Average Error: 30.9 → 0.0
Time: 8.7s
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 r1100475 = x;
        double r1100476 = r1100475 * r1100475;
        double r1100477 = 1.0;
        double r1100478 = r1100476 - r1100477;
        double r1100479 = sqrt(r1100478);
        double r1100480 = r1100475 + r1100479;
        double r1100481 = log(r1100480);
        return r1100481;
}
↓
double f(double x) {
        double r1100482 = x;
        double r1100483 = 1.0;
        double r1100484 = r1100483 + r1100482;
        double r1100485 = sqrt(r1100484);
        double r1100486 = r1100482 - r1100483;
        double r1100487 = sqrt(r1100486);
        double r1100488 = r1100485 * r1100487;
        double r1100489 = r1100482 + r1100488;
        double r1100490 = log(r1100489);
        return r1100490;
}

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-prod0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2019154 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))