Average Error: 0.1 → 0.0
Time: 2.0s
Precision: 64
\[\frac{x \cdot y}{2} - \frac{z}{8}\]
↓
\[\mathsf{fma}\left(\frac{y}{2}, x, -\frac{z}{8}\right)\]
\frac{x \cdot y}{2} - \frac{z}{8}
↓
\mathsf{fma}\left(\frac{y}{2}, x, -\frac{z}{8}\right)
double f(double x, double y, double z) {
        double r225219 = x;
        double r225220 = y;
        double r225221 = r225219 * r225220;
        double r225222 = 2.0;
        double r225223 = r225221 / r225222;
        double r225224 = z;
        double r225225 = 8.0;
        double r225226 = r225224 / r225225;
        double r225227 = r225223 - r225226;
        return r225227;
}
↓
double f(double x, double y, double z) {
        double r225228 = y;
        double r225229 = 2.0;
        double r225230 = r225228 / r225229;
        double r225231 = x;
        double r225232 = z;
        double r225233 = 8.0;
        double r225234 = r225232 / r225233;
        double r225235 = -r225234;
        double r225236 = fma(r225230, r225231, r225235);
        return r225236;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 0.1

    \[\frac{x \cdot y}{2} - \frac{z}{8}\]
  2. Using strategy rm

  3. Applied *-un-lft-identity0.1

    \[\leadsto \frac{x \cdot y}{2} - \color{blue}{1 \cdot \frac{z}{8}}\]

  4. Applied *-un-lft-identity0.1

    \[\leadsto \color{blue}{1 \cdot \frac{x \cdot y}{2}} - 1 \cdot \frac{z}{8}\]

  5. Applied distribute-lft-out--0.1

    \[\leadsto \color{blue}{1 \cdot \left(\frac{x \cdot y}{2} - \frac{z}{8}\right)}\]

  6. Simplified0.0

    \[\leadsto 1 \cdot \color{blue}{\mathsf{fma}\left(\frac{y}{2}, x, -\frac{z}{8}\right)}\]

  7. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(\frac{y}{2}, x, -\frac{z}{8}\right)\]

Reproduce

herbie shell --seed 2020042 +o rules:numerics
(FPCore (x y z)
  :name "Diagrams.Solve.Polynomial:quartForm  from diagrams-solve-0.1, D"
  :precision binary64
  (- (/ (* x y) 2) (/ z 8)))