Average Error: 40.2 → 1.1
Time: 19.7s
Precision: 64
Internal Precision: 128
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\frac{e^{x}}{x + \left(x \cdot \frac{1}{6} + \frac{1}{2}\right) \cdot \left(x \cdot x\right)}\]

Error

Bits error versus x

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Your Program's Arguments

Results

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Target

Original40.2
Target39.7
Herbie1.1
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Initial program 40.2

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

  2. Initial simplification40.2

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

  3. Taylor expanded around 0 11.2

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

  4. Simplified1.1

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

  5. Final simplification1.1

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

Runtime

Time bar (total: 19.7s)Debug logProfile

herbie shell --seed 2018304 
(FPCore (x)
  :name "expq2 (section 3.11)"

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

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