Average Error: 40.8 → 38.2
Time: 49.5s
Precision: 64
Internal Precision: 128
\[\frac{e^{x} - 1}{x}\]
↓
\[\left(\frac{1}{8} \cdot x + \left(\frac{1}{2} + \frac{2}{x}\right)\right) \cdot \left(\sqrt{\sqrt{e^{x}}} \cdot \sqrt{\sqrt{e^{x}}}\right) - \left(\frac{1}{8} \cdot x + \left(\frac{1}{2} + \frac{2}{x}\right)\right)\]

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original40.8
Target62.2
Herbie38.2
\[\begin{array}{l} \mathbf{if}\;x \lt 1 \land x \gt -1:\\ \;\;\;\;\frac{e^{x} - 1}{\log \left(e^{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x} - 1}{x}\\ \end{array}\]

Derivation

  1. Initial program 40.8

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

  3. Applied add-sqr-sqrt40.8

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

  4. Applied difference-of-sqr-140.8

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

  5. Taylor expanded around 0 39.1

    \[\leadsto \frac{\color{blue}{\left(\frac{1}{2} \cdot x + \left(\frac{1}{8} \cdot {x}^{2} + 2\right)\right)} \cdot \left(\sqrt{e^{x}} - 1\right)}{x}\]

  6. Taylor expanded around inf 38.2

    \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sqrt{e^{x}} + \left(\frac{1}{8} \cdot \left(x \cdot \sqrt{e^{x}}\right) + 2 \cdot \left(\frac{1}{x} \cdot \sqrt{e^{x}}\right)\right)\right) - \left(\frac{1}{8} \cdot x + \left(2 \cdot \frac{1}{x} + \frac{1}{2}\right)\right)}\]

  7. Simplified38.2

    \[\leadsto \color{blue}{\sqrt{e^{x}} \cdot \left(\left(\frac{2}{x} + \frac{1}{2}\right) + \frac{1}{8} \cdot x\right) - \left(\left(\frac{2}{x} + \frac{1}{2}\right) + \frac{1}{8} \cdot x\right)}\]
  8. Using strategy rm

  9. Applied add-sqr-sqrt38.2

    \[\leadsto \sqrt{\color{blue}{\sqrt{e^{x}} \cdot \sqrt{e^{x}}}} \cdot \left(\left(\frac{2}{x} + \frac{1}{2}\right) + \frac{1}{8} \cdot x\right) - \left(\left(\frac{2}{x} + \frac{1}{2}\right) + \frac{1}{8} \cdot x\right)\]

  10. Applied sqrt-prod38.2

    \[\leadsto \color{blue}{\left(\sqrt{\sqrt{e^{x}}} \cdot \sqrt{\sqrt{e^{x}}}\right)} \cdot \left(\left(\frac{2}{x} + \frac{1}{2}\right) + \frac{1}{8} \cdot x\right) - \left(\left(\frac{2}{x} + \frac{1}{2}\right) + \frac{1}{8} \cdot x\right)\]

  11. Final simplification38.2

    \[\leadsto \left(\frac{1}{8} \cdot x + \left(\frac{1}{2} + \frac{2}{x}\right)\right) \cdot \left(\sqrt{\sqrt{e^{x}}} \cdot \sqrt{\sqrt{e^{x}}}\right) - \left(\frac{1}{8} \cdot x + \left(\frac{1}{2} + \frac{2}{x}\right)\right)\]

Runtime

Time bar (total: 49.5s)Debug logProfile

herbie shell --seed 2018255 
(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))