Average Error: 0.0 → 0.0
Time: 32.8s
Precision: 64
\[\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)\]
↓
\[\log \left((\left(\sqrt[3]{\frac{1}{x}} \cdot \sqrt[3]{\frac{1}{x}}\right) \cdot \left(\sqrt[3]{\frac{1}{x}}\right) + \left(\frac{\sqrt{1 - x \cdot x}}{x}\right))_*\right)\]
\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)
↓
\log \left((\left(\sqrt[3]{\frac{1}{x}} \cdot \sqrt[3]{\frac{1}{x}}\right) \cdot \left(\sqrt[3]{\frac{1}{x}}\right) + \left(\frac{\sqrt{1 - x \cdot x}}{x}\right))_*\right)
double f(double x) {
        double r12283219 = 1.0;
        double r12283220 = x;
        double r12283221 = r12283219 / r12283220;
        double r12283222 = r12283220 * r12283220;
        double r12283223 = r12283219 - r12283222;
        double r12283224 = sqrt(r12283223);
        double r12283225 = r12283224 / r12283220;
        double r12283226 = r12283221 + r12283225;
        double r12283227 = log(r12283226);
        return r12283227;
}
↓
double f(double x) {
        double r12283228 = 1.0;
        double r12283229 = x;
        double r12283230 = r12283228 / r12283229;
        double r12283231 = cbrt(r12283230);
        double r12283232 = r12283231 * r12283231;
        double r12283233 = r12283229 * r12283229;
        double r12283234 = r12283228 - r12283233;
        double r12283235 = sqrt(r12283234);
        double r12283236 = r12283235 / r12283229;
        double r12283237 = fma(r12283232, r12283231, r12283236);
        double r12283238 = log(r12283237);
        return r12283238;
}

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. Using strategy rm

  3. Applied add-cube-cbrt0.0

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

  4. Applied fma-def0.0

    \[\leadsto \log \color{blue}{\left((\left(\sqrt[3]{\frac{1}{x}} \cdot \sqrt[3]{\frac{1}{x}}\right) \cdot \left(\sqrt[3]{\frac{1}{x}}\right) + \left(\frac{\sqrt{1 - x \cdot x}}{x}\right))_*\right)}\]

  5. Final simplification0.0

    \[\leadsto \log \left((\left(\sqrt[3]{\frac{1}{x}} \cdot \sqrt[3]{\frac{1}{x}}\right) \cdot \left(\sqrt[3]{\frac{1}{x}}\right) + \left(\frac{\sqrt{1 - x \cdot x}}{x}\right))_*\right)\]

Reproduce

herbie shell --seed 2019119 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  (log (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))))