Average Error: 40.4 → 0.7
Time: 27.2s
Precision: 64
Internal Precision: 128
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{x} - 1 \le -0.024443497198627262:\\ \;\;\;\;\left(\sqrt[3]{\frac{e^{x}}{e^{x} - 1}} \cdot \sqrt[3]{\frac{e^{x}}{e^{x} - 1}}\right) \cdot \sqrt[3]{\left(e^{x + x} + \left(e^{x} + 1\right)\right) \cdot \frac{e^{x}}{{\left(e^{x}\right)}^{3} - 1}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{2} + \frac{1}{x}\right) + \frac{1}{12} \cdot x\\ \end{array}\]

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 (- (exp x) 1) < -0.024443497198627262

    1. Initial program 0.0

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

    3. Applied add-cube-cbrt0.0

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

    5. Applied flip3--0.0

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

    6. Applied associate-/r/0.0

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

    7. Simplified0.0

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

    if -0.024443497198627262 < (- (exp x) 1)

    1. Initial program 60.1

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

    2. 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

    \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} - 1 \le -0.024443497198627262:\\ \;\;\;\;\left(\sqrt[3]{\frac{e^{x}}{e^{x} - 1}} \cdot \sqrt[3]{\frac{e^{x}}{e^{x} - 1}}\right) \cdot \sqrt[3]{\left(e^{x + x} + \left(e^{x} + 1\right)\right) \cdot \frac{e^{x}}{{\left(e^{x}\right)}^{3} - 1}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{2} + \frac{1}{x}\right) + \frac{1}{12} \cdot x\\ \end{array}\]

Runtime

Time bar (total: 27.2s)Debug logProfile

herbie shell --seed 2018274 
(FPCore (x)
  :name "expq2 (section 3.11)"

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

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