Average Error: 30.8 → 0.0
Time: 19.1s
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 r1351141 = x;
        double r1351142 = r1351141 * r1351141;
        double r1351143 = 1.0;
        double r1351144 = r1351142 - r1351143;
        double r1351145 = sqrt(r1351144);
        double r1351146 = r1351141 + r1351145;
        double r1351147 = log(r1351146);
        return r1351147;
}
↓
double f(double x) {
        double r1351148 = x;
        double r1351149 = 1.0;
        double r1351150 = r1351149 + r1351148;
        double r1351151 = sqrt(r1351150);
        double r1351152 = r1351148 - r1351149;
        double r1351153 = sqrt(r1351152);
        double r1351154 = r1351151 * r1351153;
        double r1351155 = r1351148 + r1351154;
        double r1351156 = log(r1351155);
        return r1351156;
}

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 2019138 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))