Average Error: 1.5 → 0.5
Time: 28.4s
Precision: 64
\[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right)\]
↓
\[1 \cdot \left(\left(e^{{\left(\left|x\right|\right)}^{2}} \cdot \left(\left(\frac{0.5}{{\left(\left|x\right|\right)}^{3}} + \frac{0.75}{{\left(\left|x\right|\right)}^{5}}\right) + \left(\frac{1.875}{{\left(\left|x\right|\right)}^{7}} + \frac{1}{\left|x\right|}\right)\right)\right) \cdot \sqrt{\frac{1}{\pi}}\right)\]
\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right)
↓
1 \cdot \left(\left(e^{{\left(\left|x\right|\right)}^{2}} \cdot \left(\left(\frac{0.5}{{\left(\left|x\right|\right)}^{3}} + \frac{0.75}{{\left(\left|x\right|\right)}^{5}}\right) + \left(\frac{1.875}{{\left(\left|x\right|\right)}^{7}} + \frac{1}{\left|x\right|}\right)\right)\right) \cdot \sqrt{\frac{1}{\pi}}\right)
double f(double x) {
        double r197105 = 1.0;
        double r197106 = atan2(1.0, 0.0);
        double r197107 = sqrt(r197106);
        double r197108 = r197105 / r197107;
        double r197109 = x;
        double r197110 = fabs(r197109);
        double r197111 = r197110 * r197110;
        double r197112 = exp(r197111);
        double r197113 = r197108 * r197112;
        double r197114 = r197105 / r197110;
        double r197115 = 2.0;
        double r197116 = r197105 / r197115;
        double r197117 = r197114 * r197114;
        double r197118 = r197117 * r197114;
        double r197119 = r197116 * r197118;
        double r197120 = r197114 + r197119;
        double r197121 = 3.0;
        double r197122 = 4.0;
        double r197123 = r197121 / r197122;
        double r197124 = r197118 * r197114;
        double r197125 = r197124 * r197114;
        double r197126 = r197123 * r197125;
        double r197127 = r197120 + r197126;
        double r197128 = 15.0;
        double r197129 = 8.0;
        double r197130 = r197128 / r197129;
        double r197131 = r197125 * r197114;
        double r197132 = r197131 * r197114;
        double r197133 = r197130 * r197132;
        double r197134 = r197127 + r197133;
        double r197135 = r197113 * r197134;
        return r197135;
}
↓
double f(double x) {
        double r197136 = 1.0;
        double r197137 = x;
        double r197138 = fabs(r197137);
        double r197139 = 2.0;
        double r197140 = pow(r197138, r197139);
        double r197141 = exp(r197140);
        double r197142 = 0.5;
        double r197143 = 3.0;
        double r197144 = pow(r197138, r197143);
        double r197145 = r197142 / r197144;
        double r197146 = 0.75;
        double r197147 = 5.0;
        double r197148 = pow(r197138, r197147);
        double r197149 = r197146 / r197148;
        double r197150 = r197145 + r197149;
        double r197151 = 1.875;
        double r197152 = 7.0;
        double r197153 = pow(r197138, r197152);
        double r197154 = r197151 / r197153;
        double r197155 = r197136 / r197138;
        double r197156 = r197154 + r197155;
        double r197157 = r197150 + r197156;
        double r197158 = r197141 * r197157;
        double r197159 = 1.0;
        double r197160 = atan2(1.0, 0.0);
        double r197161 = r197159 / r197160;
        double r197162 = sqrt(r197161);
        double r197163 = r197158 * r197162;
        double r197164 = r197136 * r197163;
        return r197164;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 1.5

    \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right)\]

  2. Simplified1.1

    \[\leadsto \color{blue}{\left(\frac{1}{\sqrt{\pi}} \cdot e^{{\left(\left|x\right|\right)}^{2}}\right) \cdot \left(\left(\frac{1}{\left|x\right|} + \frac{1}{\frac{2}{{\left(\frac{1}{\left|x\right|}\right)}^{3}}}\right) + \frac{1}{\left|x\right|} \cdot \left(\left(\frac{3}{4} \cdot \frac{1}{\left|x\right|}\right) \cdot {\left(\frac{1}{\left|x\right|}\right)}^{3} + \frac{15}{8} \cdot {\left(\frac{1 \cdot 1}{{\left(\left|x\right|\right)}^{2}}\right)}^{3}\right)\right)}\]
  3. Using strategy rm

  4. Applied div-inv1.1

    \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{{\left(\left|x\right|\right)}^{2}}\right) \cdot \left(\left(\frac{1}{\left|x\right|} + \frac{1}{\frac{2}{{\left(\frac{1}{\left|x\right|}\right)}^{3}}}\right) + \frac{1}{\left|x\right|} \cdot \left(\left(\frac{3}{4} \cdot \frac{1}{\left|x\right|}\right) \cdot {\left(\frac{1}{\left|x\right|}\right)}^{3} + \frac{15}{8} \cdot {\color{blue}{\left(\left(1 \cdot 1\right) \cdot \frac{1}{{\left(\left|x\right|\right)}^{2}}\right)}}^{3}\right)\right)\]

