Average Error: 0.1 → 0.0
Time: 3.2s
Precision: 64
\[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5\]
↓
\[\mathsf{fma}\left(y + z, x + x, \mathsf{fma}\left(5, y, t \cdot x\right)\right)\]
x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5
↓
\mathsf{fma}\left(y + z, x + x, \mathsf{fma}\left(5, y, t \cdot x\right)\right)
double f(double x, double y, double z, double t) {
        double r158717 = x;
        double r158718 = y;
        double r158719 = z;
        double r158720 = r158718 + r158719;
        double r158721 = r158720 + r158719;
        double r158722 = r158721 + r158718;
        double r158723 = t;
        double r158724 = r158722 + r158723;
        double r158725 = r158717 * r158724;
        double r158726 = 5.0;
        double r158727 = r158718 * r158726;
        double r158728 = r158725 + r158727;
        return r158728;
}
↓
double f(double x, double y, double z, double t) {
        double r158729 = y;
        double r158730 = z;
        double r158731 = r158729 + r158730;
        double r158732 = x;
        double r158733 = r158732 + r158732;
        double r158734 = 5.0;
        double r158735 = t;
        double r158736 = r158735 * r158732;
        double r158737 = fma(r158734, r158729, r158736);
        double r158738 = fma(r158731, r158733, r158737);
        return r158738;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Derivation

  1. Initial program 0.1

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

  2. Simplified0.1

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

  3. Taylor expanded around 0 0.1

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

  4. Simplified0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2020057 +o rules:numerics
(FPCore (x y z t)
  :name "Graphics.Rendering.Plot.Render.Plot.Legend:renderLegendOutside from plot-0.2.3.4, B"
  :precision binary64
  (+ (* x (+ (+ (+ (+ y z) z) y) t)) (* y 5)))