Average Error: 0.2 → 0.1
Time: 39.8s
Precision: 64
\[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right|\]
↓
\[\left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\frac{{\left(\left|x\right|\right)}^{7}}{21}, 1, \mathsf{fma}\left({\left(\left|x\right|\right)}^{3}, \frac{2}{3}, \left|x\right| \cdot \left(2 + \frac{1}{5} \cdot {\left(\left|x\right|\right)}^{4}\right)\right)\right)\right|\]
\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|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\frac{{\left(\left|x\right|\right)}^{7}}{21}, 1, \mathsf{fma}\left({\left(\left|x\right|\right)}^{3}, \frac{2}{3}, \left|x\right| \cdot \left(2 + \frac{1}{5} \cdot {\left(\left|x\right|\right)}^{4}\right)\right)\right)\right|
double f(double x) {
        double r28774 = 1.0;
        double r28775 = atan2(1.0, 0.0);
        double r28776 = sqrt(r28775);
        double r28777 = r28774 / r28776;
        double r28778 = 2.0;
        double r28779 = x;
        double r28780 = fabs(r28779);
        double r28781 = r28778 * r28780;
        double r28782 = 3.0;
        double r28783 = r28778 / r28782;
        double r28784 = r28780 * r28780;
        double r28785 = r28784 * r28780;
        double r28786 = r28783 * r28785;
        double r28787 = r28781 + r28786;
        double r28788 = 5.0;
        double r28789 = r28774 / r28788;
        double r28790 = r28785 * r28780;
        double r28791 = r28790 * r28780;
        double r28792 = r28789 * r28791;
        double r28793 = r28787 + r28792;
        double r28794 = 21.0;
        double r28795 = r28774 / r28794;
        double r28796 = r28791 * r28780;
        double r28797 = r28796 * r28780;
        double r28798 = r28795 * r28797;
        double r28799 = r28793 + r28798;
        double r28800 = r28777 * r28799;
        double r28801 = fabs(r28800);
        return r28801;
}
↓
double f(double x) {
        double r28802 = 1.0;
        double r28803 = atan2(1.0, 0.0);
        double r28804 = sqrt(r28803);
        double r28805 = r28802 / r28804;
        double r28806 = x;
        double r28807 = fabs(r28806);
        double r28808 = 7.0;
        double r28809 = pow(r28807, r28808);
        double r28810 = 21.0;
        double r28811 = r28809 / r28810;
        double r28812 = 3.0;
        double r28813 = pow(r28807, r28812);
        double r28814 = 2.0;
        double r28815 = 3.0;
        double r28816 = r28814 / r28815;
        double r28817 = 5.0;
        double r28818 = r28802 / r28817;
        double r28819 = 4.0;
        double r28820 = pow(r28807, r28819);
        double r28821 = r28818 * r28820;
        double r28822 = r28814 + r28821;
        double r28823 = r28807 * r28822;
        double r28824 = fma(r28813, r28816, r28823);
        double r28825 = fma(r28811, r28802, r28824);
        double r28826 = r28805 * r28825;
        double r28827 = fabs(r28826);
        return r28827;
}

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 div-inv0.2

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

  4. Applied associate-*l*0.2

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

  5. Simplified0.1

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

  6. Simplified0.1

    \[\leadsto \color{blue}{\left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\frac{{\left(\left|x\right|\right)}^{7}}{21}, 1, \mathsf{fma}\left({\left(\left|x\right|\right)}^{3}, \frac{2}{3}, \left|x\right| \cdot \left(2 + \frac{1}{5} \cdot {\left(\left|x\right|\right)}^{4}\right)\right)\right)\right|}\]

  7. Final simplification0.1

    \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\frac{{\left(\left|x\right|\right)}^{7}}{21}, 1, \mathsf{fma}\left({\left(\left|x\right|\right)}^{3}, \frac{2}{3}, \left|x\right| \cdot \left(2 + \frac{1}{5} \cdot {\left(\left|x\right|\right)}^{4}\right)\right)\right)\right|\]

Reproduce

herbie shell --seed 2019310 +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)))))))