Jmat.Real.erf

Average Error: 13.6 → 2.0

Time: 42.6s

Precision: 64

Math

\[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]

↓

\[\sqrt{\log \left(e^{1 - e^{\left|x\right| \cdot \left(-\left|x\right|\right)} \cdot \left(\left(0.254829592 + \left(-0.284496736 + \left(\left(-1.453152027 + \left(\left(1 - \left|x\right| \cdot 0.3275911\right) \cdot \frac{1}{1 - \left(\left|x\right| \cdot 0.3275911\right) \cdot \left(\left|x\right| \cdot 0.3275911\right)}\right) \cdot 1.061405429\right) \cdot \frac{1}{\left|x\right| \cdot 0.3275911 + 1} + 1.421413741\right) \cdot \frac{1}{\left|x\right| \cdot 0.3275911 + 1}\right) \cdot \frac{1}{\left|x\right| \cdot 0.3275911 + 1}\right) \cdot \frac{1}{\left|x\right| \cdot 0.3275911 + 1}\right)}\right)} \cdot \sqrt{\frac{1 - \left(e^{\left|x\right| \cdot \left(-\left|x\right|\right)} \cdot \left(\left(0.254829592 + \left(-0.284496736 + \left(\left(-1.453152027 + \left(\left(1 - \left|x\right| \cdot 0.3275911\right) \cdot \frac{1}{1 - \left(\left|x\right| \cdot 0.3275911\right) \cdot \left(\left|x\right| \cdot 0.3275911\right)}\right) \cdot 1.061405429\right) \cdot \frac{1}{\left|x\right| \cdot 0.3275911 + 1} + 1.421413741\right) \cdot \frac{1}{\left|x\right| \cdot 0.3275911 + 1}\right) \cdot \frac{1}{\left|x\right| \cdot 0.3275911 + 1}\right) \cdot \frac{1}{\left|x\right| \cdot 0.3275911 + 1}\right)\right) \cdot \left(e^{\left|x\right| \cdot \left(-\left|x\right|\right)} \cdot \left(\left(0.254829592 + \left(-0.284496736 + \left(\left(-1.453152027 + \left(\left(1 - \left|x\right| \cdot 0.3275911\right) \cdot \frac{1}{1 - \left(\left|x\right| \cdot 0.3275911\right) \cdot \left(\left|x\right| \cdot 0.3275911\right)}\right) \cdot 1.061405429\right) \cdot \frac{1}{\left|x\right| \cdot 0.3275911 + 1} + 1.421413741\right) \cdot \frac{1}{\left|x\right| \cdot 0.3275911 + 1}\right) \cdot \frac{1}{\left|x\right| \cdot 0.3275911 + 1}\right) \cdot \frac{1}{\left|x\right| \cdot 0.3275911 + 1}\right)\right)}{1 + e^{\left|x\right| \cdot \left(-\left|x\right|\right)} \cdot \left(\left(0.254829592 + \left(-0.284496736 + \left(\left(-1.453152027 + \left(\left(1 - \left|x\right| \cdot 0.3275911\right) \cdot \frac{1}{1 - \left(\left|x\right| \cdot 0.3275911\right) \cdot \left(\left|x\right| \cdot 0.3275911\right)}\right) \cdot 1.061405429\right) \cdot \frac{1}{\left|x\right| \cdot 0.3275911 + 1} + 1.421413741\right) \cdot \frac{1}{\left|x\right| \cdot 0.3275911 + 1}\right) \cdot \frac{1}{\left|x\right| \cdot 0.3275911 + 1}\right) \cdot \frac{1}{\left|x\right| \cdot 0.3275911 + 1}\right)}}\]

C

double f(double x) {
        double r4565622 = 1.0;
        double r4565623 = 0.3275911;
        double r4565624 = x;
        double r4565625 = fabs(r4565624);
        double r4565626 = r4565623 * r4565625;
        double r4565627 = r4565622 + r4565626;
        double r4565628 = r4565622 / r4565627;
        double r4565629 = 0.254829592;
        double r4565630 = -0.284496736;
        double r4565631 = 1.421413741;
        double r4565632 = -1.453152027;
        double r4565633 = 1.061405429;
        double r4565634 = r4565628 * r4565633;
        double r4565635 = r4565632 + r4565634;
        double r4565636 = r4565628 * r4565635;
        double r4565637 = r4565631 + r4565636;
        double r4565638 = r4565628 * r4565637;
        double r4565639 = r4565630 + r4565638;
        double r4565640 = r4565628 * r4565639;
        double r4565641 = r4565629 + r4565640;
        double r4565642 = r4565628 * r4565641;
        double r4565643 = r4565625 * r4565625;
        double r4565644 = -r4565643;
        double r4565645 = exp(r4565644);
        double r4565646 = r4565642 * r4565645;
        double r4565647 = r4565622 - r4565646;
        return r4565647;
}

