Average Error: 0.1 → 0.0
Time: 1.0s
Precision: 64
\[\left(\left(\left(\left(x + y\right) + y\right) + x\right) + z\right) + x\]
↓
\[\mathsf{fma}\left(3, x, \mathsf{fma}\left(2, y, z\right)\right)\]
\left(\left(\left(\left(x + y\right) + y\right) + x\right) + z\right) + x
↓
\mathsf{fma}\left(3, x, \mathsf{fma}\left(2, y, z\right)\right)
double f(double x, double y, double z) {
        double r139388 = x;
        double r139389 = y;
        double r139390 = r139388 + r139389;
        double r139391 = r139390 + r139389;
        double r139392 = r139391 + r139388;
        double r139393 = z;
        double r139394 = r139392 + r139393;
        double r139395 = r139394 + r139388;
        return r139395;
}
↓
double f(double x, double y, double z) {
        double r139396 = 3.0;
        double r139397 = x;
        double r139398 = 2.0;
        double r139399 = y;
        double r139400 = z;
        double r139401 = fma(r139398, r139399, r139400);
        double r139402 = fma(r139396, r139397, r139401);
        return r139402;
}

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

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2020001 +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))