NMSE Section 6.1 mentioned, A

Average Error: 29.6 → 1.0

Time: 1.5m

Precision: 64

• could not determine a ground truth for program body

  1. x = -1.1000495125221241e+294
  2. eps = -2.8192679536690592e-123

Math

\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]

↓

\[\begin{array}{l} \mathbf{if}\;x \le 44.840790515394986:\\ \;\;\;\;\frac{\frac{\sqrt[3]{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{3} \cdot x, x \cdot x, 2\right), \mathsf{fma}\left(\frac{2}{3} \cdot x, x \cdot x, 2\right) \cdot \mathsf{fma}\left(\frac{2}{3} \cdot x, x \cdot x, 2\right), \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(-x\right)\right)\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{3} \cdot x, x \cdot x, 2\right), \mathsf{fma}\left(\frac{2}{3} \cdot x, x \cdot x, 2\right) \cdot \mathsf{fma}\left(\frac{2}{3} \cdot x, x \cdot x, 2\right), \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(-x\right)\right)\right)\right), \mathsf{fma}\left(\frac{2}{3} \cdot x, x \cdot x, 2\right), \left(x \cdot \left(-x\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{3} \cdot x, x \cdot x, 2\right), \mathsf{fma}\left(\frac{2}{3} \cdot x, x \cdot x, 2\right) \cdot \mathsf{fma}\left(\frac{2}{3} \cdot x, x \cdot x, 2\right), \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(-x\right)\right)\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{3} \cdot x, x \cdot x, 2\right), \mathsf{fma}\left(\frac{2}{3} \cdot x, x \cdot x, 2\right) \cdot \mathsf{fma}\left(\frac{2}{3} \cdot x, x \cdot x, 2\right), \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(-x\right)\right)\right)\right)\right)\right)}}{\sqrt[3]{\mathsf{fma}\left(x \cdot x, x \cdot x + \mathsf{fma}\left(\frac{2}{3} \cdot x, x \cdot x, 2\right), \mathsf{fma}\left(\frac{2}{3} \cdot x, x \cdot x, 2\right) \cdot \mathsf{fma}\left(\frac{2}{3} \cdot x, x \cdot x, 2\right)\right) \cdot \mathsf{fma}\left(x \cdot x, x \cdot x + \mathsf{fma}\left(\frac{2}{3} \cdot x, x \cdot x, 2\right), \mathsf{fma}\left(\frac{2}{3} \cdot x, x \cdot x, 2\right) \cdot \mathsf{fma}\left(\frac{2}{3} \cdot x, x \cdot x, 2\right)\right)}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log \left(\left(e^{x \cdot \left(-1 - \varepsilon\right)} - \frac{e^{x \cdot \left(-1 - \varepsilon\right)}}{\varepsilon}\right) + \left(\frac{e^{x \cdot \left(\varepsilon + -1\right)}}{\varepsilon} + e^{x \cdot \left(\varepsilon + -1\right)}\right)\right)}}{2}\\ \end{array}\]

C

double f(double x, double eps) {
        double r2352269 = 1.0;
        double r2352270 = eps;
        double r2352271 = r2352269 / r2352270;
        double r2352272 = r2352269 + r2352271;
        double r2352273 = r2352269 - r2352270;
        double r2352274 = x;
        double r2352275 = r2352273 * r2352274;
        double r2352276 = -r2352275;
        double r2352277 = exp(r2352276);
        double r2352278 = r2352272 * r2352277;
        double r2352279 = r2352271 - r2352269;
        double r2352280 = r2352269 + r2352270;
        double r2352281 = r2352280 * r2352274;
        double r2352282 = -r2352281;
        double r2352283 = exp(r2352282);
        double r2352284 = r2352279 * r2352283;
        double r2352285 = r2352278 - r2352284;
        double r2352286 = 2.0;
        double r2352287 = r2352285 / r2352286;
        return r2352287;
}

↓

double f(double x, double eps) {
        double r2352288 = x;
        double r2352289 = 44.840790515394986;
        bool r2352290 = r2352288 <= r2352289;
        double r2352291 = 0.6666666666666666;
        double r2352292 = r2352291 * r2352288;
        double r2352293 = r2352288 * r2352288;
        double r2352294 = 2.0;
        double r2352295 = fma(r2352292, r2352293, r2352294);
        double r2352296 = r2352295 * r2352295;
        double r2352297 = -r2352288;
        double r2352298 = r2352288 * r2352297;
        double r2352299 = r2352293 * r2352298;
        double r2352300 = r2352293 * r2352299;
        double r2352301 = fma(r2352295, r2352296, r2352300);
        double r2352302 = r2352301 * r2352301;
        double r2352303 = r2352298 * r2352302;
        double r2352304 = fma(r2352302, r2352295, r2352303);
        double r2352305 = cbrt(r2352304);
        double r2352306 = r2352293 + r2352295;
        double r2352307 = fma(r2352293, r2352306, r2352296);
        double r2352308 = r2352307 * r2352307;
        double r2352309 = cbrt(r2352308);
        double r2352310 = r2352305 / r2352309;
        double r2352311 = r2352310 / r2352294;
        double r2352312 = -1.0;
        double r2352313 = eps;
        double r2352314 = r2352312 - r2352313;
        double r2352315 = r2352288 * r2352314;
        double r2352316 = exp(r2352315);
        double r2352317 = r2352316 / r2352313;
        double r2352318 = r2352316 - r2352317;
        double r2352319 = r2352313 + r2352312;
        double r2352320 = r2352288 * r2352319;
        double r2352321 = exp(r2352320);
        double r2352322 = r2352321 / r2352313;
        double r2352323 = r2352322 + r2352321;
        double r2352324 = r2352318 + r2352323;
        double r2352325 = log(r2352324);
        double r2352326 = exp(r2352325);
        double r2352327 = r2352326 / r2352294;
        double r2352328 = r2352290 ? r2352311 : r2352327;
        return r2352328;
}

