Jmat.Real.erf

Average Error: 13.6 → 12.9

Time: 34.1s

Precision: 64

Math

\[1 - \left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(0.2548295919999999936678136691625695675611 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-0.2844967359999999723108032867457950487733 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(1.421413741000000063863240029604639858007 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-1.453152027000000012790792425221297889948 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot 1.061405428999999900341322245367337018251\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]

↓

\[\frac{{1}^{3} - \sqrt{{\left(\frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)} \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(1.061405428999999900341322245367337018251, \frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}, -1.453152027000000012790792425221297889948\right), \frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}, 1.421413741000000063863240029604639858007\right), \frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}, -0.2844967359999999723108032867457950487733\right), \frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}, 0.2548295919999999936678136691625695675611\right)}{e^{{\left(\left|x\right|\right)}^{2}}}\right)}^{3}} \cdot \sqrt{{\left(\frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)} \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(1.061405428999999900341322245367337018251, \frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}, -1.453152027000000012790792425221297889948\right), \frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}, 1.421413741000000063863240029604639858007\right), \frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}, -0.2844967359999999723108032867457950487733\right), \frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}, 0.2548295919999999936678136691625695675611\right)}{e^{{\left(\left|x\right|\right)}^{2}}}\right)}^{3}}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(1.061405428999999900341322245367337018251, \frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}, -1.453152027000000012790792425221297889948\right), \frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}, 1.421413741000000063863240029604639858007\right), \frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}, -0.2844967359999999723108032867457950487733\right), \frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}, 0.2548295919999999936678136691625695675611\right), \frac{\frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}}{e^{{\left(\left|x\right|\right)}^{2}}}, 1\right), \frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)} \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(1.061405428999999900341322245367337018251, \frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}, -1.453152027000000012790792425221297889948\right), \frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}, 1.421413741000000063863240029604639858007\right), \frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}, -0.2844967359999999723108032867457950487733\right), \frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}, 0.2548295919999999936678136691625695675611\right)}{e^{{\left(\left|x\right|\right)}^{2}}}, 1 \cdot 1\right)}\]

C

double f(double x) {
        double r154897 = 1.0;
        double r154898 = 0.3275911;
        double r154899 = x;
        double r154900 = fabs(r154899);
        double r154901 = r154898 * r154900;
        double r154902 = r154897 + r154901;
        double r154903 = r154897 / r154902;
        double r154904 = 0.254829592;
        double r154905 = -0.284496736;
        double r154906 = 1.421413741;
        double r154907 = -1.453152027;
        double r154908 = 1.061405429;
        double r154909 = r154903 * r154908;
        double r154910 = r154907 + r154909;
        double r154911 = r154903 * r154910;
        double r154912 = r154906 + r154911;
        double r154913 = r154903 * r154912;
        double r154914 = r154905 + r154913;
        double r154915 = r154903 * r154914;
        double r154916 = r154904 + r154915;
        double r154917 = r154903 * r154916;
        double r154918 = r154900 * r154900;
        double r154919 = -r154918;
        double r154920 = exp(r154919);
        double r154921 = r154917 * r154920;
        double r154922 = r154897 - r154921;
        return r154922;
}

↓

double f(double x) {
        double r154923 = 1.0;
        double r154924 = 3.0;
        double r154925 = pow(r154923, r154924);
        double r154926 = 0.3275911;
        double r154927 = x;
        double r154928 = fabs(r154927);
        double r154929 = fma(r154926, r154928, r154923);
        double r154930 = r154923 / r154929;
        double r154931 = 1.061405429;
        double r154932 = -1.453152027;
        double r154933 = fma(r154931, r154930, r154932);
        double r154934 = 1.421413741;
        double r154935 = fma(r154933, r154930, r154934);
        double r154936 = -0.284496736;
        double r154937 = fma(r154935, r154930, r154936);
        double r154938 = 0.254829592;
        double r154939 = fma(r154937, r154930, r154938);
        double r154940 = 2.0;
        double r154941 = pow(r154928, r154940);
        double r154942 = exp(r154941);
        double r154943 = r154939 / r154942;
        double r154944 = r154930 * r154943;
        double r154945 = pow(r154944, r154924);
        double r154946 = sqrt(r154945);
        double r154947 = r154946 * r154946;
        double r154948 = r154925 - r154947;
        double r154949 = r154930 / r154942;
        double r154950 = fma(r154939, r154949, r154923);
        double r154951 = r154923 * r154923;
        double r154952 = fma(r154950, r154944, r154951);
        double r154953 = r154948 / r154952;
        return r154953;
}

