Average Error: 39.9 → 0.7
Time: 52.0s
Precision: 64
Internal Precision: 1408
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.0021267026330599303:\\ \;\;\;\;\frac{1}{\sqrt{e^{x}} + 1} \cdot \frac{e^{x}}{\sqrt{e^{x}} - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} + \left(\frac{1}{x} + \frac{1}{12} \cdot x\right)\\ \end{array}\]

Error

Bits error versus x

Target

Original39.9
Target39.5
Herbie0.7
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -0.0021267026330599303

    1. Initial program 0.0

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

    3. Applied add-sqr-sqrt0.0

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

    4. Applied difference-of-sqr-10.0

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

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

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

    6. Applied times-frac0.0

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

    if -0.0021267026330599303 < x

    1. Initial program 60.0

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

    2. Taylor expanded around 0 1.0

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

Runtime

Time bar (total: 52.0s)Debug logProfile

herbie shell --seed '#(1063027428 1192549564 1443466578 604016274 3637110559 1698629644)' 
(FPCore (x)
  :name "expq2 (section 3.11)"

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

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