Average Error: 31.5 → 0.1
Time: 9.0s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(x + \sqrt{\sqrt{x + \sqrt{1}}} \cdot \left(\sqrt{\sqrt{x + \sqrt{1}}} \cdot \sqrt{x - \sqrt{1}}\right)\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(x + \sqrt{\sqrt{x + \sqrt{1}}} \cdot \left(\sqrt{\sqrt{x + \sqrt{1}}} \cdot \sqrt{x - \sqrt{1}}\right)\right)
double f(double x) {
        double r64296 = x;
        double r64297 = r64296 * r64296;
        double r64298 = 1.0;
        double r64299 = r64297 - r64298;
        double r64300 = sqrt(r64299);
        double r64301 = r64296 + r64300;
        double r64302 = log(r64301);
        return r64302;
}
↓
double f(double x) {
        double r64303 = x;
        double r64304 = 1.0;
        double r64305 = sqrt(r64304);
        double r64306 = r64303 + r64305;
        double r64307 = sqrt(r64306);
        double r64308 = sqrt(r64307);
        double r64309 = r64303 - r64305;
        double r64310 = sqrt(r64309);
        double r64311 = r64308 * r64310;
        double r64312 = r64308 * r64311;
        double r64313 = r64303 + r64312;
        double r64314 = log(r64313);
        return r64314;
}

Error

Bits error versus x

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Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.5

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

  3. Applied add-sqr-sqrt31.5

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

  4. Applied difference-of-squares31.5

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

  5. Applied sqrt-prod0.1

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

  7. Applied add-sqr-sqrt0.1

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

  8. Applied sqrt-prod0.1

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

  9. Applied associate-*l*0.1

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

  10. Final simplification0.1

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

Reproduce

herbie shell --seed 2020045 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1)))))