Jmat.Real.erf

Average Error: 13.4 → 12.7

Time: 4.3m

Precision: 64

Math

\[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]

↓

\[\left(\sqrt[3]{\sqrt[3]{1 - \frac{\frac{\frac{\frac{1.421413741 + \frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{e^{\left|x\right| \cdot \left|x\right|}}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}} \cdot \log \left(e^{\sqrt[3]{1 - \frac{\frac{\frac{\frac{1.421413741 + \frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{e^{\left|x\right| \cdot \left|x\right|}}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}} \cdot \sqrt[3]{1 - \frac{\frac{\frac{\frac{1.421413741 + \frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{e^{\left|x\right| \cdot \left|x\right|}}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}}\right)} \cdot \sqrt[3]{\sqrt[3]{\log \left(e^{1 - \frac{\frac{\frac{\frac{1.421413741 + \frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{e^{\left|x\right| \cdot \left|x\right|}}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}\right)} \cdot \left(\sqrt[3]{\log \left(e^{1 - \frac{\frac{\frac{\frac{1.421413741 + \frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{e^{\left|x\right| \cdot \left|x\right|}}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}\right)} \cdot \sqrt[3]{\log \left(e^{1 - \frac{\frac{\frac{\frac{1.421413741 + \frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{e^{\left|x\right| \cdot \left|x\right|}}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}\right)}\right)}\right) \cdot \sqrt[3]{\log \left(e^{1 - \frac{\frac{\frac{\frac{1.421413741 + \frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{e^{\left|x\right| \cdot \left|x\right|}}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}\right)}\]

C

double f(double x) {
        double r5665318 = 1.0;
        double r5665319 = 0.3275911;
        double r5665320 = x;
        double r5665321 = fabs(r5665320);
        double r5665322 = r5665319 * r5665321;
        double r5665323 = r5665318 + r5665322;
        double r5665324 = r5665318 / r5665323;
        double r5665325 = 0.254829592;
        double r5665326 = -0.284496736;
        double r5665327 = 1.421413741;
        double r5665328 = -1.453152027;
        double r5665329 = 1.061405429;
        double r5665330 = r5665324 * r5665329;
        double r5665331 = r5665328 + r5665330;
        double r5665332 = r5665324 * r5665331;
        double r5665333 = r5665327 + r5665332;
        double r5665334 = r5665324 * r5665333;
        double r5665335 = r5665326 + r5665334;
        double r5665336 = r5665324 * r5665335;
        double r5665337 = r5665325 + r5665336;
        double r5665338 = r5665324 * r5665337;
        double r5665339 = r5665321 * r5665321;
        double r5665340 = -r5665339;
        double r5665341 = exp(r5665340);
        double r5665342 = r5665338 * r5665341;
        double r5665343 = r5665318 - r5665342;
        return r5665343;
}

↓

double f(double x) {
        double r5665344 = 1.0;
        double r5665345 = 1.421413741;
        double r5665346 = 1.061405429;
        double r5665347 = x;
        double r5665348 = fabs(r5665347);
        double r5665349 = 0.3275911;
        double r5665350 = fma(r5665348, r5665349, r5665344);
        double r5665351 = r5665346 / r5665350;
        double r5665352 = -1.453152027;
        double r5665353 = r5665351 + r5665352;
        double r5665354 = r5665353 / r5665350;
        double r5665355 = r5665345 + r5665354;
        double r5665356 = r5665355 / r5665350;
        double r5665357 = -0.284496736;
        double r5665358 = r5665356 + r5665357;
        double r5665359 = r5665358 / r5665350;
        double r5665360 = 0.254829592;
        double r5665361 = r5665359 + r5665360;
        double r5665362 = r5665348 * r5665348;
        double r5665363 = exp(r5665362);
        double r5665364 = r5665361 / r5665363;
        double r5665365 = r5665364 / r5665350;
        double r5665366 = r5665344 - r5665365;
        double r5665367 = cbrt(r5665366);
        double r5665368 = r5665367 * r5665367;
        double r5665369 = exp(r5665368);
        double r5665370 = log(r5665369);
        double r5665371 = r5665367 * r5665370;
        double r5665372 = cbrt(r5665371);
        double r5665373 = exp(r5665366);
        double r5665374 = log(r5665373);
        double r5665375 = cbrt(r5665374);
        double r5665376 = r5665375 * r5665375;
        double r5665377 = r5665375 * r5665376;
        double r5665378 = cbrt(r5665377);
        double r5665379 = r5665372 * r5665378;
        double r5665380 = r5665379 * r5665375;
        return r5665380;
}

