Average Error: 40.2 → 0.6
Time: 24.1s
Precision: 64
Internal Precision: 1408
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{-x} \le 1.0015501802622442:\\ \;\;\;\;\frac{1}{2} + \left(\frac{1}{x} + \frac{1}{12} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x}}{\frac{\log \left(e^{e^{x + x} - 1}\right)}{e^{x} + 1}}\\ \end{array}\]

Error

Bits error versus x

Target

Original40.2
Target39.8
Herbie0.6
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Split input into 2 regimes

  2. if (exp (- x)) < 1.0015501802622442

    1. Initial program 60.1

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

    2. Taylor expanded around 0 0.8

      \[\leadsto \color{blue}{\frac{1}{2} + \left(\frac{1}{x} + \frac{1}{12} \cdot x\right)}\]

    if 1.0015501802622442 < (exp (- x))

    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-log-exp0.0

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

Runtime

Time bar (total: 24.1s)Debug logProfile

herbie shell --seed '#(1064397287 3527694221 3797617954 1138343853 2854031332 1153838279)' 
(FPCore (x)
  :name "expq2 (section 3.11)"

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

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