Average Error: 39.5 → 0.5
Time: 1.0m
Precision: 64
Internal Precision: 1344
\[\frac{e^{x} - 1}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{e^{x} - 1}{x} \le 4.73460717902886 \cdot 10^{-310} \lor \neg \left(\frac{e^{x} - 1}{x} \le 0.32747442527855913\right):\\ \;\;\;\;\frac{\left({x}^{3} \cdot \frac{1}{6} + \frac{1}{2} \cdot {x}^{2}\right) + x}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\sqrt[3]{e^{x} - 1} \cdot \sqrt[3]{e^{x} + 1}\right) \cdot \left(\sqrt[3]{e^{x + x} - 1} \cdot \sqrt[3]{e^{x + x} - 1}\right)}{e^{x} + 1}}{x}\\ \end{array}\]

Error

Bits error versus x

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Your Program's Arguments

Results

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Target

Original39.5
Target38.6
Herbie0.5
\[\begin{array}{l} \mathbf{if}\;x \lt 1 \land x \gt -1:\\ \;\;\;\;\frac{e^{x} - 1}{\log \left(e^{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x} - 1}{x}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (- (exp x) 1) x) < 4.73460717902886e-310 or 0.32747442527855913 < (/ (- (exp x) 1) x)

    1. Initial program 59.4

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

    2. Taylor expanded around 0 0.8

      \[\leadsto \frac{\color{blue}{x + \left(\frac{1}{6} \cdot {x}^{3} + \frac{1}{2} \cdot {x}^{2}\right)}}{x}\]

    if 4.73460717902886e-310 < (/ (- (exp x) 1) x) < 0.32747442527855913

    1. Initial program 0

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

    3. Applied flip--0.0

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

    4. Applied simplify0.0

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

    6. Applied add-cube-cbrt0.0

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

    7. Applied simplify0.0

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

    9. Applied difference-of-sqr-10.0

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

    10. Applied cbrt-prod0.0

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

  4. Applied simplify0.5

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\frac{e^{x} - 1}{x} \le 4.73460717902886 \cdot 10^{-310} \lor \neg \left(\frac{e^{x} - 1}{x} \le 0.32747442527855913\right):\\ \;\;\;\;\frac{\left({x}^{3} \cdot \frac{1}{6} + \frac{1}{2} \cdot {x}^{2}\right) + x}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\sqrt[3]{e^{x} - 1} \cdot \sqrt[3]{e^{x} + 1}\right) \cdot \left(\sqrt[3]{e^{x + x} - 1} \cdot \sqrt[3]{e^{x + x} - 1}\right)}{e^{x} + 1}}{x}\\ \end{array}}\]

Runtime

Time bar (total: 1.0m)Debug logProfile

herbie shell --seed 2018296 
(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))