Error

Bits error versus x

Derivation

1. Initial program 13.4

   \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
2. Using strategy rm

3. Applied add-log-exp 13.4

   \[\leadsto \color{blue}{\log \left(e^{1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}}\right)}\]

4. Simplified 13.4

   \[\leadsto \log \color{blue}{\left(e^{1 - \frac{\frac{\frac{\frac{1.421413741 + \frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{e^{\left|x\right| \cdot \left|x\right|}}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}\right)}\]
5. Using strategy rm

6. Applied add-cube-cbrt 13.4

   \[\leadsto \color{blue}{\left(\sqrt[3]{\log \left(e^{1 - \frac{\frac{\frac{\frac{1.421413741 + \frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{e^{\left|x\right| \cdot \left|x\right|}}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}\right)} \cdot \sqrt[3]{\log \left(e^{1 - \frac{\frac{\frac{\frac{1.421413741 + \frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{e^{\left|x\right| \cdot \left|x\right|}}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}\right)}\right) \cdot \sqrt[3]{\log \left(e^{1 - \frac{\frac{\frac{\frac{1.421413741 + \frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{e^{\left|x\right| \cdot \left|x\right|}}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}\right)}}\]
7. Using strategy rm

8. Applied add-cube-cbrt 13.4

   \[\leadsto \left(\sqrt[3]{\log \left(e^{\color{blue}{\left(\sqrt[3]{1 - \frac{\frac{\frac{\frac{1.421413741 + \frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{e^{\left|x\right| \cdot \left|x\right|}}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}} \cdot \sqrt[3]{1 - \frac{\frac{\frac{\frac{1.421413741 + \frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{e^{\left|x\right| \cdot \left|x\right|}}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}\right) \cdot \sqrt[3]{1 - \frac{\frac{\frac{\frac{1.421413741 + \frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{e^{\left|x\right| \cdot \left|x\right|}}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}}}\right)} \cdot \sqrt[3]{\log \left(e^{1 - \frac{\frac{\frac{\frac{1.421413741 + \frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{e^{\left|x\right| \cdot \left|x\right|}}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}\right)}\right) \cdot \sqrt[3]{\log \left(e^{1 - \frac{\frac{\frac{\frac{1.421413741 + \frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{e^{\left|x\right| \cdot \left|x\right|}}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}\right)}\]

9. Applied exp-prod 13.4

   \[\leadsto \left(\sqrt[3]{\log \color{blue}{\left({\left(e^{\sqrt[3]{1 - \frac{\frac{\frac{\frac{1.421413741 + \frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{e^{\left|x\right| \cdot \left|x\right|}}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}} \cdot \sqrt[3]{1 - \frac{\frac{\frac{\frac{1.421413741 + \frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{e^{\left|x\right| \cdot \left|x\right|}}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}}\right)}^{\left(\sqrt[3]{1 - \frac{\frac{\frac{\frac{1.421413741 + \frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{e^{\left|x\right| \cdot \left|x\right|}}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}\right)}\right)}} \cdot \sqrt[3]{\log \left(e^{1 - \frac{\frac{\frac{\frac{1.421413741 + \frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{e^{\left|x\right| \cdot \left|x\right|}}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}\right)}\right) \cdot \sqrt[3]{\log \left(e^{1 - \frac{\frac{\frac{\frac{1.421413741 + \frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{e^{\left|x\right| \cdot \left|x\right|}}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}\right)}\]

10. Applied log-pow 12.7

    \[\leadsto \left(\sqrt[3]{\color{blue}{\sqrt[3]{1 - \frac{\frac{\frac{\frac{1.421413741 + \frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{e^{\left|x\right| \cdot \left|x\right|}}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}} \cdot \log \left(e^{\sqrt[3]{1 - \frac{\frac{\frac{\frac{1.421413741 + \frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{e^{\left|x\right| \cdot \left|x\right|}}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}} \cdot \sqrt[3]{1 - \frac{\frac{\frac{\frac{1.421413741 + \frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{e^{\left|x\right| \cdot \left|x\right|}}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}}\right)}} \cdot \sqrt[3]{\log \left(e^{1 - \frac{\frac{\frac{\frac{1.421413741 + \frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{e^{\left|x\right| \cdot \left|x\right|}}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}\right)}\right) \cdot \sqrt[3]{\log \left(e^{1 - \frac{\frac{\frac{\frac{1.421413741 + \frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{e^{\left|x\right| \cdot \left|x\right|}}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}\right)}\]
11. Using strategy rm

