Average Error: 29.4 → 0.0
Time: 1.9m
Precision: 64
Internal Precision: 1344
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.00716280177925632:\\ \;\;\;\;\log \left(e^{\frac{2}{1 + e^{-2 \cdot x}}}\right) - 1\\ \mathbf{elif}\;x \le 0.006653438178205924:\\ \;\;\;\;\left(\frac{2}{15} \cdot {x}^{5} + x\right) - {x}^{3} \cdot \frac{1}{3}\\ \mathbf{else}:\\ \;\;\;\;(\left(\frac{2}{{1}^{3} + {\left(e^{-2 \cdot x}\right)}^{3}}\right) \cdot \left(\left(e^{-2 \cdot x} \cdot e^{-2 \cdot x} - e^{-2 \cdot x}\right) + 1\right) + \left(-1\right))_*\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

Derivation

  1. Split input into 3 regimes

  2. if x < -0.00716280177925632

    1. Initial program 0.0

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

    3. Applied add-log-exp0.0

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

    if -0.00716280177925632 < x < 0.006653438178205924

    1. Initial program 59.2

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

    2. Taylor expanded around 0 0.0

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

    if 0.006653438178205924 < x

    1. Initial program 0.0

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

    3. Applied flip3-+0.0

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

    4. Applied associate-/r/0.0

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

    5. Applied fma-neg0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.00716280177925632:\\ \;\;\;\;\log \left(e^{\frac{2}{1 + e^{-2 \cdot x}}}\right) - 1\\ \mathbf{elif}\;x \le 0.006653438178205924:\\ \;\;\;\;\left(\frac{2}{15} \cdot {x}^{5} + x\right) - {x}^{3} \cdot \frac{1}{3}\\ \mathbf{else}:\\ \;\;\;\;(\left(\frac{2}{{1}^{3} + {\left(e^{-2 \cdot x}\right)}^{3}}\right) \cdot \left(\left(e^{-2 \cdot x} \cdot e^{-2 \cdot x} - e^{-2 \cdot x}\right) + 1\right) + \left(-1\right))_*\\ \end{array}\]

Runtime

Time bar (total: 1.9m)Debug logProfile

herbie shell --seed 2018220 +o rules:numerics
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))