Average Error: 0.2 → 0.2
Time: 22.1s
Precision: 64
\[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right|\]
↓
\[\left|\mathsf{fma}\left({\left(\left|x\right|\right)}^{5}, 0.2000000000000000111022302462515654042363, \mathsf{fma}\left({\left(\left|x\right|\right)}^{7}, 0.04761904761904761640423089374962728470564, \mathsf{fma}\left({\left(\left|x\right|\right)}^{3}, 0.6666666666666666296592325124947819858789, 2 \cdot \left|x\right|\right)\right)\right) \cdot \left(1 \cdot \sqrt{\frac{1}{\pi}}\right)\right|\]
\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|\mathsf{fma}\left({\left(\left|x\right|\right)}^{5}, 0.2000000000000000111022302462515654042363, \mathsf{fma}\left({\left(\left|x\right|\right)}^{7}, 0.04761904761904761640423089374962728470564, \mathsf{fma}\left({\left(\left|x\right|\right)}^{3}, 0.6666666666666666296592325124947819858789, 2 \cdot \left|x\right|\right)\right)\right) \cdot \left(1 \cdot \sqrt{\frac{1}{\pi}}\right)\right|
double f(double x) {
        double r164063 = 1.0;
        double r164064 = atan2(1.0, 0.0);
        double r164065 = sqrt(r164064);
        double r164066 = r164063 / r164065;
        double r164067 = 2.0;
        double r164068 = x;
        double r164069 = fabs(r164068);
        double r164070 = r164067 * r164069;
        double r164071 = 3.0;
        double r164072 = r164067 / r164071;
        double r164073 = r164069 * r164069;
        double r164074 = r164073 * r164069;
        double r164075 = r164072 * r164074;
        double r164076 = r164070 + r164075;
        double r164077 = 5.0;
        double r164078 = r164063 / r164077;
        double r164079 = r164074 * r164069;
        double r164080 = r164079 * r164069;
        double r164081 = r164078 * r164080;
        double r164082 = r164076 + r164081;
        double r164083 = 21.0;
        double r164084 = r164063 / r164083;
        double r164085 = r164080 * r164069;
        double r164086 = r164085 * r164069;
        double r164087 = r164084 * r164086;
        double r164088 = r164082 + r164087;
        double r164089 = r164066 * r164088;
        double r164090 = fabs(r164089);
        return r164090;
}
↓
double f(double x) {
        double r164091 = x;
        double r164092 = fabs(r164091);
        double r164093 = 5.0;
        double r164094 = pow(r164092, r164093);
        double r164095 = 0.2;
        double r164096 = 7.0;
        double r164097 = pow(r164092, r164096);
        double r164098 = 0.047619047619047616;
        double r164099 = 3.0;
        double r164100 = pow(r164092, r164099);
        double r164101 = 0.6666666666666666;
        double r164102 = 2.0;
        double r164103 = r164102 * r164092;
        double r164104 = fma(r164100, r164101, r164103);
        double r164105 = fma(r164097, r164098, r164104);
        double r164106 = fma(r164094, r164095, r164105);
        double r164107 = 1.0;
        double r164108 = 1.0;
        double r164109 = atan2(1.0, 0.0);
        double r164110 = r164108 / r164109;
        double r164111 = sqrt(r164110);
        double r164112 = r164107 * r164111;
        double r164113 = r164106 * r164112;
        double r164114 = fabs(r164113);
        return r164114;
}

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. Simplified0.2

    \[\leadsto \color{blue}{\left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({\left(\left|x\right|\right)}^{3}, \mathsf{fma}\left(\frac{1}{5}, \left|x\right| \cdot \left|x\right|, \frac{2}{3}\right), \left|x\right| \cdot \mathsf{fma}\left({\left(\left|x\right|\right)}^{6}, \frac{1}{21}, 2\right)\right)\right|}\]

  3. Taylor expanded around 0 0.2

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

  4. Simplified0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2019179 +o rules:numerics
(FPCore (x)
  :name "Jmat.Real.erfi, branch x less than or equal to 0.5"
  (fabs (* (/ 1.0 (sqrt PI)) (+ (+ (+ (* 2.0 (fabs x)) (* (/ 2.0 3.0) (* (* (fabs x) (fabs x)) (fabs x)))) (* (/ 1.0 5.0) (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)))) (* (/ 1.0 21.0) (* (* (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)) (fabs x)) (fabs x)))))))