Average Error: 0.5 → 0.5
Time: 57.0s
Precision: 64
\[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right)\]
↓
\[x1 + \left(3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - x2 \cdot 2\right) - x1}{1 + x1 \cdot x1} + \left(x1 + \left(x1 \cdot \left(x1 \cdot x1\right) + \left(\left(1 + x1 \cdot x1\right) \cdot \left(\left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + x2 \cdot 2\right) - x1}{1 + x1 \cdot x1} - 3\right) \cdot \left(\left(x1 \cdot 2\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + x2 \cdot 2\right) - x1}{1 + x1 \cdot x1}\right) + \left(\mathsf{fma}\left(6, -1, 6\right) \cdot \left(x1 \cdot x1\right) + \left(x1 \cdot x1\right) \cdot \mathsf{fma}\left(4, \frac{\left(\left(3 \cdot x1\right) \cdot x1 + x2 \cdot 2\right) - x1}{1 + x1 \cdot x1}, \left(-\sqrt{6}\right) \cdot \sqrt{6}\right)\right)\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + x2 \cdot 2\right) - x1}{1 + x1 \cdot x1}\right)\right)\right)\right)\]
x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right)
↓
x1 + \left(3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - x2 \cdot 2\right) - x1}{1 + x1 \cdot x1} + \left(x1 + \left(x1 \cdot \left(x1 \cdot x1\right) + \left(\left(1 + x1 \cdot x1\right) \cdot \left(\left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + x2 \cdot 2\right) - x1}{1 + x1 \cdot x1} - 3\right) \cdot \left(\left(x1 \cdot 2\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + x2 \cdot 2\right) - x1}{1 + x1 \cdot x1}\right) + \left(\mathsf{fma}\left(6, -1, 6\right) \cdot \left(x1 \cdot x1\right) + \left(x1 \cdot x1\right) \cdot \mathsf{fma}\left(4, \frac{\left(\left(3 \cdot x1\right) \cdot x1 + x2 \cdot 2\right) - x1}{1 + x1 \cdot x1}, \left(-\sqrt{6}\right) \cdot \sqrt{6}\right)\right)\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + x2 \cdot 2\right) - x1}{1 + x1 \cdot x1}\right)\right)\right)\right)
double f(double x1, double x2) {
        double r2911676 = x1;
        double r2911677 = 2.0;
        double r2911678 = r2911677 * r2911676;
        double r2911679 = 3.0;
        double r2911680 = r2911679 * r2911676;
        double r2911681 = r2911680 * r2911676;
        double r2911682 = x2;
        double r2911683 = r2911677 * r2911682;
        double r2911684 = r2911681 + r2911683;
        double r2911685 = r2911684 - r2911676;
        double r2911686 = r2911676 * r2911676;
        double r2911687 = 1.0;
        double r2911688 = r2911686 + r2911687;
        double r2911689 = r2911685 / r2911688;
        double r2911690 = r2911678 * r2911689;
        double r2911691 = r2911689 - r2911679;
        double r2911692 = r2911690 * r2911691;
        double r2911693 = 4.0;
        double r2911694 = r2911693 * r2911689;
        double r2911695 = 6.0;
        double r2911696 = r2911694 - r2911695;
        double r2911697 = r2911686 * r2911696;
        double r2911698 = r2911692 + r2911697;
        double r2911699 = r2911698 * r2911688;
        double r2911700 = r2911681 * r2911689;
        double r2911701 = r2911699 + r2911700;
        double r2911702 = r2911686 * r2911676;
        double r2911703 = r2911701 + r2911702;
        double r2911704 = r2911703 + r2911676;
        double r2911705 = r2911681 - r2911683;
        double r2911706 = r2911705 - r2911676;
        double r2911707 = r2911706 / r2911688;
        double r2911708 = r2911679 * r2911707;
        double r2911709 = r2911704 + r2911708;
        double r2911710 = r2911676 + r2911709;
        return r2911710;
}
↓
double f(double x1, double x2) {
        double r2911711 = x1;
        double r2911712 = 3.0;
        double r2911713 = r2911712 * r2911711;
        double r2911714 = r2911713 * r2911711;
        double r2911715 = x2;
        double r2911716 = 2.0;
        double r2911717 = r2911715 * r2911716;
        double r2911718 = r2911714 - r2911717;
        double r2911719 = r2911718 - r2911711;
        double r2911720 = 1.0;
        double r2911721 = r2911711 * r2911711;
        double r2911722 = r2911720 + r2911721;
        double r2911723 = r2911719 / r2911722;
        double r2911724 = r2911712 * r2911723;
        double r2911725 = r2911711 * r2911721;
        double r2911726 = r2911714 + r2911717;
        double r2911727 = r2911726 - r2911711;
        double r2911728 = r2911727 / r2911722;
        double r2911729 = r2911728 - r2911712;
        double r2911730 = r2911711 * r2911716;
        double r2911731 = r2911730 * r2911728;
        double r2911732 = r2911729 * r2911731;
        double r2911733 = 6.0;
        double r2911734 = -1.0;
        double r2911735 = fma(r2911733, r2911734, r2911733);
        double r2911736 = r2911735 * r2911721;
        double r2911737 = 4.0;
        double r2911738 = sqrt(r2911733);
        double r2911739 = -r2911738;
        double r2911740 = r2911739 * r2911738;
        double r2911741 = fma(r2911737, r2911728, r2911740);
        double r2911742 = r2911721 * r2911741;
        double r2911743 = r2911736 + r2911742;
        double r2911744 = r2911732 + r2911743;
        double r2911745 = r2911722 * r2911744;
        double r2911746 = r2911714 * r2911728;
        double r2911747 = r2911745 + r2911746;
        double r2911748 = r2911725 + r2911747;
        double r2911749 = r2911711 + r2911748;
        double r2911750 = r2911724 + r2911749;
        double r2911751 = r2911711 + r2911750;
        return r2911751;
}

