Average Error: 0.2 → 0.2
Time: 11.8s
Precision: 64
\[\frac{x}{1 + \sqrt{x + 1}}\]
↓
\[\frac{x}{1 + \sqrt{\sqrt{1 + x}} \cdot \sqrt{\sqrt{1 + x}}}\]
\frac{x}{1 + \sqrt{x + 1}}
↓
\frac{x}{1 + \sqrt{\sqrt{1 + x}} \cdot \sqrt{\sqrt{1 + x}}}
double f(double x) {
        double r114761 = x;
        double r114762 = 1.0;
        double r114763 = r114761 + r114762;
        double r114764 = sqrt(r114763);
        double r114765 = r114762 + r114764;
        double r114766 = r114761 / r114765;
        return r114766;
}
↓
double f(double x) {
        double r114767 = x;
        double r114768 = 1.0;
        double r114769 = r114768 + r114767;
        double r114770 = sqrt(r114769);
        double r114771 = sqrt(r114770);
        double r114772 = r114771 * r114771;
        double r114773 = r114768 + r114772;
        double r114774 = r114767 / r114773;
        return r114774;
}

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. Simplified0.2

    \[\leadsto \color{blue}{\frac{x}{1 + \sqrt{1 + x}}}\]
  3. Using strategy rm

  4. Applied add-sqr-sqrt0.2

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

  5. Applied sqrt-prod0.2

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

  6. Final simplification0.2

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

Reproduce

herbie shell --seed 2019194 
(FPCore (x)
  :name "Numeric.Log:$clog1p from log-domain-0.10.2.1, B"
  (/ x (+ 1.0 (sqrt (+ x 1.0)))))