Average Error: 29.6 → 0.9
Time: 15.0s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -0.001826657808704238751540405161222224705853:\\ \;\;\;\;\frac{e^{2 \cdot \left(a \cdot x\right)} - 1 \cdot 1}{e^{a \cdot x} + 1}\\ \mathbf{else}:\\ \;\;\;\;a \cdot x\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -0.001826657808704238751540405161222224705853:\\
\;\;\;\;\frac{e^{2 \cdot \left(a \cdot x\right)} - 1 \cdot 1}{e^{a \cdot x} + 1}\\

\mathbf{else}:\\
\;\;\;\;a \cdot x\\

\end{array}
double f(double a, double x) {
        double r104645 = a;
        double r104646 = x;
        double r104647 = r104645 * r104646;
        double r104648 = exp(r104647);
        double r104649 = 1.0;
        double r104650 = r104648 - r104649;
        return r104650;
}
↓
double f(double a, double x) {
        double r104651 = a;
        double r104652 = x;
        double r104653 = r104651 * r104652;
        double r104654 = -0.0018266578087042388;
        bool r104655 = r104653 <= r104654;
        double r104656 = 2.0;
        double r104657 = r104656 * r104653;
        double r104658 = exp(r104657);
        double r104659 = 1.0;
        double r104660 = r104659 * r104659;
        double r104661 = r104658 - r104660;
        double r104662 = exp(r104653);
        double r104663 = r104662 + r104659;
        double r104664 = r104661 / r104663;
        double r104665 = r104655 ? r104664 : r104653;
        return r104665;
}

Error

Bits error versus a

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.6
Target0.2
Herbie0.9
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt 0.1000000000000000055511151231257827021182:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* a x) < -0.0018266578087042388

    1. Initial program 0.0

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

    3. Applied flip--0.0

      \[\leadsto \color{blue}{\frac{e^{a \cdot x} \cdot e^{a \cdot x} - 1 \cdot 1}{e^{a \cdot x} + 1}}\]

    4. Simplified0.0

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

    if -0.0018266578087042388 < (* a x)

    1. Initial program 44.8

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

    2. Taylor expanded around 0 14.4

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

    3. Simplified11.4

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

    4. Taylor expanded around 0 8.4

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

    5. Simplified4.7

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

    6. Taylor expanded around 0 1.3

      \[\leadsto \color{blue}{a \cdot x}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -0.001826657808704238751540405161222224705853:\\ \;\;\;\;\frac{e^{2 \cdot \left(a \cdot x\right)} - 1 \cdot 1}{e^{a \cdot x} + 1}\\ \mathbf{else}:\\ \;\;\;\;a \cdot x\\ \end{array}\]

Reproduce

herbie shell --seed 2019209 +o rules:numerics
(FPCore (a x)
  :name "expax (section 3.5)"
  :precision binary64
  :herbie-expected 14

  :herbie-target
  (if (< (fabs (* a x)) 0.10000000000000001) (* (* a x) (+ 1 (+ (/ (* a x) 2) (/ (pow (* a x) 2) 6)))) (- (exp (* a x)) 1))

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