12. Applied add-cube-cbrt 12.7

    \[\leadsto \left(\sqrt[3]{\sqrt[3]{1 - \frac{\frac{\frac{\frac{1.421413741 + \frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{e^{\left|x\right| \cdot \left|x\right|}}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}} \cdot \log \left(e^{\sqrt[3]{1 - \frac{\frac{\frac{\frac{1.421413741 + \frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{e^{\left|x\right| \cdot \left|x\right|}}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}} \cdot \sqrt[3]{1 - \frac{\frac{\frac{\frac{1.421413741 + \frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{e^{\left|x\right| \cdot \left|x\right|}}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}}\right)} \cdot \sqrt[3]{\color{blue}{\left(\sqrt[3]{\log \left(e^{1 - \frac{\frac{\frac{\frac{1.421413741 + \frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{e^{\left|x\right| \cdot \left|x\right|}}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}\right)} \cdot \sqrt[3]{\log \left(e^{1 - \frac{\frac{\frac{\frac{1.421413741 + \frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{e^{\left|x\right| \cdot \left|x\right|}}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}\right)}\right) \cdot \sqrt[3]{\log \left(e^{1 - \frac{\frac{\frac{\frac{1.421413741 + \frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{e^{\left|x\right| \cdot \left|x\right|}}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}\right)}}}\right) \cdot \sqrt[3]{\log \left(e^{1 - \frac{\frac{\frac{\frac{1.421413741 + \frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{e^{\left|x\right| \cdot \left|x\right|}}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}\right)}\]

13. Final simplification 12.7

    \[\leadsto \left(\sqrt[3]{\sqrt[3]{1 - \frac{\frac{\frac{\frac{1.421413741 + \frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{e^{\left|x\right| \cdot \left|x\right|}}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}} \cdot \log \left(e^{\sqrt[3]{1 - \frac{\frac{\frac{\frac{1.421413741 + \frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{e^{\left|x\right| \cdot \left|x\right|}}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}} \cdot \sqrt[3]{1 - \frac{\frac{\frac{\frac{1.421413741 + \frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{e^{\left|x\right| \cdot \left|x\right|}}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}}\right)} \cdot \sqrt[3]{\sqrt[3]{\log \left(e^{1 - \frac{\frac{\frac{\frac{1.421413741 + \frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{e^{\left|x\right| \cdot \left|x\right|}}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}\right)} \cdot \left(\sqrt[3]{\log \left(e^{1 - \frac{\frac{\frac{\frac{1.421413741 + \frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{e^{\left|x\right| \cdot \left|x\right|}}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}\right)} \cdot \sqrt[3]{\log \left(e^{1 - \frac{\frac{\frac{\frac{1.421413741 + \frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{e^{\left|x\right| \cdot \left|x\right|}}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}\right)}\right)}\right) \cdot \sqrt[3]{\log \left(e^{1 - \frac{\frac{\frac{\frac{1.421413741 + \frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{e^{\left|x\right| \cdot \left|x\right|}}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}\right)}\]

Reproduce

herbie shell --seed 2019143 +o rules:numerics
(FPCore (x)
  :name "Jmat.Real.erf"
  (- 1 (* (* (/ 1 (+ 1 (* 0.3275911 (fabs x)))) (+ 0.254829592 (* (/ 1 (+ 1 (* 0.3275911 (fabs x)))) (+ -0.284496736 (* (/ 1 (+ 1 (* 0.3275911 (fabs x)))) (+ 1.421413741 (* (/ 1 (+ 1 (* 0.3275911 (fabs x)))) (+ -1.453152027 (* (/ 1 (+ 1 (* 0.3275911 (fabs x)))) 1.061405429))))))))) (exp (- (* (fabs x) (fabs x)))))))