Average Error: 0.1 → 0.1
Time: 4.8s
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 r152564 = x;
        double r152565 = y;
        double r152566 = z;
        double r152567 = r152565 + r152566;
        double r152568 = r152567 + r152566;
        double r152569 = r152568 + r152565;
        double r152570 = t;
        double r152571 = r152569 + r152570;
        double r152572 = r152564 * r152571;
        double r152573 = 5.0;
        double r152574 = r152565 * r152573;
        double r152575 = r152572 + r152574;
        return r152575;
}
↓
double f(double x, double y, double z, double t) {
        double r152576 = y;
        double r152577 = z;
        double r152578 = r152576 + r152577;
        double r152579 = x;
        double r152580 = r152579 + r152579;
        double r152581 = 5.0;
        double r152582 = t;
        double r152583 = r152582 * r152579;
        double r152584 = fma(r152581, r152576, r152583);
        double r152585 = fma(r152578, r152580, r152584);
        return r152585;
}

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 2019347 +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)))