Average Error: 0.1 → 0.0
Time: 2.6s
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 r645244 = x;
        double r645245 = y;
        double r645246 = z;
        double r645247 = r645245 + r645246;
        double r645248 = r645244 * r645247;
        double r645249 = 5.0;
        double r645250 = r645246 * r645249;
        double r645251 = r645248 + r645250;
        return r645251;
}
↓
double f(double x, double y, double z) {
        double r645252 = x;
        double r645253 = z;
        double r645254 = 5.0;
        double r645255 = y;
        double r645256 = r645252 * r645255;
        double r645257 = fma(r645254, r645253, r645256);
        double r645258 = fma(r645252, r645253, r645257);
        return r645258;
}

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. Using strategy rm

  3. Applied distribute-rgt-in0.1

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

  4. Applied associate-+l+0.1

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

  5. Simplified0.1

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

  6. Taylor expanded around 0 0.1

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

  7. Simplified0.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2020057 +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)))