Average Error: 0.0 → 0.0
Time: 5.3s
Precision: 64
\[\frac{x \cdot y}{2} - \frac{z}{8}\]
↓
\[\mathsf{fma}\left(y, \frac{x}{2}, -\frac{z}{{\left(\sqrt[3]{8}\right)}^{3}}\right) + \left(\frac{z}{{\left(\sqrt[3]{8}\right)}^{3}} - \frac{z}{{\left(\sqrt[3]{8}\right)}^{3}}\right)\]
\frac{x \cdot y}{2} - \frac{z}{8}
↓
\mathsf{fma}\left(y, \frac{x}{2}, -\frac{z}{{\left(\sqrt[3]{8}\right)}^{3}}\right) + \left(\frac{z}{{\left(\sqrt[3]{8}\right)}^{3}} - \frac{z}{{\left(\sqrt[3]{8}\right)}^{3}}\right)
double f(double x, double y, double z) {
        double r17453 = x;
        double r17454 = y;
        double r17455 = r17453 * r17454;
        double r17456 = 2.0;
        double r17457 = r17455 / r17456;
        double r17458 = z;
        double r17459 = 8.0;
        double r17460 = r17458 / r17459;
        double r17461 = r17457 - r17460;
        return r17461;
}
↓
double f(double x, double y, double z) {
        double r17462 = y;
        double r17463 = x;
        double r17464 = 2.0;
        double r17465 = r17463 / r17464;
        double r17466 = z;
        double r17467 = 8.0;
        double r17468 = cbrt(r17467);
        double r17469 = 3.0;
        double r17470 = pow(r17468, r17469);
        double r17471 = r17466 / r17470;
        double r17472 = -r17471;
        double r17473 = fma(r17462, r17465, r17472);
        double r17474 = r17471 - r17471;
        double r17475 = r17473 + r17474;
        return r17475;
}

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-sqrt28.2

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

  7. Applied prod-diff28.2

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

  8. Simplified0.0

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

  9. Simplified0.0

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

  10. Final simplification0.0

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

Reproduce

herbie shell --seed 2019315 +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)))