Average Error: 0.1 → 0.0
Time: 6.2s
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 r167647 = x;
        double r167648 = y;
        double r167649 = z;
        double r167650 = r167648 + r167649;
        double r167651 = r167650 + r167649;
        double r167652 = r167651 + r167648;
        double r167653 = t;
        double r167654 = r167652 + r167653;
        double r167655 = r167647 * r167654;
        double r167656 = 5.0;
        double r167657 = r167648 * r167656;
        double r167658 = r167655 + r167657;
        return r167658;
}
↓
double f(double x, double y, double z, double t) {
        double r167659 = y;
        double r167660 = z;
        double r167661 = r167659 + r167660;
        double r167662 = x;
        double r167663 = r167662 + r167662;
        double r167664 = 5.0;
        double r167665 = t;
        double r167666 = r167665 * r167662;
        double r167667 = fma(r167664, r167659, r167666);
        double r167668 = fma(r167661, r167663, r167667);
        return r167668;
}

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 2020083 +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)))