Average Error: 0.2 → 0.2
Time: 3.7s
Precision: 64
\[\frac{x}{1 + \sqrt{x + 1}}\]
↓
\[\frac{x}{1 + \sqrt{\sqrt{x + 1}} \cdot \sqrt{\sqrt{x + 1}}}\]
\frac{x}{1 + \sqrt{x + 1}}
↓
\frac{x}{1 + \sqrt{\sqrt{x + 1}} \cdot \sqrt{\sqrt{x + 1}}}
double f(double x) {
        double r112556 = x;
        double r112557 = 1.0;
        double r112558 = r112556 + r112557;
        double r112559 = sqrt(r112558);
        double r112560 = r112557 + r112559;
        double r112561 = r112556 / r112560;
        return r112561;
}
↓
double f(double x) {
        double r112562 = x;
        double r112563 = 1.0;
        double r112564 = r112562 + r112563;
        double r112565 = sqrt(r112564);
        double r112566 = sqrt(r112565);
        double r112567 = r112566 * r112566;
        double r112568 = r112563 + r112567;
        double r112569 = r112562 / r112568;
        return r112569;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.2

    \[\frac{x}{1 + \sqrt{x + 1}}\]
  2. Using strategy rm

  3. Applied add-sqr-sqrt0.2

    \[\leadsto \frac{x}{1 + \sqrt{\color{blue}{\sqrt{x + 1} \cdot \sqrt{x + 1}}}}\]

  4. Applied sqrt-prod0.2

    \[\leadsto \frac{x}{1 + \color{blue}{\sqrt{\sqrt{x + 1}} \cdot \sqrt{\sqrt{x + 1}}}}\]

  5. Final simplification0.2

    \[\leadsto \frac{x}{1 + \sqrt{\sqrt{x + 1}} \cdot \sqrt{\sqrt{x + 1}}}\]

Reproduce

herbie shell --seed 2019362 +o rules:numerics
(FPCore (x)
  :name "Numeric.Log:$clog1p from log-domain-0.10.2.1, B"
  :precision binary64
  (/ x (+ 1 (sqrt (+ x 1)))))