Average Error: 32.0 → 0.4
Time: 24.2s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log 2 - \left(\frac{0.25}{x \cdot x} + \left(\frac{0.09375}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)} - \log x\right)\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log 2 - \left(\frac{0.25}{x \cdot x} + \left(\frac{0.09375}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)} - \log x\right)\right)
double f(double x) {
        double r2546953 = x;
        double r2546954 = r2546953 * r2546953;
        double r2546955 = 1.0;
        double r2546956 = r2546954 - r2546955;
        double r2546957 = sqrt(r2546956);
        double r2546958 = r2546953 + r2546957;
        double r2546959 = log(r2546958);
        return r2546959;
}
↓
double f(double x) {
        double r2546960 = 2.0;
        double r2546961 = log(r2546960);
        double r2546962 = 0.25;
        double r2546963 = x;
        double r2546964 = r2546963 * r2546963;
        double r2546965 = r2546962 / r2546964;
        double r2546966 = 0.09375;
        double r2546967 = r2546964 * r2546964;
        double r2546968 = r2546966 / r2546967;
        double r2546969 = log(r2546963);
        double r2546970 = r2546968 - r2546969;
        double r2546971 = r2546965 + r2546970;
        double r2546972 = r2546961 - r2546971;
        return r2546972;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 32.0

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

  2. Taylor expanded around inf 0.4

    \[\leadsto \color{blue}{\log 2 - \left(\log \left(\frac{1}{x}\right) + \left(0.09375 \cdot \frac{1}{{x}^{4}} + 0.25 \cdot \frac{1}{{x}^{2}}\right)\right)}\]

  3. Simplified0.4

    \[\leadsto \color{blue}{\log 2 - \left(\left(\frac{0.09375}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)} - \log x\right) + \frac{0.25}{x \cdot x}\right)}\]

  4. Final simplification0.4

    \[\leadsto \log 2 - \left(\frac{0.25}{x \cdot x} + \left(\frac{0.09375}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)} - \log x\right)\right)\]

Reproduce

herbie shell --seed 2019168 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1.0)))))