Average Error: 28.5 → 0.0
Time: 25.2s
Precision: 64
Internal Precision: 1344
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.007155681498333566:\\ \;\;\;\;\left(\sqrt{\frac{2}{e^{-2 \cdot x} + 1}} - 1\right) \cdot \left(\sqrt{\sqrt{\frac{2}{e^{-2 \cdot x} + 1}} + 1} \cdot \sqrt{\sqrt{\frac{2}{e^{-2 \cdot x} + 1}} + 1}\right)\\ \mathbf{elif}\;x \le 0.0070712511510923086:\\ \;\;\;\;\left(x + {x}^{5} \cdot \frac{2}{15}\right) - {x}^{3} \cdot \frac{1}{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{e^{-2 \cdot x} + 1} - 1\\ \end{array}\]

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Derivation

  1. Split input into 3 regimes

  2. if x < -0.007155681498333566

    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-sqr-sqrt0.0

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

    5. Applied difference-of-sqr-10.0

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

    7. Applied add-sqr-sqrt0.0

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

    if -0.007155681498333566 < x < 0.0070712511510923086

    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 0.0070712511510923086 < 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. Taylor expanded around -inf 0.0

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

    4. Simplified0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.007155681498333566:\\ \;\;\;\;\left(\sqrt{\frac{2}{e^{-2 \cdot x} + 1}} - 1\right) \cdot \left(\sqrt{\sqrt{\frac{2}{e^{-2 \cdot x} + 1}} + 1} \cdot \sqrt{\sqrt{\frac{2}{e^{-2 \cdot x} + 1}} + 1}\right)\\ \mathbf{elif}\;x \le 0.0070712511510923086:\\ \;\;\;\;\left(x + {x}^{5} \cdot \frac{2}{15}\right) - {x}^{3} \cdot \frac{1}{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{e^{-2 \cdot x} + 1} - 1\\ \end{array}\]

Runtime

Time bar (total: 25.2s)Debug logProfile

herbie shell --seed 2018234 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))