Average Error: 28.9 → 0.5
Time: 8.0s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -0.0361596251241738179:\\ \;\;\;\;\frac{e^{2 \cdot \left(x \cdot a\right)} - 1 \cdot 1}{e^{a \cdot x} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \left(\left(a \cdot \left|x\right|\right) \cdot \left(a \cdot \left|x\right|\right)\right) + a \cdot x\\ \end{array}\]
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}:\\
\;\;\;\;\frac{1}{2} \cdot \left(\left(a \cdot \left|x\right|\right) \cdot \left(a \cdot \left|x\right|\right)\right) + a \cdot x\\

\end{array}
double f(double a, double x) {
        double r102407 = a;
        double r102408 = x;
        double r102409 = r102407 * r102408;
        double r102410 = exp(r102409);
        double r102411 = 1.0;
        double r102412 = r102410 - r102411;
        return r102412;
}
↓
double f(double a, double x) {
        double r102413 = a;
        double r102414 = x;
        double r102415 = r102413 * r102414;
        double r102416 = -0.03615962512417382;
        bool r102417 = r102415 <= r102416;
        double r102418 = 2.0;
        double r102419 = r102414 * r102413;
        double r102420 = r102418 * r102419;
        double r102421 = exp(r102420);
        double r102422 = 1.0;
        double r102423 = r102422 * r102422;
        double r102424 = r102421 - r102423;
        double r102425 = exp(r102415);
        double r102426 = r102425 + r102422;
        double r102427 = r102424 / r102426;
        double r102428 = 0.5;
        double r102429 = fabs(r102414);
        double r102430 = r102413 * r102429;
        double r102431 = r102430 * r102430;
        double r102432 = r102428 * r102431;
        double r102433 = r102432 + r102415;
        double r102434 = r102417 ? r102427 : r102433;
        return r102434;
}

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

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

    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 add-sqr-sqrt8.3

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

    7. Applied add-sqr-sqrt36.7

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

    8. Applied unpow-prod-down36.7

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

    9. Applied unswap-sqr34.9

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

    10. Simplified34.9

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

    11. Simplified0.7

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

  4. Final simplification0.5

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

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))