Average Error: 40.0 → 0.3
Time: 17.3s
Precision: 64
Internal Precision: 128
\[\frac{e^{x} - 1}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.003318031049799763:\\ \;\;\;\;\frac{\frac{-1 + \left(e^{x} \cdot e^{x}\right) \cdot \left(\left(e^{x} \cdot e^{x}\right) \cdot \left(e^{x} \cdot e^{x}\right)\right)}{\left(e^{x} + 1\right) \cdot \left(\left(e^{x} \cdot e^{x}\right) \cdot \left(e^{x} \cdot e^{x}\right) + \left(1 + e^{x} \cdot e^{x}\right)\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{e^{x}} + 1}{x \cdot \left(\frac{-1}{2} + x \cdot \frac{1}{24}\right) + 2}\\ \end{array}\]

Error

Bits error versus x

Target

Original40.0
Target39.3
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;x \lt 1 \land x \gt -1:\\ \;\;\;\;\frac{e^{x} - 1}{\log \left(e^{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x} - 1}{x}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -0.003318031049799763

    1. Initial program 0.0

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

    3. Applied flip--0.0

      \[\leadsto \frac{\color{blue}{\frac{e^{x} \cdot e^{x} - 1 \cdot 1}{e^{x} + 1}}}{x}\]
    4. Using strategy rm

    5. Applied flip3--0.0

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

    6. Applied associate-/l/0.0

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

    7. Simplified0.0

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

    if -0.003318031049799763 < x

    1. Initial program 60.0

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

    3. Applied add-sqr-sqrt60.1

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

    4. Applied difference-of-sqr-160.1

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

    5. Applied associate-/l*60.1

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

    6. Taylor expanded around 0 0.4

      \[\leadsto \frac{\sqrt{e^{x}} + 1}{\color{blue}{\left(\frac{1}{24} \cdot {x}^{2} + 2\right) - \frac{1}{2} \cdot x}}\]

    7. Simplified0.4

      \[\leadsto \frac{\sqrt{e^{x}} + 1}{\color{blue}{x \cdot \left(\frac{-1}{2} + \frac{1}{24} \cdot x\right) + 2}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.003318031049799763:\\ \;\;\;\;\frac{\frac{-1 + \left(e^{x} \cdot e^{x}\right) \cdot \left(\left(e^{x} \cdot e^{x}\right) \cdot \left(e^{x} \cdot e^{x}\right)\right)}{\left(e^{x} + 1\right) \cdot \left(\left(e^{x} \cdot e^{x}\right) \cdot \left(e^{x} \cdot e^{x}\right) + \left(1 + e^{x} \cdot e^{x}\right)\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{e^{x}} + 1}{x \cdot \left(\frac{-1}{2} + x \cdot \frac{1}{24}\right) + 2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019068 
(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))