Average Error: 0.0 → 0.0
Time: 14.3s
Precision: 64
\[\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)\]
↓
\[\log \left(\sqrt{\frac{1}{x} + \frac{\sqrt{\mathsf{fma}\left(-x, x, 1\right)}}{x}}\right) + \log \left(\sqrt{\frac{1}{x} + \frac{\sqrt{\mathsf{fma}\left(-x, x, 1\right)}}{x}}\right)\]
\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)
↓
\log \left(\sqrt{\frac{1}{x} + \frac{\sqrt{\mathsf{fma}\left(-x, x, 1\right)}}{x}}\right) + \log \left(\sqrt{\frac{1}{x} + \frac{\sqrt{\mathsf{fma}\left(-x, x, 1\right)}}{x}}\right)
double f(double x) {
        double r76336 = 1.0;
        double r76337 = x;
        double r76338 = r76336 / r76337;
        double r76339 = r76337 * r76337;
        double r76340 = r76336 - r76339;
        double r76341 = sqrt(r76340);
        double r76342 = r76341 / r76337;
        double r76343 = r76338 + r76342;
        double r76344 = log(r76343);
        return r76344;
}
↓
double f(double x) {
        double r76345 = 1.0;
        double r76346 = x;
        double r76347 = r76345 / r76346;
        double r76348 = -r76346;
        double r76349 = fma(r76348, r76346, r76345);
        double r76350 = sqrt(r76349);
        double r76351 = r76350 / r76346;
        double r76352 = r76347 + r76351;
        double r76353 = sqrt(r76352);
        double r76354 = log(r76353);
        double r76355 = r76354 + r76354;
        return r76355;
}

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-sqr-sqrt0.0

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

  4. Applied log-prod0.0

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

  5. Simplified0.0

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

  6. Simplified0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2019179 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  (log (+ (/ 1.0 x) (/ (sqrt (- 1.0 (* x x))) x))))