Average Error: 32.5 → 0.1
Time: 16.0s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(x + \left(\sqrt{x + \sqrt{1}} \cdot \sqrt{\sqrt{x} + \sqrt{\sqrt{1}}}\right) \cdot \sqrt{\sqrt{x} - \sqrt{\sqrt{1}}}\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(x + \left(\sqrt{x + \sqrt{1}} \cdot \sqrt{\sqrt{x} + \sqrt{\sqrt{1}}}\right) \cdot \sqrt{\sqrt{x} - \sqrt{\sqrt{1}}}\right)
double f(double x) {
        double r90079 = x;
        double r90080 = r90079 * r90079;
        double r90081 = 1.0;
        double r90082 = r90080 - r90081;
        double r90083 = sqrt(r90082);
        double r90084 = r90079 + r90083;
        double r90085 = log(r90084);
        return r90085;
}
↓
double f(double x) {
        double r90086 = x;
        double r90087 = 1.0;
        double r90088 = sqrt(r90087);
        double r90089 = r90086 + r90088;
        double r90090 = sqrt(r90089);
        double r90091 = sqrt(r90086);
        double r90092 = sqrt(r90088);
        double r90093 = r90091 + r90092;
        double r90094 = sqrt(r90093);
        double r90095 = r90090 * r90094;
        double r90096 = r90091 - r90092;
        double r90097 = sqrt(r90096);
        double r90098 = r90095 * r90097;
        double r90099 = r90086 + r90098;
        double r90100 = log(r90099);
        return r90100;
}

Error

Bits error versus x

Try it out

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. Using strategy rm

  7. Applied add-sqr-sqrt0.1

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

  8. Applied sqrt-prod0.1

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

  9. Applied add-sqr-sqrt0.1

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

  10. Applied difference-of-squares0.1

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

  11. Applied sqrt-prod0.1

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

  12. Applied associate-*r*0.1

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

  13. Final simplification0.1

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

Reproduce

herbie shell --seed 2019235 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1)))))