Average Error: 29.0 → 0.0
Time: 18.0s
Precision: 64
Internal Precision: 1344
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.005667309583127926:\\ \;\;\;\;\log \left(\sqrt{e^{\frac{2}{e^{-2 \cdot x} + 1} - 1}}\right) + \log \left(\sqrt{e^{\frac{2}{e^{-2 \cdot x} + 1} - 1}}\right)\\ \mathbf{elif}\;x \le 0.007374663675913499:\\ \;\;\;\;\left(\frac{2}{15} \cdot {x}^{5} + x\right) - {x}^{3} \cdot \frac{1}{3}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\log \left(\sqrt{e^{\frac{2}{e^{-2 \cdot x} + 1} - 1}}\right)\right)} + \log \left(\sqrt{e^{\frac{2}{e^{-2 \cdot x} + 1} - 1}}\right)\\ \end{array}\]

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 < -0.005667309583127926

    1. Initial program 0.0

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

    2. Initial simplification0.0

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

    4. Applied add-log-exp0.0

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

    6. Applied add-sqr-sqrt0.0

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

    7. Applied log-prod0.0

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

    if -0.005667309583127926 < x < 0.007374663675913499

    1. Initial program 59.0

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

    2. Initial simplification59.0

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

    3. Taylor expanded around 0 0.0

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

    if 0.007374663675913499 < x

    1. Initial program 0.0

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

    2. Initial simplification0.0

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

    4. Applied add-log-exp0.0

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

    6. Applied add-sqr-sqrt0.0

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

    7. Applied log-prod0.0

      \[\leadsto \color{blue}{\log \left(\sqrt{e^{\frac{2}{1 + e^{-2 \cdot x}} - 1}}\right) + \log \left(\sqrt{e^{\frac{2}{1 + e^{-2 \cdot x}} - 1}}\right)}\]
    8. Using strategy rm

    9. Applied add-exp-log0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.005667309583127926:\\ \;\;\;\;\log \left(\sqrt{e^{\frac{2}{e^{-2 \cdot x} + 1} - 1}}\right) + \log \left(\sqrt{e^{\frac{2}{e^{-2 \cdot x} + 1} - 1}}\right)\\ \mathbf{elif}\;x \le 0.007374663675913499:\\ \;\;\;\;\left(\frac{2}{15} \cdot {x}^{5} + x\right) - {x}^{3} \cdot \frac{1}{3}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\log \left(\sqrt{e^{\frac{2}{e^{-2 \cdot x} + 1} - 1}}\right)\right)} + \log \left(\sqrt{e^{\frac{2}{e^{-2 \cdot x} + 1} - 1}}\right)\\ \end{array}\]

Runtime

Time bar (total: 18.0s)Debug logProfile

herbie shell --seed 2018249 +o rules:numerics
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))