Average Error: 0.1 → 0.1
Time: 3.1s
Precision: binary64
\[\left(x \cdot y + z\right) \cdot y + t \]
↓
\[\mathsf{fma}\left(y, z, t + y \cdot \left(y \cdot x\right)\right) \]
\left(x \cdot y + z\right) \cdot y + t
↓
\mathsf{fma}\left(y, z, t + y \cdot \left(y \cdot x\right)\right)

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

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. Taylor expanded in y around 0 4.4

    \[\leadsto \color{blue}{y \cdot z + \left(t + {y}^{2} \cdot x\right)} \]

  4. Applied unpow2_binary644.4

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

  5. Applied associate-*l*_binary640.1

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

  6. Applied fma-def_binary640.1

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

  7. Final simplification0.1

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

Reproduce

herbie shell --seed 2022068 
(FPCore (x y z t)
  :name "Language.Haskell.HsColour.ColourHighlight:unbase from hscolour-1.23"
  :precision binary64
  (+ (* (+ (* x y) z) y) t))