Error

Bits error versus x

Derivation

1. Initial program 13.6

   \[1 - \left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(0.2548295919999999936678136691625695675611 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-0.2844967359999999723108032867457950487733 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(1.421413741000000063863240029604639858007 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-1.453152027000000012790792425221297889948 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot 1.061405428999999900341322245367337018251\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
2. Using strategy rm

3. Applied add-sqr-sqrt 13.6

   \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(0.2548295919999999936678136691625695675611 + \frac{1}{\color{blue}{\sqrt{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \sqrt{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}}} \cdot \left(-0.2844967359999999723108032867457950487733 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(1.421413741000000063863240029604639858007 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-1.453152027000000012790792425221297889948 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot 1.061405428999999900341322245367337018251\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]

4. Applied add-sqr-sqrt 13.6

   \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(0.2548295919999999936678136691625695675611 + \frac{\color{blue}{\sqrt{1} \cdot \sqrt{1}}}{\sqrt{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \sqrt{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}} \cdot \left(-0.2844967359999999723108032867457950487733 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(1.421413741000000063863240029604639858007 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-1.453152027000000012790792425221297889948 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot 1.061405428999999900341322245367337018251\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]

5. Applied times-frac 13.6

   \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(0.2548295919999999936678136691625695675611 + \color{blue}{\left(\frac{\sqrt{1}}{\sqrt{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}} \cdot \frac{\sqrt{1}}{\sqrt{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}}\right)} \cdot \left(-0.2844967359999999723108032867457950487733 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(1.421413741000000063863240029604639858007 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-1.453152027000000012790792425221297889948 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot 1.061405428999999900341322245367337018251\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]

6. Applied associate-*l* 13.6

   \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(0.2548295919999999936678136691625695675611 + \color{blue}{\frac{\sqrt{1}}{\sqrt{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}} \cdot \left(\frac{\sqrt{1}}{\sqrt{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}} \cdot \left(-0.2844967359999999723108032867457950487733 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(1.421413741000000063863240029604639858007 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-1.453152027000000012790792425221297889948 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot 1.061405428999999900341322245367337018251\right)\right)\right)\right)}\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]

7. Simplified 13.6

   \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(0.2548295919999999936678136691625695675611 + \frac{\sqrt{1}}{\sqrt{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}} \cdot \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(1.061405428999999900341322245367337018251, \frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}, -1.453152027000000012790792425221297889948\right), \frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}, 1.421413741000000063863240029604639858007\right), \frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}, -0.2844967359999999723108032867457950487733\right) \cdot \sqrt{1}}{\sqrt{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}}}\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
8. Using strategy rm

