Average Error: 0.1 → 0.0
Time: 1.4s
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 r150694 = x;
        double r150695 = y;
        double r150696 = z;
        double r150697 = r150695 + r150696;
        double r150698 = r150697 + r150696;
        double r150699 = r150698 + r150695;
        double r150700 = t;
        double r150701 = r150699 + r150700;
        double r150702 = r150694 * r150701;
        double r150703 = 5.0;
        double r150704 = r150695 * r150703;
        double r150705 = r150702 + r150704;
        return r150705;
}
↓
double f(double x, double y, double z, double t) {
        double r150706 = y;
        double r150707 = z;
        double r150708 = r150706 + r150707;
        double r150709 = x;
        double r150710 = r150709 + r150709;
        double r150711 = 5.0;
        double r150712 = t;
        double r150713 = r150712 * r150709;
        double r150714 = fma(r150711, r150706, r150713);
        double r150715 = fma(r150708, r150710, r150714);
        return r150715;
}

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 2020001 +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)))