Average Error: 40.4 → 0.6
Time: 17.6s
Precision: 64
Internal Precision: 128
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.0016102972301056966:\\ \;\;\;\;\left(\sqrt[3]{e^{x} + 1} \cdot \left(\sqrt[3]{e^{x} + 1} \cdot \sqrt[3]{e^{x} + 1}\right)\right) \cdot \frac{e^{x}}{e^{x} \cdot e^{x} - 1}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{2} + \frac{1}{x}\right) + x \cdot \frac{1}{12}\\ \end{array}\]

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original40.4
Target40.0
Herbie0.6
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -0.0016102972301056966

    1. Initial program 0.0

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

    3. Applied flip--0.0

      \[\leadsto \frac{e^{x}}{\color{blue}{\frac{e^{x} \cdot e^{x} - 1 \cdot 1}{e^{x} + 1}}}\]

    4. Applied associate-/r/0.0

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

    6. Applied add-cube-cbrt0.0

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

    if -0.0016102972301056966 < x

    1. Initial program 60.0

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

    2. Taylor expanded around 0 0.8

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.0016102972301056966:\\ \;\;\;\;\left(\sqrt[3]{e^{x} + 1} \cdot \left(\sqrt[3]{e^{x} + 1} \cdot \sqrt[3]{e^{x} + 1}\right)\right) \cdot \frac{e^{x}}{e^{x} \cdot e^{x} - 1}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{2} + \frac{1}{x}\right) + x \cdot \frac{1}{12}\\ \end{array}\]

Runtime

Time bar (total: 17.6s)Debug logProfile

herbie shell --seed 2018348 
(FPCore (x)
  :name "expq2 (section 3.11)"

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

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