↓

double f(double x) {
        double r4565648 = 1.0;
        double r4565649 = x;
        double r4565650 = fabs(r4565649);
        double r4565651 = -r4565650;
        double r4565652 = r4565650 * r4565651;
        double r4565653 = exp(r4565652);
        double r4565654 = 0.254829592;
        double r4565655 = -0.284496736;
        double r4565656 = -1.453152027;
        double r4565657 = 0.3275911;
        double r4565658 = r4565650 * r4565657;
        double r4565659 = r4565648 - r4565658;
        double r4565660 = r4565658 * r4565658;
        double r4565661 = r4565648 - r4565660;
        double r4565662 = r4565648 / r4565661;
        double r4565663 = r4565659 * r4565662;
        double r4565664 = 1.061405429;
        double r4565665 = r4565663 * r4565664;
        double r4565666 = r4565656 + r4565665;
        double r4565667 = r4565658 + r4565648;
        double r4565668 = r4565648 / r4565667;
        double r4565669 = r4565666 * r4565668;
        double r4565670 = 1.421413741;
        double r4565671 = r4565669 + r4565670;
        double r4565672 = r4565671 * r4565668;
        double r4565673 = r4565655 + r4565672;
        double r4565674 = r4565673 * r4565668;
        double r4565675 = r4565654 + r4565674;
        double r4565676 = r4565675 * r4565668;
        double r4565677 = r4565653 * r4565676;
        double r4565678 = r4565648 - r4565677;
        double r4565679 = exp(r4565678);
        double r4565680 = log(r4565679);
        double r4565681 = sqrt(r4565680);
        double r4565682 = r4565677 * r4565677;
        double r4565683 = r4565648 - r4565682;
        double r4565684 = r4565648 + r4565677;
        double r4565685 = r4565683 / r4565684;
        double r4565686 = sqrt(r4565685);
        double r4565687 = r4565681 * r4565686;
        return r4565687;
}

Error

Bits error versus x

Derivation

1. Initial program 13.6

   \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
2. Using strategy rm

3. Applied flip-+ 13.6

   \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{\color{blue}{\frac{1 \cdot 1 - \left(0.3275911 \cdot \left|x\right|\right) \cdot \left(0.3275911 \cdot \left|x\right|\right)}{1 - 0.3275911 \cdot \left|x\right|}}} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]

4. Applied associate-/r/ 13.6

   \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \color{blue}{\left(\frac{1}{1 \cdot 1 - \left(0.3275911 \cdot \left|x\right|\right) \cdot \left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(1 - 0.3275911 \cdot \left|x\right|\right)\right)} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
5. Using strategy rm

6. Applied add-sqr-sqrt 13.6

   \[\leadsto \color{blue}{\sqrt{1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \left(\frac{1}{1 \cdot 1 - \left(0.3275911 \cdot \left|x\right|\right) \cdot \left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(1 - 0.3275911 \cdot \left|x\right|\right)\right) \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}} \cdot \sqrt{1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \left(\frac{1}{1 \cdot 1 - \left(0.3275911 \cdot \left|x\right|\right) \cdot \left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(1 - 0.3275911 \cdot \left|x\right|\right)\right) \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}}}\]
7. Using strategy rm

8. Applied add-log-exp 10.0

   \[\leadsto \sqrt{1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \left(\frac{1}{1 \cdot 1 - \left(0.3275911 \cdot \left|x\right|\right) \cdot \left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(1 - 0.3275911 \cdot \left|x\right|\right)\right) \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}} \cdot \sqrt{\color{blue}{\log \left(e^{1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \left(\frac{1}{1 \cdot 1 - \left(0.3275911 \cdot \left|x\right|\right) \cdot \left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(1 - 0.3275911 \cdot \left|x\right|\right)\right) \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}}\right)}}\]
9. Using strategy rm

10. Applied flip-- 2.0

    \[\leadsto \sqrt{\color{blue}{\frac{1 \cdot 1 - \left(\left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \left(\frac{1}{1 \cdot 1 - \left(0.3275911 \cdot \left|x\right|\right) \cdot \left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(1 - 0.3275911 \cdot \left|x\right|\right)\right) \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \left(\frac{1}{1 \cdot 1 - \left(0.3275911 \cdot \left|x\right|\right) \cdot \left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(1 - 0.3275911 \cdot \left|x\right|\right)\right) \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\right)}{1 + \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \left(\frac{1}{1 \cdot 1 - \left(0.3275911 \cdot \left|x\right|\right) \cdot \left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(1 - 0.3275911 \cdot \left|x\right|\right)\right) \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}}}} \cdot \sqrt{\log \left(e^{1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \left(\frac{1}{1 \cdot 1 - \left(0.3275911 \cdot \left|x\right|\right) \cdot \left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(1 - 0.3275911 \cdot \left|x\right|\right)\right) \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}}\right)}\]

