Average Error: 0.0 → 0.0
Time: 5.5s
Precision: 64
\[\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)\]
↓
\[\log \left(\frac{1}{x} + \frac{\sqrt{\sqrt{1} + x}}{\frac{x}{\sqrt{\sqrt{1} - x}}}\right)\]
\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)
↓
\log \left(\frac{1}{x} + \frac{\sqrt{\sqrt{1} + x}}{\frac{x}{\sqrt{\sqrt{1} - x}}}\right)
double f(double x) {
        double r79923 = 1.0;
        double r79924 = x;
        double r79925 = r79923 / r79924;
        double r79926 = r79924 * r79924;
        double r79927 = r79923 - r79926;
        double r79928 = sqrt(r79927);
        double r79929 = r79928 / r79924;
        double r79930 = r79925 + r79929;
        double r79931 = log(r79930);
        return r79931;
}
↓
double f(double x) {
        double r79932 = 1.0;
        double r79933 = x;
        double r79934 = r79932 / r79933;
        double r79935 = sqrt(r79932);
        double r79936 = r79935 + r79933;
        double r79937 = sqrt(r79936);
        double r79938 = r79935 - r79933;
        double r79939 = sqrt(r79938);
        double r79940 = r79933 / r79939;
        double r79941 = r79937 / r79940;
        double r79942 = r79934 + r79941;
        double r79943 = log(r79942);
        return r79943;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.0

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

  3. Applied add-sqr-sqrt0.0

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

  4. Applied difference-of-squares0.0

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

  5. Applied sqrt-prod0.0

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

  6. Applied associate-/l*0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2020081 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  :precision binary64
  (log (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))))