Average Error: 29.4 → 0.0
Time: 33.6s
Precision: 64
Internal Precision: 128
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.00750627820210216:\\ \;\;\;\;\frac{\sqrt[3]{\frac{2}{e^{-2 \cdot x} + 1} \cdot \frac{2}{e^{-2 \cdot x} + 1} - 1} \cdot \sqrt[3]{\frac{2}{e^{-2 \cdot x} + 1} \cdot \frac{2}{e^{-2 \cdot x} + 1} - 1}}{\sqrt[3]{1 + \frac{2}{e^{-2 \cdot x} + 1}} \cdot \sqrt[3]{1 + \frac{2}{e^{-2 \cdot x} + 1}}} \cdot \frac{\sqrt[3]{\frac{2}{e^{-2 \cdot x} + 1} \cdot \frac{2}{e^{-2 \cdot x} + 1} - 1}}{\sqrt[3]{1 + \frac{2}{e^{-2 \cdot x} + 1}}}\\ \mathbf{elif}\;x \le 0.0062129209029333134:\\ \;\;\;\;\left({x}^{5} \cdot \frac{2}{15} + \left(\left(x \cdot x\right) \cdot \frac{-1}{3}\right) \cdot x\right) + x\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{2}{e^{-2 \cdot x} + 1} \cdot \frac{2}{e^{-2 \cdot x} + 1} - 1} \cdot \sqrt[3]{\frac{2}{e^{-2 \cdot x} + 1} \cdot \frac{2}{e^{-2 \cdot x} + 1} - 1}}{\sqrt[3]{1 + \frac{2}{e^{-2 \cdot x} + 1}} \cdot \sqrt[3]{1 + \frac{2}{e^{-2 \cdot x} + 1}}} \cdot \frac{\sqrt[3]{\frac{2}{e^{-2 \cdot x} + 1} \cdot \frac{2}{e^{-2 \cdot x} + 1} - 1}}{\sqrt[3]{1 + \frac{2}{e^{-2 \cdot x} + 1}}}\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

Derivation

  1. Split input into 2 regimes

  2. if x < -0.00750627820210216 or 0.0062129209029333134 < x

    1. Initial program 0.0

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

    3. Applied flip--0.0

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

    5. Applied add-cube-cbrt0.0

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

    6. Applied add-cube-cbrt0.0

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

    7. Applied times-frac0.0

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

    if -0.00750627820210216 < x < 0.0062129209029333134

    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

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.00750627820210216:\\ \;\;\;\;\frac{\sqrt[3]{\frac{2}{e^{-2 \cdot x} + 1} \cdot \frac{2}{e^{-2 \cdot x} + 1} - 1} \cdot \sqrt[3]{\frac{2}{e^{-2 \cdot x} + 1} \cdot \frac{2}{e^{-2 \cdot x} + 1} - 1}}{\sqrt[3]{1 + \frac{2}{e^{-2 \cdot x} + 1}} \cdot \sqrt[3]{1 + \frac{2}{e^{-2 \cdot x} + 1}}} \cdot \frac{\sqrt[3]{\frac{2}{e^{-2 \cdot x} + 1} \cdot \frac{2}{e^{-2 \cdot x} + 1} - 1}}{\sqrt[3]{1 + \frac{2}{e^{-2 \cdot x} + 1}}}\\ \mathbf{elif}\;x \le 0.0062129209029333134:\\ \;\;\;\;\left({x}^{5} \cdot \frac{2}{15} + \left(\left(x \cdot x\right) \cdot \frac{-1}{3}\right) \cdot x\right) + x\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{2}{e^{-2 \cdot x} + 1} \cdot \frac{2}{e^{-2 \cdot x} + 1} - 1} \cdot \sqrt[3]{\frac{2}{e^{-2 \cdot x} + 1} \cdot \frac{2}{e^{-2 \cdot x} + 1} - 1}}{\sqrt[3]{1 + \frac{2}{e^{-2 \cdot x} + 1}} \cdot \sqrt[3]{1 + \frac{2}{e^{-2 \cdot x} + 1}}} \cdot \frac{\sqrt[3]{\frac{2}{e^{-2 \cdot x} + 1} \cdot \frac{2}{e^{-2 \cdot x} + 1} - 1}}{\sqrt[3]{1 + \frac{2}{e^{-2 \cdot x} + 1}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019051 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))