Average Error: 32.9 → 0.1
Time: 53.7s
Precision: 64
Internal Precision: 1408
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -2.8896079110395453 \cdot 10^{-06}:\\ \;\;\;\;\sqrt[3]{{\left(e^{x \cdot a} - 1\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{2} + \frac{1}{6} \cdot \left(x \cdot a\right)\right) \cdot \left(\left(x \cdot a\right) \cdot \left(x \cdot a\right)\right) + x \cdot a\\ \end{array}\]

Error

Bits error versus a

Bits error versus x

Target

Original32.9
Target5.9
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) < -2.8896079110395453e-06

    1. Initial program 0.2

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

    3. Applied add-cbrt-cube0.2

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

    4. Applied simplify0.2

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

    if -2.8896079110395453e-06 < (* a x)

    1. Initial program 47.1

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

    3. Applied add-cbrt-cube47.1

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

    4. Applied simplify47.1

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

    5. Taylor expanded around 0 49.5

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

    6. Applied simplify0.1

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

Runtime

Time bar (total: 53.7s)Debug logProfile

herbie shell --seed '#(1063027428 1192549564 1443466578 604016274 3637110559 1698629644)' 
(FPCore (a x)
  :name "expax (section 3.5)"

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

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