Average Error: 29.5 → 0.1
Time: 30.0s
Precision: 64
Internal Precision: 1344
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{2}{1 + e^{-2 \cdot x}} - 1 \le -9.486460909500204 \cdot 10^{-11}:\\ \;\;\;\;e^{\log 2 - \log_* (1 + e^{-2 \cdot x})} - 1\\ \mathbf{if}\;\frac{2}{1 + e^{-2 \cdot x}} - 1 \le 0.0624999983817706:\\ \;\;\;\;\left(\frac{2}{15} \cdot {x}^{5} + x\right) - \frac{1}{3} \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;e^{\log 2 - \log_* (1 + e^{-2 \cdot x})} - 1\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

Derivation

  1. Split input into 2 regimes

  2. if (- (/ 2 (+ 1 (exp (* -2 x)))) 1) < -9.486460909500204e-11 or 0.0624999983817706 < (- (/ 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 -9.486460909500204e-11 < (- (/ 2 (+ 1 (exp (* -2 x)))) 1) < 0.0624999983817706

    1. Initial program 59.5

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

    2. Taylor expanded around 0 0.0

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

Runtime

Time bar (total: 30.0s)Debug logProfile

herbie shell --seed '#(1072330854 3074818769 591214268 3603999196 3863745332 3332387116)' +o rules:numerics
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))