Average Error: 0.5 → 0.4
Time: 25.4s
Precision: 64
\[\sqrt{x - 1} \cdot \sqrt{x}\]
↓
\[x - (\left(\sqrt{\frac{1}{2}}\right) \cdot \left(\sqrt{\frac{1}{2}}\right) + \left(\frac{\frac{1}{8}}{x}\right))_*\]
double f(double x) {
        double r414278 = x;
        double r414279 = 1.0;
        double r414280 = r414278 - r414279;
        double r414281 = sqrt(r414280);
        double r414282 = sqrt(r414278);
        double r414283 = r414281 * r414282;
        return r414283;
}
↓
double f(double x) {
        double r414284 = x;
        double r414285 = 0.5;
        double r414286 = sqrt(r414285);
        double r414287 = 0.125;
        double r414288 = r414287 / r414284;
        double r414289 = fma(r414286, r414286, r414288);
        double r414290 = r414284 - r414289;
        return r414290;
}


\sqrt{x - 1} \cdot \sqrt{x}
↓
x - (\left(\sqrt{\frac{1}{2}}\right) \cdot \left(\sqrt{\frac{1}{2}}\right) + \left(\frac{\frac{1}{8}}{x}\right))_*

Error

Bits error versus x

Derivation

  1. Initial program 0.5

    \[\sqrt{x - 1} \cdot \sqrt{x}\]

  2. Taylor expanded around inf 0.4

    \[\leadsto \color{blue}{x - \left(\frac{1}{8} \cdot \frac{1}{x} + \frac{1}{2}\right)}\]

  3. Simplified0.4

    \[\leadsto \color{blue}{x - \left(\frac{1}{2} - \frac{\frac{-1}{8}}{x}\right)}\]
  4. Using strategy rm

  5. Applied *-un-lft-identity0.4

    \[\leadsto x - \left(\frac{1}{2} - \color{blue}{1 \cdot \frac{\frac{-1}{8}}{x}}\right)\]

  6. Applied add-sqr-sqrt0.4

    \[\leadsto x - \left(\color{blue}{\sqrt{\frac{1}{2}} \cdot \sqrt{\frac{1}{2}}} - 1 \cdot \frac{\frac{-1}{8}}{x}\right)\]

  7. Applied prod-diff0.4

    \[\leadsto x - \color{blue}{\left((\left(\sqrt{\frac{1}{2}}\right) \cdot \left(\sqrt{\frac{1}{2}}\right) + \left(-\frac{\frac{-1}{8}}{x} \cdot 1\right))_* + (\left(-\frac{\frac{-1}{8}}{x}\right) \cdot 1 + \left(\frac{\frac{-1}{8}}{x} \cdot 1\right))_*\right)}\]

  8. Applied associate--r+0.4

    \[\leadsto \color{blue}{\left(x - (\left(\sqrt{\frac{1}{2}}\right) \cdot \left(\sqrt{\frac{1}{2}}\right) + \left(-\frac{\frac{-1}{8}}{x} \cdot 1\right))_*\right) - (\left(-\frac{\frac{-1}{8}}{x}\right) \cdot 1 + \left(\frac{\frac{-1}{8}}{x} \cdot 1\right))_*}\]

  9. Simplified0.4

    \[\leadsto \left(x - (\left(\sqrt{\frac{1}{2}}\right) \cdot \left(\sqrt{\frac{1}{2}}\right) + \left(-\frac{\frac{-1}{8}}{x} \cdot 1\right))_*\right) - \color{blue}{0}\]

  10. Final simplification0.4

    \[\leadsto x - (\left(\sqrt{\frac{1}{2}}\right) \cdot \left(\sqrt{\frac{1}{2}}\right) + \left(\frac{\frac{1}{8}}{x}\right))_*\]

Reproduce

herbie shell --seed 2019101 +o rules:numerics
(FPCore (x)
  :name "sqrt times"
  (* (sqrt (- x 1)) (sqrt x)))