Average Error: 39.6 → 0.4
Time: 12.6s
Precision: 64
Internal Precision: 1344
\[\frac{e^{x} - 1}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.0001319153847279274:\\ \;\;\;\;\frac{-1 + e^{x + x}}{\left(e^{x} + 1\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{1}{6} \cdot x + \frac{1}{2}\right) \cdot x\\ \end{array}\]

Error

Bits error versus x

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Your Program's Arguments

Results

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Target

Original39.6
Target38.8
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;x \lt 1 \land x \gt -1:\\ \;\;\;\;\frac{e^{x} - 1}{\log \left(e^{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x} - 1}{x}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -0.0001319153847279274

    1. Initial program 0.0

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

    2. Initial simplification0.0

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

    4. Applied flip--0.0

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

    5. Applied associate-/l/0.0

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

    6. Simplified0.0

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

    8. Applied prod-exp0.0

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

    if -0.0001319153847279274 < x

    1. Initial program 60.1

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

    2. Initial simplification60.1

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

    3. Taylor expanded around 0 0.5

      \[\leadsto \color{blue}{\frac{1}{2} \cdot x + \left(\frac{1}{6} \cdot {x}^{2} + 1\right)}\]

    4. Simplified0.5

      \[\leadsto \color{blue}{\left(\frac{1}{6} \cdot x + \frac{1}{2}\right) \cdot x + 1}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.0001319153847279274:\\ \;\;\;\;\frac{-1 + e^{x + x}}{\left(e^{x} + 1\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{1}{6} \cdot x + \frac{1}{2}\right) \cdot x\\ \end{array}\]

Runtime

Time bar (total: 12.6s)Debug logProfile

herbie shell --seed 2018234 
(FPCore (x)
  :name "Kahan's exp quotient"

  :herbie-target
  (if (and (< x 1) (> x -1)) (/ (- (exp x) 1) (log (exp x))) (/ (- (exp x) 1) x))

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