Average Error: 0.1 → 0.1
Time: 2.4s
Precision: 64
\[x \cdot \left(1 - x \cdot y\right)\]
↓
\[\left(x \cdot 1 + x \cdot \left(-x \cdot y\right)\right) + x \cdot \mathsf{fma}\left(-y, x, y \cdot x\right)\]
x \cdot \left(1 - x \cdot y\right)
↓
\left(x \cdot 1 + x \cdot \left(-x \cdot y\right)\right) + x \cdot \mathsf{fma}\left(-y, x, y \cdot x\right)
double f(double x, double y) {
        double r42991 = x;
        double r42992 = 1.0;
        double r42993 = y;
        double r42994 = r42991 * r42993;
        double r42995 = r42992 - r42994;
        double r42996 = r42991 * r42995;
        return r42996;
}
↓
double f(double x, double y) {
        double r42997 = x;
        double r42998 = 1.0;
        double r42999 = r42997 * r42998;
        double r43000 = y;
        double r43001 = r42997 * r43000;
        double r43002 = -r43001;
        double r43003 = r42997 * r43002;
        double r43004 = r42999 + r43003;
        double r43005 = -r43000;
        double r43006 = r43000 * r42997;
        double r43007 = fma(r43005, r42997, r43006);
        double r43008 = r42997 * r43007;
        double r43009 = r43004 + r43008;
        return r43009;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 0.1

    \[x \cdot \left(1 - x \cdot y\right)\]
  2. Using strategy rm

  3. Applied add-sqr-sqrt0.1

    \[\leadsto x \cdot \left(\color{blue}{\sqrt{1} \cdot \sqrt{1}} - x \cdot y\right)\]

  4. Applied prod-diff0.1

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

  5. Applied distribute-lft-in0.1

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

  6. Simplified0.1

    \[\leadsto \color{blue}{x \cdot \left(1 - x \cdot y\right)} + x \cdot \mathsf{fma}\left(-y, x, y \cdot x\right)\]
  7. Using strategy rm

  8. Applied sub-neg0.1

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

  9. Applied distribute-lft-in0.1

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

  10. Final simplification0.1

    \[\leadsto \left(x \cdot 1 + x \cdot \left(-x \cdot y\right)\right) + x \cdot \mathsf{fma}\left(-y, x, y \cdot x\right)\]

Reproduce

herbie shell --seed 2020060 +o rules:numerics
(FPCore (x y)
  :name "Numeric.SpecFunctions:log1p from math-functions-0.1.5.2, A"
  :precision binary64
  (* x (- 1 (* x y))))