Jmat.Real.erf

Average Error: 13.8 → 13.8

Time: 26.6s

Precision: 64

Math

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

↓

\[{\left(e^{\sqrt[3]{\log \left(1 - \left(\frac{1}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)} \cdot \left(\left(\frac{1.0614054289999999}{{\left(\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)\right)}^{4}} + \left(\left(\frac{1.42141374100000006}{{\left(\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)\right)}^{2}} + 0.25482959199999999\right) - \frac{1.45315202700000001}{{\left(\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)\right)}^{3}}\right)\right) - \frac{0.284496735999999972}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}\right)\right) \cdot e^{-{\left(\left|x\right|\right)}^{2}}\right)} \cdot \sqrt[3]{\log \left(1 - \left(\frac{1}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)} \cdot \left(\left(\frac{1.0614054289999999}{{\left(\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)\right)}^{4}} + \left(\left(\frac{1.42141374100000006}{{\left(\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)\right)}^{2}} + 0.25482959199999999\right) - \frac{1.45315202700000001}{{\left(\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)\right)}^{3}}\right)\right) - \frac{0.284496735999999972}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}\right)\right) \cdot e^{-{\left(\left|x\right|\right)}^{2}}\right)}}\right)}^{\left(\sqrt[3]{\log \left(1 - \left(\frac{1}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)} \cdot \left(\left(\frac{1.0614054289999999}{{\left(\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)\right)}^{4}} + \left(\left(\frac{1.42141374100000006}{{\left(\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)\right)}^{2}} + 0.25482959199999999\right) - \frac{1.45315202700000001}{{\left(\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)\right)}^{3}}\right)\right) - \frac{0.284496735999999972}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}\right)\right) \cdot e^{-{\left(\left|x\right|\right)}^{2}}\right)}\right)}\]

C

double f(double x) {
        double r200194 = 1.0;
        double r200195 = 0.3275911;
        double r200196 = x;
        double r200197 = fabs(r200196);
        double r200198 = r200195 * r200197;
        double r200199 = r200194 + r200198;
        double r200200 = r200194 / r200199;
        double r200201 = 0.254829592;
        double r200202 = -0.284496736;
        double r200203 = 1.421413741;
        double r200204 = -1.453152027;
        double r200205 = 1.061405429;
        double r200206 = r200200 * r200205;
        double r200207 = r200204 + r200206;
        double r200208 = r200200 * r200207;
        double r200209 = r200203 + r200208;
        double r200210 = r200200 * r200209;
        double r200211 = r200202 + r200210;
        double r200212 = r200200 * r200211;
        double r200213 = r200201 + r200212;
        double r200214 = r200200 * r200213;
        double r200215 = r200197 * r200197;
        double r200216 = -r200215;
        double r200217 = exp(r200216);
        double r200218 = r200214 * r200217;
        double r200219 = r200194 - r200218;
        return r200219;
}

↓

double f(double x) {
        double r200220 = 1.0;
        double r200221 = x;
        double r200222 = fabs(r200221);
        double r200223 = 0.3275911;
        double r200224 = fma(r200222, r200223, r200220);
        double r200225 = r200220 / r200224;
        double r200226 = 1.061405429;
        double r200227 = fma(r200223, r200222, r200220);
        double r200228 = 4.0;
        double r200229 = pow(r200227, r200228);
        double r200230 = r200226 / r200229;
        double r200231 = 1.421413741;
        double r200232 = 2.0;
        double r200233 = pow(r200224, r200232);
        double r200234 = r200231 / r200233;
        double r200235 = 0.254829592;
        double r200236 = r200234 + r200235;
        double r200237 = 1.453152027;
        double r200238 = 3.0;
        double r200239 = pow(r200227, r200238);
        double r200240 = r200237 / r200239;
        double r200241 = r200236 - r200240;
        double r200242 = r200230 + r200241;
        double r200243 = 0.284496736;
        double r200244 = r200243 / r200224;
        double r200245 = r200242 - r200244;
        double r200246 = r200225 * r200245;
        double r200247 = pow(r200222, r200232);
        double r200248 = -r200247;
        double r200249 = exp(r200248);
        double r200250 = r200246 * r200249;
        double r200251 = r200220 - r200250;
        double r200252 = log(r200251);
        double r200253 = cbrt(r200252);
        double r200254 = r200253 * r200253;
        double r200255 = exp(r200254);
        double r200256 = pow(r200255, r200253);
        return r200256;
}

