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 r52222 = x;
        double r52223 = 1.0;
        double r52224 = y;
        double r52225 = r52222 * r52224;
        double r52226 = r52223 - r52225;
        double r52227 = r52222 * r52226;
        return r52227;
}
↓
double f(double x, double y) {
        double r52228 = x;
        double r52229 = 1.0;
        double r52230 = r52228 * r52229;
        double r52231 = y;
        double r52232 = r52228 * r52231;
        double r52233 = -r52232;
        double r52234 = r52228 * r52233;
        double r52235 = r52230 + r52234;
        double r52236 = -r52231;
        double r52237 = r52231 * r52228;
        double r52238 = fma(r52236, r52228, r52237);
        double r52239 = r52228 * r52238;
        double r52240 = r52235 + r52239;
        return r52240;
}

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))))