  5. Applied unpow-prod-down1.1

    \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{{\left(\left|x\right|\right)}^{2}}\right) \cdot \left(\left(\frac{1}{\left|x\right|} + \frac{1}{\frac{2}{{\left(\frac{1}{\left|x\right|}\right)}^{3}}}\right) + \frac{1}{\left|x\right|} \cdot \left(\left(\frac{3}{4} \cdot \frac{1}{\left|x\right|}\right) \cdot {\left(\frac{1}{\left|x\right|}\right)}^{3} + \frac{15}{8} \cdot \color{blue}{\left({\left(1 \cdot 1\right)}^{3} \cdot {\left(\frac{1}{{\left(\left|x\right|\right)}^{2}}\right)}^{3}\right)}\right)\right)\]

  6. Simplified1.1

    \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{{\left(\left|x\right|\right)}^{2}}\right) \cdot \left(\left(\frac{1}{\left|x\right|} + \frac{1}{\frac{2}{{\left(\frac{1}{\left|x\right|}\right)}^{3}}}\right) + \frac{1}{\left|x\right|} \cdot \left(\left(\frac{3}{4} \cdot \frac{1}{\left|x\right|}\right) \cdot {\left(\frac{1}{\left|x\right|}\right)}^{3} + \frac{15}{8} \cdot \left(\color{blue}{{1}^{6}} \cdot {\left(\frac{1}{{\left(\left|x\right|\right)}^{2}}\right)}^{3}\right)\right)\right)\]

  7. Simplified0.7

    \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{{\left(\left|x\right|\right)}^{2}}\right) \cdot \left(\left(\frac{1}{\left|x\right|} + \frac{1}{\frac{2}{{\left(\frac{1}{\left|x\right|}\right)}^{3}}}\right) + \frac{1}{\left|x\right|} \cdot \left(\left(\frac{3}{4} \cdot \frac{1}{\left|x\right|}\right) \cdot {\left(\frac{1}{\left|x\right|}\right)}^{3} + \frac{15}{8} \cdot \left({1}^{6} \cdot \color{blue}{\frac{1}{{\left(\left|x\right|\right)}^{6}}}\right)\right)\right)\]

  8. Taylor expanded around 0 0.6

    \[\leadsto \color{blue}{1 \cdot \left(\left(e^{{\left(\left|x\right|\right)}^{2}} \cdot \left(0.5 \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}} + \left(0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}} + \left(1 \cdot \frac{1}{\left|x\right|} + 1.875 \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}}\right)\right)\right)\right) \cdot \sqrt{\frac{1}{\pi}}\right)}\]
  9. Using strategy rm

  10. Applied *-un-lft-identity0.6

    \[\leadsto 1 \cdot \left(\left(\color{blue}{\left(1 \cdot e^{{\left(\left|x\right|\right)}^{2}}\right)} \cdot \left(0.5 \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}} + \left(0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}} + \left(1 \cdot \frac{1}{\left|x\right|} + 1.875 \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}}\right)\right)\right)\right) \cdot \sqrt{\frac{1}{\pi}}\right)\]

  11. Applied associate-*l*0.6

    \[\leadsto 1 \cdot \left(\color{blue}{\left(1 \cdot \left(e^{{\left(\left|x\right|\right)}^{2}} \cdot \left(0.5 \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}} + \left(0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}} + \left(1 \cdot \frac{1}{\left|x\right|} + 1.875 \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}}\right)\right)\right)\right)\right)} \cdot \sqrt{\frac{1}{\pi}}\right)\]

  12. Simplified0.5

    \[\leadsto 1 \cdot \left(\left(1 \cdot \color{blue}{\left(e^{{\left(\left|x\right|\right)}^{2}} \cdot \left(\left(\frac{0.5}{{\left(\left|x\right|\right)}^{3}} + \frac{0.75}{{\left(\left|x\right|\right)}^{5}}\right) + \left(\frac{1.875}{{\left(\left|x\right|\right)}^{7}} + \frac{1}{\left|x\right|}\right)\right)\right)}\right) \cdot \sqrt{\frac{1}{\pi}}\right)\]

  13. Final simplification0.5

    \[\leadsto 1 \cdot \left(\left(e^{{\left(\left|x\right|\right)}^{2}} \cdot \left(\left(\frac{0.5}{{\left(\left|x\right|\right)}^{3}} + \frac{0.75}{{\left(\left|x\right|\right)}^{5}}\right) + \left(\frac{1.875}{{\left(\left|x\right|\right)}^{7}} + \frac{1}{\left|x\right|}\right)\right)\right) \cdot \sqrt{\frac{1}{\pi}}\right)\]

Reproduce

herbie shell --seed 2020043 
(FPCore (x)
  :name "Jmat.Real.erfi, branch x greater than or equal to 5"
  :precision binary64
  (* (* (/ 1 (sqrt PI)) (exp (* (fabs x) (fabs x)))) (+ (+ (+ (/ 1 (fabs x)) (* (/ 1 2) (* (* (/ 1 (fabs x)) (/ 1 (fabs x))) (/ 1 (fabs x))))) (* (/ 3 4) (* (* (* (* (/ 1 (fabs x)) (/ 1 (fabs x))) (/ 1 (fabs x))) (/ 1 (fabs x))) (/ 1 (fabs x))))) (* (/ 15 8) (* (* (* (* (* (* (/ 1 (fabs x)) (/ 1 (fabs x))) (/ 1 (fabs x))) (/ 1 (fabs x))) (/ 1 (fabs x))) (/ 1 (fabs x))) (/ 1 (fabs x)))))))