Average Error: 0.0 → 0.0
Time: 3.4s
Precision: 64
\[\frac{x \cdot y}{2} - \frac{z}{8}\]
↓
\[\mathsf{fma}\left(x, \frac{y}{2}, -\frac{z}{{\left(\sqrt[3]{8}\right)}^{3}}\right) + \left(\left(-\frac{z}{{\left(\sqrt[3]{8}\right)}^{3}}\right) + \frac{z}{{\left(\sqrt[3]{8}\right)}^{3}}\right)\]
\frac{x \cdot y}{2} - \frac{z}{8}
↓
\mathsf{fma}\left(x, \frac{y}{2}, -\frac{z}{{\left(\sqrt[3]{8}\right)}^{3}}\right) + \left(\left(-\frac{z}{{\left(\sqrt[3]{8}\right)}^{3}}\right) + \frac{z}{{\left(\sqrt[3]{8}\right)}^{3}}\right)
double f(double x, double y, double z) {
        double r20445 = x;
        double r20446 = y;
        double r20447 = r20445 * r20446;
        double r20448 = 2.0;
        double r20449 = r20447 / r20448;
        double r20450 = z;
        double r20451 = 8.0;
        double r20452 = r20450 / r20451;
        double r20453 = r20449 - r20452;
        return r20453;
}
↓
double f(double x, double y, double z) {
        double r20454 = x;
        double r20455 = y;
        double r20456 = 2.0;
        double r20457 = r20455 / r20456;
        double r20458 = z;
        double r20459 = 8.0;
        double r20460 = cbrt(r20459);
        double r20461 = 3.0;
        double r20462 = pow(r20460, r20461);
        double r20463 = r20458 / r20462;
        double r20464 = -r20463;
        double r20465 = fma(r20454, r20457, r20464);
        double r20466 = r20464 + r20463;
        double r20467 = r20465 + r20466;
        return r20467;
}

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-cbrt0.0

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

  4. Applied add-cube-cbrt0.8

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

  5. Applied times-frac0.8

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

  6. Applied add-sqr-sqrt28.7

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

  7. Applied prod-diff28.7

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

  8. Simplified0.0

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

  9. Simplified0.0

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

  10. Final simplification0.0

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

Reproduce

herbie shell --seed 2019310 +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)))