Average Error: 61.5 → 0.0
Time: 16.0s
Precision: 64
Internal Precision: 1408
\[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
↓
\[-\left(\frac{1}{2} \cdot {x}^{2} + \left(1 + x\right)\right)\]

Error

Bits error versus x

Target

Original61.5
Target0.3
Herbie0.0
\[-\left(\left(\left(1 + x\right) + \frac{x \cdot x}{2}\right) + \frac{5}{12} \cdot {x}^{3}\right)\]

Derivation

  1. Initial program 61.5

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

  2. Taylor expanded around 0 0.0

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

Runtime

Time bar (total: 16.0s)Debug log

herbie shell --seed '#(151349756 408087815 228312487 2538703040 1980610373 1250971417)' 
(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))))