Average Error: 28.6 → 0.0
Time: 29.8s
Precision: 64
Internal Precision: 1344
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.007463228408313171:\\ \;\;\;\;(e^{\log \left(e^{\log 2 - \log_* (1 + e^{-2 \cdot x})}\right)} - 1)^*\\ \mathbf{elif}\;x \le 0.006774998496990162:\\ \;\;\;\;\left(\frac{2}{15} \cdot {x}^{5} + x\right) - {x}^{3} \cdot \frac{1}{3}\\ \mathbf{else}:\\ \;\;\;\;(e^{\sqrt[3]{\left(\left(\log 2 - \log_* (1 + e^{-2 \cdot x})\right) \cdot \left(\log 2 - \log_* (1 + e^{-2 \cdot x})\right)\right) \cdot \left(\log 2 - \log_* (1 + e^{-2 \cdot x})\right)}} - 1)^*\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

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Results

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Derivation

  1. Split input into 3 regimes

  2. if x < -0.007463228408313171

    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

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

    7. Applied add-log-exp0.0

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

    if -0.007463228408313171 < x < 0.006774998496990162

    1. Initial program 58.7

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

    2. Taylor expanded around 0 0.0

      \[\leadsto \color{blue}{\left(x + \frac{2}{15} \cdot {x}^{5}\right) - \frac{1}{3} \cdot {x}^{3}}\]

    if 0.006774998496990162 < 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

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

    7. Applied add-cbrt-cube0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.007463228408313171:\\ \;\;\;\;(e^{\log \left(e^{\log 2 - \log_* (1 + e^{-2 \cdot x})}\right)} - 1)^*\\ \mathbf{elif}\;x \le 0.006774998496990162:\\ \;\;\;\;\left(\frac{2}{15} \cdot {x}^{5} + x\right) - {x}^{3} \cdot \frac{1}{3}\\ \mathbf{else}:\\ \;\;\;\;(e^{\sqrt[3]{\left(\left(\log 2 - \log_* (1 + e^{-2 \cdot x})\right) \cdot \left(\log 2 - \log_* (1 + e^{-2 \cdot x})\right)\right) \cdot \left(\log 2 - \log_* (1 + e^{-2 \cdot x})\right)}} - 1)^*\\ \end{array}\]

Runtime

Time bar (total: 29.8s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes28.60.00.028.6100%
herbie shell --seed 2018296 +o rules:numerics
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))