Average Error: 40.5 → 1.1
Time: 23.3s
Precision: 64
Internal Precision: 1344
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\frac{\sqrt[3]{e^{x}} \cdot \sqrt[3]{e^{x}}}{\frac{x + \left(\frac{1}{6} \cdot x + \frac{1}{2}\right) \cdot \left(x \cdot x\right)}{\sqrt[3]{e^{x}}}}\]

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
Herbie1.1
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Initial program 40.5

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

  2. Initial simplification40.5

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

  3. Taylor expanded around 0 11.3

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

  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. Using strategy rm

  6. Applied add-cube-cbrt1.1

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

  7. Applied associate-/l*1.1

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

  8. Final simplification1.1

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

Runtime

Time bar (total: 23.3s)Debug logProfile

herbie shell --seed 2018252 
(FPCore (x)
  :name "expq2 (section 3.11)"

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

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