Average Error: 40.3 → 0.8
Time: 18.0s
Precision: 64
Internal Precision: 1344
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{x} \le 2.762281475111177 \cdot 10^{-131}:\\ \;\;\;\;\frac{-e^{x}}{1 - e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} + \left(\frac{1}{x} + \frac{1}{12} \cdot x\right)\\ \end{array}\]

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original40.3
Target39.9
Herbie0.8
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Split input into 2 regimes

  2. if (exp x) < 2.762281475111177e-131

    1. Initial program 0

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

    3. Applied frac-2neg0

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

    4. Applied simplify0

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

    if 2.762281475111177e-131 < (exp x)

    1. Initial program 59.6

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

    2. Taylor expanded around 0 1.2

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

Runtime

Time bar (total: 18.0s)Debug logProfile

herbie shell --seed 2018206 
(FPCore (x)
  :name "expq2 (section 3.11)"

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

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