expax (section 3.5)

Average Error: 30.0 → 10.1

Time: 8.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -6.7079265592921415 \cdot 10^{-14}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\left({\left(\sqrt{e^{a \cdot x}}\right)}^{3} + {\left(\sqrt{1}\right)}^{3}\right) \cdot \left({\left(\sqrt{e^{a \cdot x}}\right)}^{3} - {\left(\sqrt{1}\right)}^{3}\right)} \cdot \sqrt[3]{{\left(e^{a \cdot x}\right)}^{3} - {1}^{3}}\right) \cdot \sqrt[3]{e^{a \cdot x} - 1}}{\sqrt[3]{\left(\sqrt{e^{a \cdot x}} \cdot \sqrt{e^{a \cdot x}} + \left(\sqrt{1} \cdot \sqrt{1} - \sqrt{e^{a \cdot x}} \cdot \sqrt{1}\right)\right) \cdot \left(\sqrt{e^{a \cdot x}} \cdot \sqrt{e^{a \cdot x}} + \left(\sqrt{1} \cdot \sqrt{1} + \sqrt{e^{a \cdot x}} \cdot \sqrt{1}\right)\right)} \cdot \sqrt[3]{e^{a \cdot x} \cdot e^{a \cdot x} + \left(1 \cdot 1 + e^{a \cdot x} \cdot 1\right)}}\\ \mathbf{elif}\;a \cdot x \le 1.91419640354660377 \cdot 10^{-41}:\\ \;\;\;\;x \cdot \left(a + \left(\frac{1}{2} \cdot {a}^{2}\right) \cdot x\right) + \frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt[3]{\left(\sqrt{e^{a \cdot x}} + \sqrt{1}\right) \cdot \left(\left({\left(\sqrt{\sqrt{e^{a \cdot x}}}\right)}^{3} + {\left(\sqrt{\sqrt{1}}\right)}^{3}\right) \cdot \left(\sqrt{\sqrt{e^{a \cdot x}}} \cdot \sqrt{\sqrt{e^{a \cdot x}}} - \sqrt{\sqrt{1}} \cdot \sqrt{\sqrt{1}}\right)\right)}}{\sqrt[3]{\left(\sqrt{\sqrt{e^{a \cdot x}}} \cdot \sqrt{\sqrt{e^{a \cdot x}}} + \left(\sqrt{\sqrt{1}} \cdot \sqrt{\sqrt{1}} - \sqrt{\sqrt{e^{a \cdot x}}} \cdot \sqrt{\sqrt{1}}\right)\right) \cdot \left(\sqrt{\sqrt{e^{a \cdot x}}} + \sqrt{\sqrt{1}}\right)}} \cdot \sqrt[3]{e^{a \cdot x} - 1}\right) \cdot \sqrt[3]{e^{a \cdot x} - 1}\\ \end{array}\]

C

double f(double a, double x) {
        double r194155 = a;
        double r194156 = x;
        double r194157 = r194155 * r194156;
        double r194158 = exp(r194157);
        double r194159 = 1.0;
        double r194160 = r194158 - r194159;
        return r194160;
}

↓

double f(double a, double x) {
        double r194161 = a;
        double r194162 = x;
        double r194163 = r194161 * r194162;
        double r194164 = -6.707926559292141e-14;
        bool r194165 = r194163 <= r194164;
        double r194166 = exp(r194163);
        double r194167 = sqrt(r194166);
        double r194168 = 3.0;
        double r194169 = pow(r194167, r194168);
        double r194170 = 1.0;
        double r194171 = sqrt(r194170);
        double r194172 = pow(r194171, r194168);
        double r194173 = r194169 + r194172;
        double r194174 = r194169 - r194172;
        double r194175 = r194173 * r194174;
        double r194176 = cbrt(r194175);
        double r194177 = pow(r194166, r194168);
        double r194178 = pow(r194170, r194168);
        double r194179 = r194177 - r194178;
        double r194180 = cbrt(r194179);
        double r194181 = r194176 * r194180;
        double r194182 = r194166 - r194170;
        double r194183 = cbrt(r194182);
        double r194184 = r194181 * r194183;
        double r194185 = r194167 * r194167;
        double r194186 = r194171 * r194171;
        double r194187 = r194167 * r194171;
        double r194188 = r194186 - r194187;
        double r194189 = r194185 + r194188;
        double r194190 = r194186 + r194187;
        double r194191 = r194185 + r194190;
        double r194192 = r194189 * r194191;
        double r194193 = cbrt(r194192);
        double r194194 = r194166 * r194166;
        double r194195 = r194170 * r194170;
        double r194196 = r194166 * r194170;
        double r194197 = r194195 + r194196;
        double r194198 = r194194 + r194197;
        double r194199 = cbrt(r194198);
        double r194200 = r194193 * r194199;
        double r194201 = r194184 / r194200;
        double r194202 = 1.9141964035466038e-41;
        bool r194203 = r194163 <= r194202;
        double r194204 = 0.5;
        double r194205 = 2.0;
        double r194206 = pow(r194161, r194205);
        double r194207 = r194204 * r194206;
        double r194208 = r194207 * r194162;
        double r194209 = r194161 + r194208;
        double r194210 = r194162 * r194209;
        double r194211 = 0.16666666666666666;
        double r194212 = pow(r194161, r194168);
        double r194213 = pow(r194162, r194168);
        double r194214 = r194212 * r194213;
        double r194215 = r194211 * r194214;
        double r194216 = r194210 + r194215;
        double r194217 = r194167 + r194171;
        double r194218 = sqrt(r194167);
        double r194219 = pow(r194218, r194168);
        double r194220 = sqrt(r194171);
        double r194221 = pow(r194220, r194168);
        double r194222 = r194219 + r194221;
        double r194223 = r194218 * r194218;
        double r194224 = r194220 * r194220;
        double r194225 = r194223 - r194224;
        double r194226 = r194222 * r194225;
        double r194227 = r194217 * r194226;
        double r194228 = cbrt(r194227);
        double r194229 = r194218 * r194220;
        double r194230 = r194224 - r194229;
        double r194231 = r194223 + r194230;
        double r194232 = r194218 + r194220;
        double r194233 = r194231 * r194232;
        double r194234 = cbrt(r194233);
        double r194235 = r194228 / r194234;
        double r194236 = r194235 * r194183;
        double r194237 = r194236 * r194183;
        double r194238 = r194203 ? r194216 : r194237;
        double r194239 = r194165 ? r194201 : r194238;
        return r194239;
}

