Average Error: 0.2 → 0.2
Time: 13.2s
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 r139752 = x;
        double r139753 = 1.0;
        double r139754 = r139752 + r139753;
        double r139755 = sqrt(r139754);
        double r139756 = r139753 + r139755;
        double r139757 = r139752 / r139756;
        return r139757;
}
↓
double f(double x) {
        double r139758 = x;
        double r139759 = 1.0;
        double r139760 = r139759 + r139758;
        double r139761 = sqrt(r139760);
        double r139762 = sqrt(r139761);
        double r139763 = r139762 * r139762;
        double r139764 = r139759 + r139763;
        double r139765 = r139758 / r139764;
        return r139765;
}

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. Simplified0.2

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

  6. Simplified0.2

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

  7. Final simplification0.2

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

Reproduce

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