Error

Bits error versus x1

Bits error versus x2

Derivation

  1. Initial program 0.5

    \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right)\]
  2. Using strategy rm

  3. Applied add-sqr-sqrt0.5

    \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - \color{blue}{\sqrt{6} \cdot \sqrt{6}}\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right)\]

  4. Applied prod-diff0.5

    \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \color{blue}{\left(\mathsf{fma}\left(4, \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}, -\sqrt{6} \cdot \sqrt{6}\right) + \mathsf{fma}\left(-\sqrt{6}, \sqrt{6}, \sqrt{6} \cdot \sqrt{6}\right)\right)}\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right)\]

  5. Applied distribute-lft-in0.5

    \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \color{blue}{\left(\left(x1 \cdot x1\right) \cdot \mathsf{fma}\left(4, \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}, -\sqrt{6} \cdot \sqrt{6}\right) + \left(x1 \cdot x1\right) \cdot \mathsf{fma}\left(-\sqrt{6}, \sqrt{6}, \sqrt{6} \cdot \sqrt{6}\right)\right)}\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right)\]

  6. Simplified0.5

    \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(\left(x1 \cdot x1\right) \cdot \mathsf{fma}\left(4, \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}, -\sqrt{6} \cdot \sqrt{6}\right) + \color{blue}{\mathsf{fma}\left(6, -1, 6\right) \cdot \left(x1 \cdot x1\right)}\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right)\]

  7. Final simplification0.5

    \[\leadsto x1 + \left(3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - x2 \cdot 2\right) - x1}{1 + x1 \cdot x1} + \left(x1 + \left(x1 \cdot \left(x1 \cdot x1\right) + \left(\left(1 + x1 \cdot x1\right) \cdot \left(\left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + x2 \cdot 2\right) - x1}{1 + x1 \cdot x1} - 3\right) \cdot \left(\left(x1 \cdot 2\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + x2 \cdot 2\right) - x1}{1 + x1 \cdot x1}\right) + \left(\mathsf{fma}\left(6, -1, 6\right) \cdot \left(x1 \cdot x1\right) + \left(x1 \cdot x1\right) \cdot \mathsf{fma}\left(4, \frac{\left(\left(3 \cdot x1\right) \cdot x1 + x2 \cdot 2\right) - x1}{1 + x1 \cdot x1}, \left(-\sqrt{6}\right) \cdot \sqrt{6}\right)\right)\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + x2 \cdot 2\right) - x1}{1 + x1 \cdot x1}\right)\right)\right)\right)\]

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (x1 x2)
  :name "Rosa's FloatVsDoubleBenchmark"
  (+ x1 (+ (+ (+ (+ (* (+ (* (* (* 2.0 x1) (/ (- (+ (* (* 3.0 x1) x1) (* 2.0 x2)) x1) (+ (* x1 x1) 1.0))) (- (/ (- (+ (* (* 3.0 x1) x1) (* 2.0 x2)) x1) (+ (* x1 x1) 1.0)) 3.0)) (* (* x1 x1) (- (* 4.0 (/ (- (+ (* (* 3.0 x1) x1) (* 2.0 x2)) x1) (+ (* x1 x1) 1.0))) 6.0))) (+ (* x1 x1) 1.0)) (* (* (* 3.0 x1) x1) (/ (- (+ (* (* 3.0 x1) x1) (* 2.0 x2)) x1) (+ (* x1 x1) 1.0)))) (* (* x1 x1) x1)) x1) (* 3.0 (/ (- (- (* (* 3.0 x1) x1) (* 2.0 x2)) x1) (+ (* x1 x1) 1.0))))))