Average Error: 0.0 → 0.0
Time: 3.4s
Precision: 64
\[\frac{x \cdot y}{2} - \frac{z}{8}\]
↓
\[\mathsf{fma}\left(x, \frac{y}{2}, -\frac{z}{8}\right)\]
\frac{x \cdot y}{2} - \frac{z}{8}
↓
\mathsf{fma}\left(x, \frac{y}{2}, -\frac{z}{8}\right)
double f(double x, double y, double z) {
        double r135098 = x;
        double r135099 = y;
        double r135100 = r135098 * r135099;
        double r135101 = 2.0;
        double r135102 = r135100 / r135101;
        double r135103 = z;
        double r135104 = 8.0;
        double r135105 = r135103 / r135104;
        double r135106 = r135102 - r135105;
        return r135106;
}
↓
double f(double x, double y, double z) {
        double r135107 = x;
        double r135108 = y;
        double r135109 = 2.0;
        double r135110 = r135108 / r135109;
        double r135111 = z;
        double r135112 = 8.0;
        double r135113 = r135111 / r135112;
        double r135114 = -r135113;
        double r135115 = fma(r135107, r135110, r135114);
        return r135115;
}

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 add-sqr-sqrt28.4

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

  5. Applied prod-diff28.4

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

  6. Simplified0.0

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

  7. Simplified0.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2019306 +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)))