * y));
}

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

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

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

Error

Bits error versus x

Bits error versus y

Target

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

Derivation

  1. Initial program 0.0

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

  2. Simplified0.0

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

  3. Taylor expanded in y around 0 0.0

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

  4. Simplified0

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

  5. Final simplification0

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

Reproduce

herbie shell --seed 2022150 
(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))