Average Error: 0.1 → 0.1
Time: 2.0s
Precision: 64
\[\frac{x \cdot x - 3}{6}\]
↓
\[\frac{x \cdot x - 3}{6}\]
\frac{x \cdot x - 3}{6}
↓
\frac{x \cdot x - 3}{6}
double f(double x) {
        double r59849 = x;
        double r59850 = r59849 * r59849;
        double r59851 = 3.0;
        double r59852 = r59850 - r59851;
        double r59853 = 6.0;
        double r59854 = r59852 / r59853;
        return r59854;
}
↓
double f(double x) {
        double r59855 = x;
        double r59856 = r59855 * r59855;
        double r59857 = 3.0;
        double r59858 = r59856 - r59857;
        double r59859 = 6.0;
        double r59860 = r59858 / r59859;
        return r59860;
}

Error

Bits error versus x

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Results

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Derivation

  1. Initial program 0.1

    \[\frac{x \cdot x - 3}{6}\]
  2. Using strategy rm

  3. Applied clear-num0.2

    \[\leadsto \color{blue}{\frac{1}{\frac{6}{x \cdot x - 3}}}\]
  4. Using strategy rm

  5. Applied *-un-lft-identity0.2

    \[\leadsto \frac{1}{\frac{6}{\color{blue}{1 \cdot \left(x \cdot x - 3\right)}}}\]

  6. Applied *-un-lft-identity0.2

    \[\leadsto \frac{1}{\frac{\color{blue}{1 \cdot 6}}{1 \cdot \left(x \cdot x - 3\right)}}\]

  7. Applied times-frac0.2

    \[\leadsto \frac{1}{\color{blue}{\frac{1}{1} \cdot \frac{6}{x \cdot x - 3}}}\]

  8. Applied add-cube-cbrt0.2

    \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\frac{1}{1} \cdot \frac{6}{x \cdot x - 3}}\]

  9. Applied times-frac0.2

    \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\frac{1}{1}} \cdot \frac{\sqrt[3]{1}}{\frac{6}{x \cdot x - 3}}}\]

  10. Simplified0.2

    \[\leadsto \color{blue}{1} \cdot \frac{\sqrt[3]{1}}{\frac{6}{x \cdot x - 3}}\]

  11. Simplified0.1

    \[\leadsto 1 \cdot \color{blue}{\frac{x \cdot x - 3}{6}}\]

  12. Final simplification0.1

    \[\leadsto \frac{x \cdot x - 3}{6}\]

Reproduce

herbie shell --seed 2020089 +o rules:numerics
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, H"
  :precision binary64
  (/ (- (* x x) 3) 6))