Average Error: 31.6 → 0.1
Time: 11.5s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(x + \left(\sqrt{x + \sqrt{1}} \cdot \left(\sqrt[3]{\sqrt{x - \sqrt{1}}} \cdot \sqrt[3]{\sqrt{x - \sqrt{1}}}\right)\right) \cdot \sqrt[3]{\sqrt{x - \sqrt{1}}}\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(x + \left(\sqrt{x + \sqrt{1}} \cdot \left(\sqrt[3]{\sqrt{x - \sqrt{1}}} \cdot \sqrt[3]{\sqrt{x - \sqrt{1}}}\right)\right) \cdot \sqrt[3]{\sqrt{x - \sqrt{1}}}\right)
double f(double x) {
        double r3147152 = x;
        double r3147153 = r3147152 * r3147152;
        double r3147154 = 1.0;
        double r3147155 = r3147153 - r3147154;
        double r3147156 = sqrt(r3147155);
        double r3147157 = r3147152 + r3147156;
        double r3147158 = log(r3147157);
        return r3147158;
}
↓
double f(double x) {
        double r3147159 = x;
        double r3147160 = 1.0;
        double r3147161 = sqrt(r3147160);
        double r3147162 = r3147159 + r3147161;
        double r3147163 = sqrt(r3147162);
        double r3147164 = r3147159 - r3147161;
        double r3147165 = sqrt(r3147164);
        double r3147166 = cbrt(r3147165);
        double r3147167 = r3147166 * r3147166;
        double r3147168 = r3147163 * r3147167;
        double r3147169 = r3147168 * r3147166;
        double r3147170 = r3147159 + r3147169;
        double r3147171 = log(r3147170);
        return r3147171;
}

Error

Bits error versus x

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Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.6

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

  3. Applied add-sqr-sqrt31.6

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

  4. Applied difference-of-squares31.6

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

  5. Applied sqrt-prod0.0

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

  7. Applied add-cube-cbrt0.1

    \[\leadsto \log \left(x + \sqrt{x + \sqrt{1}} \cdot \color{blue}{\left(\left(\sqrt[3]{\sqrt{x - \sqrt{1}}} \cdot \sqrt[3]{\sqrt{x - \sqrt{1}}}\right) \cdot \sqrt[3]{\sqrt{x - \sqrt{1}}}\right)}\right)\]

  8. Applied associate-*r*0.1

    \[\leadsto \log \left(x + \color{blue}{\left(\sqrt{x + \sqrt{1}} \cdot \left(\sqrt[3]{\sqrt{x - \sqrt{1}}} \cdot \sqrt[3]{\sqrt{x - \sqrt{1}}}\right)\right) \cdot \sqrt[3]{\sqrt{x - \sqrt{1}}}}\right)\]

  9. Final simplification0.1

    \[\leadsto \log \left(x + \left(\sqrt{x + \sqrt{1}} \cdot \left(\sqrt[3]{\sqrt{x - \sqrt{1}}} \cdot \sqrt[3]{\sqrt{x - \sqrt{1}}}\right)\right) \cdot \sqrt[3]{\sqrt{x - \sqrt{1}}}\right)\]

Reproduce

herbie shell --seed 2019172 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1.0)))))