Jmat.Real.erf

Average Error: 13.8 → 13.0

Time: 16.0s

Precision: 64

Math

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

↓

\[\frac{\left(\sqrt[3]{\mathsf{fma}\left(-{\left(\frac{\sqrt{1}}{\sqrt{1 + 0.32759110000000002 \cdot \left|x\right|}} \cdot \frac{\sqrt{1}}{\sqrt{1 + 0.32759110000000002 \cdot \left|x\right|}}\right)}^{3} \cdot \mathsf{log1p}\left(\mathsf{expm1}\left({\left(\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)\right)}^{3}\right)\right), {\left(\frac{1}{e^{{\left(\left|x\right|\right)}^{2}}}\right)}^{3}, {1}^{3}\right)} \cdot \sqrt[3]{\mathsf{fma}\left(-{\left(\frac{\sqrt{1}}{\sqrt{1 + 0.32759110000000002 \cdot \left|x\right|}} \cdot \frac{\sqrt{1}}{\sqrt{1 + 0.32759110000000002 \cdot \left|x\right|}}\right)}^{3} \cdot \mathsf{log1p}\left(\mathsf{expm1}\left({\left(\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)\right)}^{3}\right)\right), {\left(\frac{1}{e^{{\left(\left|x\right|\right)}^{2}}}\right)}^{3}, {1}^{3}\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(-{\left(\frac{\sqrt{1}}{\sqrt{1 + 0.32759110000000002 \cdot \left|x\right|}} \cdot \frac{\sqrt{1}}{\sqrt{1 + 0.32759110000000002 \cdot \left|x\right|}}\right)}^{3} \cdot \mathsf{log1p}\left(\mathsf{expm1}\left({\left(\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)\right)}^{3}\right)\right), {\left(\frac{1}{e^{{\left(\left|x\right|\right)}^{2}}}\right)}^{3}, {1}^{3}\right)}}{\frac{1 \cdot \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) \cdot 1}{\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)}}{e^{{\left(\left|x\right|\right)}^{2}}} \cdot \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) \cdot 1}{\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)}, e^{-{\left(\left|x\right|\right)}^{2}}, 1\right) + 1 \cdot 1}\]

C

double f(double x) {
        double r184171 = 1.0;
        double r184172 = 0.3275911;
        double r184173 = x;
        double r184174 = fabs(r184173);
        double r184175 = r184172 * r184174;
        double r184176 = r184171 + r184175;
        double r184177 = r184171 / r184176;
        double r184178 = 0.254829592;
        double r184179 = -0.284496736;
        double r184180 = 1.421413741;
        double r184181 = -1.453152027;
        double r184182 = 1.061405429;
        double r184183 = r184177 * r184182;
        double r184184 = r184181 + r184183;
        double r184185 = r184177 * r184184;
        double r184186 = r184180 + r184185;
        double r184187 = r184177 * r184186;
        double r184188 = r184179 + r184187;
        double r184189 = r184177 * r184188;
        double r184190 = r184178 + r184189;
        double r184191 = r184177 * r184190;
        double r184192 = r184174 * r184174;
        double r184193 = -r184192;
        double r184194 = exp(r184193);
        double r184195 = r184191 * r184194;
        double r184196 = r184171 - r184195;
        return r184196;
}

