Average Error: 40.5 → 0.9
Time: 10.4s
Precision: 64
Internal Precision: 1344
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{e^{x}}{e^{x} - 1} \le 1.000496496154137:\\ \;\;\;\;\frac{e^{x}}{e^{x} - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{12} \cdot x + \left(\frac{1}{2} + \frac{1}{x}\right)\\ \end{array}\]

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original40.5
Target40.1
Herbie0.9
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (exp x) (- (exp x) 1)) < 1.000496496154137

    1. Initial program 1.2

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

    2. Initial simplification1.2

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

    3. Taylor expanded around inf 1.2

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

    if 1.000496496154137 < (/ (exp x) (- (exp x) 1))

    1. Initial program 61.0

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

    2. Initial simplification61.0

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

    3. Taylor expanded around 0 0.7

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

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{x}}{e^{x} - 1} \le 1.000496496154137:\\ \;\;\;\;\frac{e^{x}}{e^{x} - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{12} \cdot x + \left(\frac{1}{2} + \frac{1}{x}\right)\\ \end{array}\]

Runtime

Time bar (total: 10.4s)Debug logProfile

herbie shell --seed 2018251 
(FPCore (x)
  :name "expq2 (section 3.11)"

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

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