Average Error: 30.9 → 0.1
Time: 12.0s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(x + \sqrt{1 + x} \cdot \sqrt{x - 1}\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(x + \sqrt{1 + x} \cdot \sqrt{x - 1}\right)
double f(double x) {
        double r996507 = x;
        double r996508 = r996507 * r996507;
        double r996509 = 1.0;
        double r996510 = r996508 - r996509;
        double r996511 = sqrt(r996510);
        double r996512 = r996507 + r996511;
        double r996513 = log(r996512);
        return r996513;
}
↓
double f(double x) {
        double r996514 = x;
        double r996515 = 1.0;
        double r996516 = r996515 + r996514;
        double r996517 = sqrt(r996516);
        double r996518 = r996514 - r996515;
        double r996519 = sqrt(r996518);
        double r996520 = r996517 * r996519;
        double r996521 = r996514 + r996520;
        double r996522 = log(r996521);
        return r996522;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 30.9

    \[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
  2. Using strategy rm

  3. Applied *-un-lft-identity30.9

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

  4. Applied difference-of-squares30.9

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

  5. Applied sqrt-prod0.1

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

  6. Final simplification0.1

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

Reproduce

herbie shell --seed 2019153 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))