Average Error: 29.0 → 0.0
Time: 15.0s
Precision: 64
Internal Precision: 1344
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.004421751435964128:\\ \;\;\;\;\log \left(e^{\frac{2}{e^{-2 \cdot x} + 1} - 1}\right)\\ \mathbf{elif}\;x \le 0.0030292719297623327:\\ \;\;\;\;(e^{(\left(\frac{-1}{2} \cdot x\right) \cdot x + \left((\frac{1}{12} \cdot \left({x}^{4}\right) + x)_*\right))_*} - 1)^*\\ \mathbf{else}:\\ \;\;\;\;(e^{\frac{{\left(\log 2\right)}^{3} - {\left(\log_* (1 + e^{-2 \cdot x})\right)}^{3}}{\left(\log_* (1 + e^{-2 \cdot x}) \cdot \log_* (1 + e^{-2 \cdot x}) + \log 2 \cdot \log_* (1 + e^{-2 \cdot x})\right) + \log 2 \cdot \log 2}} - 1)^*\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

Derivation

  1. Split input into 3 regimes

  2. if x < -0.004421751435964128

    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-log-exp0.0

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

    if -0.004421751435964128 < x < 0.0030292719297623327

    1. Initial program 59.2

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

    2. Initial simplification59.2

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

    4. Applied add-exp-log59.2

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

    5. Applied expm1-def59.2

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

    6. Simplified59.1

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

    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

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

    if 0.0030292719297623327 < 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

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

    8. Applied flip3--0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.004421751435964128:\\ \;\;\;\;\log \left(e^{\frac{2}{e^{-2 \cdot x} + 1} - 1}\right)\\ \mathbf{elif}\;x \le 0.0030292719297623327:\\ \;\;\;\;(e^{(\left(\frac{-1}{2} \cdot x\right) \cdot x + \left((\frac{1}{12} \cdot \left({x}^{4}\right) + x)_*\right))_*} - 1)^*\\ \mathbf{else}:\\ \;\;\;\;(e^{\frac{{\left(\log 2\right)}^{3} - {\left(\log_* (1 + e^{-2 \cdot x})\right)}^{3}}{\left(\log_* (1 + e^{-2 \cdot x}) \cdot \log_* (1 + e^{-2 \cdot x}) + \log 2 \cdot \log_* (1 + e^{-2 \cdot x})\right) + \log 2 \cdot \log 2}} - 1)^*\\ \end{array}\]

Runtime

Time bar (total: 15.0s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes29.00.00.029.0100%
herbie shell --seed 2018263 +o rules:numerics
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))