Average Error: 29.6 → 0.1
Time: 34.3s
Precision: 64
Internal Precision: 1344
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{2}{1 + e^{-2 \cdot x}} - 1 \le -1.2592515247767267 \cdot 10^{-05}:\\ \;\;\;\;e^{\log 2 - \log_* (1 + e^{-2 \cdot x})} - 1\\ \mathbf{if}\;\frac{2}{1 + e^{-2 \cdot x}} - 1 \le 4.372124356457306 \cdot 10^{-08}:\\ \;\;\;\;\left(x + \frac{2}{15} \cdot {x}^{5}\right) - \frac{1}{3} \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;e^{\log 2 - \log_* (1 + e^{-2 \cdot x})} - 1\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

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Results

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Derivation

  1. Split input into 2 regimes

  2. if (- (/ 2 (+ 1 (exp (* -2 x)))) 1) < -1.2592515247767267e-05 or 4.372124356457306e-08 < (- (/ 2 (+ 1 (exp (* -2 x)))) 1)

    1. Initial program 0.2

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

    3. Applied add-exp-log0.2

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

    4. Applied simplify0.2

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

    if -1.2592515247767267e-05 < (- (/ 2 (+ 1 (exp (* -2 x)))) 1) < 4.372124356457306e-08

    1. Initial program 59.6

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

    2. Taylor expanded around 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.

Runtime

Time bar (total: 34.3s)Debug logProfile

herbie shell --seed 2019053 +o rules:numerics
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))