expax (section 3.5)

Average Error: 28.6 → 14.0

Time: 19.7s

Precision: 64

Math

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

↓

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

C

double f(double a, double x) {
        double r4872905 = a;
        double r4872906 = x;
        double r4872907 = r4872905 * r4872906;
        double r4872908 = exp(r4872907);
        double r4872909 = 1.0;
        double r4872910 = r4872908 - r4872909;
        return r4872910;
}

↓

double f(double a, double x) {
        double r4872911 = x;
        double r4872912 = -4.7697460024798494e+75;
        bool r4872913 = r4872911 <= r4872912;
        double r4872914 = 3.0;
        double r4872915 = a;
        double r4872916 = r4872914 * r4872915;
        double r4872917 = r4872916 * r4872911;
        double r4872918 = r4872914 * r4872917;
        double r4872919 = exp(r4872918);
        double r4872920 = 1.0;
        double r4872921 = r4872920 * r4872920;
        double r4872922 = r4872921 * r4872921;
        double r4872923 = r4872922 * r4872922;
        double r4872924 = r4872920 * r4872923;
        double r4872925 = r4872919 - r4872924;
        double r4872926 = exp(r4872917);
        double r4872927 = r4872920 * r4872926;
        double r4872928 = fma(r4872921, r4872921, r4872927);
        double r4872929 = r4872921 * r4872928;
        double r4872930 = fma(r4872926, r4872926, r4872929);
        double r4872931 = r4872925 / r4872930;
        double r4872932 = r4872915 * r4872911;
        double r4872933 = exp(r4872932);
        double r4872934 = r4872920 + r4872933;
        double r4872935 = r4872934 * r4872920;
        double r4872936 = fma(r4872933, r4872933, r4872935);
        double r4872937 = r4872931 / r4872936;
        double r4872938 = -2.7735030981717545e-34;
        bool r4872939 = r4872911 <= r4872938;
        double r4872940 = 0.16666666666666666;
        double r4872941 = r4872932 * r4872932;
        double r4872942 = r4872932 * r4872941;
        double r4872943 = 0.5;
        double r4872944 = fma(r4872943, r4872941, r4872932);
        double r4872945 = fma(r4872940, r4872942, r4872944);
        double r4872946 = -2.2530088269998953e-43;
        bool r4872947 = r4872911 <= r4872946;
        double r4872948 = 1.61596322380751e+101;
        bool r4872949 = r4872911 <= r4872948;
        double r4872950 = 9.0;
        double r4872951 = r4872950 * r4872932;
        double r4872952 = r4872951 + r4872951;
        double r4872953 = r4872952 + r4872951;
        double r4872954 = exp(r4872953);
        double r4872955 = r4872923 * r4872923;
        double r4872956 = r4872923 * r4872955;
        double r4872957 = r4872920 * r4872921;
        double r4872958 = r4872956 * r4872957;
        double r4872959 = r4872954 - r4872958;
        double r4872960 = exp(r4872951);
        double r4872961 = r4872960 + r4872924;
        double r4872962 = r4872961 * r4872924;
        double r4872963 = fma(r4872960, r4872960, r4872962);
        double r4872964 = r4872959 / r4872963;
        double r4872965 = r4872964 / r4872930;
        double r4872966 = r4872965 / r4872936;
        double r4872967 = r4872949 ? r4872945 : r4872966;
        double r4872968 = r4872947 ? r4872937 : r4872967;
        double r4872969 = r4872939 ? r4872945 : r4872968;
        double r4872970 = r4872913 ? r4872937 : r4872969;
        return r4872970;
}

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 3 regimes

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

   1. Initial program 17.9

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

   3. Applied flip3-- 17.9

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

   4. Simplified 17.9

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

   5. Simplified 17.9

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

   7. Applied flip3-- 17.9

      \[\leadsto \frac{\color{blue}{\frac{{\left(e^{x \cdot \mathsf{fma}\left(2, a, a\right)}\right)}^{3} - {\left(\left(1 \cdot 1\right) \cdot 1\right)}^{3}}{e^{x \cdot \mathsf{fma}\left(2, a, a\right)} \cdot e^{x \cdot \mathsf{fma}\left(2, a, a\right)} + \left(\left(\left(1 \cdot 1\right) \cdot 1\right) \cdot \left(\left(1 \cdot 1\right) \cdot 1\right) + e^{x \cdot \mathsf{fma}\left(2, a, a\right)} \cdot \left(\left(1 \cdot 1\right) \cdot 1\right)\right)}}}{\mathsf{fma}\left(e^{x \cdot a}, e^{x \cdot a}, 1 \cdot \left(1 + e^{x \cdot a}\right)\right)}\]

