Average Error: 28.9 → 0.5
Time: 12.2s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -0.0361596251241738179:\\ \;\;\;\;\frac{e^{2 \cdot \left(x \cdot a\right)} - 1 \cdot 1}{e^{a \cdot x} + 1}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(x + \frac{1}{2} \cdot \left(x \cdot \left(a \cdot x\right)\right)\right)\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -0.0361596251241738179:\\
\;\;\;\;\frac{e^{2 \cdot \left(x \cdot a\right)} - 1 \cdot 1}{e^{a \cdot x} + 1}\\

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

\end{array}
double f(double a, double x) {
        double r116433 = a;
        double r116434 = x;
        double r116435 = r116433 * r116434;
        double r116436 = exp(r116435);
        double r116437 = 1.0;
        double r116438 = r116436 - r116437;
        return r116438;
}
↓
double f(double a, double x) {
        double r116439 = a;
        double r116440 = x;
        double r116441 = r116439 * r116440;
        double r116442 = -0.03615962512417382;
        bool r116443 = r116441 <= r116442;
        double r116444 = 2.0;
        double r116445 = r116440 * r116439;
        double r116446 = r116444 * r116445;
        double r116447 = exp(r116446);
        double r116448 = 1.0;
        double r116449 = r116448 * r116448;
        double r116450 = r116447 - r116449;
        double r116451 = exp(r116441);
        double r116452 = r116451 + r116448;
        double r116453 = r116450 / r116452;
        double r116454 = 0.5;
        double r116455 = r116440 * r116441;
        double r116456 = r116454 * r116455;
        double r116457 = r116440 + r116456;
        double r116458 = r116439 * r116457;
        double r116459 = r116443 ? r116453 : r116458;
        return r116459;
}

Error

Bits error versus a

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original28.9
Target0.2
Herbie0.5
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt 0.10000000000000001:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* a x) < -0.03615962512417382

    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 -0.03615962512417382 < (* a x)

    1. Initial program 43.6

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

    2. Taylor expanded around 0 14.5

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

    3. Simplified7.9

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

    4. Taylor expanded around 0 8.3

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

    6. Applied unpow28.3

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

    7. Applied associate-*l*4.7

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

    8. Simplified0.7

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

    9. Taylor expanded around inf 8.3

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

    10. Simplified0.7

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -0.0361596251241738179:\\ \;\;\;\;\frac{e^{2 \cdot \left(x \cdot a\right)} - 1 \cdot 1}{e^{a \cdot x} + 1}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(x + \frac{1}{2} \cdot \left(x \cdot \left(a \cdot x\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(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))