Average Error: 39.9 → 0.2
Time: 14.8s
Precision: 64
Internal Precision: 1344
\[\frac{e^{x} - 1}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.0001571173812753324:\\ \;\;\;\;\frac{{\left(e^{x}\right)}^{3} - 1}{\left(e^{x} \cdot e^{x} + \left(e^{x} + 1\right)\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{1}{2} + \frac{1}{6} \cdot x\right) \cdot \left(x \cdot x\right) + x}{x}\\ \end{array}\]

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original39.9
Target39.0
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;x \lt 1 \land x \gt -1:\\ \;\;\;\;\frac{e^{x} - 1}{\log \left(e^{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x} - 1}{x}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -0.0001571173812753324

    1. Initial program 0.1

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

    3. Applied flip3--0.0

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

    4. Applied associate-/l/0.0

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

    5. Simplified0.0

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

    if -0.0001571173812753324 < x

    1. Initial program 60.2

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

    2. Taylor expanded around 0 0.3

      \[\leadsto \frac{\color{blue}{x + \left(\frac{1}{6} \cdot {x}^{3} + \frac{1}{2} \cdot {x}^{2}\right)}}{x}\]

    3. Simplified0.3

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.0001571173812753324:\\ \;\;\;\;\frac{{\left(e^{x}\right)}^{3} - 1}{\left(e^{x} \cdot e^{x} + \left(e^{x} + 1\right)\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{1}{2} + \frac{1}{6} \cdot x\right) \cdot \left(x \cdot x\right) + x}{x}\\ \end{array}\]

Runtime

Time bar (total: 14.8s)Debug logProfile

herbie shell --seed 2018221 
(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))