Average Error: 29.4 → 0.3
Time: 46.4s
Precision: 64
Internal Precision: 1344
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{2}{1 + e^{-2 \cdot x}} - 1 \le -6.9003174869058 \cdot 10^{-310}:\\ \;\;\;\;(\left(\sqrt{\frac{2}{1 + e^{-2 \cdot x}}}\right) \cdot \left(\sqrt{\frac{2}{1 + e^{-2 \cdot x}}}\right) + \left(-1\right))_*\\ \mathbf{if}\;\frac{2}{1 + e^{-2 \cdot x}} - 1 \le 0.0004068705389923145:\\ \;\;\;\;\left(x + \frac{2}{15} \cdot {x}^{5}\right) - \frac{1}{3} \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;(\left(\frac{2}{{\left(e^{-2 \cdot x}\right)}^{3} + {1}^{3}}\right) \cdot \left(\left(e^{-2 \cdot x} \cdot e^{-2 \cdot x} - e^{-2 \cdot x}\right) + 1\right) + \left(-1\right))_*\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

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 fma-neg1.2

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

    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 flip3-+0.1

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

    4. Applied associate-/r/0.1

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

    5. Applied fma-neg0.1

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

  4. Applied simplify0.3

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\frac{2}{1 + e^{-2 \cdot x}} - 1 \le -6.9003174869058 \cdot 10^{-310}:\\ \;\;\;\;(\left(\sqrt{\frac{2}{1 + e^{-2 \cdot x}}}\right) \cdot \left(\sqrt{\frac{2}{1 + e^{-2 \cdot x}}}\right) + \left(-1\right))_*\\ \mathbf{if}\;\frac{2}{1 + e^{-2 \cdot x}} - 1 \le 0.0004068705389923145:\\ \;\;\;\;\left(x + \frac{2}{15} \cdot {x}^{5}\right) - \frac{1}{3} \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;(\left(\frac{2}{{\left(e^{-2 \cdot x}\right)}^{3} + {1}^{3}}\right) \cdot \left(\left(e^{-2 \cdot x} \cdot e^{-2 \cdot x} - e^{-2 \cdot x}\right) + 1\right) + \left(-1\right))_*\\ \end{array}}\]

Runtime

Time bar (total: 46.4s)Debug logProfile

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