   8. Simplified 17.8

      \[\leadsto \frac{\frac{\color{blue}{e^{3 \cdot \left(\left(3 \cdot a\right) \cdot x\right)} - \left(\left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right) \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)\right) \cdot 1}}{e^{x \cdot \mathsf{fma}\left(2, a, a\right)} \cdot e^{x \cdot \mathsf{fma}\left(2, a, a\right)} + \left(\left(\left(1 \cdot 1\right) \cdot 1\right) \cdot \left(\left(1 \cdot 1\right) \cdot 1\right) + e^{x \cdot \mathsf{fma}\left(2, a, a\right)} \cdot \left(\left(1 \cdot 1\right) \cdot 1\right)\right)}}{\mathsf{fma}\left(e^{x \cdot a}, e^{x \cdot a}, 1 \cdot \left(1 + e^{x \cdot a}\right)\right)}\]

   9. Simplified 17.8

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

   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

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{6}, \left(a \cdot x\right) \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right), \mathsf{fma}\left(\frac{1}{2}, \left(a \cdot x\right) \cdot \left(a \cdot x\right), x \cdot a\right)\right)}\]

   if 1.61596322380751e+101 < x

   1. Initial program 14.6

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

   3. Applied flip3-- 14.6

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

   4. Simplified 14.5

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

   5. Simplified 14.5

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

   7. Applied flip3-- 14.6

      \[\leadsto \frac{\color{blue}{\frac{{\left(e^{x \cdot \mathsf{fma}\left(2, a, a\right)}\right)}^{3} - {\left(\left(1 \cdot 1\right) \cdot 1\right)}^{3}}{e^{x \cdot \mathsf{fma}\left(2, a, a\right)} \cdot e^{x \cdot \mathsf{fma}\left(2, a, a\right)} + \left(\left(\left(1 \cdot 1\right) \cdot 1\right) \cdot \left(\left(1 \cdot 1\right) \cdot 1\right) + e^{x \cdot \mathsf{fma}\left(2, a, a\right)} \cdot \left(\left(1 \cdot 1\right) \cdot 1\right)\right)}}}{\mathsf{fma}\left(e^{x \cdot a}, e^{x \cdot a}, 1 \cdot \left(1 + e^{x \cdot a}\right)\right)}\]

   8. Simplified 14.6

      \[\leadsto \frac{\frac{\color{blue}{e^{3 \cdot \left(\left(3 \cdot a\right) \cdot x\right)} - \left(\left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right) \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)\right) \cdot 1}}{e^{x \cdot \mathsf{fma}\left(2, a, a\right)} \cdot e^{x \cdot \mathsf{fma}\left(2, a, a\right)} + \left(\left(\left(1 \cdot 1\right) \cdot 1\right) \cdot \left(\left(1 \cdot 1\right) \cdot 1\right) + e^{x \cdot \mathsf{fma}\left(2, a, a\right)} \cdot \left(\left(1 \cdot 1\right) \cdot 1\right)\right)}}{\mathsf{fma}\left(e^{x \cdot a}, e^{x \cdot a}, 1 \cdot \left(1 + e^{x \cdot a}\right)\right)}\]

   9. Simplified 14.6

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

   11. Applied flip3-- 14.6

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

   12. Simplified 14.6

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

   13. Simplified 14.6

       \[\leadsto \frac{\frac{\frac{e^{\left(x \cdot a\right) \cdot 9 + \left(\left(x \cdot a\right) \cdot 9 + \left(x \cdot a\right) \cdot 9\right)} - \left(\left(\left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right) \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)\right) \cdot \left(\left(\left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right) \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)\right) \cdot \left(\left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right) \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)\right)\right)\right) \cdot \left(1 \cdot \left(1 \cdot 1\right)\right)}{\color{blue}{\mathsf{fma}\left(e^{\left(x \cdot a\right) \cdot 9}, e^{\left(x \cdot a\right) \cdot 9}, \left(1 \cdot \left(\left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right) \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)\right)\right) \cdot \left(e^{\left(x \cdot a\right) \cdot 9} + 1 \cdot \left(\left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right) \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)\right)\right)\right)}}}{\mathsf{fma}\left(e^{\left(3 \cdot a\right) \cdot x}, e^{\left(3 \cdot a\right) \cdot x}, \left(1 \cdot 1\right) \cdot \mathsf{fma}\left(1 \cdot 1, 1 \cdot 1, 1 \cdot e^{\left(3 \cdot a\right) \cdot x}\right)\right)}}{\mathsf{fma}\left(e^{x \cdot a}, e^{x \cdot a}, 1 \cdot \left(1 + e^{x \cdot a}\right)\right)}\]
3. Recombined 3 regimes into one program.

4. Final simplification 14.0

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

Reproduce

herbie shell --seed 2019169 +o rules:numerics
(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))