Average Error: 29.5 → 0.0
Time: 21.8s
Precision: 64
Internal Precision: 1344
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{2}{1 + e^{-2 \cdot x}} - 1 \le -0.0001403873805974133:\\ \;\;\;\;\log \left(e^{\frac{2}{1 + e^{-2 \cdot x}} - 1}\right)\\ \mathbf{if}\;\frac{2}{1 + e^{-2 \cdot x}} - 1 \le 0.00016709790383551137:\\ \;\;\;\;x - \left(\frac{1}{3} \cdot x\right) \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\frac{2}{1 + e^{-2 \cdot x}} - 1}\right)\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

Derivation

  1. Split input into 2 regimes

  2. if (- (/ 2 (+ 1 (exp (* -2 x)))) 1) < -0.0001403873805974133 or 0.00016709790383551137 < (- (/ 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 add-log-exp0.1

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

    if -0.0001403873805974133 < (- (/ 2 (+ 1 (exp (* -2 x)))) 1) < 0.00016709790383551137

    1. Initial program 59.3

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

    2. Taylor expanded around 0 59.3

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

    3. Applied simplify0.0

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

Runtime

Time bar (total: 21.8s)Debug logProfile

herbie shell --seed '#(1072107073 2127697367 3936270018 2300570620 2134894798 4023771849)' 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))