Average Error: 0.1 → 0.1
Time: 2.0s
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 r46030 = x;
        double r46031 = 1.0;
        double r46032 = y;
        double r46033 = r46030 * r46032;
        double r46034 = r46031 - r46033;
        double r46035 = r46030 * r46034;
        return r46035;
}
↓
double f(double x, double y) {
        double r46036 = x;
        double r46037 = 1.0;
        double r46038 = y;
        double r46039 = r46036 * r46038;
        double r46040 = r46037 - r46039;
        double r46041 = r46036 * r46040;
        double r46042 = -r46038;
        double r46043 = r46038 * r46036;
        double r46044 = fma(r46042, r46036, r46043);
        double r46045 = r46036 * r46044;
        double r46046 = r46041 + r46045;
        return r46046;
}

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-cube-cbrt0.1

    \[\leadsto x \cdot \left(\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}} - x \cdot y\right)\]

  4. Applied prod-diff0.1

    \[\leadsto x \cdot \color{blue}{\left(\mathsf{fma}\left(\sqrt[3]{1} \cdot \sqrt[3]{1}, \sqrt[3]{1}, -y \cdot x\right) + \mathsf{fma}\left(-y, x, y \cdot x\right)\right)}\]

  5. Applied distribute-lft-in0.1

    \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(\sqrt[3]{1} \cdot \sqrt[3]{1}, \sqrt[3]{1}, -y \cdot x\right) + x \cdot \mathsf{fma}\left(-y, x, y \cdot x\right)}\]

  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 2019346 +o rules:numerics
(FPCore (x y)
  :name "Numeric.SpecFunctions:log1p from math-functions-0.1.5.2, A"
  :precision binary64
  (* x (- 1 (* x y))))