Average Error: 32.0 → 0.2
Time: 7.4s
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 r64744 = x;
        double r64745 = r64744 * r64744;
        double r64746 = 1.0;
        double r64747 = r64745 - r64746;
        double r64748 = sqrt(r64747);
        double r64749 = r64744 + r64748;
        double r64750 = log(r64749);
        return r64750;
}
↓
double f(double x) {
        double r64751 = x;
        double r64752 = 0.5;
        double r64753 = r64752 / r64751;
        double r64754 = r64751 - r64753;
        double r64755 = 0.125;
        double r64756 = 3.0;
        double r64757 = pow(r64751, r64756);
        double r64758 = r64755 / r64757;
        double r64759 = r64754 - r64758;
        double r64760 = r64751 + r64759;
        double r64761 = log(r64760);
        return r64761;
}

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 0.2

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

  3. Simplified0.2

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

  4. Final simplification0.2

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

Reproduce

herbie shell --seed 2019325 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1)))))