Jmat.Real.erf

Average Error: 13.7 → 13.7

Time: 8.6s

Precision: 64

Math

\[1 - \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(0.25482959199999999 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-0.284496735999999972 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(1.42141374100000006 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-1.45315202700000001 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot 1.0614054289999999\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]

↓

\[\log \left(\sqrt{e^{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, 1.0614054289999999, -1.45315202700000001\right), 1.42141374100000006\right), -0.284496735999999972\right), 0.25482959199999999\right)}{e^{\left|x\right| \cdot \left|x\right|}}, \frac{-\sqrt{1}}{\left(\sqrt[3]{\sqrt{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}}\right) \cdot \left(\sqrt[3]{\sqrt{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}}\right)} \cdot \frac{\sqrt{1}}{\sqrt[3]{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}}, 1\right)}}\right) + \log \left(\sqrt{e^{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, 1.0614054289999999, -1.45315202700000001\right), 1.42141374100000006\right), -0.284496735999999972\right), 0.25482959199999999\right)}{e^{\left|x\right| \cdot \left|x\right|}}, \frac{-\sqrt{1}}{\left(\sqrt[3]{\sqrt{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}}\right) \cdot \left(\sqrt[3]{\sqrt{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}}\right)} \cdot \frac{\sqrt{1}}{\sqrt[3]{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}}, 1\right)}}\right)\]

C

double f(double x) {
        double r145892 = 1.0;
        double r145893 = 0.3275911;
        double r145894 = x;
        double r145895 = fabs(r145894);
        double r145896 = r145893 * r145895;
        double r145897 = r145892 + r145896;
        double r145898 = r145892 / r145897;
        double r145899 = 0.254829592;
        double r145900 = -0.284496736;
        double r145901 = 1.421413741;
        double r145902 = -1.453152027;
        double r145903 = 1.061405429;
        double r145904 = r145898 * r145903;
        double r145905 = r145902 + r145904;
        double r145906 = r145898 * r145905;
        double r145907 = r145901 + r145906;
        double r145908 = r145898 * r145907;
        double r145909 = r145900 + r145908;
        double r145910 = r145898 * r145909;
        double r145911 = r145899 + r145910;
        double r145912 = r145898 * r145911;
        double r145913 = r145895 * r145895;
        double r145914 = -r145913;
        double r145915 = exp(r145914);
        double r145916 = r145912 * r145915;
        double r145917 = r145892 - r145916;
        return r145917;
}

↓

double f(double x) {
        double r145918 = 1.0;
        double r145919 = 0.3275911;
        double r145920 = x;
        double r145921 = fabs(r145920);
        double r145922 = r145919 * r145921;
        double r145923 = r145918 + r145922;
        double r145924 = r145918 / r145923;
        double r145925 = 1.061405429;
        double r145926 = -1.453152027;
        double r145927 = fma(r145924, r145925, r145926);
        double r145928 = 1.421413741;
        double r145929 = fma(r145924, r145927, r145928);
        double r145930 = -0.284496736;
        double r145931 = fma(r145924, r145929, r145930);
        double r145932 = 0.254829592;
        double r145933 = fma(r145924, r145931, r145932);
        double r145934 = r145921 * r145921;
        double r145935 = exp(r145934);
        double r145936 = r145933 / r145935;
        double r145937 = sqrt(r145918);
        double r145938 = -r145937;
        double r145939 = fma(r145921, r145919, r145918);
        double r145940 = sqrt(r145939);
        double r145941 = cbrt(r145940);
        double r145942 = r145941 * r145941;
        double r145943 = r145942 * r145942;
        double r145944 = r145938 / r145943;
        double r145945 = cbrt(r145939);
        double r145946 = r145937 / r145945;
        double r145947 = r145944 * r145946;
        double r145948 = fma(r145936, r145947, r145918);
        double r145949 = exp(r145948);
        double r145950 = sqrt(r145949);
        double r145951 = log(r145950);
        double r145952 = r145951 + r145951;
        return r145952;
}

Error

Bits error versus x

Derivation

1. Initial program 13.7

   \[1 - \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(0.25482959199999999 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-0.284496735999999972 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(1.42141374100000006 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-1.45315202700000001 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot 1.0614054289999999\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]

2. Simplified 13.7

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, 1.0614054289999999, -1.45315202700000001\right), 1.42141374100000006\right), -0.284496735999999972\right), 0.25482959199999999\right)}{e^{\left|x\right| \cdot \left|x\right|}}, \frac{-1}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}, 1\right)}\]
3. Using strategy rm

