Average Error: 0.1 → 0.0
Time: 4.8s
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 r111452 = x;
        double r111453 = y;
        double r111454 = r111452 + r111453;
        double r111455 = r111454 + r111453;
        double r111456 = r111455 + r111452;
        double r111457 = z;
        double r111458 = r111456 + r111457;
        double r111459 = r111458 + r111452;
        return r111459;
}
↓
double f(double x, double y, double z) {
        double r111460 = 3.0;
        double r111461 = x;
        double r111462 = y;
        double r111463 = 2.0;
        double r111464 = z;
        double r111465 = fma(r111462, r111463, r111464);
        double r111466 = fma(r111460, r111461, r111465);
        return r111466;
}

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 2019322 +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))