Average Error: 0.0 → 0.4
Time: 4.6s
Precision: 64
\[\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)\]
↓
\[\left(\left(\log \left(\sqrt{1} + 1\right) - \frac{1}{2} \cdot \frac{{x}^{2}}{\sqrt{1} \cdot \left(\sqrt{1} + 1\right)}\right) - \mathsf{fma}\left(\frac{1}{8}, \frac{{x}^{4}}{{\left(\sqrt{1}\right)}^{3} \cdot \left(\sqrt{1} + 1\right)}, \log x\right)\right) - \frac{\frac{1}{8}}{{\left(\sqrt{1} + 1\right)}^{2}} \cdot \frac{{x}^{4}}{1}\]
\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)
↓
\left(\left(\log \left(\sqrt{1} + 1\right) - \frac{1}{2} \cdot \frac{{x}^{2}}{\sqrt{1} \cdot \left(\sqrt{1} + 1\right)}\right) - \mathsf{fma}\left(\frac{1}{8}, \frac{{x}^{4}}{{\left(\sqrt{1}\right)}^{3} \cdot \left(\sqrt{1} + 1\right)}, \log x\right)\right) - \frac{\frac{1}{8}}{{\left(\sqrt{1} + 1\right)}^{2}} \cdot \frac{{x}^{4}}{1}
double f(double x) {
        double r61889 = 1.0;
        double r61890 = x;
        double r61891 = r61889 / r61890;
        double r61892 = r61890 * r61890;
        double r61893 = r61889 - r61892;
        double r61894 = sqrt(r61893);
        double r61895 = r61894 / r61890;
        double r61896 = r61891 + r61895;
        double r61897 = log(r61896);
        return r61897;
}
↓
double f(double x) {
        double r61898 = 1.0;
        double r61899 = sqrt(r61898);
        double r61900 = r61899 + r61898;
        double r61901 = log(r61900);
        double r61902 = 0.5;
        double r61903 = x;
        double r61904 = 2.0;
        double r61905 = pow(r61903, r61904);
        double r61906 = r61899 * r61900;
        double r61907 = r61905 / r61906;
        double r61908 = r61902 * r61907;
        double r61909 = r61901 - r61908;
        double r61910 = 0.125;
        double r61911 = 4.0;
        double r61912 = pow(r61903, r61911);
        double r61913 = 3.0;
        double r61914 = pow(r61899, r61913);
        double r61915 = r61914 * r61900;
        double r61916 = r61912 / r61915;
        double r61917 = log(r61903);
        double r61918 = fma(r61910, r61916, r61917);
        double r61919 = r61909 - r61918;
        double r61920 = pow(r61900, r61904);
        double r61921 = r61910 / r61920;
        double r61922 = r61912 / r61898;
        double r61923 = r61921 * r61922;
        double r61924 = r61919 - r61923;
        return r61924;
}

Error

Bits error versus x

Derivation

  1. Initial program 0.0

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

  2. Taylor expanded around 0 0.4

    \[\leadsto \color{blue}{\log \left(\sqrt{1} + 1\right) - \left(\frac{1}{2} \cdot \frac{{x}^{2}}{\sqrt{1} \cdot \left(\sqrt{1} + 1\right)} + \left(\frac{1}{8} \cdot \frac{{x}^{4}}{{\left(\sqrt{1}\right)}^{3} \cdot \left(\sqrt{1} + 1\right)} + \left(\log x + \frac{1}{8} \cdot \frac{{x}^{4}}{{\left(\sqrt{1}\right)}^{2} \cdot {\left(\sqrt{1} + 1\right)}^{2}}\right)\right)\right)}\]

  3. Simplified0.4

    \[\leadsto \color{blue}{\left(\left(\log \left(\sqrt{1} + 1\right) - \frac{1}{2} \cdot \frac{{x}^{2}}{\sqrt{1} \cdot \left(\sqrt{1} + 1\right)}\right) - \mathsf{fma}\left(\frac{1}{8}, \frac{{x}^{4}}{{\left(\sqrt{1}\right)}^{3} \cdot \left(\sqrt{1} + 1\right)}, \log x\right)\right) - \frac{\frac{1}{8}}{{\left(\sqrt{1} + 1\right)}^{2}} \cdot \frac{{x}^{4}}{1}}\]

  4. Final simplification0.4

    \[\leadsto \left(\left(\log \left(\sqrt{1} + 1\right) - \frac{1}{2} \cdot \frac{{x}^{2}}{\sqrt{1} \cdot \left(\sqrt{1} + 1\right)}\right) - \mathsf{fma}\left(\frac{1}{8}, \frac{{x}^{4}}{{\left(\sqrt{1}\right)}^{3} \cdot \left(\sqrt{1} + 1\right)}, \log x\right)\right) - \frac{\frac{1}{8}}{{\left(\sqrt{1} + 1\right)}^{2}} \cdot \frac{{x}^{4}}{1}\]

Reproduce

herbie shell --seed 2020021 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  :precision binary64
  (log (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))))