Average Error: 40.0 → 0.4
Time: 1.1m
Precision: 64
Internal Precision: 1344
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{1 - e^{-x}} \le 1.6584526274552116:\\ \;\;\;\;\frac{1}{1 - e^{-x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} + \left(\frac{1}{x} + \frac{1}{12} \cdot x\right)\\ \end{array}\]

Error

Bits error versus x

Target

Original40.0
Target39.5
Herbie0.4
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ 1 (- 1 (exp (- x)))) < 1.6584526274552116

    1. Initial program 2.2

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

    3. Applied clear-num2.2

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

    4. Applied simplify1.0

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

    if 1.6584526274552116 < (/ 1 (- 1 (exp (- x))))

    1. Initial program 61.2

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

    2. Taylor expanded around 0 0.1

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

Runtime

Time bar (total: 1.1m)Debug logProfile

herbie shell --seed '#(1071725047 233389029 2036512464 3988615230 2972226563 1111574017)' 
(FPCore (x)
  :name "expq2 (section 3.11)"

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

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