Average Error: 29.0 → 0.2
Time: 1.3m
Precision: 64
Internal Precision: 1408
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{2}{1 + e^{-2 \cdot x}} - 1 \le -1.8697655389918403 \cdot 10^{-05}:\\ \;\;\;\;e^{\log 2 - \log_* (1 + e^{-2 \cdot x})} - 1\\ \mathbf{if}\;\frac{2}{1 + e^{-2 \cdot x}} - 1 \le 2.1139312522677756 \cdot 10^{-11}:\\ \;\;\;\;\left(\frac{2}{15} \cdot {x}^{5} + x\right) - \frac{1}{3} \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;e^{\log 2 - \log_* (1 + e^{-2 \cdot x})} - 1\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

Derivation

  1. Split input into 3 regimes

  2. if (- (/ 2 (+ 1 (exp (* -2 x)))) 1) < -1.8697655389918403e-05

    1. Initial program 0.1

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

    3. Applied add-exp-log0.1

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

    4. Applied simplify0.1

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

    if -1.8697655389918403e-05 < (- (/ 2 (+ 1 (exp (* -2 x)))) 1) < 2.1139312522677756e-11

    1. Initial program 59.8

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

    2. Taylor expanded around 0 0

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

    if 2.1139312522677756e-11 < (- (/ 2 (+ 1 (exp (* -2 x)))) 1)

    1. Initial program 0.6

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

    3. Applied add-exp-log0.6

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

    4. Applied simplify0.6

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

Runtime

Time bar (total: 1.3m)Debug logProfile

herbie shell --seed '#(1064173506 2580572819 2847706409 4129882574 1125180799 1845288547)' +o rules:numerics
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))