Average Error: 29.5 → 0.2
Time: 11.8s
Precision: 64
Internal Precision: 1344
\[{e}^{x} - 1\]
\[\begin{array}{l} \mathbf{if}\;x \le -7.06650157794332 \cdot 10^{-07}:\\ \;\;\;\;\log \left(e^{{e}^{x} - 1}\right)\\ \mathbf{elif}\;x \le 1.2248036861576476 \cdot 10^{-06}:\\ \;\;\;\;x \cdot \log e + \left(\frac{1}{2} + \log e \cdot \left(x \cdot \frac{1}{6}\right)\right) \cdot \left(\left(x \cdot \log e\right) \cdot \left(x \cdot \log e\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{\log \left(e^{{e}^{x} - 1}\right)} \cdot \sqrt[3]{\log \left(e^{{e}^{x} - 1}\right)}\right) \cdot \sqrt[3]{\log \left(e^{{e}^{x} - 1}\right)}\\ \end{array}\]

Error

Bits error versus e

Bits error versus x

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Results

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Derivation

  1. Split input into 3 regimes
  2. if x < -7.06650157794332e-07

    1. Initial program 0.1

      \[{e}^{x} - 1\]
    2. Initial simplification0.1

      \[\leadsto {e}^{x} - 1\]
    3. Using strategy rm
    4. Applied add-log-exp0.1

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

    if -7.06650157794332e-07 < x < 1.2248036861576476e-06

    1. Initial program 59.2

      \[{e}^{x} - 1\]
    2. Initial simplification59.2

      \[\leadsto {e}^{x} - 1\]
    3. Taylor expanded around 0 0.3

      \[\leadsto \color{blue}{\log e \cdot x + \left(\frac{1}{2} \cdot \left({\left(\log e\right)}^{2} \cdot {x}^{2}\right) + \frac{1}{6} \cdot \left({\left(\log e\right)}^{3} \cdot {x}^{3}\right)\right)}\]
    4. Simplified0.3

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

    if 1.2248036861576476e-06 < x

    1. Initial program 0.1

      \[{e}^{x} - 1\]
    2. Initial simplification0.1

      \[\leadsto {e}^{x} - 1\]
    3. Using strategy rm
    4. Applied add-log-exp0.2

      \[\leadsto \color{blue}{\log \left(e^{{e}^{x} - 1}\right)}\]
    5. Using strategy rm
    6. Applied add-cube-cbrt0.3

      \[\leadsto \color{blue}{\left(\sqrt[3]{\log \left(e^{{e}^{x} - 1}\right)} \cdot \sqrt[3]{\log \left(e^{{e}^{x} - 1}\right)}\right) \cdot \sqrt[3]{\log \left(e^{{e}^{x} - 1}\right)}}\]
  3. Recombined 3 regimes into one program.
  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -7.06650157794332 \cdot 10^{-07}:\\ \;\;\;\;\log \left(e^{{e}^{x} - 1}\right)\\ \mathbf{elif}\;x \le 1.2248036861576476 \cdot 10^{-06}:\\ \;\;\;\;x \cdot \log e + \left(\frac{1}{2} + \log e \cdot \left(x \cdot \frac{1}{6}\right)\right) \cdot \left(\left(x \cdot \log e\right) \cdot \left(x \cdot \log e\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{\log \left(e^{{e}^{x} - 1}\right)} \cdot \sqrt[3]{\log \left(e^{{e}^{x} - 1}\right)}\right) \cdot \sqrt[3]{\log \left(e^{{e}^{x} - 1}\right)}\\ \end{array}\]

Runtime

Time bar (total: 11.8s)Debug log

herbie shell --seed '#(2775764126 3555076145 3898259844 1891440260 2599947619 1948460636)' 
(FPCore (e x)
  :name "e^x - 1"
  (- (pow e x) 1))