Average Error: 0.2 → 0.2
Time: 25.2s
Precision: 64
\[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right|\]
↓
\[\left|\left(1 \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \mathsf{fma}\left(0.6666666666666666296592325124947819858789, {\left(\left|x\right|\right)}^{3}, \mathsf{fma}\left(0.2000000000000000111022302462515654042363, {\left(\left|x\right|\right)}^{5}, \mathsf{fma}\left(2, \left|x\right|, 0.04761904761904761640423089374962728470564 \cdot {\left(\left|x\right|\right)}^{7}\right)\right)\right)\right|\]
\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right|
↓
\left|\left(1 \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \mathsf{fma}\left(0.6666666666666666296592325124947819858789, {\left(\left|x\right|\right)}^{3}, \mathsf{fma}\left(0.2000000000000000111022302462515654042363, {\left(\left|x\right|\right)}^{5}, \mathsf{fma}\left(2, \left|x\right|, 0.04761904761904761640423089374962728470564 \cdot {\left(\left|x\right|\right)}^{7}\right)\right)\right)\right|
double f(double x) {
        double r109117 = 1.0;
        double r109118 = atan2(1.0, 0.0);
        double r109119 = sqrt(r109118);
        double r109120 = r109117 / r109119;
        double r109121 = 2.0;
        double r109122 = x;
        double r109123 = fabs(r109122);
        double r109124 = r109121 * r109123;
        double r109125 = 3.0;
        double r109126 = r109121 / r109125;
        double r109127 = r109123 * r109123;
        double r109128 = r109127 * r109123;
        double r109129 = r109126 * r109128;
        double r109130 = r109124 + r109129;
        double r109131 = 5.0;
        double r109132 = r109117 / r109131;
        double r109133 = r109128 * r109123;
        double r109134 = r109133 * r109123;
        double r109135 = r109132 * r109134;
        double r109136 = r109130 + r109135;
        double r109137 = 21.0;
        double r109138 = r109117 / r109137;
        double r109139 = r109134 * r109123;
        double r109140 = r109139 * r109123;
        double r109141 = r109138 * r109140;
        double r109142 = r109136 + r109141;
        double r109143 = r109120 * r109142;
        double r109144 = fabs(r109143);
        return r109144;
}
↓
double f(double x) {
        double r109145 = 1.0;
        double r109146 = 1.0;
        double r109147 = atan2(1.0, 0.0);
        double r109148 = r109146 / r109147;
        double r109149 = sqrt(r109148);
        double r109150 = r109145 * r109149;
        double r109151 = 0.6666666666666666;
        double r109152 = x;
        double r109153 = fabs(r109152);
        double r109154 = 3.0;
        double r109155 = pow(r109153, r109154);
        double r109156 = 0.2;
        double r109157 = 5.0;
        double r109158 = pow(r109153, r109157);
        double r109159 = 2.0;
        double r109160 = 0.047619047619047616;
        double r109161 = 7.0;
        double r109162 = pow(r109153, r109161);
        double r109163 = r109160 * r109162;
        double r109164 = fma(r109159, r109153, r109163);
        double r109165 = fma(r109156, r109158, r109164);
        double r109166 = fma(r109151, r109155, r109165);
        double r109167 = r109150 * r109166;
        double r109168 = fabs(r109167);
        return r109168;
}

Error

Bits error versus x

Derivation

  1. Initial program 0.2

    \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right|\]

  2. Taylor expanded around 0 0.2

    \[\leadsto \left|\color{blue}{1 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(0.6666666666666666296592325124947819858789 \cdot {\left(\left|x\right|\right)}^{3} + \left(0.2000000000000000111022302462515654042363 \cdot {\left(\left|x\right|\right)}^{5} + \left(2 \cdot \left|x\right| + 0.04761904761904761640423089374962728470564 \cdot {\left(\left|x\right|\right)}^{7}\right)\right)\right)\right)}\right|\]

  3. Simplified0.2

    \[\leadsto \left|\color{blue}{\left(1 \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \mathsf{fma}\left(0.6666666666666666296592325124947819858789, {\left(\left|x\right|\right)}^{3}, \mathsf{fma}\left(0.2000000000000000111022302462515654042363, {\left(\left|x\right|\right)}^{5}, \mathsf{fma}\left(2, \left|x\right|, 0.04761904761904761640423089374962728470564 \cdot {\left(\left|x\right|\right)}^{7}\right)\right)\right)}\right|\]

  4. Final simplification0.2

    \[\leadsto \left|\left(1 \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \mathsf{fma}\left(0.6666666666666666296592325124947819858789, {\left(\left|x\right|\right)}^{3}, \mathsf{fma}\left(0.2000000000000000111022302462515654042363, {\left(\left|x\right|\right)}^{5}, \mathsf{fma}\left(2, \left|x\right|, 0.04761904761904761640423089374962728470564 \cdot {\left(\left|x\right|\right)}^{7}\right)\right)\right)\right|\]

Reproduce

herbie shell --seed 2019325 +o rules:numerics
(FPCore (x)
  :name "Jmat.Real.erfi, branch x less than or equal to 0.5"
  :precision binary64
  (fabs (* (/ 1 (sqrt PI)) (+ (+ (+ (* 2 (fabs x)) (* (/ 2 3) (* (* (fabs x) (fabs x)) (fabs x)))) (* (/ 1 5) (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)))) (* (/ 1 21) (* (* (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)) (fabs x)) (fabs x)))))))