Hyperbolic arc-(co)secant

Average Error: 0.1 → 0.2

Time: 14.1s

Precision: 64

Math

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

↓

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

C

double f(double x) {
        double r79996 = 1.0;
        double r79997 = x;
        double r79998 = r79996 / r79997;
        double r79999 = r79997 * r79997;
        double r80000 = r79996 - r79999;
        double r80001 = sqrt(r80000);
        double r80002 = r80001 / r79997;
        double r80003 = r79998 + r80002;
        double r80004 = log(r80003);
        return r80004;
}

↓

double f(double x) {
        double r80005 = x;
        double r80006 = sqrt(r80005);
        double r80007 = log(r80006);
        double r80008 = -r80007;
        double r80009 = 1.0;
        double r80010 = r80009 / r80006;
        double r80011 = -r80005;
        double r80012 = fma(r80005, r80011, r80009);
        double r80013 = sqrt(r80012);
        double r80014 = r80013 / r80006;
        double r80015 = r80010 + r80014;
        double r80016 = log(r80015);
        double r80017 = r80008 + r80016;
        return r80017;
}

Error

Bits error versus x

Derivation

1. Initial program 0.1

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

3. Applied add-sqr-sqrt 0.1

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

4. Applied *-un-lft-identity 0.1

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

5. Applied times-frac 0.1

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

6. Applied add-sqr-sqrt 0.1

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

7. Applied *-un-lft-identity 0.1

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

8. Applied times-frac 0.1

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

9. Applied distribute-lft-out 0.1

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

10. Applied log-prod 0.2

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

11. Simplified 0.2

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

12. Simplified 0.2

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

13. Final simplification 0.2

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

Reproduce

herbie shell --seed 2019195 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  (log (+ (/ 1.0 x) (/ (sqrt (- 1.0 (* x x))) x))))