Average Error: 29.0 → 0.0
Time: 4.4m
Precision: 64
Internal Precision: 1344
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.006883813457963097 \lor \neg \left(x \le 0.006317687962113081\right):\\ \;\;\;\;\frac{\frac{8}{{\left({\left(e^{-2}\right)}^{x} + 1\right)}^{3}} - 1}{\frac{\sqrt[3]{64}}{{\left({\left(e^{-2}\right)}^{x} + 1\right)}^{2}} + \left(\frac{\sqrt[3]{8}}{{\left(e^{-2}\right)}^{x} + 1} + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(x + {x}^{5} \cdot \frac{2}{15}\right) - \frac{1}{3} \cdot {x}^{3}\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if x < -0.006883813457963097 or 0.006317687962113081 < x

    1. Initial program 0.0

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

    3. Applied add-cbrt-cube0.0

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

    5. Applied flip3--0.0

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

    6. Taylor expanded around inf 0.0

      \[\leadsto \color{blue}{\frac{8 \cdot \frac{1}{{\left(e^{-2 \cdot x} + 1\right)}^{3}} - 1}{\frac{1}{e^{-2 \cdot x} + 1} \cdot {8}^{\frac{1}{3}} + \left(\frac{1}{{\left(e^{-2 \cdot x} + 1\right)}^{2}} \cdot {64}^{\frac{1}{3}} + 1\right)}}\]

    7. Simplified0.0

      \[\leadsto \color{blue}{\frac{\frac{8}{{\left(1 + {\left(e^{-2}\right)}^{x}\right)}^{3}} - 1}{\frac{\sqrt[3]{64}}{{\left(1 + {\left(e^{-2}\right)}^{x}\right)}^{2}} + \left(\frac{\sqrt[3]{8}}{1 + {\left(e^{-2}\right)}^{x}} + 1\right)}}\]

    if -0.006883813457963097 < x < 0.006317687962113081

    1. Initial program 59.0

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

    2. Taylor expanded around 0 0.0

      \[\leadsto \color{blue}{\left(x + \frac{2}{15} \cdot {x}^{5}\right) - \frac{1}{3} \cdot {x}^{3}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.006883813457963097 \lor \neg \left(x \le 0.006317687962113081\right):\\ \;\;\;\;\frac{\frac{8}{{\left({\left(e^{-2}\right)}^{x} + 1\right)}^{3}} - 1}{\frac{\sqrt[3]{64}}{{\left({\left(e^{-2}\right)}^{x} + 1\right)}^{2}} + \left(\frac{\sqrt[3]{8}}{{\left(e^{-2}\right)}^{x} + 1} + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(x + {x}^{5} \cdot \frac{2}{15}\right) - \frac{1}{3} \cdot {x}^{3}\\ \end{array}\]

Runtime

Time bar (total: 4.4m)Debug logProfile

herbie shell --seed 2018221 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))