Average Error: 39.8 → 0.5
Time: 1.1m
Precision: 64
Internal Precision: 1344
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{x} \le 0.9991732492398515:\\ \;\;\;\;\frac{e^{x}}{\frac{\sqrt[3]{{\left(e^{x} \cdot e^{x} - 1\right)}^{3}}}{e^{x} + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} + \left(\frac{1}{x} + \frac{1}{12} \cdot x\right)\\ \end{array}\]

Error

Bits error versus x

Target

Original39.8
Target39.4
Herbie0.5
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Split input into 2 regimes

  2. if (exp x) < 0.9991732492398515

    1. Initial program 0.0

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

    3. Applied flip--0.0

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

    4. Applied simplify0.0

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

    6. Applied add-cbrt-cube0.0

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

    7. Applied simplify0.0

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

    if 0.9991732492398515 < (exp x)

    1. Initial program 60.4

      \[\frac{e^{x}}{e^{x} - 1}\]

    2. Taylor expanded around 0 0.8

      \[\leadsto \color{blue}{\frac{1}{2} + \left(\frac{1}{x} + \frac{1}{12} \cdot x\right)}\]
  3. Recombined 2 regimes into one program.

Runtime

Time bar (total: 1.1m)Debug logProfile

herbie shell --seed '#(1071373924 2949776965 1885069702 3247780810 90874544 2263903749)' 
(FPCore (x)
  :name "expq2 (section 3.11)"

  :herbie-target
  (/ 1 (- 1 (exp (- x))))

  (/ (exp x) (- (exp x) 1)))