Average Error: 29.6 → 0.2
Time: 25.3s
Precision: 64
Internal Precision: 1408
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;1 + e^{-2 \cdot x} \le 1.99527027072904:\\ \;\;\;\;\frac{\frac{2}{\sqrt[3]{1 + e^{-2 \cdot x}} \cdot \sqrt[3]{1 + e^{-2 \cdot x}}}}{\sqrt[3]{1 + e^{-2 \cdot x}}} - 1\\ \mathbf{if}\;1 + e^{-2 \cdot x} \le 2.0000000000929075:\\ \;\;\;\;\left(\frac{2}{15} \cdot {x}^{5} + x\right) - \frac{1}{3} \cdot {x}^{3}\\ \mathbf{if}\;1 + e^{-2 \cdot x} \le +\infty:\\ \;\;\;\;\left(\frac{\sqrt{\frac{2}{\sqrt[3]{1 + e^{-2 \cdot x}} \cdot \sqrt[3]{1 + e^{-2 \cdot x}}}}}{\sqrt[3]{\sqrt{1 + e^{-2 \cdot x}}}} + 1\right) \cdot \left(\left(\sqrt[3]{\frac{\sqrt{\frac{2}{\sqrt[3]{1 + e^{-2 \cdot x}} \cdot \sqrt[3]{1 + e^{-2 \cdot x}}}}}{\sqrt[3]{\sqrt{1 + e^{-2 \cdot x}}}} - 1} \cdot \sqrt[3]{\frac{\sqrt{\frac{2}{\sqrt[3]{1 + e^{-2 \cdot x}} \cdot \sqrt[3]{1 + e^{-2 \cdot x}}}}}{\sqrt[3]{\sqrt{1 + e^{-2 \cdot x}}}} - 1}\right) \cdot \sqrt[3]{\frac{\sqrt{\frac{2}{\sqrt[3]{1 + e^{-2 \cdot x}} \cdot \sqrt[3]{1 + e^{-2 \cdot x}}}}}{\sqrt[3]{\sqrt{1 + e^{-2 \cdot x}}}} - 1}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{\frac{2}{\sqrt[3]{1 + e^{-2 \cdot x}} \cdot \sqrt[3]{1 + e^{-2 \cdot x}}}}}{\sqrt[3]{\sqrt{1 + e^{-2 \cdot x}}}} + 1\right) \cdot \left(\left(\sqrt[3]{\frac{\sqrt{\frac{2}{\sqrt[3]{1 + e^{-2 \cdot x}} \cdot \sqrt[3]{1 + e^{-2 \cdot x}}}}}{\sqrt[3]{\sqrt{1 + e^{-2 \cdot x}}}} - 1} \cdot \sqrt[3]{\frac{\sqrt{\frac{2}{\sqrt[3]{1 + e^{-2 \cdot x}} \cdot \sqrt[3]{1 + e^{-2 \cdot x}}}}}{\sqrt[3]{\sqrt{1 + e^{-2 \cdot x}}}} - 1}\right) \cdot \sqrt[3]{\frac{\sqrt{\frac{2}{\sqrt[3]{1 + e^{-2 \cdot x}} \cdot \sqrt[3]{1 + e^{-2 \cdot x}}}}}{\sqrt[3]{\sqrt{1 + e^{-2 \cdot x}}}} - 1}\right)\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

Derivation

  1. Split input into 3 regimes

  2. if (+ 1 (exp (* -2 x))) < 1.99527027072904

    1. Initial program 0.0

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

    3. Applied add-cube-cbrt0.1

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

    4. Applied associate-/r*0.1

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

    if 1.99527027072904 < (+ 1 (exp (* -2 x))) < 2.0000000000929075

    1. Initial program 59.5

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

    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 2.0000000000929075 < (+ 1 (exp (* -2 x)))

    1. Initial program 0.6

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

    3. Applied add-cube-cbrt0.6

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

    4. Applied associate-/r*0.6

      \[\leadsto \color{blue}{\frac{\frac{2}{\sqrt[3]{1 + e^{-2 \cdot x}} \cdot \sqrt[3]{1 + e^{-2 \cdot x}}}}{\sqrt[3]{1 + e^{-2 \cdot x}}}} - 1\]
    5. Using strategy rm

    6. Applied add-sqr-sqrt0.6

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

    7. Applied cbrt-prod0.6

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

    8. Applied add-sqr-sqrt0.6

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

    9. Applied times-frac0.6

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

    10. Applied difference-of-sqr-10.6

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

    12. Applied add-cube-cbrt0.6

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

Runtime

Time bar (total: 25.3s)Debug logProfile

herbie shell --seed '#(1070131407 1246090267 3027482374 2150728003 2026520792 2347815650)' 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))