Average Error: 0.0 → 0.0
Time: 11.7s
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 r88571 = 1.0;
        double r88572 = x;
        double r88573 = r88571 / r88572;
        double r88574 = r88572 * r88572;
        double r88575 = r88571 - r88574;
        double r88576 = sqrt(r88575);
        double r88577 = r88576 / r88572;
        double r88578 = r88573 + r88577;
        double r88579 = log(r88578);
        return r88579;
}
↓
double f(double x) {
        double r88580 = 1.0;
        double r88581 = x;
        double r88582 = r88580 / r88581;
        double r88583 = -r88581;
        double r88584 = fma(r88581, r88583, r88580);
        double r88585 = sqrt(r88584);
        double r88586 = r88585 / r88581;
        double r88587 = r88582 + r88586;
        double r88588 = sqrt(r88587);
        double r88589 = log(r88588);
        double r88590 = r88589 + r88589;
        return r88590;
}

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 2019194 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  (log (+ (/ 1.0 x) (/ (sqrt (- 1.0 (* x x))) x))))