expax (section 3.5)

Average Error: 28.6 → 14.0

Time: 1.1m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -4.769746002479849413485688169289897765182 \cdot 10^{75}:\\ \;\;\;\;\frac{\frac{\frac{e^{\left(a \cdot x\right) \cdot 27} - \left(\left(\left(\left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right) \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)\right) \cdot 1\right) \cdot \left(\left(\left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right) \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)\right) \cdot 1\right)\right) \cdot \left(\left(\left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right) \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)\right) \cdot 1\right)}{e^{x \cdot \left(a \cdot 9\right)} \cdot \left(e^{x \cdot \left(a \cdot 9\right)} + \left(\left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right) \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)\right) \cdot 1\right) + \left(\left(\left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right) \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)\right) \cdot 1\right) \cdot \left(\left(\left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right) \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)\right) \cdot 1\right)}}{\left(e^{\left(a \cdot x\right) \cdot 3} + \left(1 \cdot 1\right) \cdot 1\right) \cdot e^{\left(a \cdot x\right) \cdot 3} + \left(\left(1 \cdot 1\right) \cdot 1\right) \cdot \left(\left(1 \cdot 1\right) \cdot 1\right)}}{\left(1 \cdot 1 + e^{a \cdot x} \cdot 1\right) + e^{a \cdot x} \cdot e^{a \cdot x}}\\ \mathbf{elif}\;x \le -2.773503098171754549439485465195885024121 \cdot 10^{-34}:\\ \;\;\;\;\left(\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \left(a \cdot x\right)\right) \cdot \frac{1}{6} + \left(a \cdot x + \left(a \cdot x\right) \cdot \left(\left(a \cdot x\right) \cdot \frac{1}{2}\right)\right)\\ \mathbf{elif}\;x \le -2.253008826999895251155572445945559152759 \cdot 10^{-43}:\\ \;\;\;\;\frac{\frac{\frac{e^{\left(a \cdot x\right) \cdot 27} - \left(\left(\left(\left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right) \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)\right) \cdot 1\right) \cdot \left(\left(\left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right) \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)\right) \cdot 1\right)\right) \cdot \left(\left(\left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right) \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)\right) \cdot 1\right)}{e^{x \cdot \left(a \cdot 9\right)} \cdot \left(e^{x \cdot \left(a \cdot 9\right)} + \left(\left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right) \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)\right) \cdot 1\right) + \left(\left(\left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right) \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)\right) \cdot 1\right) \cdot \left(\left(\left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right) \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)\right) \cdot 1\right)}}{\left(e^{\left(a \cdot x\right) \cdot 3} + \left(1 \cdot 1\right) \cdot 1\right) \cdot e^{\left(a \cdot x\right) \cdot 3} + \left(\left(1 \cdot 1\right) \cdot 1\right) \cdot \left(\left(1 \cdot 1\right) \cdot 1\right)}}{\left(1 \cdot 1 + e^{a \cdot x} \cdot 1\right) + e^{a \cdot x} \cdot e^{a \cdot x}}\\ \mathbf{elif}\;x \le 1.615963223807510020381636500169761159394 \cdot 10^{101}:\\ \;\;\;\;\left(\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \left(a \cdot x\right)\right) \cdot \frac{1}{6} + \left(a \cdot x + \left(a \cdot x\right) \cdot \left(\left(a \cdot x\right) \cdot \frac{1}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{e^{\left(a \cdot x\right) \cdot 27} - \left(\left(\left(\left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right) \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)\right) \cdot 1\right) \cdot \left(\left(\left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right) \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)\right) \cdot 1\right)\right) \cdot \left(\left(\left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right) \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)\right) \cdot 1\right)}{e^{x \cdot \left(a \cdot 9\right)} \cdot \left(e^{x \cdot \left(a \cdot 9\right)} + \left(\left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right) \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)\right) \cdot 1\right) + \left(\left(\left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right) \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)\right) \cdot 1\right) \cdot \left(\left(\left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right) \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)\right) \cdot 1\right)}}{\left(e^{\left(a \cdot x\right) \cdot 3} + \left(1 \cdot 1\right) \cdot 1\right) \cdot e^{\left(a \cdot x\right) \cdot 3} + \left(\left(1 \cdot 1\right) \cdot 1\right) \cdot \left(\left(1 \cdot 1\right) \cdot 1\right)}}{\left(1 \cdot 1 + e^{a \cdot x} \cdot 1\right) + e^{a \cdot x} \cdot e^{a \cdot x}}\\ \end{array}\]

