Average Error: 0.1 → 0.0
Time: 6.4s
Precision: 64
\[\left(\left(\left(\left(x + y\right) + y\right) + x\right) + z\right) + x\]
↓
\[\mathsf{fma}\left(3, x, \mathsf{fma}\left(y, 2, z\right)\right)\]
\left(\left(\left(\left(x + y\right) + y\right) + x\right) + z\right) + x
↓
\mathsf{fma}\left(3, x, \mathsf{fma}\left(y, 2, z\right)\right)
double f(double x, double y, double z) {
        double r123197 = x;
        double r123198 = y;
        double r123199 = r123197 + r123198;
        double r123200 = r123199 + r123198;
        double r123201 = r123200 + r123197;
        double r123202 = z;
        double r123203 = r123201 + r123202;
        double r123204 = r123203 + r123197;
        return r123204;
}
↓
double f(double x, double y, double z) {
        double r123205 = 3.0;
        double r123206 = x;
        double r123207 = y;
        double r123208 = 2.0;
        double r123209 = z;
        double r123210 = fma(r123207, r123208, r123209);
        double r123211 = fma(r123205, r123206, r123210);
        return r123211;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 0.1

    \[\left(\left(\left(\left(x + y\right) + y\right) + x\right) + z\right) + x\]

  2. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(3, x, \mathsf{fma}\left(y, 2, z\right)\right)}\]

  3. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(3, x, \mathsf{fma}\left(y, 2, z\right)\right)\]

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (x y z)
  :name "Graphics.Rendering.Plot.Render.Plot.Legend:renderLegendInside from plot-0.2.3.4"
  :precision binary64
  (+ (+ (+ (+ (+ x y) y) x) z) x))