4. Applied add-log-exp 13.7

   \[\leadsto \color{blue}{\log \left(e^{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, 1.0614054289999999, -1.45315202700000001\right), 1.42141374100000006\right), -0.284496735999999972\right), 0.25482959199999999\right)}{e^{\left|x\right| \cdot \left|x\right|}}, \frac{-1}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}, 1\right)}\right)}\]
5. Using strategy rm

6. Applied add-cube-cbrt 13.7

   \[\leadsto \log \left(e^{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, 1.0614054289999999, -1.45315202700000001\right), 1.42141374100000006\right), -0.284496735999999972\right), 0.25482959199999999\right)}{e^{\left|x\right| \cdot \left|x\right|}}, \frac{-1}{\color{blue}{\left(\sqrt[3]{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}}}, 1\right)}\right)\]

7. Applied add-sqr-sqrt 13.7

   \[\leadsto \log \left(e^{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, 1.0614054289999999, -1.45315202700000001\right), 1.42141374100000006\right), -0.284496735999999972\right), 0.25482959199999999\right)}{e^{\left|x\right| \cdot \left|x\right|}}, \frac{-\color{blue}{\sqrt{1} \cdot \sqrt{1}}}{\left(\sqrt[3]{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}}, 1\right)}\right)\]

8. Applied distribute-lft-neg-in 13.7

   \[\leadsto \log \left(e^{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, 1.0614054289999999, -1.45315202700000001\right), 1.42141374100000006\right), -0.284496735999999972\right), 0.25482959199999999\right)}{e^{\left|x\right| \cdot \left|x\right|}}, \frac{\color{blue}{\left(-\sqrt{1}\right) \cdot \sqrt{1}}}{\left(\sqrt[3]{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}}, 1\right)}\right)\]

9. Applied times-frac 13.7

   \[\leadsto \log \left(e^{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, 1.0614054289999999, -1.45315202700000001\right), 1.42141374100000006\right), -0.284496735999999972\right), 0.25482959199999999\right)}{e^{\left|x\right| \cdot \left|x\right|}}, \color{blue}{\frac{-\sqrt{1}}{\sqrt[3]{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}} \cdot \frac{\sqrt{1}}{\sqrt[3]{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}}}, 1\right)}\right)\]
10. Using strategy rm

11. Applied add-sqr-sqrt 13.7

    \[\leadsto \log \left(e^{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, 1.0614054289999999, -1.45315202700000001\right), 1.42141374100000006\right), -0.284496735999999972\right), 0.25482959199999999\right)}{e^{\left|x\right| \cdot \left|x\right|}}, \frac{-\sqrt{1}}{\sqrt[3]{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)} \cdot \sqrt[3]{\color{blue}{\sqrt{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)} \cdot \sqrt{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}}}} \cdot \frac{\sqrt{1}}{\sqrt[3]{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}}, 1\right)}\right)\]

12. Applied cbrt-prod 13.7

    \[\leadsto \log \left(e^{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, 1.0614054289999999, -1.45315202700000001\right), 1.42141374100000006\right), -0.284496735999999972\right), 0.25482959199999999\right)}{e^{\left|x\right| \cdot \left|x\right|}}, \frac{-\sqrt{1}}{\sqrt[3]{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)} \cdot \color{blue}{\left(\sqrt[3]{\sqrt{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}}\right)}} \cdot \frac{\sqrt{1}}{\sqrt[3]{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}}, 1\right)}\right)\]

13. Applied add-sqr-sqrt 13.7

    \[\leadsto \log \left(e^{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, 1.0614054289999999, -1.45315202700000001\right), 1.42141374100000006\right), -0.284496735999999972\right), 0.25482959199999999\right)}{e^{\left|x\right| \cdot \left|x\right|}}, \frac{-\sqrt{1}}{\sqrt[3]{\color{blue}{\sqrt{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)} \cdot \sqrt{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}}} \cdot \left(\sqrt[3]{\sqrt{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}}\right)} \cdot \frac{\sqrt{1}}{\sqrt[3]{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}}, 1\right)}\right)\]

14. Applied cbrt-prod 13.7

    \[\leadsto \log \left(e^{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, 1.0614054289999999, -1.45315202700000001\right), 1.42141374100000006\right), -0.284496735999999972\right), 0.25482959199999999\right)}{e^{\left|x\right| \cdot \left|x\right|}}, \frac{-\sqrt{1}}{\color{blue}{\left(\sqrt[3]{\sqrt{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}}\right)} \cdot \left(\sqrt[3]{\sqrt{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}}\right)} \cdot \frac{\sqrt{1}}{\sqrt[3]{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}}, 1\right)}\right)\]

