Average Error: 0.0 → 0
Time: 3.4s
Precision: 64
\[x \cdot y - x\]
↓
\[\mathsf{fma}\left(x, y, -x\right) + \left(x - x\right)\]
x \cdot y - x
↓
\mathsf{fma}\left(x, y, -x\right) + \left(x - x\right)
double f(double x, double y) {
        double r164261 = x;
        double r164262 = y;
        double r164263 = r164261 * r164262;
        double r164264 = r164263 - r164261;
        return r164264;
}
↓
double f(double x, double y) {
        double r164265 = x;
        double r164266 = y;
        double r164267 = -r164265;
        double r164268 = fma(r164265, r164266, r164267);
        double r164269 = r164265 - r164265;
        double r164270 = r164268 + r164269;
        return r164270;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 0.0

    \[x \cdot y - x\]
  2. Using strategy rm

  3. Applied add-cube-cbrt0.8

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

  4. Applied prod-diff0.8

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

  5. Simplified0.0

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

  6. Simplified0

    \[\leadsto \mathsf{fma}\left(x, y, -x\right) + \color{blue}{\left(x - x\right)}\]

  7. Final simplification0

    \[\leadsto \mathsf{fma}\left(x, y, -x\right) + \left(x - x\right)\]

Reproduce

herbie shell --seed 2019196 +o rules:numerics
(FPCore (x y)
  :name "Data.Histogram.Bin.LogBinD:$cbinSizeN from histogram-fill-0.8.4.1"
  (- (* x y) x))