Average Error: 0.0 → 0
Time: 3.2s
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 r123469 = x;
        double r123470 = y;
        double r123471 = r123469 * r123470;
        double r123472 = 2.0;
        double r123473 = r123471 / r123472;
        double r123474 = z;
        double r123475 = 8.0;
        double r123476 = r123474 / r123475;
        double r123477 = r123473 - r123476;
        return r123477;
}
↓
double f(double x, double y, double z) {
        double r123478 = x;
        double r123479 = y;
        double r123480 = 2.0;
        double r123481 = r123479 / r123480;
        double r123482 = z;
        double r123483 = 8.0;
        double r123484 = r123482 / r123483;
        double r123485 = -r123484;
        double r123486 = fma(r123478, r123481, r123485);
        double r123487 = 0.0;
        double r123488 = r123484 * r123487;
        double r123489 = r123486 + r123488;
        return r123489;
}

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 2019305 +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)))