Average Error: 29.8 → 0.1
Time: 1.2m
Precision: 64
Internal Precision: 1344
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{2}{1 + e^{-2 \cdot x}} - 1 \le -4.178633382720398 \cdot 10^{-05}:\\ \;\;\;\;\frac{2}{(e^{\log_* (1 + \left(1 + e^{-2 \cdot x}\right))} - 1)^*} - 1\\ \mathbf{if}\;\frac{2}{1 + e^{-2 \cdot x}} - 1 \le 1.0646512524247942 \cdot 10^{-06}:\\ \;\;\;\;\left(\frac{2}{15} \cdot {x}^{5} + x\right) - \frac{1}{3} \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;(\left(\frac{2}{{1}^{3} + {\left(e^{-2 \cdot x}\right)}^{3}}\right) \cdot \left(\left(e^{-2 \cdot x} \cdot e^{-2 \cdot x} - e^{-2 \cdot x}\right) + 1\right) + \left(-1\right))_*\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

Derivation

  1. Split input into 3 regimes

  2. if (- (/ 2 (+ 1 (exp (* -2 x)))) 1) < -4.178633382720398e-05

    1. Initial program 0.1

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

    3. Applied expm1-log1p-u0.1

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

    if -4.178633382720398e-05 < (- (/ 2 (+ 1 (exp (* -2 x)))) 1) < 1.0646512524247942e-06

    1. Initial program 59.3

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

    2. Taylor expanded around 0 0

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

    if 1.0646512524247942e-06 < (- (/ 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-+0.2

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

    4. Applied associate-/r/0.2

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

    5. Applied fma-neg0.2

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

  4. Applied simplify0.1

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\frac{2}{1 + e^{-2 \cdot x}} - 1 \le -4.178633382720398 \cdot 10^{-05}:\\ \;\;\;\;\frac{2}{(e^{\log_* (1 + \left(1 + e^{-2 \cdot x}\right))} - 1)^*} - 1\\ \mathbf{if}\;\frac{2}{1 + e^{-2 \cdot x}} - 1 \le 1.0646512524247942 \cdot 10^{-06}:\\ \;\;\;\;\left(\frac{2}{15} \cdot {x}^{5} + x\right) - \frac{1}{3} \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;(\left(\frac{2}{{1}^{3} + {\left(e^{-2 \cdot x}\right)}^{3}}\right) \cdot \left(\left(e^{-2 \cdot x} \cdot e^{-2 \cdot x} - e^{-2 \cdot x}\right) + 1\right) + \left(-1\right))_*\\ \end{array}}\]

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed 2018206 +o rules:numerics
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))