Error

Bits error versus x

Bits error versus eps

Derivation

1. Split input into 2 regimes

2. if x < 44.840790515394986

   1. Initial program 39.0

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]

   2. Simplified 39.0

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

   3. Taylor expanded around 0 1.2

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

   4. Simplified 1.2

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{2}{3}, \left(x \cdot x\right) \cdot x, 2\right) - x \cdot x}}{2}\]
   5. Using strategy rm

   6. Applied add-cbrt-cube 1.2

      \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\left(\mathsf{fma}\left(\frac{2}{3}, \left(x \cdot x\right) \cdot x, 2\right) - x \cdot x\right) \cdot \left(\mathsf{fma}\left(\frac{2}{3}, \left(x \cdot x\right) \cdot x, 2\right) - x \cdot x\right)\right) \cdot \left(\mathsf{fma}\left(\frac{2}{3}, \left(x \cdot x\right) \cdot x, 2\right) - x \cdot x\right)}}}{2}\]
   7. Using strategy rm

   8. Applied flip3-- 1.2

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

   9. Applied flip3-- 1.2

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

   10. Applied associate-*l/ 1.2

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

   11. Applied frac-times 1.2

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

   12. Applied cbrt-div 1.2

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

   13. Simplified 1.2

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

   14. Simplified 1.2

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

   if 44.840790515394986 < x

   1. Initial program 0.2

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]

   2. Simplified 0.2

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

   4. Applied add-exp-log 0.2

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

4. Final simplification 1.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 44.840790515394986:\\ \;\;\;\;\frac{\frac{\sqrt[3]{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{3} \cdot x, x \cdot x, 2\right), \mathsf{fma}\left(\frac{2}{3} \cdot x, x \cdot x, 2\right) \cdot \mathsf{fma}\left(\frac{2}{3} \cdot x, x \cdot x, 2\right), \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(-x\right)\right)\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{3} \cdot x, x \cdot x, 2\right), \mathsf{fma}\left(\frac{2}{3} \cdot x, x \cdot x, 2\right) \cdot \mathsf{fma}\left(\frac{2}{3} \cdot x, x \cdot x, 2\right), \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(-x\right)\right)\right)\right), \mathsf{fma}\left(\frac{2}{3} \cdot x, x \cdot x, 2\right), \left(x \cdot \left(-x\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{3} \cdot x, x \cdot x, 2\right), \mathsf{fma}\left(\frac{2}{3} \cdot x, x \cdot x, 2\right) \cdot \mathsf{fma}\left(\frac{2}{3} \cdot x, x \cdot x, 2\right), \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(-x\right)\right)\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{3} \cdot x, x \cdot x, 2\right), \mathsf{fma}\left(\frac{2}{3} \cdot x, x \cdot x, 2\right) \cdot \mathsf{fma}\left(\frac{2}{3} \cdot x, x \cdot x, 2\right), \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(-x\right)\right)\right)\right)\right)\right)}}{\sqrt[3]{\mathsf{fma}\left(x \cdot x, x \cdot x + \mathsf{fma}\left(\frac{2}{3} \cdot x, x \cdot x, 2\right), \mathsf{fma}\left(\frac{2}{3} \cdot x, x \cdot x, 2\right) \cdot \mathsf{fma}\left(\frac{2}{3} \cdot x, x \cdot x, 2\right)\right) \cdot \mathsf{fma}\left(x \cdot x, x \cdot x + \mathsf{fma}\left(\frac{2}{3} \cdot x, x \cdot x, 2\right), \mathsf{fma}\left(\frac{2}{3} \cdot x, x \cdot x, 2\right) \cdot \mathsf{fma}\left(\frac{2}{3} \cdot x, x \cdot x, 2\right)\right)}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log \left(\left(e^{x \cdot \left(-1 - \varepsilon\right)} - \frac{e^{x \cdot \left(-1 - \varepsilon\right)}}{\varepsilon}\right) + \left(\frac{e^{x \cdot \left(\varepsilon + -1\right)}}{\varepsilon} + e^{x \cdot \left(\varepsilon + -1\right)}\right)\right)}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019135 +o rules:numerics
(FPCore (x eps)
  :name "NMSE Section 6.1 mentioned, A"
  (/ (- (* (+ 1 (/ 1 eps)) (exp (- (* (- 1 eps) x)))) (* (- (/ 1 eps) 1) (exp (- (* (+ 1 eps) x))))) 2))