↓

double f(double x) {
        double r184197 = 1.0;
        double r184198 = sqrt(r184197);
        double r184199 = 0.3275911;
        double r184200 = x;
        double r184201 = fabs(r184200);
        double r184202 = r184199 * r184201;
        double r184203 = r184197 + r184202;
        double r184204 = sqrt(r184203);
        double r184205 = r184198 / r184204;
        double r184206 = r184205 * r184205;
        double r184207 = 3.0;
        double r184208 = pow(r184206, r184207);
        double r184209 = r184197 / r184203;
        double r184210 = 1.061405429;
        double r184211 = -1.453152027;
        double r184212 = fma(r184209, r184210, r184211);
        double r184213 = 1.421413741;
        double r184214 = fma(r184209, r184212, r184213);
        double r184215 = -0.284496736;
        double r184216 = fma(r184209, r184214, r184215);
        double r184217 = 0.254829592;
        double r184218 = fma(r184209, r184216, r184217);
        double r184219 = pow(r184218, r184207);
        double r184220 = expm1(r184219);
        double r184221 = log1p(r184220);
        double r184222 = r184208 * r184221;
        double r184223 = -r184222;
        double r184224 = 1.0;
        double r184225 = 2.0;
        double r184226 = pow(r184201, r184225);
        double r184227 = exp(r184226);
        double r184228 = r184224 / r184227;
        double r184229 = pow(r184228, r184207);
        double r184230 = pow(r184197, r184207);
        double r184231 = fma(r184223, r184229, r184230);
        double r184232 = cbrt(r184231);
        double r184233 = r184232 * r184232;
        double r184234 = r184233 * r184232;
        double r184235 = r184218 * r184197;
        double r184236 = fma(r184199, r184201, r184197);
        double r184237 = r184235 / r184236;
        double r184238 = r184224 * r184237;
        double r184239 = r184238 / r184227;
        double r184240 = -r184226;
        double r184241 = exp(r184240);
        double r184242 = fma(r184237, r184241, r184197);
        double r184243 = r184239 * r184242;
        double r184244 = r184197 * r184197;
        double r184245 = r184243 + r184244;
        double r184246 = r184234 / r184245;
        return r184246;
}

Error

Bits error versus x

Derivation

1. Initial program 13.8

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

3. Applied add-sqr-sqrt 13.8

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

4. Applied add-sqr-sqrt 13.8

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

5. Applied add-sqr-sqrt 13.8

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

6. Applied times-frac 13.8

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

7. Applied unswap-sqr 13.8

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

9. Applied flip3-- 13.8

   \[\leadsto \color{blue}{\frac{{1}^{3} - {\left(\left(\left(\frac{\sqrt{1}}{\sqrt{1 + 0.32759110000000002 \cdot \left|x\right|}} \cdot \sqrt{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) \cdot \left(\frac{\sqrt{1}}{\sqrt{1 + 0.32759110000000002 \cdot \left|x\right|}} \cdot \sqrt{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|}\right)}^{3}}{1 \cdot 1 + \left(\left(\left(\left(\frac{\sqrt{1}}{\sqrt{1 + 0.32759110000000002 \cdot \left|x\right|}} \cdot \sqrt{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) \cdot \left(\frac{\sqrt{1}}{\sqrt{1 + 0.32759110000000002 \cdot \left|x\right|}} \cdot \sqrt{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|}\right) \cdot \left(\left(\left(\frac{\sqrt{1}}{\sqrt{1 + 0.32759110000000002 \cdot \left|x\right|}} \cdot \sqrt{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) \cdot \left(\frac{\sqrt{1}}{\sqrt{1 + 0.32759110000000002 \cdot \left|x\right|}} \cdot \sqrt{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|}\right) + 1 \cdot \left(\left(\left(\frac{\sqrt{1}}{\sqrt{1 + 0.32759110000000002 \cdot \left|x\right|}} \cdot \sqrt{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) \cdot \left(\frac{\sqrt{1}}{\sqrt{1 + 0.32759110000000002 \cdot \left|x\right|}} \cdot \sqrt{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|}\right)\right)}}\]

