Average Error: 28.9 → 0.0
Time: 7.9s
Precision: 64
Internal Precision: 1344
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.003319952615108916 \lor \neg \left(x \le 0.002925096837680471\right):\\ \;\;\;\;(e^{\log 2 - \log_* (1 + e^{-2 \cdot x})} - 1)^*\\ \mathbf{else}:\\ \;\;\;\;(e^{(\left(\frac{-1}{2} \cdot x\right) \cdot x + \left((\frac{1}{12} \cdot \left({x}^{4}\right) + x)_*\right))_*} - 1)^*\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

Derivation

  1. Split input into 2 regimes

  2. if x < -0.003319952615108916 or 0.002925096837680471 < x

    1. Initial program 0.0

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1\]

    2. Initial simplification0.0

      \[\leadsto \frac{2}{1 + e^{-2 \cdot x}} - 1\]
    3. Using strategy rm

    4. Applied add-exp-log0.0

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

    5. Applied expm1-def0.0

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

    6. Simplified0.0

      \[\leadsto (e^{\color{blue}{\log 2 - \log_* (1 + e^{-2 \cdot x})}} - 1)^*\]

    if -0.003319952615108916 < x < 0.002925096837680471

    1. Initial program 59.1

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1\]

    2. Initial simplification59.1

      \[\leadsto \frac{2}{1 + e^{-2 \cdot x}} - 1\]
    3. Using strategy rm

    4. Applied add-exp-log59.1

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

    5. Applied expm1-def59.1

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

    6. Simplified59.0

      \[\leadsto (e^{\color{blue}{\log 2 - \log_* (1 + e^{-2 \cdot x})}} - 1)^*\]

    7. Taylor expanded around 0 0.0

      \[\leadsto (e^{\color{blue}{\left(x + \frac{1}{12} \cdot {x}^{4}\right) - \frac{1}{2} \cdot {x}^{2}}} - 1)^*\]

    8. Simplified0.0

      \[\leadsto (e^{\color{blue}{(\left(x \cdot \frac{-1}{2}\right) \cdot x + \left((\frac{1}{12} \cdot \left({x}^{4}\right) + x)_*\right))_*}} - 1)^*\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.003319952615108916 \lor \neg \left(x \le 0.002925096837680471\right):\\ \;\;\;\;(e^{\log 2 - \log_* (1 + e^{-2 \cdot x})} - 1)^*\\ \mathbf{else}:\\ \;\;\;\;(e^{(\left(\frac{-1}{2} \cdot x\right) \cdot x + \left((\frac{1}{12} \cdot \left({x}^{4}\right) + x)_*\right))_*} - 1)^*\\ \end{array}\]

Runtime

Time bar (total: 7.9s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes28.90.00.028.8100%
herbie shell --seed 2018297 +o rules:numerics
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))