11. Final simplification 2.0

    \[\leadsto \sqrt{\log \left(e^{1 - e^{\left|x\right| \cdot \left(-\left|x\right|\right)} \cdot \left(\left(0.254829592 + \left(-0.284496736 + \left(\left(-1.453152027 + \left(\left(1 - \left|x\right| \cdot 0.3275911\right) \cdot \frac{1}{1 - \left(\left|x\right| \cdot 0.3275911\right) \cdot \left(\left|x\right| \cdot 0.3275911\right)}\right) \cdot 1.061405429\right) \cdot \frac{1}{\left|x\right| \cdot 0.3275911 + 1} + 1.421413741\right) \cdot \frac{1}{\left|x\right| \cdot 0.3275911 + 1}\right) \cdot \frac{1}{\left|x\right| \cdot 0.3275911 + 1}\right) \cdot \frac{1}{\left|x\right| \cdot 0.3275911 + 1}\right)}\right)} \cdot \sqrt{\frac{1 - \left(e^{\left|x\right| \cdot \left(-\left|x\right|\right)} \cdot \left(\left(0.254829592 + \left(-0.284496736 + \left(\left(-1.453152027 + \left(\left(1 - \left|x\right| \cdot 0.3275911\right) \cdot \frac{1}{1 - \left(\left|x\right| \cdot 0.3275911\right) \cdot \left(\left|x\right| \cdot 0.3275911\right)}\right) \cdot 1.061405429\right) \cdot \frac{1}{\left|x\right| \cdot 0.3275911 + 1} + 1.421413741\right) \cdot \frac{1}{\left|x\right| \cdot 0.3275911 + 1}\right) \cdot \frac{1}{\left|x\right| \cdot 0.3275911 + 1}\right) \cdot \frac{1}{\left|x\right| \cdot 0.3275911 + 1}\right)\right) \cdot \left(e^{\left|x\right| \cdot \left(-\left|x\right|\right)} \cdot \left(\left(0.254829592 + \left(-0.284496736 + \left(\left(-1.453152027 + \left(\left(1 - \left|x\right| \cdot 0.3275911\right) \cdot \frac{1}{1 - \left(\left|x\right| \cdot 0.3275911\right) \cdot \left(\left|x\right| \cdot 0.3275911\right)}\right) \cdot 1.061405429\right) \cdot \frac{1}{\left|x\right| \cdot 0.3275911 + 1} + 1.421413741\right) \cdot \frac{1}{\left|x\right| \cdot 0.3275911 + 1}\right) \cdot \frac{1}{\left|x\right| \cdot 0.3275911 + 1}\right) \cdot \frac{1}{\left|x\right| \cdot 0.3275911 + 1}\right)\right)}{1 + e^{\left|x\right| \cdot \left(-\left|x\right|\right)} \cdot \left(\left(0.254829592 + \left(-0.284496736 + \left(\left(-1.453152027 + \left(\left(1 - \left|x\right| \cdot 0.3275911\right) \cdot \frac{1}{1 - \left(\left|x\right| \cdot 0.3275911\right) \cdot \left(\left|x\right| \cdot 0.3275911\right)}\right) \cdot 1.061405429\right) \cdot \frac{1}{\left|x\right| \cdot 0.3275911 + 1} + 1.421413741\right) \cdot \frac{1}{\left|x\right| \cdot 0.3275911 + 1}\right) \cdot \frac{1}{\left|x\right| \cdot 0.3275911 + 1}\right) \cdot \frac{1}{\left|x\right| \cdot 0.3275911 + 1}\right)}}\]

Reproduce

herbie shell --seed 2019133 +o rules:numerics
(FPCore (x)
  :name "Jmat.Real.erf"
  (- 1 (* (* (/ 1 (+ 1 (* 0.3275911 (fabs x)))) (+ 0.254829592 (* (/ 1 (+ 1 (* 0.3275911 (fabs x)))) (+ -0.284496736 (* (/ 1 (+ 1 (* 0.3275911 (fabs x)))) (+ 1.421413741 (* (/ 1 (+ 1 (* 0.3275911 (fabs x)))) (+ -1.453152027 (* (/ 1 (+ 1 (* 0.3275911 (fabs x)))) 1.061405429))))))))) (exp (- (* (fabs x) (fabs x)))))))