Average Error: 0.2 → 0.2
Time: 11.9s
Precision: 64
\[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right|\]
↓
\[\left|1 \cdot \mathsf{fma}\left(\frac{1}{\sqrt{\pi}}, \mathsf{fma}\left(\left(\frac{1}{5} \cdot {\left(\left|x\right|\right)}^{3}\right) \cdot \left|x\right|, \left|x\right|, \mathsf{fma}\left(\left|x\right|, 2, {\left(\left|x\right|\right)}^{3} \cdot \frac{2}{3}\right)\right), \frac{1}{\sqrt{\pi}} \cdot \left(\left(\frac{1}{21} \cdot {\left(\left|x\right|\right)}^{6}\right) \cdot \left|x\right|\right)\right)\right|\]
\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|1 \cdot \mathsf{fma}\left(\frac{1}{\sqrt{\pi}}, \mathsf{fma}\left(\left(\frac{1}{5} \cdot {\left(\left|x\right|\right)}^{3}\right) \cdot \left|x\right|, \left|x\right|, \mathsf{fma}\left(\left|x\right|, 2, {\left(\left|x\right|\right)}^{3} \cdot \frac{2}{3}\right)\right), \frac{1}{\sqrt{\pi}} \cdot \left(\left(\frac{1}{21} \cdot {\left(\left|x\right|\right)}^{6}\right) \cdot \left|x\right|\right)\right)\right|
double f(double x) {
        double r138095 = 1.0;
        double r138096 = atan2(1.0, 0.0);
        double r138097 = sqrt(r138096);
        double r138098 = r138095 / r138097;
        double r138099 = 2.0;
        double r138100 = x;
        double r138101 = fabs(r138100);
        double r138102 = r138099 * r138101;
        double r138103 = 3.0;
        double r138104 = r138099 / r138103;
        double r138105 = r138101 * r138101;
        double r138106 = r138105 * r138101;
        double r138107 = r138104 * r138106;
        double r138108 = r138102 + r138107;
        double r138109 = 5.0;
        double r138110 = r138095 / r138109;
        double r138111 = r138106 * r138101;
        double r138112 = r138111 * r138101;
        double r138113 = r138110 * r138112;
        double r138114 = r138108 + r138113;
        double r138115 = 21.0;
        double r138116 = r138095 / r138115;
        double r138117 = r138112 * r138101;
        double r138118 = r138117 * r138101;
        double r138119 = r138116 * r138118;
        double r138120 = r138114 + r138119;
        double r138121 = r138098 * r138120;
        double r138122 = fabs(r138121);
        return r138122;
}
↓
double f(double x) {
        double r138123 = 1.0;
        double r138124 = 1.0;
        double r138125 = atan2(1.0, 0.0);
        double r138126 = sqrt(r138125);
        double r138127 = r138124 / r138126;
        double r138128 = 5.0;
        double r138129 = r138123 / r138128;
        double r138130 = x;
        double r138131 = fabs(r138130);
        double r138132 = 3.0;
        double r138133 = pow(r138131, r138132);
        double r138134 = r138129 * r138133;
        double r138135 = r138134 * r138131;
        double r138136 = 2.0;
        double r138137 = 3.0;
        double r138138 = r138136 / r138137;
        double r138139 = r138133 * r138138;
        double r138140 = fma(r138131, r138136, r138139);
        double r138141 = fma(r138135, r138131, r138140);
        double r138142 = 21.0;
        double r138143 = r138123 / r138142;
        double r138144 = 6.0;
        double r138145 = pow(r138131, r138144);
        double r138146 = r138143 * r138145;
        double r138147 = r138146 * r138131;
        double r138148 = r138127 * r138147;
        double r138149 = fma(r138127, r138141, r138148);
        double r138150 = r138123 * r138149;
        double r138151 = fabs(r138150);
        return r138151;
}

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. Using strategy rm

  3. Applied associate-*r*0.2

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

  4. Simplified0.2

    \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \color{blue}{\left(\frac{1}{21} \cdot {\left(\left|x\right|\right)}^{6}\right)} \cdot \left|x\right|\right)\right|\]
  5. Using strategy rm

  6. Applied div-inv0.2

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

  7. Applied associate-*l*0.2

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

  8. Simplified0.2

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

  9. Final simplification0.2

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

Reproduce

herbie shell --seed 2020039 +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)))))))