Average Error: 32.6 → 0.0
Time: 2.6s
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 r41749 = x;
        double r41750 = r41749 * r41749;
        double r41751 = 1.0;
        double r41752 = r41750 - r41751;
        double r41753 = sqrt(r41752);
        double r41754 = r41749 + r41753;
        double r41755 = log(r41754);
        return r41755;
}
↓
double f(double x) {
        double r41756 = x;
        double r41757 = 1.0;
        double r41758 = sqrt(r41757);
        double r41759 = r41756 + r41758;
        double r41760 = sqrt(r41759);
        double r41761 = r41756 - r41758;
        double r41762 = sqrt(r41761);
        double r41763 = r41760 * r41762;
        double r41764 = r41756 + r41763;
        double r41765 = log(r41764);
        return r41765;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 32.6

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

  3. Applied add-sqr-sqrt32.6

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

  4. Applied difference-of-squares32.6

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

  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 2019354 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1)))))