Average Error: 29.4 → 0.0
Time: 12.4s
Precision: 64
Internal Precision: 128
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.0071335074373506824:\\ \;\;\;\;\sqrt[3]{\frac{2}{e^{-2 \cdot x} + 1} - 1} \cdot \left(\sqrt[3]{\frac{2}{e^{-2 \cdot x} + 1} - 1} \cdot \sqrt[3]{\frac{2}{e^{-2 \cdot x} + 1} - 1}\right)\\ \mathbf{elif}\;x \le 0.008092968034014079:\\ \;\;\;\;(\left(x \cdot \left(x \cdot \frac{-1}{3}\right)\right) \cdot x + \left((\frac{2}{15} \cdot \left({x}^{5}\right) + x)_*\right))_*\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{2}{e^{-2 \cdot x} + 1} - 1} \cdot \left(\sqrt[3]{\frac{2}{e^{-2 \cdot x} + 1} - 1} \cdot \sqrt[3]{\frac{2}{e^{-2 \cdot x} + 1} - 1}\right)\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

Derivation

  1. Split input into 2 regimes

  2. if x < -0.0071335074373506824 or 0.008092968034014079 < x

    1. Initial program 0.0

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

    2. Taylor expanded around -inf 0.0

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

    3. Simplified0.0

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

    5. Applied add-cube-cbrt0.0

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

    if -0.0071335074373506824 < x < 0.008092968034014079

    1. Initial program 59.1

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

    2. Taylor expanded around -inf 59.1

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

    3. Simplified59.1

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

    4. Taylor expanded around 0 0.0

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

    5. Simplified0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.0071335074373506824:\\ \;\;\;\;\sqrt[3]{\frac{2}{e^{-2 \cdot x} + 1} - 1} \cdot \left(\sqrt[3]{\frac{2}{e^{-2 \cdot x} + 1} - 1} \cdot \sqrt[3]{\frac{2}{e^{-2 \cdot x} + 1} - 1}\right)\\ \mathbf{elif}\;x \le 0.008092968034014079:\\ \;\;\;\;(\left(x \cdot \left(x \cdot \frac{-1}{3}\right)\right) \cdot x + \left((\frac{2}{15} \cdot \left({x}^{5}\right) + x)_*\right))_*\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{2}{e^{-2 \cdot x} + 1} - 1} \cdot \left(\sqrt[3]{\frac{2}{e^{-2 \cdot x} + 1} - 1} \cdot \sqrt[3]{\frac{2}{e^{-2 \cdot x} + 1} - 1}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019068 +o rules:numerics
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))