Average Error: 60.9 → 0.4
Time: 2.0m
Precision: 64
Internal Precision: 1408
\[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
↓
\[\log \left(e^{\frac{\left(\left(-x\right) \cdot x\right) \cdot \left(\frac{1}{3} \cdot x + \frac{1}{2}\right) + \left(-x\right)}{x + \left(x \cdot x\right) \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right)}}\right)\]

Error

Bits error versus x

Target

Original60.9
Target0.3
Herbie0.4
\[-\left(\left(\left(1 + x\right) + \frac{x \cdot x}{2}\right) + \frac{5}{12} \cdot {x}^{3}\right)\]

Derivation

  1. Initial program 60.9

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

  2. Taylor expanded around 0 60.3

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

  3. Taylor expanded around 0 0.4

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

  4. Applied simplify0.4

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

  6. Applied add-log-exp0.4

    \[\leadsto \color{blue}{\log \left(e^{\frac{\left(\left(-x\right) \cdot x\right) \cdot \left(\frac{1}{3} \cdot x + \frac{1}{2}\right) + \left(-x\right)}{x + \left(x \cdot x\right) \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right)}}\right)}\]
  7. Removed slow pow expressions.

Runtime

Time bar (total: 2.0m)Debug logProfile

herbie shell --seed '#(1063185673 2139736501 2393378123 1907444849 1070993796 1007244912)' 
(FPCore (x)
  :name "qlog (example 3.10)"
  :pre (and (< -1 x) (< x 1))

  :herbie-target
  (- (+ (+ (+ 1 x) (/ (* x x) 2)) (* 5/12 (pow x 3))))

  (/ (log (- 1 x)) (log (+ 1 x))))