Average Error: 39.8 → 0.6
Time: 8.1s
Precision: 64
Internal Precision: 1344
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.0011897565413541344:\\ \;\;\;\;\frac{\sqrt{e^{x}}}{\sqrt[3]{\frac{e^{x} \cdot e^{x} - 1}{e^{x} + 1}} \cdot \sqrt[3]{\log \left(e^{e^{x} - 1}\right)}} \cdot \frac{\sqrt{e^{x}}}{\sqrt[3]{e^{x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{2} + \frac{1}{x}\right) + \frac{1}{12} \cdot x\\ \end{array}\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original39.8
Target39.4
Herbie0.6
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -0.0011897565413541344

    1. Initial program 0.0

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

    3. Applied add-cube-cbrt0.0

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

    4. Applied add-sqr-sqrt0.0

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

    5. Applied times-frac0.0

      \[\leadsto \color{blue}{\frac{\sqrt{e^{x}}}{\sqrt[3]{e^{x} - 1} \cdot \sqrt[3]{e^{x} - 1}} \cdot \frac{\sqrt{e^{x}}}{\sqrt[3]{e^{x} - 1}}}\]
    6. Using strategy rm

    7. Applied add-log-exp0.0

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

    9. Applied flip--0.0

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

    if -0.0011897565413541344 < x

    1. Initial program 60.2

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

    2. Taylor expanded around 0 0.9

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.0011897565413541344:\\ \;\;\;\;\frac{\sqrt{e^{x}}}{\sqrt[3]{\frac{e^{x} \cdot e^{x} - 1}{e^{x} + 1}} \cdot \sqrt[3]{\log \left(e^{e^{x} - 1}\right)}} \cdot \frac{\sqrt{e^{x}}}{\sqrt[3]{e^{x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{2} + \frac{1}{x}\right) + \frac{1}{12} \cdot x\\ \end{array}\]

Runtime

Time bar (total: 8.1s)Debug logProfile

herbie shell --seed 2018225 
(FPCore (x)
  :name "expq2 (section 3.11)"

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

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