Average Error: 29.5 → 0.0
Time: 38.6s
Precision: 64
Internal Precision: 128
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.006968916312078497 \lor \neg \left(x \le 0.005263739581790171\right):\\ \;\;\;\;(e^{\log 2 - \log_* (1 + e^{-2 \cdot x})} - 1)^*\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{15} \cdot {x}^{5} + x\right) - {x}^{3} \cdot \frac{1}{3}\\ \end{array}\]

Error

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Bits error versus y

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Results

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Derivation

  1. Split input into 2 regimes

  2. if x < -0.006968916312078497 or 0.005263739581790171 < x

    1. Initial program 0.0

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

    3. Applied add-exp-log0.0

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

    4. Applied expm1-def0.0

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

    5. Simplified0.0

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

    if -0.006968916312078497 < x < 0.005263739581790171

    1. Initial program 58.9

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

    2. Taylor expanded around 0 0.0

      \[\leadsto \color{blue}{\left(x + \frac{2}{15} \cdot {x}^{5}\right) - \frac{1}{3} \cdot {x}^{3}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.006968916312078497 \lor \neg \left(x \le 0.005263739581790171\right):\\ \;\;\;\;(e^{\log 2 - \log_* (1 + e^{-2 \cdot x})} - 1)^*\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{15} \cdot {x}^{5} + x\right) - {x}^{3} \cdot \frac{1}{3}\\ \end{array}\]

Runtime

Time bar (total: 38.6s)Debug logProfile

herbie shell --seed 2018348 +o rules:numerics
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))