Average Error: 29.3 → 0.1
Time: 1.6m
Precision: 64
Internal Precision: 1344
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{2}{1 + e^{-2 \cdot x}} - 1 \le -0.0007462943001403419:\\ \;\;\;\;\log \left(e^{\frac{2}{1 + e^{-2 \cdot x}} - 1}\right)\\ \mathbf{if}\;\frac{2}{1 + e^{-2 \cdot x}} - 1 \le 3.291559972895376 \cdot 10^{-09}:\\ \;\;\;\;\left(x + \frac{2}{15} \cdot {x}^{5}\right) - \frac{1}{3} \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\frac{2}{1 + e^{-2 \cdot x}} - 1}\right)\\ \end{array}\]

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) < -0.0007462943001403419 or 3.291559972895376e-09 < (- (/ 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-log-exp0.2

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

    if -0.0007462943001403419 < (- (/ 2 (+ 1 (exp (* -2 x)))) 1) < 3.291559972895376e-09

    1. Initial program 59.6

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

    2. 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.

Runtime

Time bar (total: 1.6m)Debug logProfile

herbie shell --seed 2020178 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))