Average Error: 61.0 → 0.5
Time: 17.1s
Precision: 64
Internal Precision: 1344
\[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
↓
\[\left(-1\right) + \left(\left(-x\right) + {x}^{2} \cdot \left(-\frac{1}{2}\right)\right)\]

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original61.0
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 61.0

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

  2. Initial simplification61.0

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

  3. Taylor expanded around 0 0.5

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

  5. Applied associate-+r+0.5

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

  6. Final simplification0.5

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

Runtime

Time bar (total: 17.1s)Debug logProfile

herbie shell --seed 2018230 
(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))))