Average Error: 28.6 → 0.1
Time: 3.6m
Precision: 64
Internal Precision: 1344
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{2}{1 + e^{-2 \cdot x}} - 1 \le -1.6888443106222433 \cdot 10^{-07}:\\ \;\;\;\;\frac{{\left(e^{\log 2 - \log_* (1 + e^{-2 \cdot x})}\right)}^{3} - {1}^{3}}{(\left(\frac{2}{e^{\log_* (1 + e^{-2 \cdot x})}}\right) \cdot \left(\frac{2}{e^{\log_* (1 + e^{-2 \cdot x})}}\right) + \left(1 + \frac{2}{e^{\log_* (1 + e^{-2 \cdot x})}}\right))_*}\\ \mathbf{if}\;\frac{2}{1 + e^{-2 \cdot x}} - 1 \le 0.0032654260743848235:\\ \;\;\;\;\left(x + \frac{2}{15} \cdot {x}^{5}\right) - \frac{1}{3} \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left((e^{\log 2 - \log_* (1 + e^{-2 \cdot x})} - 1)^*\right)}\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

Derivation

  1. Split input into 3 regimes

  2. if (- (/ 2 (+ 1 (exp (* -2 x)))) 1) < -1.6888443106222433e-07

    1. Initial program 0.2

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

    3. Applied add-exp-log0.2

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

    4. Applied simplify0.2

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

    6. Applied flip3--0.2

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

    7. Applied simplify0.2

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

    if -1.6888443106222433e-07 < (- (/ 2 (+ 1 (exp (* -2 x)))) 1) < 0.0032654260743848235

    1. Initial program 59.1

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

    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 0.0032654260743848235 < (- (/ 2 (+ 1 (exp (* -2 x)))) 1)

    1. Initial program 0.0

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

    3. Applied add-exp-log0.0

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

    4. Applied simplify0.0

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

    6. Applied add-exp-log0.0

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

    7. Applied simplify0.0

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

Runtime

Time bar (total: 3.6m)Debug logProfile

herbie shell --seed 2018296 +o rules:numerics
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))