Average Error: 29.6 → 0.0
Time: 9.5s
Precision: 64
Internal Precision: 1344
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.007425741325783795 \lor \neg \left(x \le 0.0066542557880153504\right):\\ \;\;\;\;\frac{2}{e^{-2 \cdot x} + 1} - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{15} \cdot {x}^{5} + x\right) - {x}^{3} \cdot \frac{1}{3}\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if x < -0.007425741325783795 or 0.0066542557880153504 < x

    1. Initial program 0.0

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

    2. Initial simplification0.0

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

    3. Taylor expanded around inf 0.0

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

    4. Simplified0.0

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

    if -0.007425741325783795 < x < 0.0066542557880153504

    1. Initial program 59.0

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

    2. Initial simplification59.0

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

    3. 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. Recombined 2 regimes into one program.

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.007425741325783795 \lor \neg \left(x \le 0.0066542557880153504\right):\\ \;\;\;\;\frac{2}{e^{-2 \cdot x} + 1} - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{15} \cdot {x}^{5} + x\right) - {x}^{3} \cdot \frac{1}{3}\\ \end{array}\]

Runtime

Time bar (total: 9.5s)Debug logProfile

herbie shell --seed 2018255 +o rules:numerics
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))