Average Error: 0.0 → 0
Time: 3.8s
Precision: 64
\[x - y \cdot y\]
↓
\[\mathsf{fma}\left(y, -y, x\right)\]
x - y \cdot y
↓
\mathsf{fma}\left(y, -y, x\right)
double f(double x, double y) {
        double r105871 = x;
        double r105872 = y;
        double r105873 = r105872 * r105872;
        double r105874 = r105871 - r105873;
        return r105874;
}
↓
double f(double x, double y) {
        double r105875 = y;
        double r105876 = -r105875;
        double r105877 = x;
        double r105878 = fma(r105875, r105876, r105877);
        return r105878;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 0.0

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

  3. Applied add-cube-cbrt0.9

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

  4. Applied prod-diff0.9

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

  5. Simplified0.0

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

  6. Simplified0

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

  7. Final simplification0

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

Reproduce

herbie shell --seed 2019325 +o rules:numerics
(FPCore (x y)
  :name "Graphics.Rasterific.Shading:$sradialGradientWithFocusShader from Rasterific-0.6.1"
  :precision binary64
  (- x (* y y)))