Error

Try it out

Target

Derivation

1. Split input into 2 regimes

2. if x < 0.00016083016722330594

   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. Simplified0.2

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

   5. Applied add-exp-log34.6

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

   6. Taylor expanded around 0 34.6

      \[\leadsto e^{\color{blue}{\left(\frac{5}{24} \cdot {x}^{2} + \log x\right) - \frac{1}{2} \cdot x}}\]
   7. Using strategy rm

   8. Applied sub-neg34.6

      \[\leadsto e^{\color{blue}{\left(\frac{5}{24} \cdot {x}^{2} + \log x\right) + \left(-\frac{1}{2} \cdot x\right)}}\]

   9. Applied exp-sum34.6

      \[\leadsto \color{blue}{e^{\frac{5}{24} \cdot {x}^{2} + \log x} \cdot e^{-\frac{1}{2} \cdot x}}\]

   10. Simplified0.2

       \[\leadsto \color{blue}{\left(x \cdot {\left(e^{\frac{5}{24}}\right)}^{\left(x \cdot x\right)}\right)} \cdot e^{-\frac{1}{2} \cdot x}\]

   if 0.00016083016722330594 < x

   1. Initial program 0.1

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

4. Final simplification0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 0.00016083016722330594:\\ \;\;\;\;e^{\frac{-1}{2} \cdot x} \cdot \left(x \cdot {\left(e^{\frac{5}{24}}\right)}^{\left(x \cdot x\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + x\right)\\ \end{array}\]

Runtime

Time bar (total: 24.3s)Debug logProfile

herbie shell --seed 2018248 
(FPCore (x)
  :name "ln(1 + x)"

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

  (log (+ 1 x)))