Average Error: 29.0 → 0.0
Time: 25.3s
Precision: 64
Internal Precision: 320
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.007578026029582469:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\ \mathbf{elif}\;x \le 0.006767654417923374:\\ \;\;\;\;\left(\frac{2}{15} \cdot {x}^{5} + x\right) - {x}^{3} \cdot \frac{1}{3}\\ \mathbf{else}:\\ \;\;\;\;(\left(e^{-2 \cdot x}\right) \cdot \left((e^{-2 \cdot x} - 1)^*\right) + 1)_* \cdot \frac{2}{1 + {\left(e^{-2 \cdot x}\right)}^{3}} - 1\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

Derivation

  1. Split input into 3 regimes

  2. if x < -0.007578026029582469

    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

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

    if -0.007578026029582469 < x < 0.006767654417923374

    1. Initial program 58.9

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

    2. Initial simplification58.9

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

    3. Taylor expanded around -inf 58.9

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

    4. 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 0.006767654417923374 < 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

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

    5. Applied flip3-+0.0

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

    6. Applied associate-/r/0.0

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

    7. Simplified0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.007578026029582469:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\ \mathbf{elif}\;x \le 0.006767654417923374:\\ \;\;\;\;\left(\frac{2}{15} \cdot {x}^{5} + x\right) - {x}^{3} \cdot \frac{1}{3}\\ \mathbf{else}:\\ \;\;\;\;(\left(e^{-2 \cdot x}\right) \cdot \left((e^{-2 \cdot x} - 1)^*\right) + 1)_* \cdot \frac{2}{1 + {\left(e^{-2 \cdot x}\right)}^{3}} - 1\\ \end{array}\]

Runtime

Time bar (total: 25.3s)Debug logProfile

herbie shell --seed 2018256 +o rules:numerics
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))