Average Error: 39.9 → 0.6
Time: 21.1s
Precision: 64
Internal Precision: 128
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.001835991498205079:\\ \;\;\;\;\frac{e^{x}}{\log \left(e^{e^{x} - 1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{2} + \frac{1}{x}\right) + \frac{1}{12} \cdot x\\ \end{array}\]

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original39.9
Target39.5
Herbie0.6
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -0.001835991498205079

    1. Initial program 0.0

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

    3. Applied add-log-exp0.0

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

    if -0.001835991498205079 < 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. 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

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.001835991498205079:\\ \;\;\;\;\frac{e^{x}}{\log \left(e^{e^{x} - 1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{2} + \frac{1}{x}\right) + \frac{1}{12} \cdot x\\ \end{array}\]

Runtime

Time bar (total: 21.1s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes21.50.60.421.199%
herbie shell --seed 2018351 
(FPCore (x)
  :name "expq2 (section 3.11)"

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

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