Average Error: 0.0 → 0.0
Time: 3.0s
Precision: 64
\[\frac{x \cdot y}{2} - \frac{z}{8}\]
↓
\[\mathsf{fma}\left(x, y \cdot \frac{1}{2}, -\frac{z}{8}\right) + \frac{z}{8} \cdot 0\]
\frac{x \cdot y}{2} - \frac{z}{8}
↓
\mathsf{fma}\left(x, y \cdot \frac{1}{2}, -\frac{z}{8}\right) + \frac{z}{8} \cdot 0
double f(double x, double y, double z) {
        double r121112 = x;
        double r121113 = y;
        double r121114 = r121112 * r121113;
        double r121115 = 2.0;
        double r121116 = r121114 / r121115;
        double r121117 = z;
        double r121118 = 8.0;
        double r121119 = r121117 / r121118;
        double r121120 = r121116 - r121119;
        return r121120;
}
↓
double f(double x, double y, double z) {
        double r121121 = x;
        double r121122 = y;
        double r121123 = 1.0;
        double r121124 = 2.0;
        double r121125 = r121123 / r121124;
        double r121126 = r121122 * r121125;
        double r121127 = z;
        double r121128 = 8.0;
        double r121129 = r121127 / r121128;
        double r121130 = -r121129;
        double r121131 = fma(r121121, r121126, r121130);
        double r121132 = 0.0;
        double r121133 = r121129 * r121132;
        double r121134 = r121131 + r121133;
        return r121134;
}

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 div-inv0.8

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

  5. Applied prod-diff0.8

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

  6. Simplified0.0

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

  7. Simplified0.0

    \[\leadsto \mathsf{fma}\left(x, y \cdot \frac{1}{2}, -\frac{z}{8}\right) + \color{blue}{\frac{z}{8} \cdot 0}\]

  8. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(x, y \cdot \frac{1}{2}, -\frac{z}{8}\right) + \frac{z}{8} \cdot 0\]

Reproduce

herbie shell --seed 2019208 +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)))