Average Error: 29.1 → 0.0
Time: 22.7s
Precision: 64
Internal Precision: 1344
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.0072194553879333535:\\ \;\;\;\;\frac{-1 + {\left(\left(\sqrt{\frac{1}{1 + e^{-2 \cdot x}}} \cdot \sqrt{2}\right) \cdot \left(\sqrt{\frac{1}{1 + e^{-2 \cdot x}}} \cdot \sqrt{2}\right)\right)}^{3}}{\left(\left(\left(\sqrt{\frac{1}{1 + e^{-2 \cdot x}}} \cdot \sqrt{2}\right) \cdot \left(\sqrt{\frac{1}{1 + e^{-2 \cdot x}}} \cdot \sqrt{2}\right)\right) \cdot \left(\left(\sqrt{\frac{1}{1 + e^{-2 \cdot x}}} \cdot \sqrt{2}\right) \cdot \left(\sqrt{\frac{1}{1 + e^{-2 \cdot x}}} \cdot \sqrt{2}\right)\right) - \left(\sqrt{\frac{1}{1 + e^{-2 \cdot x}}} \cdot \sqrt{2}\right) \cdot \left(\sqrt{\frac{1}{1 + e^{-2 \cdot x}}} \cdot \left(-\sqrt{2}\right)\right)\right) + 1}\\ \mathbf{elif}\;x \le 0.00794597355349963:\\ \;\;\;\;\left(x + {x}^{5} \cdot \frac{2}{15}\right) - \frac{1}{3} \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right) \cdot \sqrt{\frac{2}{1 + e^{-2 \cdot x}}} - \left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right)}{\sqrt{\frac{2}{1 + e^{-2 \cdot x}}} - 1}\\ \end{array}\]

Error

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Bits error versus y

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Results

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Derivation

  1. Split input into 3 regimes

  2. if x < -0.0072194553879333535

    1. Initial program 0.0

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

    2. Initial simplification0.0

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

    4. Applied add-sqr-sqrt0.0

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

    5. Applied difference-of-sqr-10.0

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

    6. Taylor expanded around -inf 0.0

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

    7. Simplified0.0

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

    9. Applied flip3-+0.0

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

    if -0.0072194553879333535 < x < 0.00794597355349963

    1. Initial program 59.2

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

    2. Initial simplification59.2

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

    3. Taylor expanded around 0 0.0

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

    if 0.00794597355349963 < x

    1. Initial program 0.0

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

    2. Initial simplification0.0

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

    4. Applied add-sqr-sqrt1.6

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

    5. Applied difference-of-sqr-11.0

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

    7. Applied flip-+1.6

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

    8. Applied associate-*l/1.6

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

    9. Simplified0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.0072194553879333535:\\ \;\;\;\;\frac{-1 + {\left(\left(\sqrt{\frac{1}{1 + e^{-2 \cdot x}}} \cdot \sqrt{2}\right) \cdot \left(\sqrt{\frac{1}{1 + e^{-2 \cdot x}}} \cdot \sqrt{2}\right)\right)}^{3}}{\left(\left(\left(\sqrt{\frac{1}{1 + e^{-2 \cdot x}}} \cdot \sqrt{2}\right) \cdot \left(\sqrt{\frac{1}{1 + e^{-2 \cdot x}}} \cdot \sqrt{2}\right)\right) \cdot \left(\left(\sqrt{\frac{1}{1 + e^{-2 \cdot x}}} \cdot \sqrt{2}\right) \cdot \left(\sqrt{\frac{1}{1 + e^{-2 \cdot x}}} \cdot \sqrt{2}\right)\right) - \left(\sqrt{\frac{1}{1 + e^{-2 \cdot x}}} \cdot \sqrt{2}\right) \cdot \left(\sqrt{\frac{1}{1 + e^{-2 \cdot x}}} \cdot \left(-\sqrt{2}\right)\right)\right) + 1}\\ \mathbf{elif}\;x \le 0.00794597355349963:\\ \;\;\;\;\left(x + {x}^{5} \cdot \frac{2}{15}\right) - \frac{1}{3} \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right) \cdot \sqrt{\frac{2}{1 + e^{-2 \cdot x}}} - \left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right)}{\sqrt{\frac{2}{1 + e^{-2 \cdot x}}} - 1}\\ \end{array}\]

Runtime

Time bar (total: 22.7s)Debug logProfile

herbie shell --seed 2018248 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))