Average Error: 29.2 → 0.0
Time: 10.4s
Precision: 64
Internal Precision: 1344
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.007549888556423157:\\ \;\;\;\;(\left(\sqrt{\frac{2}{1 + e^{-2 \cdot x}}}\right) \cdot \left(\sqrt{\frac{2}{1 + e^{-2 \cdot x}}}\right) + -1)_*\\ \mathbf{elif}\;x \le 0.006967479385908131:\\ \;\;\;\;\left(\frac{2}{15} \cdot {x}^{5} - {x}^{3} \cdot \frac{1}{3}\right) + x\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

Derivation

  1. Split input into 3 regimes

  2. if x < -0.007549888556423157

    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. Taylor expanded around -inf 0.0

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

    5. Applied add-sqr-sqrt0.0

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

    6. Applied fma-neg0.0

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

    if -0.007549888556423157 < x < 0.006967479385908131

    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 -inf 59.2

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

    4. Taylor expanded around 0 0.0

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

    6. Applied associate--l+0.0

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

    if 0.006967479385908131 < 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. Taylor expanded around -inf 0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.007549888556423157:\\ \;\;\;\;(\left(\sqrt{\frac{2}{1 + e^{-2 \cdot x}}}\right) \cdot \left(\sqrt{\frac{2}{1 + e^{-2 \cdot x}}}\right) + -1)_*\\ \mathbf{elif}\;x \le 0.006967479385908131:\\ \;\;\;\;\left(\frac{2}{15} \cdot {x}^{5} - {x}^{3} \cdot \frac{1}{3}\right) + x\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\ \end{array}\]

Runtime

Time bar (total: 10.4s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes29.20.00.029.2100%
herbie shell --seed 2018285 +o rules:numerics
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))