Average Error: 31.7 → 0.1
Time: 9.0s
Precision: 64
Internal precision: 1408
\[e^{a \cdot x} - 1\]
⬇
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -3.2061054256324616 \cdot 10^{-05}:\\ \;\;\;\;{\left(\sqrt[3]{\sqrt{e^{a \cdot x}} + 1} \cdot \sqrt[3]{\sqrt{e^{a \cdot x}} - 1}\right)}^3\\ \mathbf{else}:\\ \;\;\;\;x \cdot a + \left(\left(x \cdot a\right) \cdot \left(x \cdot a\right)\right) \cdot \left(\frac{1}{2} + \left(a \cdot \frac{1}{6}\right) \cdot x\right)\\ \end{array}\]

Error

Bits error versus a

Bits error versus x

Target

Original31.7
Comparison6.0
Herbie0.1
\[ \begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt \frac{1}{10}:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^2}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array} \]

Derivation

  1. Split input into 2 regimes.

  2. if (* a x) < -3.2061054256324616e-05

    1. Initial program 0.1

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

    3. Applied add-cube-cbrt 0.1

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

    5. Applied add-sqr-sqrt 0.1

      \[\leadsto {\left(\sqrt[3]{\color{blue}{{\left(\sqrt{e^{a \cdot x}}\right)}^2} - 1}\right)}^3\]

    6. Applied difference-of-sqr-1 0.1

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

    7. Applied cbrt-prod 0.1

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

    if -3.2061054256324616e-05 < (* a x)

    1. Initial program 47.0

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

    3. Applied add-cube-cbrt 47.0

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

    4. Applied taylor 49.3

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

    5. Taylor expanded around 0 49.3

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

    6. Applied simplify 0.1

      \[\leadsto \color{blue}{x \cdot a + \left(\left(x \cdot a\right) \cdot \left(x \cdot a\right)\right) \cdot \left(\frac{1}{2} + \left(a \cdot \frac{1}{6}\right) \cdot x\right)}\]
  3. Recombined 2 regimes into one program.
  4. Removed slow pow expressions

Runtime

Time bar (total: 9.0s) Debug logProfile

Please include this information when filing a bug report:

herbie shell --seed '#(1067488128 1186562765 67428877 2948030040 4169093828 3206893112)'
(FPCore (a x)
  :name "expax (section 3.5)"

  :target
  (if (< (fabs (* a x)) 1/10) (* (* a x) (+ 1 (+ (/ (* a x) 2) (/ (sqr (* a x)) 6)))) (- (exp (* a x)) 1))

  (- (exp (* a x)) 1))