Average Error: 29.2 → 0.0
Time: 24.1s
Precision: 64
Internal Precision: 128
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.007549888556423157:\\ \;\;\;\;\left(\sqrt{\frac{2}{e^{-2 \cdot x} + 1}} + 1\right) \cdot \left(\sqrt{\frac{2}{e^{-2 \cdot x} + 1}} - 1\right)\\ \mathbf{elif}\;x \le 0.006967479385908131:\\ \;\;\;\;\left(\frac{2}{15} \cdot {x}^{5} - {x}^{3} \cdot \frac{1}{3}\right) + x\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{e^{-2 \cdot x} + 1} - 1\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if x < -0.007549888556423157

    1. Initial program 0.0

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

    3. Applied add-sqr-sqrt0.0

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

    4. Applied difference-of-sqr-10.0

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

    if -0.007549888556423157 < x < 0.006967479385908131

    1. Initial program 59.2

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

    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. Using strategy rm

    4. Applied associate--l+0.0

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

    if 0.006967479385908131 < x

    1. Initial program 0.0

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

    2. Taylor expanded around inf 0.0

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

    3. Simplified0.0

      \[\leadsto \color{blue}{\frac{2}{1 + e^{-2 \cdot x}} - 1}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.007549888556423157:\\ \;\;\;\;\left(\sqrt{\frac{2}{e^{-2 \cdot x} + 1}} + 1\right) \cdot \left(\sqrt{\frac{2}{e^{-2 \cdot x} + 1}} - 1\right)\\ \mathbf{elif}\;x \le 0.006967479385908131:\\ \;\;\;\;\left(\frac{2}{15} \cdot {x}^{5} - {x}^{3} \cdot \frac{1}{3}\right) + x\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{e^{-2 \cdot x} + 1} - 1\\ \end{array}\]

Runtime

Time bar (total: 24.1s)Debug logProfile

herbie shell --seed 2018285 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))