Average Error: 0.1 → 0.1
Time: 2.6s
Precision: binary64
\[\left(x \cdot y + z\right) \cdot y + t \]
↓
\[y \cdot \left(x \cdot y\right) + \mathsf{fma}\left(y, z, t\right) \]
\left(x \cdot y + z\right) \cdot y + t
↓
y \cdot \left(x \cdot y\right) + \mathsf{fma}\left(y, z, t\right)

(FPCore (x y z t) :precision binary64 (+ (* (+ (* x y) z) y) t))
↓
(FPCore (x y z t) :precision binary64 (+ (* y (* x y)) (fma y z t)))
double code(double x, double y, double z, double t) {
	return (((x * y) + z) * y) + t;
}
↓
double code(double x, double y, double z, double t) {
	return (y * (x * y)) + fma(y, z, t);
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Derivation

  1. Initial program 0.1

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

  2. Simplified0.1

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

  3. Applied fma-udef_binary640.1

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

  4. Applied fma-udef_binary640.1

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

  5. Applied distribute-rgt-in_binary640.1

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

  6. Applied associate-+l+_binary640.1

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

  7. Simplified0.1

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

  8. Final simplification0.1

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

Reproduce

herbie shell --seed 2021313 
(FPCore (x y z t)
  :name "Language.Haskell.HsColour.ColourHighlight:unbase from hscolour-1.23"
  :precision binary64
  (+ (* (+ (* x y) z) y) t))