Average Error: 0.1 → 0.1
Time: 17.8s
Precision: 64
\[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5\]
↓
\[\mathsf{fma}\left(x, \mathsf{fma}\left(2, y + z, t\right), y \cdot 5\right)\]
x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5
↓
\mathsf{fma}\left(x, \mathsf{fma}\left(2, y + z, t\right), y \cdot 5\right)
double f(double x, double y, double z, double t) {
        double r106058 = x;
        double r106059 = y;
        double r106060 = z;
        double r106061 = r106059 + r106060;
        double r106062 = r106061 + r106060;
        double r106063 = r106062 + r106059;
        double r106064 = t;
        double r106065 = r106063 + r106064;
        double r106066 = r106058 * r106065;
        double r106067 = 5.0;
        double r106068 = r106059 * r106067;
        double r106069 = r106066 + r106068;
        return r106069;
}
↓
double f(double x, double y, double z, double t) {
        double r106070 = x;
        double r106071 = 2.0;
        double r106072 = y;
        double r106073 = z;
        double r106074 = r106072 + r106073;
        double r106075 = t;
        double r106076 = fma(r106071, r106074, r106075);
        double r106077 = 5.0;
        double r106078 = r106072 * r106077;
        double r106079 = fma(r106070, r106076, r106078);
        return r106079;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Derivation

  1. Initial program 0.1

    \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5\]

  2. Simplified0.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(2, y + z, t\right), y \cdot 5\right)}\]

  3. Final simplification0.1

    \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(2, y + z, t\right), y \cdot 5\right)\]

Reproduce

herbie shell --seed 2019322 +o rules:numerics
(FPCore (x y z t)
  :name "Graphics.Rendering.Plot.Render.Plot.Legend:renderLegendOutside from plot-0.2.3.4, B"
  :precision binary64
  (+ (* x (+ (+ (+ (+ y z) z) y) t)) (* y 5)))