ln(1 + x)

Average Error: 39.3 → 0.3

Time: 11.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le 6.37915373787209162315678737109614360179 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{-1}{2}}{1} \cdot \frac{x \cdot x}{1} + \left(1 \cdot x + \log 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \log \left(x + 1\right) + \log \left(\sqrt{x + 1}\right)\\ \end{array}\]

C

double f(double x) {
        double r54948 = 1.0;
        double r54949 = x;
        double r54950 = r54948 + r54949;
        double r54951 = log(r54950);
        return r54951;
}

↓

double f(double x) {
        double r54952 = x;
        double r54953 = 6.379153737872092e-06;
        bool r54954 = r54952 <= r54953;
        double r54955 = -0.5;
        double r54956 = 1.0;
        double r54957 = r54955 / r54956;
        double r54958 = r54952 * r54952;
        double r54959 = r54958 / r54956;
        double r54960 = r54957 * r54959;
        double r54961 = r54956 * r54952;
        double r54962 = log(r54956);
        double r54963 = r54961 + r54962;
        double r54964 = r54960 + r54963;
        double r54965 = 0.5;
        double r54966 = r54952 + r54956;
        double r54967 = log(r54966);
        double r54968 = r54965 * r54967;
        double r54969 = sqrt(r54966);
        double r54970 = log(r54969);
        double r54971 = r54968 + r54970;
        double r54972 = r54954 ? r54964 : r54971;
        return r54972;
}

Error

Bits error versus x

Target

Original	39.3
Target	0.2
Herbie	0.3

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

Derivation

1. Split input into 2 regimes

2. if x < 6.379153737872092e-06

   1. Initial program 59.0

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

   2. Simplified 59.0

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

   3. Taylor expanded around 0 0.4

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

   4. Simplified 0.4

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

   if 6.379153737872092e-06 < x

   1. Initial program 0.1

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

   2. Simplified 0.1

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

   4. Applied add-sqr-sqrt 0.1

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

   5. Applied log-prod 0.1

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

   6. Simplified 0.1

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

   7. Simplified 0.1

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

   9. Applied pow1/2 0.1

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

   10. Applied log-pow 0.1

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

   11. Simplified 0.1

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 6.37915373787209162315678737109614360179 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{-1}{2}}{1} \cdot \frac{x \cdot x}{1} + \left(1 \cdot x + \log 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \log \left(x + 1\right) + \log \left(\sqrt{x + 1}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019195 
(FPCore (x)
  :name "ln(1 + x)"

  :herbie-target
  (if (== (+ 1.0 x) 1.0) x (/ (* x (log (+ 1.0 x))) (- (+ 1.0 x) 1.0)))

  (log (+ 1.0 x)))