Average Error: 29.3 → 0.1
Time: 3.8m
Precision: 64
Internal Precision: 1344
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{2}{1 + e^{-2 \cdot x}} - 1 \le -0.0007462943001403419:\\ \;\;\;\;\frac{\frac{2}{\sqrt{1 + e^{-2 \cdot x}}}}{\sqrt{1 + e^{-2 \cdot x}}} - 1\\ \mathbf{if}\;\frac{2}{1 + e^{-2 \cdot x}} - 1 \le 3.291559972895376 \cdot 10^{-09}:\\ \;\;\;\;\left(x + \frac{2}{15} \cdot {x}^{5}\right) - \frac{1}{3} \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{(\left(\frac{2}{1 + e^{-2 \cdot x}}\right) \cdot \left(\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}}\right) + \left(-1\right))_*}{\frac{2}{1 + e^{-2 \cdot x}} + (\left(\frac{2}{1 + e^{-2 \cdot x}}\right) \cdot \left(\frac{2}{1 + e^{-2 \cdot x}}\right) + 1)_*}\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

Derivation

  1. Split input into 3 regimes

  2. if (- (/ 2 (+ 1 (exp (* -2 x)))) 1) < -0.0007462943001403419

    1. Initial program 0.0

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

    3. Applied add-sqr-sqrt0.0

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

    4. Applied associate-/r*0.1

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

    if -0.0007462943001403419 < (- (/ 2 (+ 1 (exp (* -2 x)))) 1) < 3.291559972895376e-09

    1. Initial program 59.6

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

    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 3.291559972895376e-09 < (- (/ 2 (+ 1 (exp (* -2 x)))) 1)

    1. Initial program 0.3

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

    3. Applied flip3--0.3

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

    4. Applied simplify0.3

      \[\leadsto \frac{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3} - {1}^{3}}{\color{blue}{(\left(\frac{2}{1 + e^{-2 \cdot x}}\right) \cdot \left(\frac{2}{1 + e^{-2 \cdot x}}\right) + 1)_* + \frac{2}{1 + e^{-2 \cdot x}}}}\]
    5. Using strategy rm

    6. Applied unpow30.3

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

    7. Applied add-cube-cbrt3.3

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

    8. Applied unpow-prod-down3.6

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

    9. Applied prod-diff3.6

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

    10. Applied simplify0.3

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

    11. Applied simplify0.3

      \[\leadsto \frac{(\left(\frac{2}{1 + e^{-2 \cdot x}}\right) \cdot \left(\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}}\right) + \left(-1\right))_* + \color{blue}{0}}{(\left(\frac{2}{1 + e^{-2 \cdot x}}\right) \cdot \left(\frac{2}{1 + e^{-2 \cdot x}}\right) + 1)_* + \frac{2}{1 + e^{-2 \cdot x}}}\]
  3. Recombined 3 regimes into one program.

  4. Applied simplify0.1

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\frac{2}{1 + e^{-2 \cdot x}} - 1 \le -0.0007462943001403419:\\ \;\;\;\;\frac{\frac{2}{\sqrt{1 + e^{-2 \cdot x}}}}{\sqrt{1 + e^{-2 \cdot x}}} - 1\\ \mathbf{if}\;\frac{2}{1 + e^{-2 \cdot x}} - 1 \le 3.291559972895376 \cdot 10^{-09}:\\ \;\;\;\;\left(x + \frac{2}{15} \cdot {x}^{5}\right) - \frac{1}{3} \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{(\left(\frac{2}{1 + e^{-2 \cdot x}}\right) \cdot \left(\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}}\right) + \left(-1\right))_*}{\frac{2}{1 + e^{-2 \cdot x}} + (\left(\frac{2}{1 + e^{-2 \cdot x}}\right) \cdot \left(\frac{2}{1 + e^{-2 \cdot x}}\right) + 1)_*}\\ \end{array}}\]

Runtime

Time bar (total: 3.8m)Debug logProfile

herbie shell --seed 2020178 +o rules:numerics
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))