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 r201777 = x;
        double r201778 = y;
        double r201779 = z;
        double r201780 = r201778 + r201779;
        double r201781 = r201780 + r201779;
        double r201782 = r201781 + r201778;
        double r201783 = t;
        double r201784 = r201782 + r201783;
        double r201785 = r201777 * r201784;
        double r201786 = 5.0;
        double r201787 = r201778 * r201786;
        double r201788 = r201785 + r201787;
        return r201788;
}
↓
double f(double x, double y, double z, double t) {
        double r201789 = y;
        double r201790 = z;
        double r201791 = r201789 + r201790;
        double r201792 = x;
        double r201793 = r201792 + r201792;
        double r201794 = 5.0;
        double r201795 = t;
        double r201796 = r201795 * r201792;
        double r201797 = fma(r201794, r201789, r201796);
        double r201798 = fma(r201791, r201793, r201797);
        return r201798;
}

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. Simplified0.0

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

  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 2020020 +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)))