Average Error: 29.5 → 0.0
Time: 56.5s
Precision: 64
Internal Precision: 1344
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{2}{1 + e^{-2 \cdot x}} - 1 \le -0.0007513068054331472:\\ \;\;\;\;\frac{\sqrt[3]{\frac{\frac{8}{1 + e^{-2 \cdot x}}}{\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)}} \cdot \sqrt[3]{\frac{\frac{8}{1 + e^{-2 \cdot x}}}{\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)}} - 1}{\sqrt[3]{\frac{\frac{8}{1 + e^{-2 \cdot x}}}{\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)}} + 1}\\ \mathbf{if}\;\frac{2}{1 + e^{-2 \cdot x}} - 1 \le 0.3751591238535792:\\ \;\;\;\;x + \left({x}^{5} \cdot \frac{2}{15} - \left(x \cdot \frac{1}{3}\right) \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{\frac{8}{1 + e^{-2 \cdot x}}}{\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)}} \cdot \sqrt[3]{\frac{\frac{8}{1 + e^{-2 \cdot x}}}{\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)}} - 1}{\sqrt[3]{\frac{\frac{8}{1 + e^{-2 \cdot x}}}{\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)}} + 1}\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

Derivation

  1. Split input into 2 regimes

  2. if (- (/ 2 (+ 1 (exp (* -2 x)))) 1) < -0.0007513068054331472 or 0.3751591238535792 < (- (/ 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

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

    4. Applied simplify0.0

      \[\leadsto \sqrt[3]{\color{blue}{\frac{\frac{8}{1 + e^{-2 \cdot x}}}{\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)}}} - 1\]
    5. Using strategy rm

    6. Applied flip--0.0

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

    7. Applied simplify0.0

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

    if -0.0007513068054331472 < (- (/ 2 (+ 1 (exp (* -2 x)))) 1) < 0.3751591238535792

    1. Initial program 59.1

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

    2. Taylor expanded around 0 0.1

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

    3. Taylor expanded around -inf 62.9

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

    4. Applied simplify0.1

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

Runtime

Time bar (total: 56.5s)Debug logProfile

herbie shell --seed '#(1071373924 2949776965 1885069702 3247780810 90874544 2263903749)' 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))