Average Error: 31.7 → 0.3
Time: 6.4s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(x + \left(\left(x - \frac{\frac{0.125}{x}}{x \cdot x}\right) - \frac{0.5}{x}\right)\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(x + \left(\left(x - \frac{\frac{0.125}{x}}{x \cdot x}\right) - \frac{0.5}{x}\right)\right)
double f(double x) {
        double r2163561 = x;
        double r2163562 = r2163561 * r2163561;
        double r2163563 = 1.0;
        double r2163564 = r2163562 - r2163563;
        double r2163565 = sqrt(r2163564);
        double r2163566 = r2163561 + r2163565;
        double r2163567 = log(r2163566);
        return r2163567;
}
↓
double f(double x) {
        double r2163568 = x;
        double r2163569 = 0.125;
        double r2163570 = r2163569 / r2163568;
        double r2163571 = r2163568 * r2163568;
        double r2163572 = r2163570 / r2163571;
        double r2163573 = r2163568 - r2163572;
        double r2163574 = 0.5;
        double r2163575 = r2163574 / r2163568;
        double r2163576 = r2163573 - r2163575;
        double r2163577 = r2163568 + r2163576;
        double r2163578 = log(r2163577);
        return r2163578;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.7

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

  2. Taylor expanded around inf 0.3

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

  3. Simplified0.3

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

  4. Final simplification0.3

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

Reproduce

herbie shell --seed 2019192 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1.0)))))