Average Error: 29.4 → 0.3
Time: 35.2s
Precision: 64
Internal Precision: 1344
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{2}{1 + e^{-2 \cdot x}} - 1 \le -6.9003174869058 \cdot 10^{-310}:\\ \;\;\;\;\left(\sqrt{\frac{2}{1 + e^{-2 \cdot x}}} + 1\right) \cdot \left(\sqrt{\frac{2}{1 + e^{-2 \cdot x}}} - 1\right)\\ \mathbf{if}\;\frac{2}{1 + e^{-2 \cdot x}} - 1 \le 0.0004068705389923145:\\ \;\;\;\;\left(\frac{2}{15} \cdot {x}^{5} + x\right) - \frac{1}{3} \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} - 1} \cdot \sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} - 1}\right) \cdot \left(\log \left(\sqrt{e^{\sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} - 1}}}\right) + \log \left(\sqrt{e^{\sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} - 1}}}\right)\right)\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

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Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if (- (/ 2 (+ 1 (exp (* -2 x)))) 1) < -6.9003174869058e-310

    1. Initial program 1.1

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

    3. Applied add-sqr-sqrt1.2

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

    4. Applied difference-of-sqr-11.2

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

    if -6.9003174869058e-310 < (- (/ 2 (+ 1 (exp (* -2 x)))) 1) < 0.0004068705389923145

    1. Initial program 59.9

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

    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 0.0004068705389923145 < (- (/ 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-cube-cbrt0.1

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

    5. Applied add-log-exp0.1

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

    7. Applied add-sqr-sqrt0.1

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

    8. Applied log-prod0.1

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

Runtime

Time bar (total: 35.2s)Debug logProfile

herbie shell --seed 2018167 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))