Average Error: 29.1 → 0.0
Time: 15.4s
Precision: 64
Internal Precision: 1344
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{2}{1 + e^{-2 \cdot x}} - 1 \le -0.0030461172554450875 \lor \neg \left(\frac{2}{1 + e^{-2 \cdot x}} - 1 \le 0.0010350707963469223\right):\\ \;\;\;\;\frac{2}{\sqrt[3]{\left(\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)\right) \cdot \left(1 + e^{-2 \cdot x}\right)}} - 1\\ \mathbf{else}:\\ \;\;\;\;\left(x + {x}^{5} \cdot \frac{2}{15}\right) - {x}^{3} \cdot \frac{1}{3}\\ \end{array}\]

Error

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Bits error versus y

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Results

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Derivation

  1. Split input into 2 regimes

  2. if (- (/ 2 (+ 1 (exp (* -2 x)))) 1) < -0.0030461172554450875 or 0.0010350707963469223 < (- (/ 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-cbrt-cube0.0

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

    if -0.0030461172554450875 < (- (/ 2 (+ 1 (exp (* -2 x)))) 1) < 0.0010350707963469223

    1. Initial program 58.8

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

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{1 + e^{-2 \cdot x}} - 1 \le -0.0030461172554450875 \lor \neg \left(\frac{2}{1 + e^{-2 \cdot x}} - 1 \le 0.0010350707963469223\right):\\ \;\;\;\;\frac{2}{\sqrt[3]{\left(\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)\right) \cdot \left(1 + e^{-2 \cdot x}\right)}} - 1\\ \mathbf{else}:\\ \;\;\;\;\left(x + {x}^{5} \cdot \frac{2}{15}\right) - {x}^{3} \cdot \frac{1}{3}\\ \end{array}\]

Runtime

Time bar (total: 15.4s)Debug logProfile

herbie shell --seed 2018217 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))