Average Error: 32.6 → 0.3
Time: 15.4s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(x + \left(\left(x - \frac{0.5}{x}\right) - \frac{0.125}{{x}^{3}}\right)\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(x + \left(\left(x - \frac{0.5}{x}\right) - \frac{0.125}{{x}^{3}}\right)\right)
double f(double x) {
        double r49162 = x;
        double r49163 = r49162 * r49162;
        double r49164 = 1.0;
        double r49165 = r49163 - r49164;
        double r49166 = sqrt(r49165);
        double r49167 = r49162 + r49166;
        double r49168 = log(r49167);
        return r49168;
}
↓
double f(double x) {
        double r49169 = x;
        double r49170 = 0.5;
        double r49171 = r49170 / r49169;
        double r49172 = r49169 - r49171;
        double r49173 = 0.125;
        double r49174 = 3.0;
        double r49175 = pow(r49169, r49174);
        double r49176 = r49173 / r49175;
        double r49177 = r49172 - r49176;
        double r49178 = r49169 + r49177;
        double r49179 = log(r49178);
        return r49179;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 32.6

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

  2. Taylor expanded around inf 0.3

    \[\leadsto \log \left(x + \color{blue}{\left(x - \left(0.5 \cdot \frac{1}{x} + 0.125 \cdot \frac{1}{{x}^{3}}\right)\right)}\right)\]

  3. Simplified0.3

    \[\leadsto \log \left(x + \color{blue}{\left(\left(x - \frac{0.5}{x}\right) - \frac{0.125}{{x}^{3}}\right)}\right)\]

  4. Final simplification0.3

    \[\leadsto \log \left(x + \left(\left(x - \frac{0.5}{x}\right) - \frac{0.125}{{x}^{3}}\right)\right)\]

Reproduce

herbie shell --seed 2019326 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1)))))