Average Error: 1.5 → 0.5
Time: 26.2s
Precision: 64
\[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right)\]
↓
\[\left(\left(\left(\frac{1.875}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + \frac{0.75}{{\left(\left|x\right|\right)}^{5}}\right)\right) + \frac{0.5}{{\left(\left|x\right|\right)}^{3}}\right) \cdot \left(e^{{\left(\left|x\right|\right)}^{2}} \cdot 1\right)\right) \cdot \sqrt{\frac{1}{\pi}}\]
\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)
↓
\left(\left(\left(\frac{1.875}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + \frac{0.75}{{\left(\left|x\right|\right)}^{5}}\right)\right) + \frac{0.5}{{\left(\left|x\right|\right)}^{3}}\right) \cdot \left(e^{{\left(\left|x\right|\right)}^{2}} \cdot 1\right)\right) \cdot \sqrt{\frac{1}{\pi}}
double f(double x) {
        double r219750 = 1.0;
        double r219751 = atan2(1.0, 0.0);
        double r219752 = sqrt(r219751);
        double r219753 = r219750 / r219752;
        double r219754 = x;
        double r219755 = fabs(r219754);
        double r219756 = r219755 * r219755;
        double r219757 = exp(r219756);
        double r219758 = r219753 * r219757;
        double r219759 = r219750 / r219755;
        double r219760 = 2.0;
        double r219761 = r219750 / r219760;
        double r219762 = r219759 * r219759;
        double r219763 = r219762 * r219759;
        double r219764 = r219761 * r219763;
        double r219765 = r219759 + r219764;
        double r219766 = 3.0;
        double r219767 = 4.0;
        double r219768 = r219766 / r219767;
        double r219769 = r219763 * r219759;
        double r219770 = r219769 * r219759;
        double r219771 = r219768 * r219770;
        double r219772 = r219765 + r219771;
        double r219773 = 15.0;
        double r219774 = 8.0;
        double r219775 = r219773 / r219774;
        double r219776 = r219770 * r219759;
        double r219777 = r219776 * r219759;
        double r219778 = r219775 * r219777;
        double r219779 = r219772 + r219778;
        double r219780 = r219758 * r219779;
        return r219780;
}
↓
double f(double x) {
        double r219781 = 1.875;
        double r219782 = x;
        double r219783 = fabs(r219782);
        double r219784 = 7.0;
        double r219785 = pow(r219783, r219784);
        double r219786 = r219781 / r219785;
        double r219787 = 1.0;
        double r219788 = r219787 / r219783;
        double r219789 = 0.75;
        double r219790 = 5.0;
        double r219791 = pow(r219783, r219790);
        double r219792 = r219789 / r219791;
        double r219793 = r219788 + r219792;
        double r219794 = r219786 + r219793;
        double r219795 = 0.5;
        double r219796 = 3.0;
        double r219797 = pow(r219783, r219796);
        double r219798 = r219795 / r219797;
        double r219799 = r219794 + r219798;
        double r219800 = 2.0;
        double r219801 = pow(r219783, r219800);
        double r219802 = exp(r219801);
        double r219803 = r219802 * r219787;
        double r219804 = r219799 * r219803;
        double r219805 = 1.0;
        double r219806 = atan2(1.0, 0.0);
        double r219807 = r219805 / r219806;
        double r219808 = sqrt(r219807);
        double r219809 = r219804 * r219808;
        return r219809;
}

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

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

  3. Taylor expanded around 0 0.7

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

  4. Taylor expanded around 0 0.6

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

  5. Simplified0.5

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

  7. Applied associate-*r*0.5

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

  8. Simplified0.5

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

  9. Final simplification0.5

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

Reproduce

herbie shell --seed 2020043 +o rules:numerics
(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)))))))