Average Error: 0.1 → 0.1
Time: 3.4s
Precision: 64
\[\left(x \cdot y + z\right) \cdot y + t\]
↓
\[\mathsf{fma}\left(x, y, z\right) \cdot y + t\]
\left(x \cdot y + z\right) \cdot y + t
↓
\mathsf{fma}\left(x, y, z\right) \cdot y + t
double f(double x, double y, double z, double t) {
        double r139076 = x;
        double r139077 = y;
        double r139078 = r139076 * r139077;
        double r139079 = z;
        double r139080 = r139078 + r139079;
        double r139081 = r139080 * r139077;
        double r139082 = t;
        double r139083 = r139081 + r139082;
        return r139083;
}
↓
double f(double x, double y, double z, double t) {
        double r139084 = x;
        double r139085 = y;
        double r139086 = z;
        double r139087 = fma(r139084, r139085, r139086);
        double r139088 = r139087 * r139085;
        double r139089 = t;
        double r139090 = r139088 + r139089;
        return r139090;
}

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

    \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x, y, z\right), y, t\right)}\]
  3. Using strategy rm

  4. Applied fma-udef0.1

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

  5. Final simplification0.1

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

Reproduce

herbie shell --seed 2020033 +o rules:numerics
(FPCore (x y z t)
  :name "Language.Haskell.HsColour.ColourHighlight:unbase from hscolour-1.23"
  :precision binary64
  (+ (* (+ (* x y) z) y) t))