Average Error: 32.4 → 0.1
Time: 21.7s
Precision: 64
Internal precision: 1408
\[e^{a \cdot x} - 1\]
⬇
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -9.689533085427536 \cdot 10^{-06}:\\ \;\;\;\;\sqrt[3]{{\left(e^{a \cdot x} - 1\right)}^3}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{2} + a \cdot \left(x \cdot \frac{1}{6}\right)\right) \cdot {\left(a \cdot x\right)}^2 + x \cdot a\\ \end{array}\]

Error

Bits error versus a

Bits error versus x

Target

Original32.4
Comparison5.9
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) < -9.689533085427536e-06

    1. Initial program 0.1

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

    3. Applied add-cbrt-cube 0.1

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

    if -9.689533085427536e-06 < (* a x)

    1. Initial program 46.9

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

    3. Applied add-cbrt-cube 46.9

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

    4. Applied taylor 49.9

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

    5. Taylor expanded around 0 49.9

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

    6. Applied simplify 0.1

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

    7. Applied simplify 0.1

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

Runtime

Time bar (total: 21.7s) Debug log

Please include this information when filing a bug report:

herbie shell --seed '#(2277612311 2645429965 1090895633 2857793080 2144184008 3989768357)'
(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))