C

double f(double a, double x) {
        double r4104549 = a;
        double r4104550 = x;
        double r4104551 = r4104549 * r4104550;
        double r4104552 = exp(r4104551);
        double r4104553 = 1.0;
        double r4104554 = r4104552 - r4104553;
        return r4104554;
}

↓

double f(double a, double x) {
        double r4104555 = x;
        double r4104556 = -4.7697460024798494e+75;
        bool r4104557 = r4104555 <= r4104556;
        double r4104558 = a;
        double r4104559 = r4104558 * r4104555;
        double r4104560 = 27.0;
        double r4104561 = r4104559 * r4104560;
        double r4104562 = exp(r4104561);
        double r4104563 = 1.0;
        double r4104564 = r4104563 * r4104563;
        double r4104565 = r4104564 * r4104564;
        double r4104566 = r4104565 * r4104565;
        double r4104567 = r4104566 * r4104563;
        double r4104568 = r4104567 * r4104567;
        double r4104569 = r4104568 * r4104567;
        double r4104570 = r4104562 - r4104569;
        double r4104571 = 9.0;
        double r4104572 = r4104558 * r4104571;
        double r4104573 = r4104555 * r4104572;
        double r4104574 = exp(r4104573);
        double r4104575 = r4104574 + r4104567;
        double r4104576 = r4104574 * r4104575;
        double r4104577 = r4104576 + r4104568;
        double r4104578 = r4104570 / r4104577;
        double r4104579 = 3.0;
        double r4104580 = r4104559 * r4104579;
        double r4104581 = exp(r4104580);
        double r4104582 = r4104564 * r4104563;
        double r4104583 = r4104581 + r4104582;
        double r4104584 = r4104583 * r4104581;
        double r4104585 = r4104582 * r4104582;
        double r4104586 = r4104584 + r4104585;
        double r4104587 = r4104578 / r4104586;
        double r4104588 = exp(r4104559);
        double r4104589 = r4104588 * r4104563;
        double r4104590 = r4104564 + r4104589;
        double r4104591 = r4104588 * r4104588;
        double r4104592 = r4104590 + r4104591;
        double r4104593 = r4104587 / r4104592;
        double r4104594 = -2.7735030981717545e-34;
        bool r4104595 = r4104555 <= r4104594;
        double r4104596 = r4104559 * r4104559;
        double r4104597 = r4104596 * r4104559;
        double r4104598 = 0.16666666666666666;
        double r4104599 = r4104597 * r4104598;
        double r4104600 = 0.5;
        double r4104601 = r4104559 * r4104600;
        double r4104602 = r4104559 * r4104601;
        double r4104603 = r4104559 + r4104602;
        double r4104604 = r4104599 + r4104603;
        double r4104605 = -2.2530088269998953e-43;
        bool r4104606 = r4104555 <= r4104605;
        double r4104607 = 1.61596322380751e+101;
        bool r4104608 = r4104555 <= r4104607;
        double r4104609 = r4104608 ? r4104604 : r4104593;
        double r4104610 = r4104606 ? r4104593 : r4104609;
        double r4104611 = r4104595 ? r4104604 : r4104610;
        double r4104612 = r4104557 ? r4104593 : r4104611;
        return r4104612;
}

Error

Bits error versus a

Bits error versus x

Target

Original	28.6
Target	0.2
Herbie	14.0

\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt 0.1000000000000000055511151231257827021182:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if x < -4.7697460024798494e+75 or -2.7735030981717545e-34 < x < -2.2530088269998953e-43 or 1.61596322380751e+101 < x

   1. Initial program 16.4

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

   3. Applied flip3-- 16.4

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

   4. Simplified 16.3

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

   6. Applied flip3-- 16.4

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

   7. Simplified 16.3

      \[\leadsto \frac{\frac{\color{blue}{e^{x \cdot \left(9 \cdot a\right)} - \left(\left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right) \cdot 1\right) \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)}}{e^{x \cdot \left(3 \cdot a\right)} \cdot e^{x \cdot \left(3 \cdot a\right)} + \left(\left(1 \cdot \left(1 \cdot 1\right)\right) \cdot \left(1 \cdot \left(1 \cdot 1\right)\right) + e^{x \cdot \left(3 \cdot a\right)} \cdot \left(1 \cdot \left(1 \cdot 1\right)\right)\right)}}{e^{a \cdot x} \cdot e^{a \cdot x} + \left(1 \cdot 1 + e^{a \cdot x} \cdot 1\right)}\]

