Input Error: 32.3b
Output Error: 12.2b
Time: 16.2s
Precision: 64b
Ground Truth: 128b
\[e^{a \cdot x} - 1\]
⬇
\[\begin{cases} \sqrt[3]{{\left(e^{a \cdot x} - 1\right)}^3} & \text{when } a \cdot x \le -4.1762420202878676 \cdot 10^{-24} \\ \left(\left({a}^3 \cdot {x}^3\right) \cdot \frac{1}{6} + x \cdot a\right) + \frac{1}{2} \cdot {\left(x \cdot a\right)}^2 & \text{otherwise} \end{cases}\]

Error

Bits error versus a
Bits error versus x

Derivation

    if (* a x) < -4.1762420202878676e-24

    1. Initial program 2.6b

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

    3. Applied add-cbrt-cube 2.6b

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

    if -4.1762420202878676e-24 < (* a x)

    1. Initial program 46.7b

      \[e^{a \cdot x} - 1\]

    2. Applied taylor 16.9b

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

    3. Taylor expanded around 0 16.9b

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

    4. Applied simplify 16.9b

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

Runtime

Total time: 16.2s Debug log

herbie --seed '#(589067380 2256569194 3321018431 1278551903 2252468623 1570104132)'
(FPCore (a x)
  :name "NMSE 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))