Average Error: 39.8 → 0.5
Time: 1.0m
Precision: 64
Internal Precision: 1344
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{1 - e^{-x}} \le 1.0308725086867097:\\ \;\;\;\;\frac{1}{1 - e^{-x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} + \left(\frac{1}{x} + \frac{1}{12} \cdot x\right)\\ \end{array}\]

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Split input into 2 regimes

  2. if (/ 1 (- 1 (exp (- x)))) < 1.0308725086867097

    1. Initial program 2.1

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

    3. Applied clear-num2.1

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

    4. Applied simplify1.2

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

    if 1.0308725086867097 < (/ 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: 1.0m)Debug logProfile

herbie shell --seed 2018195 
(FPCore (x)
  :name "expq2 (section 3.11)"

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

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