Average Error: 28.7 → 0.0
Time: 37.9s
Precision: 64
Internal Precision: 128
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.0068219427175233716:\\ \;\;\;\;(e^{\log 2 - \log_* (1 + e^{-2 \cdot x})} - 1)^*\\ \mathbf{elif}\;x \le 0.006602655597416138:\\ \;\;\;\;(\left(\frac{-1}{3} \cdot x\right) \cdot \left(x \cdot x\right) + \left((\frac{2}{15} \cdot \left({x}^{5}\right) + x)_*\right))_*\\ \mathbf{else}:\\ \;\;\;\;(e^{\log 2 - \log_* (1 + e^{-2 \cdot x})} - 1)^*\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

Derivation

  1. Split input into 2 regimes

  2. if x < -0.0068219427175233716 or 0.006602655597416138 < x

    1. Initial program 0.0

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

    3. Applied add-exp-log0.0

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

    4. Applied add-exp-log0.0

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

    5. Applied div-exp0.0

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

    6. Applied expm1-def0.0

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

    7. Simplified0.0

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

    if -0.0068219427175233716 < x < 0.006602655597416138

    1. Initial program 58.8

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

    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. Simplified0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.0068219427175233716:\\ \;\;\;\;(e^{\log 2 - \log_* (1 + e^{-2 \cdot x})} - 1)^*\\ \mathbf{elif}\;x \le 0.006602655597416138:\\ \;\;\;\;(\left(\frac{-1}{3} \cdot x\right) \cdot \left(x \cdot x\right) + \left((\frac{2}{15} \cdot \left({x}^{5}\right) + x)_*\right))_*\\ \mathbf{else}:\\ \;\;\;\;(e^{\log 2 - \log_* (1 + e^{-2 \cdot x})} - 1)^*\\ \end{array}\]

Reproduce

herbie shell --seed 2019088 +o rules:numerics
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))