Average Error: 61.3 → 0.0
Time: 9.1s
Precision: 64
Internal precision: 1408
\[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
\[-\frac{{\left(\frac{1}{2} \cdot {x}^2\right)}^3 + {\left(1 + x\right)}^3}{\sqrt[3]{{\left({\left(\frac{1}{2} \cdot {x}^2\right)}^2 + \left({\left(1 + x\right)}^2 - \left(\frac{1}{2} \cdot {x}^2\right) \cdot \left(1 + x\right)\right)\right)}^3}}\]

Error

Bits error versus x

Target

Original61.3
Comparison0.4
Herbie0.0
\[ -\left(\left(\left(1 + x\right) + \frac{{x}^2}{2}\right) + \frac{5}{12} \cdot {x}^{3}\right) \]

Derivation

  1. Initial program 61.3

    \[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
  2. Applied taylor 0.0

    \[\leadsto -\left(\frac{1}{2} \cdot {x}^2 + \left(1 + x\right)\right)\]
  3. Taylor expanded around 0 0.0

    \[\leadsto \color{blue}{-\left(\frac{1}{2} \cdot {x}^2 + \left(1 + x\right)\right)}\]
  4. Using strategy rm
  5. Applied add-cbrt-cube 0.0

    \[\leadsto -\color{blue}{\sqrt[3]{{\left(\frac{1}{2} \cdot {x}^2 + \left(1 + x\right)\right)}^3}}\]
  6. Using strategy rm
  7. Applied flip3-+ 0.0

    \[\leadsto -\sqrt[3]{{\color{blue}{\left(\frac{{\left(\frac{1}{2} \cdot {x}^2\right)}^{3} + {\left(1 + x\right)}^{3}}{{\left(\frac{1}{2} \cdot {x}^2\right)}^2 + \left({\left(1 + x\right)}^2 - \left(\frac{1}{2} \cdot {x}^2\right) \cdot \left(1 + x\right)\right)}\right)}}^3}\]
  8. Applied cube-div 0.0

    \[\leadsto -\sqrt[3]{\color{blue}{\frac{{\left({\left(\frac{1}{2} \cdot {x}^2\right)}^{3} + {\left(1 + x\right)}^{3}\right)}^3}{{\left({\left(\frac{1}{2} \cdot {x}^2\right)}^2 + \left({\left(1 + x\right)}^2 - \left(\frac{1}{2} \cdot {x}^2\right) \cdot \left(1 + x\right)\right)\right)}^3}}}\]
  9. Applied cbrt-div 0.0

    \[\leadsto -\color{blue}{\frac{\sqrt[3]{{\left({\left(\frac{1}{2} \cdot {x}^2\right)}^{3} + {\left(1 + x\right)}^{3}\right)}^3}}{\sqrt[3]{{\left({\left(\frac{1}{2} \cdot {x}^2\right)}^2 + \left({\left(1 + x\right)}^2 - \left(\frac{1}{2} \cdot {x}^2\right) \cdot \left(1 + x\right)\right)\right)}^3}}}\]
  10. Applied simplify 0.0

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

Runtime

Time bar (total: 9.1s) Debug logProfile

Please include this information when filing a bug report:

herbie shell --seed '#(1066500295 745726447 3908002351 725592315 4114972361 2368915013)'
(FPCore (x)
  :name "qlog (example 3.10)"
  :pre (and (< -1 x) (< x 1))

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

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