10. Simplified 13.7

    \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-{\left(\frac{\sqrt{1}}{\sqrt{1 + 0.32759110000000002 \cdot \left|x\right|}} \cdot \frac{\sqrt{1}}{\sqrt{1 + 0.32759110000000002 \cdot \left|x\right|}}\right)}^{3} \cdot {\left(\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)\right)}^{3}, {\left(\frac{1}{e^{{\left(\left|x\right|\right)}^{2}}}\right)}^{3}, {1}^{3}\right)}}{1 \cdot 1 + \left(\left(\left(\left(\frac{\sqrt{1}}{\sqrt{1 + 0.32759110000000002 \cdot \left|x\right|}} \cdot \sqrt{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) \cdot \left(\frac{\sqrt{1}}{\sqrt{1 + 0.32759110000000002 \cdot \left|x\right|}} \cdot \sqrt{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|}\right) \cdot \left(\left(\left(\frac{\sqrt{1}}{\sqrt{1 + 0.32759110000000002 \cdot \left|x\right|}} \cdot \sqrt{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) \cdot \left(\frac{\sqrt{1}}{\sqrt{1 + 0.32759110000000002 \cdot \left|x\right|}} \cdot \sqrt{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|}\right) + 1 \cdot \left(\left(\left(\frac{\sqrt{1}}{\sqrt{1 + 0.32759110000000002 \cdot \left|x\right|}} \cdot \sqrt{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) \cdot \left(\frac{\sqrt{1}}{\sqrt{1 + 0.32759110000000002 \cdot \left|x\right|}} \cdot \sqrt{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|}\right)\right)}\]

11. Simplified 13.7

    \[\leadsto \frac{\mathsf{fma}\left(-{\left(\frac{\sqrt{1}}{\sqrt{1 + 0.32759110000000002 \cdot \left|x\right|}} \cdot \frac{\sqrt{1}}{\sqrt{1 + 0.32759110000000002 \cdot \left|x\right|}}\right)}^{3} \cdot {\left(\mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, 1.0614054289999999, -1.45315202700000001\right), 1.42141374100000006\right), -0.284496735999999972\right), 0.25482959199999999\right)\right)}^{3}, {\left(\frac{1}{e^{{\left(\left|x\right|\right)}^{2}}}\right)}^{3}, {1}^{3}\right)}{\color{blue}{\frac{1 \cdot \frac{\mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, 1.0614054289999999, -1.45315202700000001\right), 1.42141374100000006\right), -0.284496735999999972\right), 0.25482959199999999\right) \cdot 1}{\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)}}{e^{{\left(\left|x\right|\right)}^{2}}} \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, 1.0614054289999999, -1.45315202700000001\right), 1.42141374100000006\right), -0.284496735999999972\right), 0.25482959199999999\right) \cdot 1}{\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)}, e^{-{\left(\left|x\right|\right)}^{2}}, 1\right) + 1 \cdot 1}}\]
12. Using strategy rm

13. Applied log1p-expm1-u 13.0

    \[\leadsto \frac{\mathsf{fma}\left(-{\left(\frac{\sqrt{1}}{\sqrt{1 + 0.32759110000000002 \cdot \left|x\right|}} \cdot \frac{\sqrt{1}}{\sqrt{1 + 0.32759110000000002 \cdot \left|x\right|}}\right)}^{3} \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left({\left(\mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, 1.0614054289999999, -1.45315202700000001\right), 1.42141374100000006\right), -0.284496735999999972\right), 0.25482959199999999\right)\right)}^{3}\right)\right)}, {\left(\frac{1}{e^{{\left(\left|x\right|\right)}^{2}}}\right)}^{3}, {1}^{3}\right)}{\frac{1 \cdot \frac{\mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, 1.0614054289999999, -1.45315202700000001\right), 1.42141374100000006\right), -0.284496735999999972\right), 0.25482959199999999\right) \cdot 1}{\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)}}{e^{{\left(\left|x\right|\right)}^{2}}} \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, 1.0614054289999999, -1.45315202700000001\right), 1.42141374100000006\right), -0.284496735999999972\right), 0.25482959199999999\right) \cdot 1}{\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)}, e^{-{\left(\left|x\right|\right)}^{2}}, 1\right) + 1 \cdot 1}\]
14. Using strategy rm

