Average Error: 0.0 → 0.0
Time: 5.6s
Precision: 64
\[\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)\]
↓
\[\log \left(\sqrt{\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}}\right) + \log \left(\sqrt{\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}}\right)\]
\log \left(\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) + \log \left(\sqrt{\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}}\right)
double f(double x) {
        double r70104 = 1.0;
        double r70105 = x;
        double r70106 = r70104 / r70105;
        double r70107 = r70105 * r70105;
        double r70108 = r70104 - r70107;
        double r70109 = sqrt(r70108);
        double r70110 = r70109 / r70105;
        double r70111 = r70106 + r70110;
        double r70112 = log(r70111);
        return r70112;
}
↓
double f(double x) {
        double r70113 = 1.0;
        double r70114 = x;
        double r70115 = r70113 / r70114;
        double r70116 = r70114 * r70114;
        double r70117 = r70113 - r70116;
        double r70118 = sqrt(r70117);
        double r70119 = r70118 / r70114;
        double r70120 = r70115 + r70119;
        double r70121 = sqrt(r70120);
        double r70122 = log(r70121);
        double r70123 = r70122 + r70122;
        return r70123;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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. Final simplification0.0

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

Reproduce

herbie shell --seed 2020059 
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  :precision binary64
  (log (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))))