Average Error: 0.1 → 0.1
Time: 3.0s
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 r120208 = x;
        double r120209 = y;
        double r120210 = r120208 * r120209;
        double r120211 = z;
        double r120212 = r120210 + r120211;
        double r120213 = r120212 * r120209;
        double r120214 = t;
        double r120215 = r120213 + r120214;
        return r120215;
}
↓
double f(double x, double y, double z, double t) {
        double r120216 = x;
        double r120217 = y;
        double r120218 = z;
        double r120219 = fma(r120216, r120217, r120218);
        double r120220 = t;
        double r120221 = fma(r120219, r120217, r120220);
        return r120221;
}

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 2020047 +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))