Average Error: 29.2 → 0.0
Time: 4.0m
Precision: 64
Internal Precision: 1344
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{2}{1 + e^{-2 \cdot x}} - 1 \le -0.003435176905733078:\\ \;\;\;\;\frac{\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}} - 1}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\\ \mathbf{if}\;\frac{2}{1 + e^{-2 \cdot x}} - 1 \le 0.0025513264927490347:\\ \;\;\;\;x + \left({x}^{5} \cdot \frac{2}{15} - \left(x \cdot \frac{1}{3}\right) \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}} - 1}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if (- (/ 2 (+ 1 (exp (* -2 x)))) 1) < -0.003435176905733078 or 0.0025513264927490347 < (- (/ 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 flip--0.0

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

    4. Applied simplify0.0

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

    if -0.003435176905733078 < (- (/ 2 (+ 1 (exp (* -2 x)))) 1) < 0.0025513264927490347

    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. Taylor expanded around -inf 62.9

      \[\leadsto \left(\color{blue}{\frac{2}{15} \cdot e^{5 \cdot \left(\log -1 - \log \left(\frac{-1}{x}\right)\right)}} + x\right) - \frac{1}{3} \cdot {x}^{3}\]

    4. Applied simplify0

      \[\leadsto \color{blue}{x + \left({x}^{5} \cdot \frac{2}{15} - \left(x \cdot \frac{1}{3}\right) \cdot \left(x \cdot x\right)\right)}\]
  3. Recombined 2 regimes into one program.

Runtime

Time bar (total: 4.0m)Debug logProfile

herbie shell --seed 2018198 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))