Average Error: 33.1 → 0.1
Time: 6.0s
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 r62208 = x;
        double r62209 = r62208 * r62208;
        double r62210 = 1.0;
        double r62211 = r62209 - r62210;
        double r62212 = sqrt(r62211);
        double r62213 = r62208 + r62212;
        double r62214 = log(r62213);
        return r62214;
}
↓
double f(double x) {
        double r62215 = x;
        double r62216 = 1.0;
        double r62217 = sqrt(r62216);
        double r62218 = r62215 + r62217;
        double r62219 = sqrt(r62218);
        double r62220 = r62215 - r62217;
        double r62221 = sqrt(r62220);
        double r62222 = r62219 * r62221;
        double r62223 = r62215 + r62222;
        double r62224 = log(r62223);
        return r62224;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 33.1

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

  3. Applied add-sqr-sqrt33.1

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

  4. Applied difference-of-squares33.1

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

  5. Applied sqrt-prod0.1

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

  6. Final simplification0.1

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

Reproduce

herbie shell --seed 2019297 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1)))))