Average Error: 40.1 → 0.3
Time: 28.8s
Precision: 64
Internal Precision: 128
\[\frac{e^{x} - 1}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.00011161306602174622:\\ \;\;\;\;\frac{\frac{\sqrt[3]{\frac{e^{x + \left(x + x\right)} \cdot e^{x + \left(x + x\right)} - 1}{1 + e^{x + \left(x + x\right)}} \cdot \left(\left(e^{x} - 1\right) \cdot \left(e^{x} - 1\right)\right)}}{\sqrt[3]{e^{x} \cdot e^{x} + \left(e^{x} + 1\right)}}}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(x \cdot \frac{1}{6} + \frac{1}{2}\right) \cdot x\\ \end{array}\]

Error

Bits error versus x

Target

Original40.1
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 < -0.00011161306602174622

    1. Initial program 0.1

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

    3. Applied add-cbrt-cube0.1

      \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\left(e^{x} - 1\right) \cdot \left(e^{x} - 1\right)\right) \cdot \left(e^{x} - 1\right)}}}{x}\]
    4. Using strategy rm

    5. Applied flip3--0.1

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

    6. Applied associate-*r/0.1

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

    7. Applied cbrt-div0.1

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

    8. Simplified0.1

      \[\leadsto \frac{\frac{\color{blue}{\sqrt[3]{\left(\left(e^{x} - 1\right) \cdot \left(e^{x} - 1\right)\right) \cdot \left(e^{\left(x + x\right) + x} - 1\right)}}}{\sqrt[3]{e^{x} \cdot e^{x} + \left(1 \cdot 1 + e^{x} \cdot 1\right)}}}{x}\]
    9. Using strategy rm

    10. Applied flip--0.1

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

    if -0.00011161306602174622 < x

    1. Initial program 60.3

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

    2. Taylor expanded around 0 0.4

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

    3. Simplified0.4

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.00011161306602174622:\\ \;\;\;\;\frac{\frac{\sqrt[3]{\frac{e^{x + \left(x + x\right)} \cdot e^{x + \left(x + x\right)} - 1}{1 + e^{x + \left(x + x\right)}} \cdot \left(\left(e^{x} - 1\right) \cdot \left(e^{x} - 1\right)\right)}}{\sqrt[3]{e^{x} \cdot e^{x} + \left(e^{x} + 1\right)}}}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(x \cdot \frac{1}{6} + \frac{1}{2}\right) \cdot x\\ \end{array}\]

Reproduce

herbie shell --seed 2019091 
(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))