Average Error: 29.3 → 0.1
Time: 1.1m
Precision: 64
Internal Precision: 1344
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{2}{1 + e^{-2 \cdot x}} - 1 \le -0.005193000106973418:\\ \;\;\;\;\frac{2}{(e^{\log_* (1 + \left(1 + e^{-2 \cdot x}\right))} - 1)^*} - 1\\ \mathbf{if}\;\frac{2}{1 + e^{-2 \cdot x}} - 1 \le 6.514835391826027 \cdot 10^{-07}:\\ \;\;\;\;\left(\frac{2}{15} \cdot {x}^{5} + x\right) - \frac{1}{3} \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{(e^{\log_* (1 + \left(1 + e^{-2 \cdot x}\right))} - 1)^*} - 1\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

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Results

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Derivation

  1. Split input into 2 regimes

  2. if (- (/ 2 (+ 1 (exp (* -2 x)))) 1) < -0.005193000106973418 or 6.514835391826027e-07 < (- (/ 2 (+ 1 (exp (* -2 x)))) 1)

    1. Initial program 0.1

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

    3. Applied expm1-log1p-u0.1

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

    if -0.005193000106973418 < (- (/ 2 (+ 1 (exp (* -2 x)))) 1) < 6.514835391826027e-07

    1. Initial program 59.2

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

    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: 1.1m)Debug logProfile

herbie shell --seed 2018166 +o rules:numerics
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))