Average Error: 0.1 → 0.1
Time: 3.3s
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 r172787 = x;
        double r172788 = y;
        double r172789 = z;
        double r172790 = r172788 + r172789;
        double r172791 = r172790 + r172789;
        double r172792 = r172791 + r172788;
        double r172793 = t;
        double r172794 = r172792 + r172793;
        double r172795 = r172787 * r172794;
        double r172796 = 5.0;
        double r172797 = r172788 * r172796;
        double r172798 = r172795 + r172797;
        return r172798;
}
↓
double f(double x, double y, double z, double t) {
        double r172799 = y;
        double r172800 = z;
        double r172801 = r172799 + r172800;
        double r172802 = x;
        double r172803 = r172802 + r172802;
        double r172804 = 5.0;
        double r172805 = t;
        double r172806 = r172805 * r172802;
        double r172807 = fma(r172804, r172799, r172806);
        double r172808 = fma(r172801, r172803, r172807);
        return r172808;
}

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. Simplified0.1

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

  5. Final simplification0.1

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

Reproduce

herbie shell --seed 2020018 +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)))