Average Error: 32.2 → 0.0
Time: 2.5s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(x + \sqrt{x + \sqrt{1}} \cdot \sqrt{x - \sqrt{1}}\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(x + \sqrt{x + \sqrt{1}} \cdot \sqrt{x - \sqrt{1}}\right)
double f(double x) {
        double r65124 = x;
        double r65125 = r65124 * r65124;
        double r65126 = 1.0;
        double r65127 = r65125 - r65126;
        double r65128 = sqrt(r65127);
        double r65129 = r65124 + r65128;
        double r65130 = log(r65129);
        return r65130;
}
↓
double f(double x) {
        double r65131 = x;
        double r65132 = 1.0;
        double r65133 = sqrt(r65132);
        double r65134 = r65131 + r65133;
        double r65135 = sqrt(r65134);
        double r65136 = r65131 - r65133;
        double r65137 = sqrt(r65136);
        double r65138 = r65135 * r65137;
        double r65139 = r65131 + r65138;
        double r65140 = log(r65139);
        return r65140;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 32.2

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

  3. Applied add-sqr-sqrt32.2

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

  4. Applied difference-of-squares32.2

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

  5. Applied sqrt-prod0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1)))))