Average Error: 29.4 → 0.4
Time: 3.3s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -1.4477752003782106 \cdot 10^{-4}:\\ \;\;\;\;\frac{{\left(e^{a \cdot x}\right)}^{3} - {1}^{3}}{e^{a \cdot x} \cdot \left(e^{a \cdot x} + 1\right) + 1 \cdot 1}\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot x + \frac{1}{2} \cdot {\left(a \cdot x\right)}^{2}\right) + \frac{1}{6} \cdot {\left(a \cdot x\right)}^{3}\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -1.4477752003782106 \cdot 10^{-4}:\\
\;\;\;\;\frac{{\left(e^{a \cdot x}\right)}^{3} - {1}^{3}}{e^{a \cdot x} \cdot \left(e^{a \cdot x} + 1\right) + 1 \cdot 1}\\

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

\end{array}
double f(double a, double x) {
        double r99375 = a;
        double r99376 = x;
        double r99377 = r99375 * r99376;
        double r99378 = exp(r99377);
        double r99379 = 1.0;
        double r99380 = r99378 - r99379;
        return r99380;
}
↓
double f(double a, double x) {
        double r99381 = a;
        double r99382 = x;
        double r99383 = r99381 * r99382;
        double r99384 = -0.00014477752003782106;
        bool r99385 = r99383 <= r99384;
        double r99386 = exp(r99383);
        double r99387 = 3.0;
        double r99388 = pow(r99386, r99387);
        double r99389 = 1.0;
        double r99390 = pow(r99389, r99387);
        double r99391 = r99388 - r99390;
        double r99392 = r99386 + r99389;
        double r99393 = r99386 * r99392;
        double r99394 = r99389 * r99389;
        double r99395 = r99393 + r99394;
        double r99396 = r99391 / r99395;
        double r99397 = 0.5;
        double r99398 = 2.0;
        double r99399 = pow(r99383, r99398);
        double r99400 = r99397 * r99399;
        double r99401 = r99383 + r99400;
        double r99402 = 0.16666666666666666;
        double r99403 = pow(r99383, r99387);
        double r99404 = r99402 * r99403;
        double r99405 = r99401 + r99404;
        double r99406 = r99385 ? r99396 : r99405;
        return r99406;
}

Error

Bits error versus a

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.4
Target0.2
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt 0.10000000000000001:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* a x) < -0.00014477752003782106

    1. Initial program 0.1

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

    3. Applied flip3--0.1

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

    4. Simplified0.1

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

    if -0.00014477752003782106 < (* a x)

    1. Initial program 44.4

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

    2. Taylor expanded around 0 14.7

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

    3. Simplified14.7

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

    5. Applied pow-prod-down5.0

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

    7. Applied distribute-lft-in5.0

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

    8. Simplified5.0

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

    9. Simplified0.5

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -1.4477752003782106 \cdot 10^{-4}:\\ \;\;\;\;\frac{{\left(e^{a \cdot x}\right)}^{3} - {1}^{3}}{e^{a \cdot x} \cdot \left(e^{a \cdot x} + 1\right) + 1 \cdot 1}\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot x + \frac{1}{2} \cdot {\left(a \cdot x\right)}^{2}\right) + \frac{1}{6} \cdot {\left(a \cdot x\right)}^{3}\\ \end{array}\]

Reproduce

herbie shell --seed 2020025 
(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))