Average Error: 38.8 → 0.2
Time: 12.8s
Precision: 64
Internal Precision: 1344
\[\log \left(\frac{1}{1 - x}\right)\]
\[\begin{array}{l} \mathbf{if}\;x \le -0.00012149033212153346:\\ \;\;\;\;\left(-\log \left(\sqrt{1 - x}\right)\right) + \left(-\log \left(\sqrt{1 - x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-\left(\left(-x\right) - \left(x \cdot x\right) \cdot \left(x \cdot \frac{1}{3} + \frac{1}{2}\right)\right)\\ \end{array}\]

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes
  2. if x < -0.00012149033212153346

    1. Initial program 0.1

      \[\log \left(\frac{1}{1 - x}\right)\]
    2. Initial simplification0.1

      \[\leadsto -\log \left(1 - x\right)\]
    3. Using strategy rm
    4. Applied add-sqr-sqrt0.1

      \[\leadsto -\log \color{blue}{\left(\sqrt{1 - x} \cdot \sqrt{1 - x}\right)}\]
    5. Applied log-prod0.1

      \[\leadsto -\color{blue}{\left(\log \left(\sqrt{1 - x}\right) + \log \left(\sqrt{1 - x}\right)\right)}\]

    if -0.00012149033212153346 < x

    1. Initial program 58.8

      \[\log \left(\frac{1}{1 - x}\right)\]
    2. Initial simplification58.8

      \[\leadsto -\log \left(1 - x\right)\]
    3. Taylor expanded around 0 0.2

      \[\leadsto -\color{blue}{\left(-\left(\frac{1}{3} \cdot {x}^{3} + \left(\frac{1}{2} \cdot {x}^{2} + x\right)\right)\right)}\]
    4. Simplified0.2

      \[\leadsto -\color{blue}{\left(\left(-x\right) - \left(x \cdot x\right) \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot x\right)\right)}\]
  3. Recombined 2 regimes into one program.
  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.00012149033212153346:\\ \;\;\;\;\left(-\log \left(\sqrt{1 - x}\right)\right) + \left(-\log \left(\sqrt{1 - x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-\left(\left(-x\right) - \left(x \cdot x\right) \cdot \left(x \cdot \frac{1}{3} + \frac{1}{2}\right)\right)\\ \end{array}\]

Runtime

Time bar (total: 12.8s)Debug log

herbie shell --seed '#(2775764126 3555076145 3898259844 1891440260 2599947619 1948460636)' 
(FPCore (x)
  :name "log(1/(1-x))"
  (log (/ 1 (- 1 x))))