Numeric.Log:$clog1p from log-domain-0.10.2.1, B

Average Error: 0.2 → 0.2

Time: 15.7s

Precision: 64

Math

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

↓

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

C

double f(double x) {
        double r7519879 = x;
        double r7519880 = 1.0;
        double r7519881 = r7519879 + r7519880;
        double r7519882 = sqrt(r7519881);
        double r7519883 = r7519880 + r7519882;
        double r7519884 = r7519879 / r7519883;
        return r7519884;
}

↓

double f(double x) {
        double r7519885 = x;
        double r7519886 = 1.0;
        double r7519887 = 1.0;
        double r7519888 = r7519885 + r7519887;
        double r7519889 = sqrt(r7519888);
        double r7519890 = r7519887 + r7519889;
        double r7519891 = r7519886 / r7519890;
        double r7519892 = r7519885 * r7519891;
        return r7519892;
}

Error

Bits error versus x

Derivation

1. Initial program 0.2

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

3. Applied div-inv 0.2

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

4. Final simplification 0.2

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

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (x)
  :name "Numeric.Log:$clog1p from log-domain-0.10.2.1, B"
  (/ x (+ 1.0 (sqrt (+ x 1.0)))))