Average Error: 60.9 → 0.5
Time: 3.4m
Precision: 64
Internal Precision: 1344
\[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
↓
\[\frac{\left(-\frac{1}{2}\right) \cdot {x}^{2} + \left(\left(-x\right) + \left(-\frac{1}{3}\right) \cdot {x}^{3}\right)}{\left(\frac{1}{3} \cdot {x}^{3} + x\right) - {x}^{2} \cdot \frac{1}{2}}\]

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

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 60.3

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

  3. Taylor expanded around 0 0.5

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

  4. Final simplification0.5

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

Runtime

Time bar (total: 3.4m)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))))