Error

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Target

Derivation

1. Split input into 2 regimes

2. if x < 0.00011767117963648764

   1. Initial program 58.7

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

   2. Initial simplification58.7

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

   3. Taylor expanded around 0 0.3

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

   4. Simplified0.2

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

   if 0.00011767117963648764 < x

   1. Initial program 0.1

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

   2. Initial simplification0.1

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

4. Final simplification0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 0.00011767117963648764:\\ \;\;\;\;x + \left(x \cdot x\right) \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2018360 
(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: 5.9s)Debug log