Average Error: 61.0 → 0.7
Time: 22.3s
Precision: 64
Internal Precision: 1344
\[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\log \left(1 - x\right)}{\log \left(x + 1\right)} \le -0.9317643058325272:\\ \;\;\;\;\frac{\log \left(1 - x\right)}{\log \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-1 - x\right) - x \cdot \left(\frac{1}{2} \cdot x\right)\\ \end{array}\]

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original61.0
Target0.3
Herbie0.7
\[-\left(\left(\left(1 + x\right) + \frac{x \cdot x}{2}\right) + \frac{5}{12} \cdot {x}^{3}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (log (- 1 x)) (log (+ 1 x))) < -0.9317643058325272

    1. Initial program 13.3

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

    if -0.9317643058325272 < (/ (log (- 1 x)) (log (+ 1 x)))

    1. Initial program 63.5

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

    2. Taylor expanded around 0 0.1

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

    3. Simplified0.1

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\log \left(1 - x\right)}{\log \left(x + 1\right)} \le -0.9317643058325272:\\ \;\;\;\;\frac{\log \left(1 - x\right)}{\log \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-1 - x\right) - x \cdot \left(\frac{1}{2} \cdot x\right)\\ \end{array}\]

Runtime

Time bar (total: 22.3s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes0.40.70.00.4-70.8%
herbie shell --seed 2018340 
(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))))