Hyperbolic arc-(co)secant

Average Error: 0.0 → 0.0

Time: 9.9s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

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

C

double f(double x) {
        double r95048 = 1.0;
        double r95049 = x;
        double r95050 = r95048 / r95049;
        double r95051 = r95049 * r95049;
        double r95052 = r95048 - r95051;
        double r95053 = sqrt(r95052);
        double r95054 = r95053 / r95049;
        double r95055 = r95050 + r95054;
        double r95056 = log(r95055);
        return r95056;
}

↓

double f(double x) {
        double r95057 = 1.0;
        double r95058 = sqrt(r95057);
        double r95059 = x;
        double r95060 = sqrt(r95059);
        double r95061 = r95058 / r95060;
        double r95062 = r95059 * r95059;
        double r95063 = r95057 - r95062;
        double r95064 = sqrt(r95063);
        double r95065 = sqrt(r95064);
        double r95066 = r95065 / r95060;
        double r95067 = hypot(r95061, r95066);
        double r95068 = log(r95067);
        double r95069 = r95057 / r95059;
        double r95070 = r95064 / r95059;
        double r95071 = r95069 + r95070;
        double r95072 = sqrt(r95071);
        double r95073 = log(r95072);
        double r95074 = r95068 + r95073;
        return r95074;
}

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-sqrt 0.0

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

4. Applied log-prod 0.0

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

6. Applied add-sqr-sqrt 0.0

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

7. Applied add-sqr-sqrt 0.0

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

8. Applied sqrt-prod 0.0

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

9. Applied times-frac 0.0

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

10. Applied add-sqr-sqrt 0.0

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

11. Applied add-sqr-sqrt 0.0

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

12. Applied times-frac 0.0

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

13. Applied hypot-def 0.0

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

14. Final simplification 0.0

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

Reproduce

herbie shell --seed 2020045 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  :precision binary64
  (log (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))))