Average Error: 40.3 → 0.3
Time: 35.4s
Precision: 64
Internal Precision: 1344
\[\frac{e^{x} - 1}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.00012351129926539475:\\ \;\;\;\;\frac{\frac{\frac{1}{\sqrt{\sqrt{1 + e^{x}}}}}{\sqrt{\sqrt{1 + e^{x}}}} \cdot \frac{e^{x + x} - 1}{\sqrt{1 + e^{x}}}}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{2} + \left(1 + {x}^{2} \cdot \frac{1}{6}\right)\\ \end{array}\]

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original40.3
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 < -0.00012351129926539475

    1. Initial program 0.0

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

    2. Initial simplification0.0

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

    4. Applied flip--0.0

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

    6. Applied add-sqr-sqrt0.0

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

    7. Applied *-un-lft-identity0.0

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

    8. Applied times-frac0.0

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

    9. Simplified0.0

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

    11. Applied add-sqr-sqrt0.0

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

    12. Applied associate-/r*0.0

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

    if -0.00012351129926539475 < x

    1. Initial program 60.0

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

    2. Initial simplification60.0

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

    3. Taylor expanded around 0 0.4

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.00012351129926539475:\\ \;\;\;\;\frac{\frac{\frac{1}{\sqrt{\sqrt{1 + e^{x}}}}}{\sqrt{\sqrt{1 + e^{x}}}} \cdot \frac{e^{x + x} - 1}{\sqrt{1 + e^{x}}}}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{2} + \left(1 + {x}^{2} \cdot \frac{1}{6}\right)\\ \end{array}\]

Runtime

Time bar (total: 35.4s)Debug logProfile

herbie shell --seed 2018242 
(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))