Average Error: 28.6 → 0.0
Time: 45.7s
Precision: 64
Internal Precision: 128
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.006828581683054248:\\ \;\;\;\;\sqrt[3]{\left(\frac{2}{e^{-2 \cdot x} + 1} - 1\right) \cdot \left(\log \left(e^{\frac{2}{e^{-2 \cdot x} + 1} - 1}\right) \cdot \log \left(e^{\frac{2}{e^{-2 \cdot x} + 1} - 1}\right)\right)}\\ \mathbf{elif}\;x \le 0.006546954388256739:\\ \;\;\;\;\left(x + x \cdot \left(\left(x \cdot x\right) \cdot \frac{-1}{3}\right)\right) + {x}^{5} \cdot \frac{2}{15}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(\frac{2}{e^{-2 \cdot x} + 1} - 1\right) \cdot \left(\log \left(e^{\frac{2}{e^{-2 \cdot x} + 1} - 1}\right) \cdot \log \left(e^{\frac{2}{e^{-2 \cdot x} + 1} - 1}\right)\right)}\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

Derivation

  1. Split input into 2 regimes

  2. if x < -0.006828581683054248 or 0.006546954388256739 < x

    1. Initial program 0.0

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

    3. Applied add-log-exp0.0

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

    4. Applied add-log-exp0.0

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

    5. Applied diff-log0.0

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

    6. Simplified0.0

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

    8. Applied add-cbrt-cube0.0

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

    9. Taylor expanded around -inf 0.0

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

    10. Simplified0.0

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

    if -0.006828581683054248 < x < 0.006546954388256739

    1. Initial program 59.0

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

    3. Applied add-log-exp59.0

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

    4. Applied add-log-exp59.0

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

    5. Applied diff-log59.0

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

    6. Simplified59.0

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

    7. Taylor expanded around 0 0.0

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

    8. Simplified0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.006828581683054248:\\ \;\;\;\;\sqrt[3]{\left(\frac{2}{e^{-2 \cdot x} + 1} - 1\right) \cdot \left(\log \left(e^{\frac{2}{e^{-2 \cdot x} + 1} - 1}\right) \cdot \log \left(e^{\frac{2}{e^{-2 \cdot x} + 1} - 1}\right)\right)}\\ \mathbf{elif}\;x \le 0.006546954388256739:\\ \;\;\;\;\left(x + x \cdot \left(\left(x \cdot x\right) \cdot \frac{-1}{3}\right)\right) + {x}^{5} \cdot \frac{2}{15}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(\frac{2}{e^{-2 \cdot x} + 1} - 1\right) \cdot \left(\log \left(e^{\frac{2}{e^{-2 \cdot x} + 1} - 1}\right) \cdot \log \left(e^{\frac{2}{e^{-2 \cdot x} + 1} - 1}\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019090 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))