Error

Bits error versus a

Bits error versus x

Target

Original	30.0
Target	0.2
Herbie	10.1

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

Derivation

1. Split input into 3 regimes

2. if (* a x) < -6.707926559292141e-14

   1. Initial program 0.8

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

   3. Applied add-cube-cbrt 0.8

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

   5. Applied add-sqr-sqrt 0.8

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

   6. Applied add-sqr-sqrt 0.8

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

   7. Applied difference-of-squares 0.8

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

   9. Applied flip3-- 0.8

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

   10. Applied cbrt-div 0.8

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

   11. Applied flip3-- 0.8

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

   12. Applied flip3-+ 0.8

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

   13. Applied frac-times 0.8

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

   14. Applied cbrt-div 0.8

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

   15. Applied associate-*l/ 0.8

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

   16. Applied frac-times 0.8

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

   17. Simplified 0.8

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

   if -6.707926559292141e-14 < (* a x) < 1.9141964035466038e-41

   1. Initial program 44.7

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

   2. Taylor expanded around 0 12.5

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

   3. Simplified 12.5

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

   if 1.9141964035466038e-41 < (* a x)

   1. Initial program 44.3

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

   3. Applied add-cube-cbrt 44.3

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

   5. Applied add-sqr-sqrt 44.3

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

   6. Applied add-sqr-sqrt 44.4

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

   7. Applied difference-of-squares 44.4

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

   9. Applied add-sqr-sqrt 44.4

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

   10. Applied sqrt-prod 44.4

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

   11. Applied add-sqr-sqrt 44.4

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

   12. Applied sqrt-prod 44.7

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

   13. Applied difference-of-squares 44.7

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

   15. Applied flip-- 44.7

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

   16. Applied flip3-+ 44.7

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

   17. Applied frac-times 44.7

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

   18. Applied associate-*r/ 44.8

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

   19. Applied cbrt-div 44.8

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

4. Final simplification 10.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -6.7079265592921415 \cdot 10^{-14}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\left({\left(\sqrt{e^{a \cdot x}}\right)}^{3} + {\left(\sqrt{1}\right)}^{3}\right) \cdot \left({\left(\sqrt{e^{a \cdot x}}\right)}^{3} - {\left(\sqrt{1}\right)}^{3}\right)} \cdot \sqrt[3]{{\left(e^{a \cdot x}\right)}^{3} - {1}^{3}}\right) \cdot \sqrt[3]{e^{a \cdot x} - 1}}{\sqrt[3]{\left(\sqrt{e^{a \cdot x}} \cdot \sqrt{e^{a \cdot x}} + \left(\sqrt{1} \cdot \sqrt{1} - \sqrt{e^{a \cdot x}} \cdot \sqrt{1}\right)\right) \cdot \left(\sqrt{e^{a \cdot x}} \cdot \sqrt{e^{a \cdot x}} + \left(\sqrt{1} \cdot \sqrt{1} + \sqrt{e^{a \cdot x}} \cdot \sqrt{1}\right)\right)} \cdot \sqrt[3]{e^{a \cdot x} \cdot e^{a \cdot x} + \left(1 \cdot 1 + e^{a \cdot x} \cdot 1\right)}}\\ \mathbf{elif}\;a \cdot x \le 1.91419640354660377 \cdot 10^{-41}:\\ \;\;\;\;x \cdot \left(a + \left(\frac{1}{2} \cdot {a}^{2}\right) \cdot x\right) + \frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt[3]{\left(\sqrt{e^{a \cdot x}} + \sqrt{1}\right) \cdot \left(\left({\left(\sqrt{\sqrt{e^{a \cdot x}}}\right)}^{3} + {\left(\sqrt{\sqrt{1}}\right)}^{3}\right) \cdot \left(\sqrt{\sqrt{e^{a \cdot x}}} \cdot \sqrt{\sqrt{e^{a \cdot x}}} - \sqrt{\sqrt{1}} \cdot \sqrt{\sqrt{1}}\right)\right)}}{\sqrt[3]{\left(\sqrt{\sqrt{e^{a \cdot x}}} \cdot \sqrt{\sqrt{e^{a \cdot x}}} + \left(\sqrt{\sqrt{1}} \cdot \sqrt{\sqrt{1}} - \sqrt{\sqrt{e^{a \cdot x}}} \cdot \sqrt{\sqrt{1}}\right)\right) \cdot \left(\sqrt{\sqrt{e^{a \cdot x}}} + \sqrt{\sqrt{1}}\right)}} \cdot \sqrt[3]{e^{a \cdot x} - 1}\right) \cdot \sqrt[3]{e^{a \cdot x} - 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020062 
(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))