Average Error: 32.5 → 0.1
Time: 2.3s
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 r34199 = x;
        double r34200 = r34199 * r34199;
        double r34201 = 1.0;
        double r34202 = r34200 - r34201;
        double r34203 = sqrt(r34202);
        double r34204 = r34199 + r34203;
        double r34205 = log(r34204);
        return r34205;
}
↓
double f(double x) {
        double r34206 = x;
        double r34207 = 1.0;
        double r34208 = sqrt(r34207);
        double r34209 = r34206 + r34208;
        double r34210 = sqrt(r34209);
        double r34211 = r34206 - r34208;
        double r34212 = sqrt(r34211);
        double r34213 = r34210 * r34212;
        double r34214 = r34206 + r34213;
        double r34215 = log(r34214);
        return r34215;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 32.5

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

  3. Applied add-sqr-sqrt32.5

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

  4. Applied difference-of-squares32.5

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

  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 2020065 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1)))))