Average Error: 0.0 → 0.0
Time: 13.5s
Precision: 64
\[\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)\]
↓
\[\log \left(\frac{1}{x} + \frac{\sqrt{\sqrt{1} + x} \cdot \sqrt{\sqrt{1} - x}}{x}\right)\]
\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)
↓
\log \left(\frac{1}{x} + \frac{\sqrt{\sqrt{1} + x} \cdot \sqrt{\sqrt{1} - x}}{x}\right)
double f(double x) {
        double r78228 = 1.0;
        double r78229 = x;
        double r78230 = r78228 / r78229;
        double r78231 = r78229 * r78229;
        double r78232 = r78228 - r78231;
        double r78233 = sqrt(r78232);
        double r78234 = r78233 / r78229;
        double r78235 = r78230 + r78234;
        double r78236 = log(r78235);
        return r78236;
}
↓
double f(double x) {
        double r78237 = 1.0;
        double r78238 = x;
        double r78239 = r78237 / r78238;
        double r78240 = sqrt(r78237);
        double r78241 = r78240 + r78238;
        double r78242 = sqrt(r78241);
        double r78243 = r78240 - r78238;
        double r78244 = sqrt(r78243);
        double r78245 = r78242 * r78244;
        double r78246 = r78245 / r78238;
        double r78247 = r78239 + r78246;
        double r78248 = log(r78247);
        return r78248;
}

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. Final simplification0.0

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

Reproduce

herbie shell --seed 2019304 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  :precision binary64
  (log (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))))