Average Error: 30.7 → 0.3
Time: 16.8s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(\frac{\frac{-1}{8}}{\left(x \cdot x\right) \cdot x} + \left(\frac{\frac{-1}{2}}{x} + 2 \cdot x\right)\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(\frac{\frac{-1}{8}}{\left(x \cdot x\right) \cdot x} + \left(\frac{\frac{-1}{2}}{x} + 2 \cdot x\right)\right)
double f(double x) {
        double r2082087 = x;
        double r2082088 = r2082087 * r2082087;
        double r2082089 = 1.0;
        double r2082090 = r2082088 - r2082089;
        double r2082091 = sqrt(r2082090);
        double r2082092 = r2082087 + r2082091;
        double r2082093 = log(r2082092);
        return r2082093;
}
↓
double f(double x) {
        double r2082094 = -0.125;
        double r2082095 = x;
        double r2082096 = r2082095 * r2082095;
        double r2082097 = r2082096 * r2082095;
        double r2082098 = r2082094 / r2082097;
        double r2082099 = -0.5;
        double r2082100 = r2082099 / r2082095;
        double r2082101 = 2.0;
        double r2082102 = r2082101 * r2082095;
        double r2082103 = r2082100 + r2082102;
        double r2082104 = r2082098 + r2082103;
        double r2082105 = log(r2082104);
        return r2082105;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 30.7

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

  2. Taylor expanded around inf 0.3

    \[\leadsto \log \color{blue}{\left(2 \cdot x - \left(\frac{1}{8} \cdot \frac{1}{{x}^{3}} + \frac{1}{2} \cdot \frac{1}{x}\right)\right)}\]

  3. Simplified0.3

    \[\leadsto \log \color{blue}{\left(\frac{\frac{-1}{8}}{x \cdot \left(x \cdot x\right)} + \left(x \cdot 2 + \frac{\frac{-1}{2}}{x}\right)\right)}\]

  4. Final simplification0.3

    \[\leadsto \log \left(\frac{\frac{-1}{8}}{\left(x \cdot x\right) \cdot x} + \left(\frac{\frac{-1}{2}}{x} + 2 \cdot x\right)\right)\]

Reproduce

herbie shell --seed 2019135 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))