   8. Simplified 16.3

      \[\leadsto \frac{\frac{e^{x \cdot \left(9 \cdot a\right)} - \left(\left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right) \cdot 1\right) \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)}{\color{blue}{e^{\left(a \cdot x\right) \cdot 3} \cdot \left(e^{\left(a \cdot x\right) \cdot 3} + 1 \cdot \left(1 \cdot 1\right)\right) + \left(1 \cdot \left(1 \cdot 1\right)\right) \cdot \left(1 \cdot \left(1 \cdot 1\right)\right)}}}{e^{a \cdot x} \cdot e^{a \cdot x} + \left(1 \cdot 1 + e^{a \cdot x} \cdot 1\right)}\]
   9. Using strategy rm

   10. Applied flip3-- 16.4

       \[\leadsto \frac{\frac{\color{blue}{\frac{{\left(e^{x \cdot \left(9 \cdot a\right)}\right)}^{3} - {\left(\left(\left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right) \cdot 1\right) \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)\right)}^{3}}{e^{x \cdot \left(9 \cdot a\right)} \cdot e^{x \cdot \left(9 \cdot a\right)} + \left(\left(\left(\left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right) \cdot 1\right) \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)\right) \cdot \left(\left(\left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right) \cdot 1\right) \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)\right) + e^{x \cdot \left(9 \cdot a\right)} \cdot \left(\left(\left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right) \cdot 1\right) \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)\right)\right)}}}{e^{\left(a \cdot x\right) \cdot 3} \cdot \left(e^{\left(a \cdot x\right) \cdot 3} + 1 \cdot \left(1 \cdot 1\right)\right) + \left(1 \cdot \left(1 \cdot 1\right)\right) \cdot \left(1 \cdot \left(1 \cdot 1\right)\right)}}{e^{a \cdot x} \cdot e^{a \cdot x} + \left(1 \cdot 1 + e^{a \cdot x} \cdot 1\right)}\]

   11. Simplified 16.3

       \[\leadsto \frac{\frac{\frac{\color{blue}{e^{\left(x \cdot a\right) \cdot 27} - \left(\left(\left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right) \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)\right) \cdot 1\right) \cdot \left(\left(\left(\left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right) \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)\right) \cdot 1\right) \cdot \left(\left(\left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right) \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)\right) \cdot 1\right)\right)}}{e^{x \cdot \left(9 \cdot a\right)} \cdot e^{x \cdot \left(9 \cdot a\right)} + \left(\left(\left(\left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right) \cdot 1\right) \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)\right) \cdot \left(\left(\left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right) \cdot 1\right) \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)\right) + e^{x \cdot \left(9 \cdot a\right)} \cdot \left(\left(\left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right) \cdot 1\right) \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)\right)\right)}}{e^{\left(a \cdot x\right) \cdot 3} \cdot \left(e^{\left(a \cdot x\right) \cdot 3} + 1 \cdot \left(1 \cdot 1\right)\right) + \left(1 \cdot \left(1 \cdot 1\right)\right) \cdot \left(1 \cdot \left(1 \cdot 1\right)\right)}}{e^{a \cdot x} \cdot e^{a \cdot x} + \left(1 \cdot 1 + e^{a \cdot x} \cdot 1\right)}\]

   12. Simplified 16.3

       \[\leadsto \frac{\frac{\frac{e^{\left(x \cdot a\right) \cdot 27} - \left(\left(\left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right) \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)\right) \cdot 1\right) \cdot \left(\left(\left(\left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right) \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)\right) \cdot 1\right) \cdot \left(\left(\left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right) \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)\right) \cdot 1\right)\right)}{\color{blue}{\left(\left(\left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right) \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)\right) \cdot 1\right) \cdot \left(\left(\left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right) \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)\right) \cdot 1\right) + e^{\left(a \cdot 9\right) \cdot x} \cdot \left(\left(\left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right) \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)\right) \cdot 1 + e^{\left(a \cdot 9\right) \cdot x}\right)}}}{e^{\left(a \cdot x\right) \cdot 3} \cdot \left(e^{\left(a \cdot x\right) \cdot 3} + 1 \cdot \left(1 \cdot 1\right)\right) + \left(1 \cdot \left(1 \cdot 1\right)\right) \cdot \left(1 \cdot \left(1 \cdot 1\right)\right)}}{e^{a \cdot x} \cdot e^{a \cdot x} + \left(1 \cdot 1 + e^{a \cdot x} \cdot 1\right)}\]

   if -4.7697460024798494e+75 < x < -2.7735030981717545e-34 or -2.2530088269998953e-43 < x < 1.61596322380751e+101

