Average Error: 0.0 → 0.0
Time: 11.6s
Precision: 64
\[\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)\]
↓
\[\log \left(\frac{\log \left(e^{\sqrt{1 - x \cdot x}}\right)}{x} + \frac{1}{x}\right)\]
\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)
↓
\log \left(\frac{\log \left(e^{\sqrt{1 - x \cdot x}}\right)}{x} + \frac{1}{x}\right)
double f(double x) {
        double r1818039 = 1.0;
        double r1818040 = x;
        double r1818041 = r1818039 / r1818040;
        double r1818042 = r1818040 * r1818040;
        double r1818043 = r1818039 - r1818042;
        double r1818044 = sqrt(r1818043);
        double r1818045 = r1818044 / r1818040;
        double r1818046 = r1818041 + r1818045;
        double r1818047 = log(r1818046);
        return r1818047;
}
↓
double f(double x) {
        double r1818048 = 1.0;
        double r1818049 = x;
        double r1818050 = r1818049 * r1818049;
        double r1818051 = r1818048 - r1818050;
        double r1818052 = sqrt(r1818051);
        double r1818053 = exp(r1818052);
        double r1818054 = log(r1818053);
        double r1818055 = r1818054 / r1818049;
        double r1818056 = r1818048 / r1818049;
        double r1818057 = r1818055 + r1818056;
        double r1818058 = log(r1818057);
        return r1818058;
}

Error

Bits error versus x

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Results

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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-log-exp0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019139 
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  (log (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))))