Average Error: 29.1 → 0.2
Time: 25.6s
Precision: 64
Internal Precision: 1344
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{2}{1 + e^{-2 \cdot x}} - 1 \le -1.0579786829598148 \cdot 10^{-12}:\\ \;\;\;\;(e^{\log_* (1 + \left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right))} - 1)^*\\ \mathbf{if}\;\frac{2}{1 + e^{-2 \cdot x}} - 1 \le 0.0006024819647004083:\\ \;\;\;\;\left(\frac{2}{15} \cdot {x}^{5} + x\right) - \frac{1}{3} \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;(e^{\log_* (1 + \left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right))} - 1)^*\\ \end{array}\]

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) < -1.0579786829598148e-12 or 0.0006024819647004083 < (- (/ 2 (+ 1 (exp (* -2 x)))) 1)

    1. Initial program 0.3

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

    3. Applied expm1-log1p-u0.3

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

    if -1.0579786829598148e-12 < (- (/ 2 (+ 1 (exp (* -2 x)))) 1) < 0.0006024819647004083

    1. Initial program 59.8

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

    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: 25.6s)Debug logProfile

herbie shell --seed 2018199 +o rules:numerics
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))