Average Error: 60.9 → 0.5
Time: 1.6m
Precision: 64
Internal Precision: 1344
\[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
↓
\[-\sqrt[3]{{\left(\left(\left(x \cdot x\right) \cdot \frac{1}{2} + x\right) + 1\right)}^{3}}\]

Error

Bits error versus x

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Results

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Target

Original60.9
Target0.3
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 60.9

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

  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-cbrt-cube0.5

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

  5. Applied simplify0.5

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

  7. Applied associate-+r+0.5

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

Runtime

Time bar (total: 1.6m)Debug logProfile

herbie shell --seed 2018214 
(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))))