15. Applied add-cube-cbrt 13.0

    \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\mathsf{fma}\left(-{\left(\frac{\sqrt{1}}{\sqrt{1 + 0.32759110000000002 \cdot \left|x\right|}} \cdot \frac{\sqrt{1}}{\sqrt{1 + 0.32759110000000002 \cdot \left|x\right|}}\right)}^{3} \cdot \mathsf{log1p}\left(\mathsf{expm1}\left({\left(\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)\right)}^{3}\right)\right), {\left(\frac{1}{e^{{\left(\left|x\right|\right)}^{2}}}\right)}^{3}, {1}^{3}\right)} \cdot \sqrt[3]{\mathsf{fma}\left(-{\left(\frac{\sqrt{1}}{\sqrt{1 + 0.32759110000000002 \cdot \left|x\right|}} \cdot \frac{\sqrt{1}}{\sqrt{1 + 0.32759110000000002 \cdot \left|x\right|}}\right)}^{3} \cdot \mathsf{log1p}\left(\mathsf{expm1}\left({\left(\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)\right)}^{3}\right)\right), {\left(\frac{1}{e^{{\left(\left|x\right|\right)}^{2}}}\right)}^{3}, {1}^{3}\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(-{\left(\frac{\sqrt{1}}{\sqrt{1 + 0.32759110000000002 \cdot \left|x\right|}} \cdot \frac{\sqrt{1}}{\sqrt{1 + 0.32759110000000002 \cdot \left|x\right|}}\right)}^{3} \cdot \mathsf{log1p}\left(\mathsf{expm1}\left({\left(\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)\right)}^{3}\right)\right), {\left(\frac{1}{e^{{\left(\left|x\right|\right)}^{2}}}\right)}^{3}, {1}^{3}\right)}}}{\frac{1 \cdot \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) \cdot 1}{\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)}}{e^{{\left(\left|x\right|\right)}^{2}}} \cdot \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) \cdot 1}{\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)}, e^{-{\left(\left|x\right|\right)}^{2}}, 1\right) + 1 \cdot 1}\]

16. Final simplification 13.0

    \[\leadsto \frac{\left(\sqrt[3]{\mathsf{fma}\left(-{\left(\frac{\sqrt{1}}{\sqrt{1 + 0.32759110000000002 \cdot \left|x\right|}} \cdot \frac{\sqrt{1}}{\sqrt{1 + 0.32759110000000002 \cdot \left|x\right|}}\right)}^{3} \cdot \mathsf{log1p}\left(\mathsf{expm1}\left({\left(\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)\right)}^{3}\right)\right), {\left(\frac{1}{e^{{\left(\left|x\right|\right)}^{2}}}\right)}^{3}, {1}^{3}\right)} \cdot \sqrt[3]{\mathsf{fma}\left(-{\left(\frac{\sqrt{1}}{\sqrt{1 + 0.32759110000000002 \cdot \left|x\right|}} \cdot \frac{\sqrt{1}}{\sqrt{1 + 0.32759110000000002 \cdot \left|x\right|}}\right)}^{3} \cdot \mathsf{log1p}\left(\mathsf{expm1}\left({\left(\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)\right)}^{3}\right)\right), {\left(\frac{1}{e^{{\left(\left|x\right|\right)}^{2}}}\right)}^{3}, {1}^{3}\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(-{\left(\frac{\sqrt{1}}{\sqrt{1 + 0.32759110000000002 \cdot \left|x\right|}} \cdot \frac{\sqrt{1}}{\sqrt{1 + 0.32759110000000002 \cdot \left|x\right|}}\right)}^{3} \cdot \mathsf{log1p}\left(\mathsf{expm1}\left({\left(\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)\right)}^{3}\right)\right), {\left(\frac{1}{e^{{\left(\left|x\right|\right)}^{2}}}\right)}^{3}, {1}^{3}\right)}}{\frac{1 \cdot \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) \cdot 1}{\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)}}{e^{{\left(\left|x\right|\right)}^{2}}} \cdot \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) \cdot 1}{\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)}, e^{-{\left(\left|x\right|\right)}^{2}}, 1\right) + 1 \cdot 1}\]

Reproduce

herbie shell --seed 2020056 +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)))))))