Average Error: 0.0 → 0
Time: 3.8s
Precision: 64
\[\frac{x \cdot y}{2} - \frac{z}{8}\]
↓
\[\mathsf{fma}\left(x, \frac{y}{2}, -\frac{z}{8}\right) + \frac{z}{8} \cdot 0\]
\frac{x \cdot y}{2} - \frac{z}{8}
↓
\mathsf{fma}\left(x, \frac{y}{2}, -\frac{z}{8}\right) + \frac{z}{8} \cdot 0
double f(double x, double y, double z) {
        double r130298 = x;
        double r130299 = y;
        double r130300 = r130298 * r130299;
        double r130301 = 2.0;
        double r130302 = r130300 / r130301;
        double r130303 = z;
        double r130304 = 8.0;
        double r130305 = r130303 / r130304;
        double r130306 = r130302 - r130305;
        return r130306;
}
↓
double f(double x, double y, double z) {
        double r130307 = x;
        double r130308 = y;
        double r130309 = 2.0;
        double r130310 = r130308 / r130309;
        double r130311 = z;
        double r130312 = 8.0;
        double r130313 = r130311 / r130312;
        double r130314 = -r130313;
        double r130315 = fma(r130307, r130310, r130314);
        double r130316 = 0.0;
        double r130317 = r130313 * r130316;
        double r130318 = r130315 + r130317;
        return r130318;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 0.0

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

  3. Applied add-cube-cbrt0.8

    \[\leadsto \frac{x \cdot y}{2} - \color{blue}{\left(\sqrt[3]{\frac{z}{8}} \cdot \sqrt[3]{\frac{z}{8}}\right) \cdot \sqrt[3]{\frac{z}{8}}}\]

  4. Applied *-un-lft-identity0.8

    \[\leadsto \frac{x \cdot y}{\color{blue}{1 \cdot 2}} - \left(\sqrt[3]{\frac{z}{8}} \cdot \sqrt[3]{\frac{z}{8}}\right) \cdot \sqrt[3]{\frac{z}{8}}\]

  5. Applied times-frac0.8

    \[\leadsto \color{blue}{\frac{x}{1} \cdot \frac{y}{2}} - \left(\sqrt[3]{\frac{z}{8}} \cdot \sqrt[3]{\frac{z}{8}}\right) \cdot \sqrt[3]{\frac{z}{8}}\]

  6. Applied prod-diff0.8

    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{1}, \frac{y}{2}, -\sqrt[3]{\frac{z}{8}} \cdot \left(\sqrt[3]{\frac{z}{8}} \cdot \sqrt[3]{\frac{z}{8}}\right)\right) + \mathsf{fma}\left(-\sqrt[3]{\frac{z}{8}}, \sqrt[3]{\frac{z}{8}} \cdot \sqrt[3]{\frac{z}{8}}, \sqrt[3]{\frac{z}{8}} \cdot \left(\sqrt[3]{\frac{z}{8}} \cdot \sqrt[3]{\frac{z}{8}}\right)\right)}\]

  7. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{y}{2}, -\frac{z}{8}\right)} + \mathsf{fma}\left(-\sqrt[3]{\frac{z}{8}}, \sqrt[3]{\frac{z}{8}} \cdot \sqrt[3]{\frac{z}{8}}, \sqrt[3]{\frac{z}{8}} \cdot \left(\sqrt[3]{\frac{z}{8}} \cdot \sqrt[3]{\frac{z}{8}}\right)\right)\]

  8. Simplified0

    \[\leadsto \mathsf{fma}\left(x, \frac{y}{2}, -\frac{z}{8}\right) + \color{blue}{\frac{z}{8} \cdot 0}\]

  9. Final simplification0

    \[\leadsto \mathsf{fma}\left(x, \frac{y}{2}, -\frac{z}{8}\right) + \frac{z}{8} \cdot 0\]

Reproduce

herbie shell --seed 2019209 +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)))