Average Error: 30.7 → 0.1
Time: 12.2s
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 r6765133 = x;
        double r6765134 = r6765133 * r6765133;
        double r6765135 = 1.0;
        double r6765136 = r6765134 - r6765135;
        double r6765137 = sqrt(r6765136);
        double r6765138 = r6765133 + r6765137;
        double r6765139 = log(r6765138);
        return r6765139;
}
↓
double f(double x) {
        double r6765140 = x;
        double r6765141 = 1.0;
        double r6765142 = r6765141 + r6765140;
        double r6765143 = sqrt(r6765142);
        double r6765144 = r6765140 - r6765141;
        double r6765145 = sqrt(r6765144);
        double r6765146 = r6765143 * r6765145;
        double r6765147 = r6765140 + r6765146;
        double r6765148 = log(r6765147);
        return r6765148;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 30.7

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

  3. Applied difference-of-sqr-130.7

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

  4. Applied sqrt-prod0.1

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

  5. Final simplification0.1

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

Reproduce

herbie shell --seed 2019107 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))