\[\frac{e^{x} - 1}{x}\]
Test:
Kahan's exp quotient
Bits:
128 bits
Bits error versus x
Time: 38.8 s
Input Error: 26.9
Output Error: 2.1
Log: ⚲
Profile: 🕒
\(\frac{x \cdot \left(\left(x \cdot \frac{1}{8} + \frac{1}{2}\right) + {x}^2 \cdot \frac{1}{48}\right)}{\frac{x}{1 + \sqrt{e^{x}}}}\)
  1. Started with
    \[\frac{e^{x} - 1}{x}\]
    26.9
  2. Using strategy rm
    26.9
  3. Applied add-sqr-sqrt to get
    \[\frac{\color{red}{e^{x}} - 1}{x} \leadsto \frac{\color{blue}{{\left(\sqrt{e^{x}}\right)}^2} - 1}{x}\]
    27.1
  4. Applied difference-of-sqr-1 to get
    \[\frac{\color{red}{{\left(\sqrt{e^{x}}\right)}^2 - 1}}{x} \leadsto \frac{\color{blue}{\left(\sqrt{e^{x}} + 1\right) \cdot \left(\sqrt{e^{x}} - 1\right)}}{x}\]
    27.1
  5. Applied taylor to get
    \[\frac{\left(\sqrt{e^{x}} + 1\right) \cdot \left(\sqrt{e^{x}} - 1\right)}{x} \leadsto \frac{\left(\sqrt{e^{x}} + 1\right) \cdot \left(\frac{1}{8} \cdot {x}^2 + \left(\frac{1}{48} \cdot {x}^{3} + \frac{1}{2} \cdot x\right)\right)}{x}\]
    2.3
  6. Taylor expanded around 0 to get
    \[\frac{\left(\sqrt{e^{x}} + 1\right) \cdot \color{red}{\left(\frac{1}{8} \cdot {x}^2 + \left(\frac{1}{48} \cdot {x}^{3} + \frac{1}{2} \cdot x\right)\right)}}{x} \leadsto \frac{\left(\sqrt{e^{x}} + 1\right) \cdot \color{blue}{\left(\frac{1}{8} \cdot {x}^2 + \left(\frac{1}{48} \cdot {x}^{3} + \frac{1}{2} \cdot x\right)\right)}}{x}\]
    2.3
  7. Applied simplify to get
    \[\frac{\left(\sqrt{e^{x}} + 1\right) \cdot \left(\frac{1}{8} \cdot {x}^2 + \left(\frac{1}{48} \cdot {x}^{3} + \frac{1}{2} \cdot x\right)\right)}{x} \leadsto \frac{\left({x}^3 \cdot \frac{1}{48} + {x}^2 \cdot \frac{1}{8}\right) + x \cdot \frac{1}{2}}{\frac{x}{\sqrt{e^{x}} + 1}}\]
    2.3
  8. Applied final simplification
  9. Applied simplify to get
    \[\color{red}{\frac{\left({x}^3 \cdot \frac{1}{48} + {x}^2 \cdot \frac{1}{8}\right) + x \cdot \frac{1}{2}}{\frac{x}{\sqrt{e^{x}} + 1}}} \leadsto \color{blue}{\frac{x \cdot \left(\left(x \cdot \frac{1}{8} + \frac{1}{2}\right) + {x}^2 \cdot \frac{1}{48}\right)}{\frac{x}{1 + \sqrt{e^{x}}}}}\]
    2.1

Original test:


(lambda ((x default))
  #:name "Kahan's exp quotient"
  (/ (- (exp x) 1) x)
  #:target
  (if (and (< x 1) (> x -1)) (/ (- (exp x) 1) (log (exp x))) (/ (- (exp x) 1) x)))