Average Error: 60.9 → 0.4
Time: 45.8s
Precision: 64
Internal Precision: 1344
\[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
↓
\[-\log \left(e^{\frac{1}{2} \cdot {x}^{2}} \cdot e^{1 + x}\right)\]

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 0.4

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

  4. Applied add-log-exp0.4

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

  5. Applied add-log-exp0.4

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

  6. Applied sum-log0.4

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

Runtime

Time bar (total: 45.8s)Debug logProfile

herbie shell --seed '#(1071852389 864846987 1238109217 3425890003 4124793586 650694553)' 
(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))))