Average Error: 40.4 → 0.7
Time: 9.0s
Precision: 64
Internal Precision: 1344
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.0019955033651246914:\\ \;\;\;\;\sqrt[3]{\frac{e^{x}}{e^{x} - 1}} \cdot \left(\left(\sqrt[3]{\frac{e^{x}}{e^{x} - 1}} \cdot \sqrt[3]{\frac{\sqrt{e^{x}}}{e^{x} - 1}}\right) \cdot \sqrt[3]{\sqrt{e^{x}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{12} \cdot x + \left(\frac{1}{x} + \frac{1}{2}\right)\\ \end{array}\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original40.4
Target39.9
Herbie0.7
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -0.0019955033651246914

    1. Initial program 0.0

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

    2. Initial simplification0.0

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

    4. Applied add-cube-cbrt0.0

      \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{e^{x}}{e^{x} - 1}} \cdot \sqrt[3]{\frac{e^{x}}{e^{x} - 1}}\right) \cdot \sqrt[3]{\frac{e^{x}}{e^{x} - 1}}}\]
    5. Using strategy rm

    6. Applied *-un-lft-identity0.0

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

    7. Applied add-sqr-sqrt0.0

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

    8. Applied times-frac0.0

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

    9. Applied cbrt-prod0.0

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

    10. Applied associate-*l*0.0

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

    11. Simplified0.0

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

    if -0.0019955033651246914 < x

    1. Initial program 60.2

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

    2. Initial simplification60.2

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

    3. Taylor expanded around 0 1.0

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.0019955033651246914:\\ \;\;\;\;\sqrt[3]{\frac{e^{x}}{e^{x} - 1}} \cdot \left(\left(\sqrt[3]{\frac{e^{x}}{e^{x} - 1}} \cdot \sqrt[3]{\frac{\sqrt{e^{x}}}{e^{x} - 1}}\right) \cdot \sqrt[3]{\sqrt{e^{x}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{12} \cdot x + \left(\frac{1}{x} + \frac{1}{2}\right)\\ \end{array}\]

Runtime

Time bar (total: 9.0s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes21.10.70.420.698.9%
herbie shell --seed 2018274 
(FPCore (x)
  :name "expq2 (section 3.11)"

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

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