Average Error: 0.1 → 0.0
Time: 1.2s
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 r123386 = x;
        double r123387 = y;
        double r123388 = r123386 + r123387;
        double r123389 = r123388 + r123387;
        double r123390 = r123389 + r123386;
        double r123391 = z;
        double r123392 = r123390 + r123391;
        double r123393 = r123392 + r123386;
        return r123393;
}
↓
double f(double x, double y, double z) {
        double r123394 = 3.0;
        double r123395 = x;
        double r123396 = 2.0;
        double r123397 = y;
        double r123398 = z;
        double r123399 = fma(r123396, r123397, r123398);
        double r123400 = fma(r123394, r123395, r123399);
        return r123400;
}

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 2020089 +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))