9. Applied flip3-- 13.6

   \[\leadsto \color{blue}{\frac{{1}^{3} - {\left(\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(0.2548295919999999936678136691625695675611 + \frac{\sqrt{1}}{\sqrt{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}} \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(1.061405428999999900341322245367337018251, \frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}, -1.453152027000000012790792425221297889948\right), \frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}, 1.421413741000000063863240029604639858007\right), \frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}, -0.2844967359999999723108032867457950487733\right) \cdot \sqrt{1}}{\sqrt{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}}\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\right)}^{3}}{1 \cdot 1 + \left(\left(\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(0.2548295919999999936678136691625695675611 + \frac{\sqrt{1}}{\sqrt{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}} \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(1.061405428999999900341322245367337018251, \frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}, -1.453152027000000012790792425221297889948\right), \frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}, 1.421413741000000063863240029604639858007\right), \frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}, -0.2844967359999999723108032867457950487733\right) \cdot \sqrt{1}}{\sqrt{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}}\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(0.2548295919999999936678136691625695675611 + \frac{\sqrt{1}}{\sqrt{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}} \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(1.061405428999999900341322245367337018251, \frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}, -1.453152027000000012790792425221297889948\right), \frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}, 1.421413741000000063863240029604639858007\right), \frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}, -0.2844967359999999723108032867457950487733\right) \cdot \sqrt{1}}{\sqrt{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}}\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\right) + 1 \cdot \left(\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(0.2548295919999999936678136691625695675611 + \frac{\sqrt{1}}{\sqrt{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}} \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(1.061405428999999900341322245367337018251, \frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}, -1.453152027000000012790792425221297889948\right), \frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}, 1.421413741000000063863240029604639858007\right), \frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}, -0.2844967359999999723108032867457950487733\right) \cdot \sqrt{1}}{\sqrt{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}}\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\right)\right)}}\]

10. Simplified 13.6

    \[\leadsto \frac{\color{blue}{{1}^{3} - {\left(\frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)} \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(1.061405428999999900341322245367337018251, \frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}, -1.453152027000000012790792425221297889948\right), \frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}, 1.421413741000000063863240029604639858007\right), \frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}, -0.2844967359999999723108032867457950487733\right), \frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}, 0.2548295919999999936678136691625695675611\right)}{e^{{\left(\left|x\right|\right)}^{2}}}\right)}^{3}}}{1 \cdot 1 + \left(\left(\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(0.2548295919999999936678136691625695675611 + \frac{\sqrt{1}}{\sqrt{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}} \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(1.061405428999999900341322245367337018251, \frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}, -1.453152027000000012790792425221297889948\right), \frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}, 1.421413741000000063863240029604639858007\right), \frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}, -0.2844967359999999723108032867457950487733\right) \cdot \sqrt{1}}{\sqrt{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}}\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(0.2548295919999999936678136691625695675611 + \frac{\sqrt{1}}{\sqrt{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}} \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(1.061405428999999900341322245367337018251, \frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}, -1.453152027000000012790792425221297889948\right), \frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}, 1.421413741000000063863240029604639858007\right), \frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}, -0.2844967359999999723108032867457950487733\right) \cdot \sqrt{1}}{\sqrt{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}}\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\right) + 1 \cdot \left(\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(0.2548295919999999936678136691625695675611 + \frac{\sqrt{1}}{\sqrt{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}} \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(1.061405428999999900341322245367337018251, \frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}, -1.453152027000000012790792425221297889948\right), \frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}, 1.421413741000000063863240029604639858007\right), \frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}, -0.2844967359999999723108032867457950487733\right) \cdot \sqrt{1}}{\sqrt{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}}\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\right)\right)}\]

11. Simplified 13.6

    \[\leadsto \frac{{1}^{3} - {\left(\frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)} \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(1.061405428999999900341322245367337018251, \frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}, -1.453152027000000012790792425221297889948\right), \frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}, 1.421413741000000063863240029604639858007\right), \frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}, -0.2844967359999999723108032867457950487733\right), \frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}, 0.2548295919999999936678136691625695675611\right)}{e^{{\left(\left|x\right|\right)}^{2}}}\right)}^{3}}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(1.061405428999999900341322245367337018251, \frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}, -1.453152027000000012790792425221297889948\right), \frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}, 1.421413741000000063863240029604639858007\right), \frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}, -0.2844967359999999723108032867457950487733\right), \frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}, 0.2548295919999999936678136691625695675611\right), \frac{\frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}}{e^{{\left(\left|x\right|\right)}^{2}}}, 1\right), \frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)} \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(1.061405428999999900341322245367337018251, \frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}, -1.453152027000000012790792425221297889948\right), \frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}, 1.421413741000000063863240029604639858007\right), \frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}, -0.2844967359999999723108032867457950487733\right), \frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}, 0.2548295919999999936678136691625695675611\right)}{e^{{\left(\left|x\right|\right)}^{2}}}, 1 \cdot 1\right)}}\]
12. Using strategy rm

