Average Error: 0.2 → 0.2
Time: 12.4s
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 r128151 = x;
        double r128152 = 1.0;
        double r128153 = r128151 + r128152;
        double r128154 = sqrt(r128153);
        double r128155 = r128152 + r128154;
        double r128156 = r128151 / r128155;
        return r128156;
}
↓
double f(double x) {
        double r128157 = x;
        double r128158 = 1.0;
        double r128159 = r128158 + r128157;
        double r128160 = sqrt(r128159);
        double r128161 = sqrt(r128160);
        double r128162 = r128161 * r128161;
        double r128163 = r128158 + r128162;
        double r128164 = r128157 / r128163;
        return r128164;
}

Error

Bits error versus x

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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)))))