Average Error: 29.5 → 0.8
Time: 3.5s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -7.1307752378243545 \cdot 10^{-9}:\\ \;\;\;\;\log \left(e^{e^{a \cdot x} - 1}\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot x + \frac{1}{6} \cdot {\left(a \cdot x\right)}^{3}\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -7.1307752378243545 \cdot 10^{-9}:\\
\;\;\;\;\log \left(e^{e^{a \cdot x} - 1}\right)\\

\mathbf{else}:\\
\;\;\;\;a \cdot x + \frac{1}{6} \cdot {\left(a \cdot x\right)}^{3}\\

\end{array}
double f(double a, double x) {
        double r101656 = a;
        double r101657 = x;
        double r101658 = r101656 * r101657;
        double r101659 = exp(r101658);
        double r101660 = 1.0;
        double r101661 = r101659 - r101660;
        return r101661;
}
↓
double f(double a, double x) {
        double r101662 = a;
        double r101663 = x;
        double r101664 = r101662 * r101663;
        double r101665 = -7.1307752378243545e-09;
        bool r101666 = r101664 <= r101665;
        double r101667 = exp(r101664);
        double r101668 = 1.0;
        double r101669 = r101667 - r101668;
        double r101670 = exp(r101669);
        double r101671 = log(r101670);
        double r101672 = 0.16666666666666666;
        double r101673 = 3.0;
        double r101674 = pow(r101664, r101673);
        double r101675 = r101672 * r101674;
        double r101676 = r101664 + r101675;
        double r101677 = r101666 ? r101671 : r101676;
        return r101677;
}

Error

Bits error versus a

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.5
Target0.2
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt 0.10000000000000001:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* a x) < -7.1307752378243545e-09

    1. Initial program 0.3

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

    3. Applied add-log-exp0.3

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

    4. Applied add-log-exp0.4

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

    5. Applied diff-log0.4

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

    6. Simplified0.4

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

    if -7.1307752378243545e-09 < (* a x)

    1. Initial program 44.5

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

    2. Taylor expanded around 0 14.2

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

    3. Simplified14.2

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

    5. Applied pow-prod-down4.2

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

    6. Taylor expanded around 0 1.0

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -7.1307752378243545 \cdot 10^{-9}:\\ \;\;\;\;\log \left(e^{e^{a \cdot x} - 1}\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot x + \frac{1}{6} \cdot {\left(a \cdot x\right)}^{3}\\ \end{array}\]

Reproduce

herbie shell --seed 2020059 
(FPCore (a x)
  :name "expax (section 3.5)"
  :precision binary64
  :herbie-expected 14

  :herbie-target
  (if (< (fabs (* a x)) 0.1) (* (* a x) (+ 1 (+ (/ (* a x) 2) (/ (pow (* a x) 2) 6)))) (- (exp (* a x)) 1))

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