Average Error: 61.1 → 0.0

Time: 28.8s

Precision: 64

Internal Precision: 1344

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

↓

\[\log \left({e}^{\left(\frac{\log_* (1 + \left(-x\right))}{\log_* (1 + x)}\right)}\right)\]

Error

The X axis uses an exponential scale — Bits error versus `x`

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In
Out

Enter valid numbers for all inputs

Target

Original	61.1
Target	0.3
Herbie	0.0

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

Derivation

Initial program 61.1
\[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
Initial simplification60.1
\[\leadsto \frac{\log \left(1 - x\right)}{\log_* (1 + x)}\]
Using strategy rm
Applied sub-neg60.1
\[\leadsto \frac{\log \color{blue}{\left(1 + \left(-x\right)\right)}}{\log_* (1 + x)}\]
Applied log1p-def0.0
\[\leadsto \frac{\color{blue}{\log_* (1 + \left(-x\right))}}{\log_* (1 + x)}\]
Using strategy rm
Applied add-log-exp0.0
\[\leadsto \color{blue}{\log \left(e^{\frac{\log_* (1 + \left(-x\right))}{\log_* (1 + x)}}\right)}\]
Using strategy rm
Applied *-un-lft-identity0.0
\[\leadsto \log \left(e^{\color{blue}{1 \cdot \frac{\log_* (1 + \left(-x\right))}{\log_* (1 + x)}}}\right)\]
Applied exp-prod0.0
\[\leadsto \log \color{blue}{\left({\left(e^{1}\right)}^{\left(\frac{\log_* (1 + \left(-x\right))}{\log_* (1 + x)}\right)}\right)}\]
Simplified0.0
\[\leadsto \log \left({\color{blue}{e}}^{\left(\frac{\log_* (1 + \left(-x\right))}{\log_* (1 + x)}\right)}\right)\]
Final simplification0.0
\[\leadsto \log \left({e}^{\left(\frac{\log_* (1 + \left(-x\right))}{\log_* (1 + x)}\right)}\right)\]

Runtime

herbie shell --seed 2018248 +o rules:numerics
(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))))