Average Error: 0.1 → 0.0
Time: 4.1s
Precision: 64
\[x \cdot \left(y + z\right) + z \cdot 5\]
↓
\[\mathsf{fma}\left(x, z, \mathsf{fma}\left(5, z, x \cdot y\right)\right)\]
x \cdot \left(y + z\right) + z \cdot 5
↓
\mathsf{fma}\left(x, z, \mathsf{fma}\left(5, z, x \cdot y\right)\right)
double f(double x, double y, double z) {
        double r1075999 = x;
        double r1076000 = y;
        double r1076001 = z;
        double r1076002 = r1076000 + r1076001;
        double r1076003 = r1075999 * r1076002;
        double r1076004 = 5.0;
        double r1076005 = r1076001 * r1076004;
        double r1076006 = r1076003 + r1076005;
        return r1076006;
}
↓
double f(double x, double y, double z) {
        double r1076007 = x;
        double r1076008 = z;
        double r1076009 = 5.0;
        double r1076010 = y;
        double r1076011 = r1076007 * r1076010;
        double r1076012 = fma(r1076009, r1076008, r1076011);
        double r1076013 = fma(r1076007, r1076008, r1076012);
        return r1076013;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original0.1
Target0.1
Herbie0.0
\[\left(x + 5\right) \cdot z + x \cdot y\]

Derivation

  1. Initial program 0.1

    \[x \cdot \left(y + z\right) + z \cdot 5\]

  2. Simplified0.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, y + z, z \cdot 5\right)}\]

  3. Taylor expanded around 0 0.1

    \[\leadsto \color{blue}{x \cdot z + \left(5 \cdot z + x \cdot y\right)}\]

  4. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, z, \mathsf{fma}\left(5, z, x \cdot y\right)\right)}\]

  5. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(x, z, \mathsf{fma}\left(5, z, x \cdot y\right)\right)\]

Reproduce

herbie shell --seed 2020025 +o rules:numerics
(FPCore (x y z)
  :name "Graphics.Rendering.Plot.Render.Plot.Legend:renderLegendOutside from plot-0.2.3.4, C"
  :precision binary64

  :herbie-target
  (+ (* (+ x 5) z) (* x y))

  (+ (* x (+ y z)) (* z 5)))