expax (section 3.5)

Average Error: 30.0 → 10.1

Time: 8.7s

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 r180279 = a;
        double r180280 = x;
        double r180281 = r180279 * r180280;
        double r180282 = exp(r180281);
        double r180283 = 1.0;
        double r180284 = r180282 - r180283;
        return r180284;
}

↓

double f(double a, double x) {
        double r180285 = a;
        double r180286 = x;
        double r180287 = r180285 * r180286;
        double r180288 = -6.707926559292141e-14;
        bool r180289 = r180287 <= r180288;
        double r180290 = exp(r180287);
        double r180291 = sqrt(r180290);
        double r180292 = 3.0;
        double r180293 = pow(r180291, r180292);
        double r180294 = 1.0;
        double r180295 = sqrt(r180294);
        double r180296 = pow(r180295, r180292);
        double r180297 = r180293 + r180296;
        double r180298 = r180293 - r180296;
        double r180299 = r180297 * r180298;
        double r180300 = cbrt(r180299);
        double r180301 = pow(r180290, r180292);
        double r180302 = pow(r180294, r180292);
        double r180303 = r180301 - r180302;
        double r180304 = cbrt(r180303);
        double r180305 = r180300 * r180304;
        double r180306 = r180290 - r180294;
        double r180307 = cbrt(r180306);
        double r180308 = r180305 * r180307;
        double r180309 = r180291 * r180291;
        double r180310 = r180295 * r180295;
        double r180311 = r180291 * r180295;
        double r180312 = r180310 - r180311;
        double r180313 = r180309 + r180312;
        double r180314 = r180310 + r180311;
        double r180315 = r180309 + r180314;
        double r180316 = r180313 * r180315;
        double r180317 = cbrt(r180316);
        double r180318 = r180290 * r180290;
        double r180319 = r180294 * r180294;
        double r180320 = r180290 * r180294;
        double r180321 = r180319 + r180320;
        double r180322 = r180318 + r180321;
        double r180323 = cbrt(r180322);
        double r180324 = r180317 * r180323;
        double r180325 = r180308 / r180324;
        double r180326 = 1.9141964035466038e-41;
        bool r180327 = r180287 <= r180326;
        double r180328 = 0.5;
        double r180329 = 2.0;
        double r180330 = pow(r180285, r180329);
        double r180331 = r180328 * r180330;
        double r180332 = r180331 * r180286;
        double r180333 = r180285 + r180332;
        double r180334 = r180286 * r180333;
        double r180335 = 0.16666666666666666;
        double r180336 = pow(r180285, r180292);
        double r180337 = pow(r180286, r180292);
        double r180338 = r180336 * r180337;
        double r180339 = r180335 * r180338;
        double r180340 = r180334 + r180339;
        double r180341 = r180291 + r180295;
        double r180342 = sqrt(r180291);
        double r180343 = pow(r180342, r180292);
        double r180344 = sqrt(r180295);
        double r180345 = pow(r180344, r180292);
        double r180346 = r180343 + r180345;
        double r180347 = r180342 * r180342;
        double r180348 = r180344 * r180344;
        double r180349 = r180347 - r180348;
        double r180350 = r180346 * r180349;
        double r180351 = r180341 * r180350;
        double r180352 = cbrt(r180351);
        double r180353 = r180342 * r180344;
        double r180354 = r180348 - r180353;
        double r180355 = r180347 + r180354;
        double r180356 = r180342 + r180344;
        double r180357 = r180355 * r180356;
        double r180358 = cbrt(r180357);
        double r180359 = r180352 / r180358;
        double r180360 = r180359 * r180307;
        double r180361 = r180360 * r180307;
        double r180362 = r180327 ? r180340 : r180361;
        double r180363 = r180289 ? r180325 : r180362;
        return r180363;
}

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))