Average Error: 39.6 → 0.4
Time: 27.2s
Precision: 64
Internal Precision: 1344
\[\frac{e^{x} - 1}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.0001524787362974112:\\ \;\;\;\;\frac{{\left(e^{x}\right)}^{3} - 1}{\left(e^{x} \cdot e^{x} + \left(e^{x} + 1\right)\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left({x}^{2} \cdot \frac{1}{36} + 1\right) + \frac{1}{6} \cdot x\right) \cdot \left(\left(\frac{1}{12} \cdot {x}^{2} + 1\right) + \frac{1}{3} \cdot x\right)\\ \end{array}\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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 < -0.0001524787362974112

    1. Initial program 0.1

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

    2. Initial simplification0.1

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

    4. Applied flip3--0.1

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

    5. Applied associate-/l/0.1

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

    6. Simplified0.1

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

    if -0.0001524787362974112 < 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. Using strategy rm

    5. Applied add-cube-cbrt0.6

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

    6. Taylor expanded around 0 0.5

      \[\leadsto \color{blue}{\left(\frac{1}{3} \cdot x + \left(\frac{1}{12} \cdot {x}^{2} + 1\right)\right)} \cdot \sqrt[3]{\frac{1}{2} \cdot x + \left(\frac{1}{6} \cdot {x}^{2} + 1\right)}\]

    7. Taylor expanded around 0 0.5

      \[\leadsto \left(\frac{1}{3} \cdot x + \left(\frac{1}{12} \cdot {x}^{2} + 1\right)\right) \cdot \color{blue}{\left(\frac{1}{6} \cdot x + \left(\frac{1}{36} \cdot {x}^{2} + 1\right)\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.0001524787362974112:\\ \;\;\;\;\frac{{\left(e^{x}\right)}^{3} - 1}{\left(e^{x} \cdot e^{x} + \left(e^{x} + 1\right)\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left({x}^{2} \cdot \frac{1}{36} + 1\right) + \frac{1}{6} \cdot x\right) \cdot \left(\left(\frac{1}{12} \cdot {x}^{2} + 1\right) + \frac{1}{3} \cdot x\right)\\ \end{array}\]

Runtime

Time bar (total: 27.2s)Debug logProfile

herbie shell --seed 2018230 
(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))