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Target

Derivation

1. Split input into 2 regimes

2. if x < 0.00014110950786400104

   1. Initial program 58.9

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

   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. Simplified0.2

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

   if 0.00014110950786400104 < x

   1. Initial program 0.1

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

   3. Applied add-sqr-sqrt0.1

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

   4. Applied log-prod0.1

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

   6. Applied add-sqr-sqrt0.1

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

   7. Applied log-prod0.1

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

   9. Applied pow1/20.1

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

   10. Applied log-pow0.1

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

4. Final simplification0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 0.00014110950786400104:\\ \;\;\;\;x + \left(x \cdot x\right) \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log \left(\sqrt{\sqrt{x + 1}}\right) + \log \left(\sqrt{\sqrt{x + 1}}\right)\right) + \log \left(x + 1\right) \cdot \frac{1}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019004 
(FPCore (x)
  :name "ln(1 + x)"

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

  (log (+ 1 x)))

Details

Time bar (total: 10.6s)Debug log