   1. Initial program 33.5

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

   2. Taylor expanded around 0 19.2

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

   3. Simplified 13.0

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

4. Final simplification 14.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4.769746002479849413485688169289897765182 \cdot 10^{75}:\\ \;\;\;\;\frac{\frac{\frac{e^{\left(a \cdot x\right) \cdot 27} - \left(\left(\left(\left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right) \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)\right) \cdot 1\right) \cdot \left(\left(\left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right) \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)\right) \cdot 1\right)\right) \cdot \left(\left(\left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right) \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)\right) \cdot 1\right)}{e^{x \cdot \left(a \cdot 9\right)} \cdot \left(e^{x \cdot \left(a \cdot 9\right)} + \left(\left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right) \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)\right) \cdot 1\right) + \left(\left(\left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right) \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)\right) \cdot 1\right) \cdot \left(\left(\left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right) \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)\right) \cdot 1\right)}}{\left(e^{\left(a \cdot x\right) \cdot 3} + \left(1 \cdot 1\right) \cdot 1\right) \cdot e^{\left(a \cdot x\right) \cdot 3} + \left(\left(1 \cdot 1\right) \cdot 1\right) \cdot \left(\left(1 \cdot 1\right) \cdot 1\right)}}{\left(1 \cdot 1 + e^{a \cdot x} \cdot 1\right) + e^{a \cdot x} \cdot e^{a \cdot x}}\\ \mathbf{elif}\;x \le -2.773503098171754549439485465195885024121 \cdot 10^{-34}:\\ \;\;\;\;\left(\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \left(a \cdot x\right)\right) \cdot \frac{1}{6} + \left(a \cdot x + \left(a \cdot x\right) \cdot \left(\left(a \cdot x\right) \cdot \frac{1}{2}\right)\right)\\ \mathbf{elif}\;x \le -2.253008826999895251155572445945559152759 \cdot 10^{-43}:\\ \;\;\;\;\frac{\frac{\frac{e^{\left(a \cdot x\right) \cdot 27} - \left(\left(\left(\left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right) \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)\right) \cdot 1\right) \cdot \left(\left(\left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right) \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)\right) \cdot 1\right)\right) \cdot \left(\left(\left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right) \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)\right) \cdot 1\right)}{e^{x \cdot \left(a \cdot 9\right)} \cdot \left(e^{x \cdot \left(a \cdot 9\right)} + \left(\left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right) \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)\right) \cdot 1\right) + \left(\left(\left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right) \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)\right) \cdot 1\right) \cdot \left(\left(\left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right) \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)\right) \cdot 1\right)}}{\left(e^{\left(a \cdot x\right) \cdot 3} + \left(1 \cdot 1\right) \cdot 1\right) \cdot e^{\left(a \cdot x\right) \cdot 3} + \left(\left(1 \cdot 1\right) \cdot 1\right) \cdot \left(\left(1 \cdot 1\right) \cdot 1\right)}}{\left(1 \cdot 1 + e^{a \cdot x} \cdot 1\right) + e^{a \cdot x} \cdot e^{a \cdot x}}\\ \mathbf{elif}\;x \le 1.615963223807510020381636500169761159394 \cdot 10^{101}:\\ \;\;\;\;\left(\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \left(a \cdot x\right)\right) \cdot \frac{1}{6} + \left(a \cdot x + \left(a \cdot x\right) \cdot \left(\left(a \cdot x\right) \cdot \frac{1}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{e^{\left(a \cdot x\right) \cdot 27} - \left(\left(\left(\left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right) \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)\right) \cdot 1\right) \cdot \left(\left(\left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right) \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)\right) \cdot 1\right)\right) \cdot \left(\left(\left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right) \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)\right) \cdot 1\right)}{e^{x \cdot \left(a \cdot 9\right)} \cdot \left(e^{x \cdot \left(a \cdot 9\right)} + \left(\left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right) \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)\right) \cdot 1\right) + \left(\left(\left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right) \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)\right) \cdot 1\right) \cdot \left(\left(\left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right) \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)\right) \cdot 1\right)}}{\left(e^{\left(a \cdot x\right) \cdot 3} + \left(1 \cdot 1\right) \cdot 1\right) \cdot e^{\left(a \cdot x\right) \cdot 3} + \left(\left(1 \cdot 1\right) \cdot 1\right) \cdot \left(\left(1 \cdot 1\right) \cdot 1\right)}}{\left(1 \cdot 1 + e^{a \cdot x} \cdot 1\right) + e^{a \cdot x} \cdot e^{a \cdot x}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019169 
(FPCore (a x)
  :name "expax (section 3.5)"
  :herbie-expected 14

  :herbie-target
  (if (< (fabs (* a x)) 0.1) (* (* a x) (+ 1.0 (+ (/ (* a x) 2.0) (/ (pow (* a x) 2.0) 6.0)))) (- (exp (* a x)) 1.0))

  (- (exp (* a x)) 1.0))