Average Error: 39.6 → 0.3
Time: 55.6s
Precision: 64
Internal Precision: 1344
\[\frac{e^{x} - 1}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.0001149485681894146:\\ \;\;\;\;\frac{\sqrt[3]{\frac{e^{x + x} - 1}{e^{x} + 1}} \cdot \sqrt[3]{\frac{e^{x + x} - 1}{e^{x} + 1}}}{\frac{x}{\sqrt[3]{e^{x} - 1}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{2} \cdot {x}^{2} + \left(\frac{1}{6} \cdot {x}^{3} + x\right)}{x}\\ \end{array}\]

Error

Bits error versus x

Target

Original39.6
Target38.8
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 < -0.0001149485681894146

    1. Initial program 0.1

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

    3. Applied flip--0.1

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

    4. Applied simplify0.1

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

    6. Applied add-cube-cbrt0.1

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

    7. Applied associate-/l*0.1

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

    8. Applied simplify0.1

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

    if -0.0001149485681894146 < x

    1. Initial program 60.4

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

    2. Taylor expanded around 0 0.4

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

Runtime

Time bar (total: 55.6s)Debug logProfile

herbie shell --seed '#(1071373924 2949776965 1885069702 3247780810 90874544 2263903749)' 
(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))