Average Error: 28.9 → 0.0
Time: 2.8m
Precision: 64
Internal Precision: 1408
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{2}{1 + e^{-2 \cdot x}} - 1 \le -0.0010361760913228334:\\ \;\;\;\;\frac{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3} - {1}^{3}}{\left(\frac{2}{1 + e^{-2 \cdot x}} + 1\right) + \frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}}}\\ \mathbf{if}\;\frac{2}{1 + e^{-2 \cdot x}} - 1 \le 0.0039062543158841865:\\ \;\;\;\;\left(\frac{2}{15} \cdot {x}^{5} + x\right) - \frac{1}{3} \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3} - {1}^{3}}{\left(\frac{2}{1 + e^{-2 \cdot x}} + 1\right) + \frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}}}\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

Derivation

  1. Split input into 2 regimes

  2. if (- (/ 2 (+ 1 (exp (* -2 x)))) 1) < -0.0010361760913228334 or 0.0039062543158841865 < (- (/ 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 flip3--0.0

      \[\leadsto \color{blue}{\frac{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3} - {1}^{3}}{\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}} + \left(1 \cdot 1 + \frac{2}{1 + e^{-2 \cdot x}} \cdot 1\right)}}\]

    4. Applied simplify0.0

      \[\leadsto \frac{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3} - {1}^{3}}{\color{blue}{\left(\frac{2}{1 + e^{-2 \cdot x}} + 1\right) + \frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}}}}\]

    if -0.0010361760913228334 < (- (/ 2 (+ 1 (exp (* -2 x)))) 1) < 0.0039062543158841865

    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: 2.8m)Debug logProfile

herbie shell --seed '#(1070386091 2509006183 1430610344 1025408621 36622005 1425925650)' 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))