15. Applied swap-sqr 13.7

    \[\leadsto \log \left(e^{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, 1.0614054289999999, -1.45315202700000001\right), 1.42141374100000006\right), -0.284496735999999972\right), 0.25482959199999999\right)}{e^{\left|x\right| \cdot \left|x\right|}}, \frac{-\sqrt{1}}{\color{blue}{\left(\sqrt[3]{\sqrt{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}}\right) \cdot \left(\sqrt[3]{\sqrt{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}}\right)}} \cdot \frac{\sqrt{1}}{\sqrt[3]{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}}, 1\right)}\right)\]
16. Using strategy rm

17. Applied add-sqr-sqrt 13.7

    \[\leadsto \log \color{blue}{\left(\sqrt{e^{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, 1.0614054289999999, -1.45315202700000001\right), 1.42141374100000006\right), -0.284496735999999972\right), 0.25482959199999999\right)}{e^{\left|x\right| \cdot \left|x\right|}}, \frac{-\sqrt{1}}{\left(\sqrt[3]{\sqrt{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}}\right) \cdot \left(\sqrt[3]{\sqrt{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}}\right)} \cdot \frac{\sqrt{1}}{\sqrt[3]{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}}, 1\right)}} \cdot \sqrt{e^{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, 1.0614054289999999, -1.45315202700000001\right), 1.42141374100000006\right), -0.284496735999999972\right), 0.25482959199999999\right)}{e^{\left|x\right| \cdot \left|x\right|}}, \frac{-\sqrt{1}}{\left(\sqrt[3]{\sqrt{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}}\right) \cdot \left(\sqrt[3]{\sqrt{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}}\right)} \cdot \frac{\sqrt{1}}{\sqrt[3]{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}}, 1\right)}}\right)}\]

18. Applied log-prod 13.7

    \[\leadsto \color{blue}{\log \left(\sqrt{e^{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, 1.0614054289999999, -1.45315202700000001\right), 1.42141374100000006\right), -0.284496735999999972\right), 0.25482959199999999\right)}{e^{\left|x\right| \cdot \left|x\right|}}, \frac{-\sqrt{1}}{\left(\sqrt[3]{\sqrt{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}}\right) \cdot \left(\sqrt[3]{\sqrt{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}}\right)} \cdot \frac{\sqrt{1}}{\sqrt[3]{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}}, 1\right)}}\right) + \log \left(\sqrt{e^{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, 1.0614054289999999, -1.45315202700000001\right), 1.42141374100000006\right), -0.284496735999999972\right), 0.25482959199999999\right)}{e^{\left|x\right| \cdot \left|x\right|}}, \frac{-\sqrt{1}}{\left(\sqrt[3]{\sqrt{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}}\right) \cdot \left(\sqrt[3]{\sqrt{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}}\right)} \cdot \frac{\sqrt{1}}{\sqrt[3]{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}}, 1\right)}}\right)}\]

19. Final simplification 13.7

    \[\leadsto \log \left(\sqrt{e^{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, 1.0614054289999999, -1.45315202700000001\right), 1.42141374100000006\right), -0.284496735999999972\right), 0.25482959199999999\right)}{e^{\left|x\right| \cdot \left|x\right|}}, \frac{-\sqrt{1}}{\left(\sqrt[3]{\sqrt{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}}\right) \cdot \left(\sqrt[3]{\sqrt{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}}\right)} \cdot \frac{\sqrt{1}}{\sqrt[3]{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}}, 1\right)}}\right) + \log \left(\sqrt{e^{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, 1.0614054289999999, -1.45315202700000001\right), 1.42141374100000006\right), -0.284496735999999972\right), 0.25482959199999999\right)}{e^{\left|x\right| \cdot \left|x\right|}}, \frac{-\sqrt{1}}{\left(\sqrt[3]{\sqrt{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}}\right) \cdot \left(\sqrt[3]{\sqrt{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}}\right)} \cdot \frac{\sqrt{1}}{\sqrt[3]{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}}, 1\right)}}\right)\]

Reproduce

herbie shell --seed 2020064 +o rules:numerics
(FPCore (x)
  :name "Jmat.Real.erf"
  :precision binary64
  (- 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)))))))