Average Error: 29.7 → 5.3
Time: 12.2s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -0.00565259617492738353:\\ \;\;\;\;\frac{e^{2 \cdot \left(x \cdot a\right)} - 1 \cdot 1}{e^{a \cdot x} + 1}\\ \mathbf{elif}\;a \cdot x \le 6.429099761039471 \cdot 10^{-102}:\\ \;\;\;\;\frac{1}{2} \cdot \left({a}^{2} \cdot {x}^{2}\right) + a \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a + x \cdot \left(\frac{1}{2} \cdot {a}^{2} + \left(\frac{1}{6} \cdot {a}^{3}\right) \cdot x\right)\right)\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -0.00565259617492738353:\\
\;\;\;\;\frac{e^{2 \cdot \left(x \cdot a\right)} - 1 \cdot 1}{e^{a \cdot x} + 1}\\

\mathbf{elif}\;a \cdot x \le 6.429099761039471 \cdot 10^{-102}:\\
\;\;\;\;\frac{1}{2} \cdot \left({a}^{2} \cdot {x}^{2}\right) + a \cdot x\\

\mathbf{else}:\\
\;\;\;\;x \cdot \left(a + x \cdot \left(\frac{1}{2} \cdot {a}^{2} + \left(\frac{1}{6} \cdot {a}^{3}\right) \cdot x\right)\right)\\

\end{array}
double f(double a, double x) {
        double r85415 = a;
        double r85416 = x;
        double r85417 = r85415 * r85416;
        double r85418 = exp(r85417);
        double r85419 = 1.0;
        double r85420 = r85418 - r85419;
        return r85420;
}
↓
double f(double a, double x) {
        double r85421 = a;
        double r85422 = x;
        double r85423 = r85421 * r85422;
        double r85424 = -0.0056525961749273835;
        bool r85425 = r85423 <= r85424;
        double r85426 = 2.0;
        double r85427 = r85422 * r85421;
        double r85428 = r85426 * r85427;
        double r85429 = exp(r85428);
        double r85430 = 1.0;
        double r85431 = r85430 * r85430;
        double r85432 = r85429 - r85431;
        double r85433 = exp(r85423);
        double r85434 = r85433 + r85430;
        double r85435 = r85432 / r85434;
        double r85436 = 6.429099761039471e-102;
        bool r85437 = r85423 <= r85436;
        double r85438 = 0.5;
        double r85439 = pow(r85421, r85426);
        double r85440 = pow(r85422, r85426);
        double r85441 = r85439 * r85440;
        double r85442 = r85438 * r85441;
        double r85443 = r85442 + r85423;
        double r85444 = r85438 * r85439;
        double r85445 = 0.16666666666666666;
        double r85446 = 3.0;
        double r85447 = pow(r85421, r85446);
        double r85448 = r85445 * r85447;
        double r85449 = r85448 * r85422;
        double r85450 = r85444 + r85449;
        double r85451 = r85422 * r85450;
        double r85452 = r85421 + r85451;
        double r85453 = r85422 * r85452;
        double r85454 = r85437 ? r85443 : r85453;
        double r85455 = r85425 ? r85435 : r85454;
        return r85455;
}

Error

Bits error versus a

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.7
Target0.2
Herbie5.3
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt 0.10000000000000001:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (* a x) < -0.0056525961749273835

    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

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

    if -0.0056525961749273835 < (* a x) < 6.429099761039471e-102

    1. Initial program 42.7

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

    2. Taylor expanded around 0 10.6

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

    3. Simplified5.4

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

    4. Taylor expanded around 0 5.6

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

    if 6.429099761039471e-102 < (* a x)

    1. Initial program 53.5

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

    2. Taylor expanded around 0 35.0

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

    3. Simplified20.8

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

  4. Final simplification5.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -0.00565259617492738353:\\ \;\;\;\;\frac{e^{2 \cdot \left(x \cdot a\right)} - 1 \cdot 1}{e^{a \cdot x} + 1}\\ \mathbf{elif}\;a \cdot x \le 6.429099761039471 \cdot 10^{-102}:\\ \;\;\;\;\frac{1}{2} \cdot \left({a}^{2} \cdot {x}^{2}\right) + a \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a + x \cdot \left(\frac{1}{2} \cdot {a}^{2} + \left(\frac{1}{6} \cdot {a}^{3}\right) \cdot x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 
(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))