Average Error: 0.1 → 0.1
Time: 4.1s
Precision: 64
\[x \cdot \left(1 - x \cdot y\right)\]
↓
\[x \cdot \left(1 - x \cdot y\right) + x \cdot \mathsf{fma}\left(-y, x, y \cdot x\right)\]
x \cdot \left(1 - x \cdot y\right)
↓
x \cdot \left(1 - x \cdot y\right) + x \cdot \mathsf{fma}\left(-y, x, y \cdot x\right)
double f(double x, double y) {
        double r75569 = x;
        double r75570 = 1.0;
        double r75571 = y;
        double r75572 = r75569 * r75571;
        double r75573 = r75570 - r75572;
        double r75574 = r75569 * r75573;
        return r75574;
}
↓
double f(double x, double y) {
        double r75575 = x;
        double r75576 = 1.0;
        double r75577 = y;
        double r75578 = r75575 * r75577;
        double r75579 = r75576 - r75578;
        double r75580 = r75575 * r75579;
        double r75581 = -r75577;
        double r75582 = r75577 * r75575;
        double r75583 = fma(r75581, r75575, r75582);
        double r75584 = r75575 * r75583;
        double r75585 = r75580 + r75584;
        return r75585;
}

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. Final simplification0.1

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

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (x y)
  :name "Numeric.SpecFunctions:log1p from math-functions-0.1.5.2, A"
  :precision binary64
  (* x (- 1 (* x y))))