Average Error: 39.8 → 0.7
Time: 17.0s
Precision: 64
Internal Precision: 128
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.0019604778091778475:\\ \;\;\;\;\frac{e^{x}}{\sqrt[3]{e^{x} - 1}} \cdot \frac{1}{\sqrt[3]{e^{x} - 1} \cdot \sqrt[3]{e^{x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x} + \frac{1}{2}\right) + \frac{1}{12} \cdot x\\ \end{array}\]

Error

Bits error versus x

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Your Program's Arguments

Results

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Target

Original39.8
Target39.3
Herbie0.7
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -0.0019604778091778475

    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 *-un-lft-identity0.0

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

    5. Applied times-frac0.0

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

    if -0.0019604778091778475 < x

    1. Initial program 60.0

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

    2. Taylor expanded around 0 1.1

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.0019604778091778475:\\ \;\;\;\;\frac{e^{x}}{\sqrt[3]{e^{x} - 1}} \cdot \frac{1}{\sqrt[3]{e^{x} - 1} \cdot \sqrt[3]{e^{x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x} + \frac{1}{2}\right) + \frac{1}{12} \cdot x\\ \end{array}\]

Runtime

Time bar (total: 17.0s)Debug logProfile

herbie shell --seed 2018349 
(FPCore (x)
  :name "expq2 (section 3.11)"

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

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