Average Error: 29.5 → 0.0
Time: 30.6s
Precision: 64
Internal Precision: 1408
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{2}{1 + e^{-2 \cdot x}} - 1 \le -0.06640644624851669:\\ \;\;\;\;\frac{\left(\sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}} - 1} \cdot \sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}} - 1}\right) \cdot \sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}} - 1}}{\left(\sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} + 1} \cdot \sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} + 1}\right) \cdot \sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} + 1}}\\ \mathbf{if}\;\frac{2}{1 + e^{-2 \cdot x}} - 1 \le 1.0876732435667694 \cdot 10^{-06}:\\ \;\;\;\;\left(\frac{2}{15} \cdot {x}^{5} + x\right) - \frac{1}{3} \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}} - 1} \cdot \sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}} - 1}\right) \cdot \sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}} - 1}}{\left(\sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} + 1} \cdot \sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} + 1}\right) \cdot \sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} + 1}}\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

Derivation

  1. Split input into 2 regimes

  2. if (- (/ 2 (+ 1 (exp (* -2 x)))) 1) < -0.06640644624851669 or 1.0876732435667694e-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 flip--0.1

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

    4. Applied simplify0.1

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

    6. Applied add-cube-cbrt0.1

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

    8. Applied add-cube-cbrt0.1

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

    if -0.06640644624851669 < (- (/ 2 (+ 1 (exp (* -2 x)))) 1) < 1.0876732435667694e-06

    1. Initial program 59.3

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

    2. Taylor expanded around 0 0.0

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

Runtime

Time bar (total: 30.6s)Debug logProfile

herbie shell --seed '#(1064269945 2896236262 301053905 1701069080 1701464310 1614783279)' 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))