Average Error: 0.1 → 0.1
Time: 4.6s
Precision: 64
\[\left(x \cdot y + z\right) \cdot y + t\]
↓
\[\mathsf{fma}\left(\mathsf{fma}\left(x, y, z\right), y, t\right)\]
\left(x \cdot y + z\right) \cdot y + t
↓
\mathsf{fma}\left(\mathsf{fma}\left(x, y, z\right), y, t\right)
double f(double x, double y, double z, double t) {
        double r145209 = x;
        double r145210 = y;
        double r145211 = r145209 * r145210;
        double r145212 = z;
        double r145213 = r145211 + r145212;
        double r145214 = r145213 * r145210;
        double r145215 = t;
        double r145216 = r145214 + r145215;
        return r145216;
}
↓
double f(double x, double y, double z, double t) {
        double r145217 = x;
        double r145218 = y;
        double r145219 = z;
        double r145220 = fma(r145217, r145218, r145219);
        double r145221 = t;
        double r145222 = fma(r145220, r145218, r145221);
        return r145222;
}

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. Final simplification0.1

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

Reproduce

herbie shell --seed 2020021 +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))