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

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

4. Applied associate-/r* 13.8

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

5. Simplified 13.8

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

6. Taylor expanded around 0 14.6

   \[\leadsto 1 - \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \color{blue}{\left(\left(1.0614054289999999 \cdot \frac{1}{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{4}} + \left(1.42141374100000006 \cdot \frac{1}{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{2}} + 0.25482959199999999\right)\right) - \left(1.45315202700000001 \cdot \frac{1}{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{3}} + 0.284496735999999972 \cdot \frac{1}{0.32759110000000002 \cdot \left|x\right| + 1}\right)\right)}\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]

7. Simplified 13.8

   \[\leadsto 1 - \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \color{blue}{\left(\left(\frac{1.0614054289999999}{{\left(\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)\right)}^{4}} + \left(\left(\frac{1.42141374100000006}{{\left(\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)\right)}^{2}} + 0.25482959199999999\right) - \frac{1.45315202700000001}{{\left(\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)\right)}^{3}}\right)\right) - \frac{0.284496735999999972}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}\right)}\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
8. Using strategy rm

9. Applied add-exp-log 13.8

   \[\leadsto \color{blue}{e^{\log \left(1 - \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(\left(\frac{1.0614054289999999}{{\left(\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)\right)}^{4}} + \left(\left(\frac{1.42141374100000006}{{\left(\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)\right)}^{2}} + 0.25482959199999999\right) - \frac{1.45315202700000001}{{\left(\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)\right)}^{3}}\right)\right) - \frac{0.284496735999999972}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\right)}}\]

10. Simplified 13.8

    \[\leadsto e^{\color{blue}{\log \left(1 - \left(\frac{1}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)} \cdot \left(\left(\frac{1.0614054289999999}{{\left(\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)\right)}^{4}} + \left(\left(\frac{1.42141374100000006}{{\left(\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)\right)}^{2}} + 0.25482959199999999\right) - \frac{1.45315202700000001}{{\left(\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)\right)}^{3}}\right)\right) - \frac{0.284496735999999972}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}\right)\right) \cdot e^{-{\left(\left|x\right|\right)}^{2}}\right)}}\]
11. Using strategy rm

12. Applied add-cube-cbrt 13.8

    \[\leadsto e^{\color{blue}{\left(\sqrt[3]{\log \left(1 - \left(\frac{1}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)} \cdot \left(\left(\frac{1.0614054289999999}{{\left(\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)\right)}^{4}} + \left(\left(\frac{1.42141374100000006}{{\left(\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)\right)}^{2}} + 0.25482959199999999\right) - \frac{1.45315202700000001}{{\left(\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)\right)}^{3}}\right)\right) - \frac{0.284496735999999972}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}\right)\right) \cdot e^{-{\left(\left|x\right|\right)}^{2}}\right)} \cdot \sqrt[3]{\log \left(1 - \left(\frac{1}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)} \cdot \left(\left(\frac{1.0614054289999999}{{\left(\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)\right)}^{4}} + \left(\left(\frac{1.42141374100000006}{{\left(\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)\right)}^{2}} + 0.25482959199999999\right) - \frac{1.45315202700000001}{{\left(\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)\right)}^{3}}\right)\right) - \frac{0.284496735999999972}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}\right)\right) \cdot e^{-{\left(\left|x\right|\right)}^{2}}\right)}\right) \cdot \sqrt[3]{\log \left(1 - \left(\frac{1}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)} \cdot \left(\left(\frac{1.0614054289999999}{{\left(\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)\right)}^{4}} + \left(\left(\frac{1.42141374100000006}{{\left(\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)\right)}^{2}} + 0.25482959199999999\right) - \frac{1.45315202700000001}{{\left(\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)\right)}^{3}}\right)\right) - \frac{0.284496735999999972}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}\right)\right) \cdot e^{-{\left(\left|x\right|\right)}^{2}}\right)}}}\]

