Average Error: 39.9 → 0.3
Time: 1.0m
Precision: 64
Internal Precision: 1344
\[\frac{e^{x} - 1}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.00010900310540885043:\\ \;\;\;\;\frac{\sqrt[3]{e^{x} \cdot e^{x} - 1} \cdot \frac{\log \left(e^{\sqrt[3]{e^{x + x} - 1} \cdot \sqrt[3]{e^{x + x} - 1}}\right)}{e^{x} + 1}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{2} \cdot {x}^{2} + \left(\frac{1}{6} \cdot {x}^{3} + x\right)}{x}\\ \end{array}\]

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original39.9
Target39.1
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 < -0.00010900310540885043

    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-log-exp0.1

      \[\leadsto \frac{\frac{\color{blue}{\log \left(e^{e^{x + x} - 1}\right)}}{e^{x} + 1}}{x}\]
    7. Using strategy rm

    8. Applied *-un-lft-identity0.1

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

    9. Applied add-cube-cbrt0.1

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

    10. Applied exp-prod0.1

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

    11. Applied log-pow0.1

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

    12. Applied times-frac0.1

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

    13. Applied simplify0.1

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

    if -0.00010900310540885043 < x

    1. Initial program 60.3

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

    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: 1.0m)Debug logProfile

herbie shell --seed 2018178 
(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))