Average Error: 31.2 → 0.1
Time: 14.1s
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 r1962955 = x;
        double r1962956 = r1962955 * r1962955;
        double r1962957 = 1.0;
        double r1962958 = r1962956 - r1962957;
        double r1962959 = sqrt(r1962958);
        double r1962960 = r1962955 + r1962959;
        double r1962961 = log(r1962960);
        return r1962961;
}
↓
double f(double x) {
        double r1962962 = x;
        double r1962963 = 1.0;
        double r1962964 = r1962963 + r1962962;
        double r1962965 = sqrt(r1962964);
        double r1962966 = r1962962 - r1962963;
        double r1962967 = sqrt(r1962966);
        double r1962968 = r1962965 * r1962967;
        double r1962969 = r1962962 + r1962968;
        double r1962970 = log(r1962969);
        return r1962970;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.2

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

  3. Applied difference-of-sqr-131.2

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

  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 2019129 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))