13. Applied exp-prod 13.8

    \[\leadsto \color{blue}{{\left(e^{\sqrt[3]{\log \left(1 - \left(\frac{1}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)} \cdot \left(\left(\frac{1.0614054289999999}{{\left(\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)\right)}^{4}} + \left(\left(\frac{1.42141374100000006}{{\left(\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)\right)}^{2}} + 0.25482959199999999\right) - \frac{1.45315202700000001}{{\left(\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)\right)}^{3}}\right)\right) - \frac{0.284496735999999972}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}\right)\right) \cdot e^{-{\left(\left|x\right|\right)}^{2}}\right)} \cdot \sqrt[3]{\log \left(1 - \left(\frac{1}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)} \cdot \left(\left(\frac{1.0614054289999999}{{\left(\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)\right)}^{4}} + \left(\left(\frac{1.42141374100000006}{{\left(\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)\right)}^{2}} + 0.25482959199999999\right) - \frac{1.45315202700000001}{{\left(\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)\right)}^{3}}\right)\right) - \frac{0.284496735999999972}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}\right)\right) \cdot e^{-{\left(\left|x\right|\right)}^{2}}\right)}}\right)}^{\left(\sqrt[3]{\log \left(1 - \left(\frac{1}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)} \cdot \left(\left(\frac{1.0614054289999999}{{\left(\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)\right)}^{4}} + \left(\left(\frac{1.42141374100000006}{{\left(\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)\right)}^{2}} + 0.25482959199999999\right) - \frac{1.45315202700000001}{{\left(\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)\right)}^{3}}\right)\right) - \frac{0.284496735999999972}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}\right)\right) \cdot e^{-{\left(\left|x\right|\right)}^{2}}\right)}\right)}}\]

14. Final simplification 13.8

    \[\leadsto {\left(e^{\sqrt[3]{\log \left(1 - \left(\frac{1}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)} \cdot \left(\left(\frac{1.0614054289999999}{{\left(\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)\right)}^{4}} + \left(\left(\frac{1.42141374100000006}{{\left(\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)\right)}^{2}} + 0.25482959199999999\right) - \frac{1.45315202700000001}{{\left(\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)\right)}^{3}}\right)\right) - \frac{0.284496735999999972}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}\right)\right) \cdot e^{-{\left(\left|x\right|\right)}^{2}}\right)} \cdot \sqrt[3]{\log \left(1 - \left(\frac{1}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)} \cdot \left(\left(\frac{1.0614054289999999}{{\left(\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)\right)}^{4}} + \left(\left(\frac{1.42141374100000006}{{\left(\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)\right)}^{2}} + 0.25482959199999999\right) - \frac{1.45315202700000001}{{\left(\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)\right)}^{3}}\right)\right) - \frac{0.284496735999999972}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}\right)\right) \cdot e^{-{\left(\left|x\right|\right)}^{2}}\right)}}\right)}^{\left(\sqrt[3]{\log \left(1 - \left(\frac{1}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)} \cdot \left(\left(\frac{1.0614054289999999}{{\left(\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)\right)}^{4}} + \left(\left(\frac{1.42141374100000006}{{\left(\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)\right)}^{2}} + 0.25482959199999999\right) - \frac{1.45315202700000001}{{\left(\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)\right)}^{3}}\right)\right) - \frac{0.284496735999999972}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}\right)\right) \cdot e^{-{\left(\left|x\right|\right)}^{2}}\right)}\right)}\]

Reproduce

herbie shell --seed 2019198 +o rules:numerics
(FPCore (x)
  :name "Jmat.Real.erf"
  (- 1.0 (* (* (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x)))) (+ 0.254829592 (* (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x)))) (+ -0.284496736 (* (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x)))) (+ 1.421413741 (* (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x)))) (+ -1.453152027 (* (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x)))) 1.061405429))))))))) (exp (- (* (fabs x) (fabs x)))))))