Average Error: 29.7 → 0.1
Time: 39.0s
Precision: 64
Internal Precision: 1344
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{2}{1 + e^{-2 \cdot x}} - 1 \le -0.0013552338004752108:\\ \;\;\;\;\sqrt[3]{{\left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right)}^{3}}\\ \mathbf{if}\;\frac{2}{1 + e^{-2 \cdot x}} - 1 \le 5.108256064763852 \cdot 10^{-08}:\\ \;\;\;\;\left(\frac{2}{15} \cdot {x}^{5} + x\right) - \frac{1}{3} \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right)}^{3}}\\ \end{array}\]

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 (- (/ 2 (+ 1 (exp (* -2 x)))) 1) < -0.0013552338004752108 or 5.108256064763852e-08 < (- (/ 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 add-cbrt-cube0.1

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

    4. Applied simplify0.1

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

    if -0.0013552338004752108 < (- (/ 2 (+ 1 (exp (* -2 x)))) 1) < 5.108256064763852e-08

    1. Initial program 59.7

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

    2. Taylor expanded around 0 0.0

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

Runtime

Time bar (total: 39.0s)Debug logProfile

herbie shell --seed 2018207 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))