Average Error: 40.5 → 0.6
Time: 10.0s
Precision: 64
Internal Precision: 1344
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.00204031999091806:\\ \;\;\;\;\frac{e^{x}}{\frac{e^{x} \cdot e^{x} - 1}{e^{x} + 1}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x} + \frac{1}{2}\right) + \frac{1}{12} \cdot x\\ \end{array}\]

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original40.5
Target40.1
Herbie0.6
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -0.00204031999091806

    1. Initial program 0.0

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

    3. Applied flip--0.0

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

    if -0.00204031999091806 < x

    1. Initial program 60.4

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

    2. Taylor expanded around 0 1.0

      \[\leadsto \frac{e^{x}}{\color{blue}{x + \left(\frac{1}{6} \cdot {x}^{3} + \frac{1}{2} \cdot {x}^{2}\right)}}\]

    3. Simplified1.0

      \[\leadsto \frac{e^{x}}{\color{blue}{x + \left(x \cdot x\right) \cdot \left(x \cdot \frac{1}{6} + \frac{1}{2}\right)}}\]

    4. Taylor expanded around 0 0.8

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.00204031999091806:\\ \;\;\;\;\frac{e^{x}}{\frac{e^{x} \cdot e^{x} - 1}{e^{x} + 1}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x} + \frac{1}{2}\right) + \frac{1}{12} \cdot x\\ \end{array}\]

Runtime

Time bar (total: 10.0s)Debug logProfile

herbie shell --seed 2018220 
(FPCore (x)
  :name "expq2 (section 3.11)"

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

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