Average Error: 0.0 → 0
Time: 5.5s
Precision: 64
\[\left(x \cdot x\right) \cdot 2 - 1\]
↓
\[\mathsf{fma}\left(x \cdot 2, x, -1\right)\]
\left(x \cdot x\right) \cdot 2 - 1
↓
\mathsf{fma}\left(x \cdot 2, x, -1\right)
double f(double x) {
        double r37078 = x;
        double r37079 = r37078 * r37078;
        double r37080 = 2.0;
        double r37081 = r37079 * r37080;
        double r37082 = 1.0;
        double r37083 = r37081 - r37082;
        return r37083;
}
↓
double f(double x) {
        double r37084 = x;
        double r37085 = 2.0;
        double r37086 = r37084 * r37085;
        double r37087 = 1.0;
        double r37088 = -r37087;
        double r37089 = fma(r37086, r37084, r37088);
        return r37089;
}

Error

Bits error versus x

Derivation

  1. Initial program 0.0

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

  3. Applied add-cube-cbrt0.0

    \[\leadsto \left(x \cdot x\right) \cdot 2 - \color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}\]

  4. Applied prod-diff0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, 2, -\sqrt[3]{1} \cdot \left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right)\right) + \mathsf{fma}\left(-\sqrt[3]{1}, \sqrt[3]{1} \cdot \sqrt[3]{1}, \sqrt[3]{1} \cdot \left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right)\right)}\]

  5. Simplified0

    \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot 2, x, -1\right)} + \mathsf{fma}\left(-\sqrt[3]{1}, \sqrt[3]{1} \cdot \sqrt[3]{1}, \sqrt[3]{1} \cdot \left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right)\right)\]

  6. Simplified0

    \[\leadsto \mathsf{fma}\left(x \cdot 2, x, -1\right) + \color{blue}{0}\]

  7. Final simplification0

    \[\leadsto \mathsf{fma}\left(x \cdot 2, x, -1\right)\]

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(FPCore (x)
  :name "Numeric.SpecFunctions:logGammaCorrection from math-functions-0.1.5.2"
  :precision binary64
  (- (* (* x x) 2) 1))