Average Error: 60.8 → 0.5
Time: 21.1s
Precision: 64
Internal Precision: 1408
\[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
↓
\[\left(x \cdot x\right) \cdot \left(-\frac{1}{2}\right) + \left(-\left(x + 1\right)\right)\]

Error

Bits error versus x

Target

Original60.8
Target0.4
Herbie0.5
\[-\left(\left(\left(1 + x\right) + \frac{x \cdot x}{2}\right) + \frac{5}{12} \cdot {x}^{3}\right)\]

Derivation

  1. Initial program 60.8

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

  2. Taylor expanded around 0 0.5

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

  4. Applied add-log-exp0.5

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

  6. Applied simplify0.5

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

Runtime

Time bar (total: 21.1s)Debug log

herbie shell --seed '#(633950927 2092594946 1442981 2827247922 2812758452 390991499)' 
(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))))