Average Error: 29.4 → 0.0
Time: 6.4m
Precision: 64
Internal Precision: 1344
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{2}{1 + e^{-2 \cdot x}} - 1 \le -0.004855436202985079:\\ \;\;\;\;\frac{\left(\frac{2}{1 + e^{-2 \cdot x}} \cdot \left(\sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}}} \cdot \sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}}}\right)\right) \cdot \sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}}} - 1}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\\ \mathbf{if}\;\frac{2}{1 + e^{-2 \cdot x}} - 1 \le 7.469456627647396 \cdot 10^{-07}:\\ \;\;\;\;\left(\frac{2}{15} \cdot {x}^{5} + x\right) - \frac{1}{3} \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\left(\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}}\right)}^{3} - {1}^{3}}{\left(\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}} + 1\right) + {\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{\left(1 + 3\right)}}}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

Derivation

  1. Split input into 3 regimes

  2. if (- (/ 2 (+ 1 (exp (* -2 x)))) 1) < -0.004855436202985079

    1. Initial program 0.0

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

    3. Applied flip--0.0

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

    4. Applied simplify0.0

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

    6. Applied add-cube-cbrt0.0

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

    7. Applied associate-*r*0.0

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

    if -0.004855436202985079 < (- (/ 2 (+ 1 (exp (* -2 x)))) 1) < 7.469456627647396e-07

    1. Initial program 59.2

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

    2. Taylor expanded around 0 0.0

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

    if 7.469456627647396e-07 < (- (/ 2 (+ 1 (exp (* -2 x)))) 1)

    1. Initial program 0.1

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

    3. Applied flip--0.1

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

    4. Applied simplify0.1

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

    6. Applied flip3--0.1

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

    7. Applied simplify0.1

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

Runtime

Time bar (total: 6.4m)Debug logProfile

herbie shell --seed '#(1071948828 1180510430 2986424009 997076509 406109801 420189285)' 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))