Average Error: 0.1 → 0.1
Time: 17.8s
Precision: 64
\[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5\]
↓
\[x \cdot \left(\left(\left(y + z\right) + z\right) + y\right) + \mathsf{fma}\left(x, t, 5 \cdot y\right)\]
x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5
↓
x \cdot \left(\left(\left(y + z\right) + z\right) + y\right) + \mathsf{fma}\left(x, t, 5 \cdot y\right)
double f(double x, double y, double z, double t) {
        double r207387 = x;
        double r207388 = y;
        double r207389 = z;
        double r207390 = r207388 + r207389;
        double r207391 = r207390 + r207389;
        double r207392 = r207391 + r207388;
        double r207393 = t;
        double r207394 = r207392 + r207393;
        double r207395 = r207387 * r207394;
        double r207396 = 5.0;
        double r207397 = r207388 * r207396;
        double r207398 = r207395 + r207397;
        return r207398;
}
↓
double f(double x, double y, double z, double t) {
        double r207399 = x;
        double r207400 = y;
        double r207401 = z;
        double r207402 = r207400 + r207401;
        double r207403 = r207402 + r207401;
        double r207404 = r207403 + r207400;
        double r207405 = r207399 * r207404;
        double r207406 = t;
        double r207407 = 5.0;
        double r207408 = r207407 * r207400;
        double r207409 = fma(r207399, r207406, r207408);
        double r207410 = r207405 + r207409;
        return r207410;
}

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. Using strategy rm

  3. Applied distribute-lft-in0.1

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

  4. Applied associate-+l+0.1

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

  5. Simplified0.1

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

  6. Final simplification0.1

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

Reproduce

herbie shell --seed 2019326 +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)))