Average Error: 0.1 → 0.0
Time: 3.5s
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 r127787 = x;
        double r127788 = y;
        double r127789 = z;
        double r127790 = r127788 + r127789;
        double r127791 = r127790 + r127789;
        double r127792 = r127791 + r127788;
        double r127793 = t;
        double r127794 = r127792 + r127793;
        double r127795 = r127787 * r127794;
        double r127796 = 5.0;
        double r127797 = r127788 * r127796;
        double r127798 = r127795 + r127797;
        return r127798;
}
↓
double f(double x, double y, double z, double t) {
        double r127799 = y;
        double r127800 = z;
        double r127801 = r127799 + r127800;
        double r127802 = x;
        double r127803 = r127802 + r127802;
        double r127804 = 5.0;
        double r127805 = t;
        double r127806 = r127805 * r127802;
        double r127807 = fma(r127804, r127799, r127806);
        double r127808 = fma(r127801, r127803, r127807);
        return r127808;
}

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 2020062 +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)))