Hyperbolic arc-(co)secant

Average Error: 0.0 → 0.2

Time: 14.2s

Precision: 64

Math

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

↓

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

C

double f(double x) {
        double r1250090 = 1.0;
        double r1250091 = x;
        double r1250092 = r1250090 / r1250091;
        double r1250093 = r1250091 * r1250091;
        double r1250094 = r1250090 - r1250093;
        double r1250095 = sqrt(r1250094);
        double r1250096 = r1250095 / r1250091;
        double r1250097 = r1250092 + r1250096;
        double r1250098 = log(r1250097);
        return r1250098;
}

↓

double f(double x) {
        double r1250099 = 1.0;
        double r1250100 = x;
        double r1250101 = sqrt(r1250100);
        double r1250102 = r1250099 / r1250101;
        double r1250103 = r1250100 * r1250100;
        double r1250104 = r1250099 - r1250103;
        double r1250105 = sqrt(r1250104);
        double r1250106 = r1250105 / r1250101;
        double r1250107 = r1250102 + r1250106;
        double r1250108 = log(r1250107);
        double r1250109 = log(r1250102);
        double r1250110 = r1250108 + r1250109;
        return r1250110;
}

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

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

4. Applied *-un-lft-identity 0.0

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

5. Applied times-frac 0.0

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

6. Applied add-sqr-sqrt 0.0

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

7. Applied *-un-lft-identity 0.0

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

8. Applied times-frac 0.0

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

9. Applied distribute-lft-out 0.0

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

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. Final simplification 0.2

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

Reproduce

herbie shell --seed 2019151 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  (log (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))))