Average Error: 0.2 → 0.1
Time: 7.0s
Precision: 64
\[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right|\]
↓
\[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \left(\left(\frac{\sqrt{1}}{\sqrt{\sqrt{\pi}}} \cdot \left(\frac{\sqrt{1}}{\sqrt{\sqrt{\pi}}} \cdot \frac{1}{21}\right)\right) \cdot {\left(\left|x\right|\right)}^{6}\right) \cdot \left|x\right|\right|\]
\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 \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(\left(\frac{\sqrt{1}}{\sqrt{\sqrt{\pi}}} \cdot \left(\frac{\sqrt{1}}{\sqrt{\sqrt{\pi}}} \cdot \frac{1}{21}\right)\right) \cdot {\left(\left|x\right|\right)}^{6}\right) \cdot \left|x\right|\right|
double f(double x) {
        double r171813 = 1.0;
        double r171814 = atan2(1.0, 0.0);
        double r171815 = sqrt(r171814);
        double r171816 = r171813 / r171815;
        double r171817 = 2.0;
        double r171818 = x;
        double r171819 = fabs(r171818);
        double r171820 = r171817 * r171819;
        double r171821 = 3.0;
        double r171822 = r171817 / r171821;
        double r171823 = r171819 * r171819;
        double r171824 = r171823 * r171819;
        double r171825 = r171822 * r171824;
        double r171826 = r171820 + r171825;
        double r171827 = 5.0;
        double r171828 = r171813 / r171827;
        double r171829 = r171824 * r171819;
        double r171830 = r171829 * r171819;
        double r171831 = r171828 * r171830;
        double r171832 = r171826 + r171831;
        double r171833 = 21.0;
        double r171834 = r171813 / r171833;
        double r171835 = r171830 * r171819;
        double r171836 = r171835 * r171819;
        double r171837 = r171834 * r171836;
        double r171838 = r171832 + r171837;
        double r171839 = r171816 * r171838;
        double r171840 = fabs(r171839);
        return r171840;
}
↓
double f(double x) {
        double r171841 = 1.0;
        double r171842 = atan2(1.0, 0.0);
        double r171843 = sqrt(r171842);
        double r171844 = r171841 / r171843;
        double r171845 = 2.0;
        double r171846 = x;
        double r171847 = fabs(r171846);
        double r171848 = r171845 * r171847;
        double r171849 = 3.0;
        double r171850 = r171845 / r171849;
        double r171851 = r171847 * r171847;
        double r171852 = r171851 * r171847;
        double r171853 = r171850 * r171852;
        double r171854 = r171848 + r171853;
        double r171855 = 5.0;
        double r171856 = r171841 / r171855;
        double r171857 = r171852 * r171847;
        double r171858 = r171857 * r171847;
        double r171859 = r171856 * r171858;
        double r171860 = r171854 + r171859;
        double r171861 = r171844 * r171860;
        double r171862 = sqrt(r171841);
        double r171863 = sqrt(r171843);
        double r171864 = r171862 / r171863;
        double r171865 = 21.0;
        double r171866 = r171841 / r171865;
        double r171867 = r171864 * r171866;
        double r171868 = r171864 * r171867;
        double r171869 = 6.0;
        double r171870 = pow(r171847, r171869);
        double r171871 = r171868 * r171870;
        double r171872 = r171871 * r171847;
        double r171873 = r171861 + r171872;
        double r171874 = fabs(r171873);
        return r171874;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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 distribute-lft-in0.2

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

  4. Simplified0.2

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

  6. Applied add-sqr-sqrt0.2

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

  7. Applied sqrt-prod0.2

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

  8. Applied add-sqr-sqrt0.2

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

  9. Applied times-frac0.2

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

  10. Applied associate-*l*0.1

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

  11. Final simplification0.1

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

Reproduce

herbie shell --seed 2020060 
(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)))))))