13. Applied add-sqr-sqrt 12.9

    \[\leadsto \frac{{1}^{3} - \color{blue}{\sqrt{{\left(\frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)} \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(1.061405428999999900341322245367337018251, \frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}, -1.453152027000000012790792425221297889948\right), \frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}, 1.421413741000000063863240029604639858007\right), \frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}, -0.2844967359999999723108032867457950487733\right), \frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}, 0.2548295919999999936678136691625695675611\right)}{e^{{\left(\left|x\right|\right)}^{2}}}\right)}^{3}} \cdot \sqrt{{\left(\frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)} \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(1.061405428999999900341322245367337018251, \frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}, -1.453152027000000012790792425221297889948\right), \frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}, 1.421413741000000063863240029604639858007\right), \frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}, -0.2844967359999999723108032867457950487733\right), \frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}, 0.2548295919999999936678136691625695675611\right)}{e^{{\left(\left|x\right|\right)}^{2}}}\right)}^{3}}}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(1.061405428999999900341322245367337018251, \frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}, -1.453152027000000012790792425221297889948\right), \frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}, 1.421413741000000063863240029604639858007\right), \frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}, -0.2844967359999999723108032867457950487733\right), \frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}, 0.2548295919999999936678136691625695675611\right), \frac{\frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}}{e^{{\left(\left|x\right|\right)}^{2}}}, 1\right), \frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)} \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(1.061405428999999900341322245367337018251, \frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}, -1.453152027000000012790792425221297889948\right), \frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}, 1.421413741000000063863240029604639858007\right), \frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}, -0.2844967359999999723108032867457950487733\right), \frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}, 0.2548295919999999936678136691625695675611\right)}{e^{{\left(\left|x\right|\right)}^{2}}}, 1 \cdot 1\right)}\]

14. Final simplification 12.9

    \[\leadsto \frac{{1}^{3} - \sqrt{{\left(\frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)} \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(1.061405428999999900341322245367337018251, \frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}, -1.453152027000000012790792425221297889948\right), \frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}, 1.421413741000000063863240029604639858007\right), \frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}, -0.2844967359999999723108032867457950487733\right), \frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}, 0.2548295919999999936678136691625695675611\right)}{e^{{\left(\left|x\right|\right)}^{2}}}\right)}^{3}} \cdot \sqrt{{\left(\frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)} \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(1.061405428999999900341322245367337018251, \frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}, -1.453152027000000012790792425221297889948\right), \frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}, 1.421413741000000063863240029604639858007\right), \frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}, -0.2844967359999999723108032867457950487733\right), \frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}, 0.2548295919999999936678136691625695675611\right)}{e^{{\left(\left|x\right|\right)}^{2}}}\right)}^{3}}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(1.061405428999999900341322245367337018251, \frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}, -1.453152027000000012790792425221297889948\right), \frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}, 1.421413741000000063863240029604639858007\right), \frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}, -0.2844967359999999723108032867457950487733\right), \frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}, 0.2548295919999999936678136691625695675611\right), \frac{\frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}}{e^{{\left(\left|x\right|\right)}^{2}}}, 1\right), \frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)} \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(1.061405428999999900341322245367337018251, \frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}, -1.453152027000000012790792425221297889948\right), \frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}, 1.421413741000000063863240029604639858007\right), \frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}, -0.2844967359999999723108032867457950487733\right), \frac{1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}, 0.2548295919999999936678136691625695675611\right)}{e^{{\left(\left|x\right|\right)}^{2}}}, 1 \cdot 1\right)}\]

Reproduce

herbie shell --seed 2019325 +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)))))))