Error

Derivation

1. Initial program 61.3b

   \[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]

2. Applied taylor 60.3b

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

3. Taylor expanded around 0 60.3b

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

4. Applied simplify 60.3b

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

5. Applied taylor 0.4b

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

6. Taylor expanded around 0 0.4b

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

7. Applied taylor 0.0b

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

8. Taylor expanded around 0 0.0b

   \[\leadsto \color{blue}{-\left(x + \left(1 + \frac{1}{2} \cdot {x}^2\right)\right)}\]
9. Removed slow pow expressions

Runtime

Total time: 19.0s Debug log

herbie --seed '#(3110316182 520928328 2340230692 2905220866 2631597852 2842409522)'
(FPCore (x)
  :name "NMSE example 3.10"
  
  :target
  (- (+ (+ (+ 1 x) (/ (sqr x) 2)) (* 5/12 (pow x 3))))(/ (log (- 1 x)) (log (+ 1 x))))