Average Error: 30.9 → 0.1
Time: 11.5s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(x + \sqrt{1 + x} \cdot \sqrt{x - 1}\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(x + \sqrt{1 + x} \cdot \sqrt{x - 1}\right)
double f(double x) {
        double r2812471 = x;
        double r2812472 = r2812471 * r2812471;
        double r2812473 = 1.0;
        double r2812474 = r2812472 - r2812473;
        double r2812475 = sqrt(r2812474);
        double r2812476 = r2812471 + r2812475;
        double r2812477 = log(r2812476);
        return r2812477;
}
↓
double f(double x) {
        double r2812478 = x;
        double r2812479 = 1.0;
        double r2812480 = r2812479 + r2812478;
        double r2812481 = sqrt(r2812480);
        double r2812482 = r2812478 - r2812479;
        double r2812483 = sqrt(r2812482);
        double r2812484 = r2812481 * r2812483;
        double r2812485 = r2812478 + r2812484;
        double r2812486 = log(r2812485);
        return r2812486;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 30.9

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

  3. Applied *-un-lft-identity30.9

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

  4. Applied difference-of-squares30.9

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

  5. Applied sqrt-prod0.1

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

  6. Final simplification0.1

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

Reproduce

herbie shell --seed 2019164 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))