Average Error: 40.3 → 0.6
Time: 11.5s
Precision: 64
Internal Precision: 128
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.0021313132355051233:\\ \;\;\;\;\frac{\frac{e^{x}}{1 + \sqrt{e^{x}}}}{\sqrt{e^{x}} - 1}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{12} + \left(\frac{1}{x} + \frac{1}{2}\right)\\ \end{array}\]

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

Derivation

  1. Split input into 2 regimes

  2. if x < -0.0021313132355051233

    1. Initial program 0.0

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

    3. Applied add-sqr-sqrt0.0

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

    4. Applied difference-of-sqr-10.0

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

    5. Applied associate-/r*0.0

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

    if -0.0021313132355051233 < x

    1. Initial program 60.1

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

    2. 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

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.0021313132355051233:\\ \;\;\;\;\frac{\frac{e^{x}}{1 + \sqrt{e^{x}}}}{\sqrt{e^{x}} - 1}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{12} + \left(\frac{1}{x} + \frac{1}{2}\right)\\ \end{array}\]

Runtime

Time bar (total: 11.5s)Debug logProfile

herbie shell --seed 2018273 
(FPCore (x)
  :name "expq2 (section 3.11)"

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

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