Average Error: 29.5 → 0.1
Time: 20.7s
Precision: 64
Internal Precision: 1344
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{2}{1 + e^{-2 \cdot x}} - 1 \le -0.26572249488984434:\\ \;\;\;\;\frac{2}{(e^{\log_* (1 + \left(1 + e^{-2 \cdot x}\right))} - 1)^*} - 1\\ \mathbf{if}\;\frac{2}{1 + e^{-2 \cdot x}} - 1 \le 0.00016709790383551137:\\ \;\;\;\;(\left({x}^{3}\right) \cdot \left(-\frac{1}{3}\right) + x)_*\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{(e^{\log_* (1 + \left(1 + e^{-2 \cdot x}\right))} - 1)^*} - 1\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

Derivation

  1. Split input into 2 regimes

  2. if (- (/ 2 (+ 1 (exp (* -2 x)))) 1) < -0.26572249488984434 or 0.00016709790383551137 < (- (/ 2 (+ 1 (exp (* -2 x)))) 1)

    1. Initial program 0.0

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

    3. Applied expm1-log1p-u0.0

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

    if -0.26572249488984434 < (- (/ 2 (+ 1 (exp (* -2 x)))) 1) < 0.00016709790383551137

    1. Initial program 59.1

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

    2. Taylor expanded around 0 59.1

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

    3. Applied simplify0.1

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

Runtime

Time bar (total: 20.7s)Debug logProfile

herbie shell --seed '#(1072107073 2127697367 3936270018 2300570620 2134894798 4023771849)' +o rules:numerics
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))