Average Error: 32.1 → 0.1
Time: 10.4s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(\left(\sqrt{\sqrt{x + \sqrt{1}}} \cdot \sqrt{x - \sqrt{1}}\right) \cdot \sqrt{\sqrt{x + \sqrt{1}}} + x\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(\left(\sqrt{\sqrt{x + \sqrt{1}}} \cdot \sqrt{x - \sqrt{1}}\right) \cdot \sqrt{\sqrt{x + \sqrt{1}}} + x\right)
double f(double x) {
        double r73415 = x;
        double r73416 = r73415 * r73415;
        double r73417 = 1.0;
        double r73418 = r73416 - r73417;
        double r73419 = sqrt(r73418);
        double r73420 = r73415 + r73419;
        double r73421 = log(r73420);
        return r73421;
}
↓
double f(double x) {
        double r73422 = x;
        double r73423 = 1.0;
        double r73424 = sqrt(r73423);
        double r73425 = r73422 + r73424;
        double r73426 = sqrt(r73425);
        double r73427 = sqrt(r73426);
        double r73428 = r73422 - r73424;
        double r73429 = sqrt(r73428);
        double r73430 = r73427 * r73429;
        double r73431 = r73430 * r73427;
        double r73432 = r73431 + r73422;
        double r73433 = log(r73432);
        return r73433;
}

Error

Bits error versus x

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Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 32.1

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

  3. Applied add-sqr-sqrt32.1

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

  4. Applied difference-of-squares32.1

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

  5. Applied sqrt-prod0.0

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

  6. Simplified0.0

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

  8. Applied add-sqr-sqrt0.0

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

  9. Applied sqrt-prod0.1

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

  10. Applied associate-*l*0.1

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

  11. Simplified0.1

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

  12. Final simplification0.1

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

Reproduce

herbie shell --seed 2019195 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1.0)))))