Average Error: 31.5 → 0.0
Time: 11.6s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(\sqrt{\mathsf{fma}\left(\sqrt{x - \sqrt{1}}, \sqrt{x + \sqrt{1}}, x\right)} \cdot \sqrt{\mathsf{fma}\left(\sqrt{x - \sqrt{1}}, \sqrt{x + \sqrt{1}}, x\right)}\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(\sqrt{\mathsf{fma}\left(\sqrt{x - \sqrt{1}}, \sqrt{x + \sqrt{1}}, x\right)} \cdot \sqrt{\mathsf{fma}\left(\sqrt{x - \sqrt{1}}, \sqrt{x + \sqrt{1}}, x\right)}\right)
double f(double x) {
        double r73749 = x;
        double r73750 = r73749 * r73749;
        double r73751 = 1.0;
        double r73752 = r73750 - r73751;
        double r73753 = sqrt(r73752);
        double r73754 = r73749 + r73753;
        double r73755 = log(r73754);
        return r73755;
}
↓
double f(double x) {
        double r73756 = x;
        double r73757 = 1.0;
        double r73758 = sqrt(r73757);
        double r73759 = r73756 - r73758;
        double r73760 = sqrt(r73759);
        double r73761 = r73756 + r73758;
        double r73762 = sqrt(r73761);
        double r73763 = fma(r73760, r73762, r73756);
        double r73764 = sqrt(r73763);
        double r73765 = r73764 * r73764;
        double r73766 = log(r73765);
        return r73766;
}

Error

Bits error versus x

Derivation

  1. Initial program 31.5

    \[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
  2. Using strategy rm

  3. Applied add-sqr-sqrt31.5

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

  4. Applied difference-of-squares31.5

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

  5. Applied sqrt-prod0.0

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

  6. Simplified0.0

    \[\leadsto \log \left(x + \color{blue}{\sqrt{\sqrt{1} + x}} \cdot \sqrt{x - \sqrt{1}}\right)\]
  7. Using strategy rm

  8. Applied add-sqr-sqrt0.0

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

  9. Simplified0.0

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

  10. Simplified0.0

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

  11. Final simplification0.0

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

Reproduce

herbie shell --seed 2019194 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1.0)))))