Average Error: 0.1 → 0.1
Time: 11.6s
Precision: 64
\[\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)\]
↓
\[\log \left(\frac{1}{x} + \frac{\sqrt{\sqrt{1 - x \cdot x}}}{\sqrt{x}} \cdot \frac{\sqrt{\sqrt{1 - x \cdot x}}}{\sqrt{x}}\right)\]
\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)
↓
\log \left(\frac{1}{x} + \frac{\sqrt{\sqrt{1 - x \cdot x}}}{\sqrt{x}} \cdot \frac{\sqrt{\sqrt{1 - x \cdot x}}}{\sqrt{x}}\right)
double f(double x) {
        double r56055 = 1.0;
        double r56056 = x;
        double r56057 = r56055 / r56056;
        double r56058 = r56056 * r56056;
        double r56059 = r56055 - r56058;
        double r56060 = sqrt(r56059);
        double r56061 = r56060 / r56056;
        double r56062 = r56057 + r56061;
        double r56063 = log(r56062);
        return r56063;
}
↓
double f(double x) {
        double r56064 = 1.0;
        double r56065 = x;
        double r56066 = r56064 / r56065;
        double r56067 = r56065 * r56065;
        double r56068 = r56064 - r56067;
        double r56069 = sqrt(r56068);
        double r56070 = sqrt(r56069);
        double r56071 = sqrt(r56065);
        double r56072 = r56070 / r56071;
        double r56073 = r56072 * r56072;
        double r56074 = r56066 + r56073;
        double r56075 = log(r56074);
        return r56075;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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-sqrt0.1

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

  4. Applied add-sqr-sqrt0.1

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

  5. Applied sqrt-prod0.1

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

  6. Applied times-frac0.1

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

  7. Final simplification0.1

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

Reproduce

herbie shell --seed 2019208 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  :precision binary64
  (log (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))))