Average Error: 31.8 → 0.0
Time: 15.5s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(\sqrt{x - \sqrt{1}} \cdot \sqrt{\sqrt{1} + x} + x\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(\sqrt{x - \sqrt{1}} \cdot \sqrt{\sqrt{1} + x} + x\right)
double f(double x) {
        double r1564109 = x;
        double r1564110 = r1564109 * r1564109;
        double r1564111 = 1.0;
        double r1564112 = r1564110 - r1564111;
        double r1564113 = sqrt(r1564112);
        double r1564114 = r1564109 + r1564113;
        double r1564115 = log(r1564114);
        return r1564115;
}
↓
double f(double x) {
        double r1564116 = x;
        double r1564117 = 1.0;
        double r1564118 = sqrt(r1564117);
        double r1564119 = r1564116 - r1564118;
        double r1564120 = sqrt(r1564119);
        double r1564121 = r1564118 + r1564116;
        double r1564122 = sqrt(r1564121);
        double r1564123 = r1564120 * r1564122;
        double r1564124 = r1564123 + r1564116;
        double r1564125 = log(r1564124);
        return r1564125;
}

Error

Bits error versus x

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Results

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Derivation

  1. Initial program 31.8

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

  3. Applied add-sqr-sqrt31.8

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

  4. Applied difference-of-squares31.8

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

  5. Applied sqrt-prod0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2019200 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1.0)))))