Average Error: 0.1 → 0.0
Time: 7.2s
Precision: 64
\[\left(\left(\left(\left(x + y\right) + y\right) + x\right) + z\right) + x\]
↓
\[\mathsf{fma}\left(x, 3, \mathsf{fma}\left(2, y, z\right)\right)\]
\left(\left(\left(\left(x + y\right) + y\right) + x\right) + z\right) + x
↓
\mathsf{fma}\left(x, 3, \mathsf{fma}\left(2, y, z\right)\right)
double f(double x, double y, double z) {
        double r7910538 = x;
        double r7910539 = y;
        double r7910540 = r7910538 + r7910539;
        double r7910541 = r7910540 + r7910539;
        double r7910542 = r7910541 + r7910538;
        double r7910543 = z;
        double r7910544 = r7910542 + r7910543;
        double r7910545 = r7910544 + r7910538;
        return r7910545;
}
↓
double f(double x, double y, double z) {
        double r7910546 = x;
        double r7910547 = 3.0;
        double r7910548 = 2.0;
        double r7910549 = y;
        double r7910550 = z;
        double r7910551 = fma(r7910548, r7910549, r7910550);
        double r7910552 = fma(r7910546, r7910547, r7910551);
        return r7910552;
}

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

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

  3. Taylor expanded around 0 0.1

    \[\leadsto \color{blue}{3 \cdot x + \left(z + 2 \cdot y\right)}\]

  4. Simplified0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019170 +o rules:numerics
(FPCore (x y z)
  :name "Graphics.Rendering.Plot.Render.Plot.Legend:renderLegendInside from plot-0.2.3.4"
  (+ (+ (+ (+ (+ x y) y) x) z) x))