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 r118819 = x;
        double r118820 = y;
        double r118821 = z;
        double r118822 = r118820 + r118821;
        double r118823 = r118822 + r118821;
        double r118824 = r118823 + r118820;
        double r118825 = t;
        double r118826 = r118824 + r118825;
        double r118827 = r118819 * r118826;
        double r118828 = 5.0;
        double r118829 = r118820 * r118828;
        double r118830 = r118827 + r118829;
        return r118830;
}
↓
double f(double x, double y, double z, double t) {
        double r118831 = y;
        double r118832 = z;
        double r118833 = r118831 + r118832;
        double r118834 = x;
        double r118835 = r118834 + r118834;
        double r118836 = 5.0;
        double r118837 = t;
        double r118838 = r118837 * r118834;
        double r118839 = fma(r118836, r118831, r118838);
        double r118840 = fma(r118833, r118835, r118839);
        return r118840;
}

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)))