ln(1 + x)

Average Error: 39.2 → 0.2

Time: 11.4s

Precision: 64

Math

\[\log \left(1 + x\right)\]

↓

\[\begin{array}{l} \mathbf{if}\;x \le 0.00014341067325748845:\\ \;\;\;\;x + \left(x \cdot \left(\frac{-1}{2} + \frac{1}{3} \cdot x\right)\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt{x + 1}\right) + \log \left(\sqrt{x + 1}\right)\\ \end{array}\]

C

double f(double x) {
        double r1100926 = 1.0;
        double r1100927 = x;
        double r1100928 = r1100926 + r1100927;
        double r1100929 = log(r1100928);
        return r1100929;
}

↓

double f(double x) {
        double r1100930 = x;
        double r1100931 = 0.00014341067325748845;
        bool r1100932 = r1100930 <= r1100931;
        double r1100933 = -0.5;
        double r1100934 = 0.3333333333333333;
        double r1100935 = r1100934 * r1100930;
        double r1100936 = r1100933 + r1100935;
        double r1100937 = r1100930 * r1100936;
        double r1100938 = r1100937 * r1100930;
        double r1100939 = r1100930 + r1100938;
        double r1100940 = 1.0;
        double r1100941 = r1100930 + r1100940;
        double r1100942 = sqrt(r1100941);
        double r1100943 = log(r1100942);
        double r1100944 = r1100943 + r1100943;
        double r1100945 = r1100932 ? r1100939 : r1100944;
        return r1100945;
}

Error

Bits error versus x

Target

Original	39.2
Target	0.2
Herbie	0.2

\[\begin{array}{l} \mathbf{if}\;1 + x = 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \log \left(1 + x\right)}{\left(1 + x\right) - 1}\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if x < 0.00014341067325748845

   1. Initial program 59.0

      \[\log \left(1 + x\right)\]

   2. Taylor expanded around 0 0.2

      \[\leadsto \color{blue}{\left(x + \frac{1}{3} \cdot {x}^{3}\right) - \frac{1}{2} \cdot {x}^{2}}\]

   3. Simplified 0.2

      \[\leadsto \color{blue}{x + \left(x \cdot \left(\frac{-1}{2} + \frac{1}{3} \cdot x\right)\right) \cdot x}\]

   if 0.00014341067325748845 < x

   1. Initial program 0.1

      \[\log \left(1 + x\right)\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt 0.1

      \[\leadsto \log \color{blue}{\left(\sqrt{1 + x} \cdot \sqrt{1 + x}\right)}\]

   4. Applied log-prod 0.1

      \[\leadsto \color{blue}{\log \left(\sqrt{1 + x}\right) + \log \left(\sqrt{1 + x}\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 0.00014341067325748845:\\ \;\;\;\;x + \left(x \cdot \left(\frac{-1}{2} + \frac{1}{3} \cdot x\right)\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt{x + 1}\right) + \log \left(\sqrt{x + 1}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019128 
(FPCore (x)
  :name "ln(1 + x)"

  :herbie-target
  (if (== (+ 1 x) 1) x (/ (* x (log (+ 1 x))) (- (+ 1 x) 1)))

  (log (+ 1 x)))