Average Error: 30.8 → 0.0
Time: 12.7s
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 r11572494 = x;
        double r11572495 = r11572494 * r11572494;
        double r11572496 = 1.0;
        double r11572497 = r11572495 - r11572496;
        double r11572498 = sqrt(r11572497);
        double r11572499 = r11572494 + r11572498;
        double r11572500 = log(r11572499);
        return r11572500;
}
↓
double f(double x) {
        double r11572501 = x;
        double r11572502 = 1.0;
        double r11572503 = r11572502 + r11572501;
        double r11572504 = sqrt(r11572503);
        double r11572505 = r11572501 - r11572502;
        double r11572506 = sqrt(r11572505);
        double r11572507 = r11572504 * r11572506;
        double r11572508 = r11572501 + r11572507;
        double r11572509 = log(r11572508);
        return r11572509;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 30.8

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

  3. Applied *-un-lft-identity30.8

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

  4. Applied difference-of-squares30.8

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

  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 2019124 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))