ln(1 + x)

Average Error: 39.2 → 0.6

Time: 3.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;1 + x \le 1.000000000000000666133814775093924254179:\\ \;\;\;\;\mathsf{fma}\left(x, 1, \log 1 - \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \log \left(1 + x\right) + \frac{1}{2} \cdot \log \left(1 + x\right)\\ \end{array}\]

C

double f(double x) {
        double r69945 = 1.0;
        double r69946 = x;
        double r69947 = r69945 + r69946;
        double r69948 = log(r69947);
        return r69948;
}

↓

double f(double x) {
        double r69949 = 1.0;
        double r69950 = x;
        double r69951 = r69949 + r69950;
        double r69952 = 1.0000000000000007;
        bool r69953 = r69951 <= r69952;
        double r69954 = log(r69949);
        double r69955 = 0.5;
        double r69956 = 2.0;
        double r69957 = pow(r69950, r69956);
        double r69958 = pow(r69949, r69956);
        double r69959 = r69957 / r69958;
        double r69960 = r69955 * r69959;
        double r69961 = r69954 - r69960;
        double r69962 = fma(r69950, r69949, r69961);
        double r69963 = log(r69951);
        double r69964 = r69955 * r69963;
        double r69965 = r69964 + r69964;
        double r69966 = r69953 ? r69962 : r69965;
        return r69966;
}

Error

Bits error versus x

Target

Original	39.2
Target	0.3
Herbie	0.6

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

Derivation

1. Split input into 2 regimes

2. if (+ 1.0 x) < 1.0000000000000007

   1. Initial program 59.5

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

   2. Taylor expanded around 0 0.3

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

   3. Simplified 0.3

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 1, \log 1 - \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}\right)}\]

   if 1.0000000000000007 < (+ 1.0 x)

   1. Initial program 1.0

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

   3. Applied add-sqr-sqrt 1.1

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

   4. Applied log-prod 1.1

      \[\leadsto \color{blue}{\log \left(\sqrt{1 + x}\right) + \log \left(\sqrt{1 + x}\right)}\]
   5. Using strategy rm

   6. Applied pow1 1.1

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

   7. Applied sqrt-pow1 1.1

      \[\leadsto \log \left(\sqrt{1 + x}\right) + \log \color{blue}{\left({\left(1 + x\right)}^{\left(\frac{1}{2}\right)}\right)}\]

   8. Applied log-pow 1.0

      \[\leadsto \log \left(\sqrt{1 + x}\right) + \color{blue}{\frac{1}{2} \cdot \log \left(1 + x\right)}\]
   9. Using strategy rm

   10. Applied pow1 1.0

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

   11. Applied sqrt-pow1 1.0

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

   12. Applied log-pow 1.0

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

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 + x \le 1.000000000000000666133814775093924254179:\\ \;\;\;\;\mathsf{fma}\left(x, 1, \log 1 - \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \log \left(1 + x\right) + \frac{1}{2} \cdot \log \left(1 + x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019354 +o rules:numerics
(FPCore (x)
  :name "ln(1 + x)"
  :precision binary64

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

  (log (+ 1 x)))