quadm (p42, negative)

Average Error: 1.5 → 1.7

Time: 22.9s

Precision: 64

Math

\[\frac{\left(\left(-b\right) - \left(\sqrt{\left(\left(b \cdot b\right) - \left(\left(4\right) \cdot \left(a \cdot c\right)\right)\right)}\right)\right)}{\left(\left(2\right) \cdot a\right)}\]

↓

\[\frac{\frac{\left(\left(-b\right) + b\right) \cdot \left(\left(-b\right) + \left(-b\right)\right) + \left(4 \cdot a\right) \cdot c}{2 \cdot a}}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\]

C

double f(double a, double b, double c) {
        double r815744 = b;
        double r815745 = -r815744;
        double r815746 = r815744 * r815744;
        double r815747 = 4.0;
        double r815748 = /* ERROR: no posit support in C */;
        double r815749 = a;
        double r815750 = c;
        double r815751 = r815749 * r815750;
        double r815752 = r815748 * r815751;
        double r815753 = r815746 - r815752;
        double r815754 = sqrt(r815753);
        double r815755 = r815745 - r815754;
        double r815756 = 2.0;
        double r815757 = /* ERROR: no posit support in C */;
        double r815758 = r815757 * r815749;
        double r815759 = r815755 / r815758;
        return r815759;
}

↓

double f(double a, double b, double c) {
        double r815760 = b;
        double r815761 = -r815760;
        double r815762 = r815761 + r815760;
        double r815763 = r815761 + r815761;
        double r815764 = r815762 * r815763;
        double r815765 = 4.0;
        double r815766 = a;
        double r815767 = r815765 * r815766;
        double r815768 = c;
        double r815769 = r815767 * r815768;
        double r815770 = r815764 + r815769;
        double r815771 = 2.0;
        double r815772 = r815771 * r815766;
        double r815773 = r815770 / r815772;
        double r815774 = r815760 * r815760;
        double r815775 = r815774 - r815769;
        double r815776 = sqrt(r815775);
        double r815777 = r815761 + r815776;
        double r815778 = r815773 / r815777;
        return r815778;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 1.5

   \[\frac{\left(\left(-b\right) - \left(\sqrt{\left(\left(b \cdot b\right) - \left(\left(4\right) \cdot \left(a \cdot c\right)\right)\right)}\right)\right)}{\left(\left(2\right) \cdot a\right)}\]
2. Using strategy rm

3. Applied p16-flip-- 2.5

   \[\leadsto \frac{\color{blue}{\left(\frac{\left(\left(\left(-b\right) \cdot \left(-b\right)\right) - \left(\left(\sqrt{\left(\left(b \cdot b\right) - \left(\left(4\right) \cdot \left(a \cdot c\right)\right)\right)}\right) \cdot \left(\sqrt{\left(\left(b \cdot b\right) - \left(\left(4\right) \cdot \left(a \cdot c\right)\right)\right)}\right)\right)\right)}{\left(\frac{\left(-b\right)}{\left(\sqrt{\left(\left(b \cdot b\right) - \left(\left(4\right) \cdot \left(a \cdot c\right)\right)\right)}\right)}\right)}\right)}}{\left(\left(2\right) \cdot a\right)}\]

4. Applied associate-/l/ 2.6

   \[\leadsto \color{blue}{\frac{\left(\left(\left(-b\right) \cdot \left(-b\right)\right) - \left(\left(\sqrt{\left(\left(b \cdot b\right) - \left(\left(4\right) \cdot \left(a \cdot c\right)\right)\right)}\right) \cdot \left(\sqrt{\left(\left(b \cdot b\right) - \left(\left(4\right) \cdot \left(a \cdot c\right)\right)\right)}\right)\right)\right)}{\left(\left(\left(2\right) \cdot a\right) \cdot \left(\frac{\left(-b\right)}{\left(\sqrt{\left(\left(b \cdot b\right) - \left(\left(4\right) \cdot \left(a \cdot c\right)\right)\right)}\right)}\right)\right)}}\]

5. Simplified 1.8

   \[\leadsto \frac{\color{blue}{\left(\frac{\left(\left(\frac{\left(-b\right)}{b}\right) \cdot \left(\frac{\left(-b\right)}{\left(-b\right)}\right)\right)}{\left(\left(4\right) \cdot \left(a \cdot c\right)\right)}\right)}}{\left(\left(\left(2\right) \cdot a\right) \cdot \left(\frac{\left(-b\right)}{\left(\sqrt{\left(\left(b \cdot b\right) - \left(\left(4\right) \cdot \left(a \cdot c\right)\right)\right)}\right)}\right)\right)}\]
6. Using strategy rm

7. Applied associate-/r* 1.7

   \[\leadsto \color{blue}{\frac{\left(\frac{\left(\frac{\left(\left(\frac{\left(-b\right)}{b}\right) \cdot \left(\frac{\left(-b\right)}{\left(-b\right)}\right)\right)}{\left(\left(4\right) \cdot \left(a \cdot c\right)\right)}\right)}{\left(\left(2\right) \cdot a\right)}\right)}{\left(\frac{\left(-b\right)}{\left(\sqrt{\left(\left(b \cdot b\right) - \left(\left(4\right) \cdot \left(a \cdot c\right)\right)\right)}\right)}\right)}}\]
8. Using strategy rm

9. Applied associate-*r* 1.7

   \[\leadsto \frac{\left(\frac{\left(\frac{\left(\left(\frac{\left(-b\right)}{b}\right) \cdot \left(\frac{\left(-b\right)}{\left(-b\right)}\right)\right)}{\color{blue}{\left(\left(\left(4\right) \cdot a\right) \cdot c\right)}}\right)}{\left(\left(2\right) \cdot a\right)}\right)}{\left(\frac{\left(-b\right)}{\left(\sqrt{\left(\left(b \cdot b\right) - \left(\left(4\right) \cdot \left(a \cdot c\right)\right)\right)}\right)}\right)}\]
10. Using strategy rm

11. Applied associate-*r* 1.7

    \[\leadsto \frac{\left(\frac{\left(\frac{\left(\left(\frac{\left(-b\right)}{b}\right) \cdot \left(\frac{\left(-b\right)}{\left(-b\right)}\right)\right)}{\left(\left(\left(4\right) \cdot a\right) \cdot c\right)}\right)}{\left(\left(2\right) \cdot a\right)}\right)}{\left(\frac{\left(-b\right)}{\left(\sqrt{\left(\left(b \cdot b\right) - \color{blue}{\left(\left(\left(4\right) \cdot a\right) \cdot c\right)}\right)}\right)}\right)}\]

12. Final simplification 1.7

    \[\leadsto \frac{\frac{\left(\left(-b\right) + b\right) \cdot \left(\left(-b\right) + \left(-b\right)\right) + \left(4 \cdot a\right) \cdot c}{2 \cdot a}}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\]

Reproduce

herbie shell --seed 2019124 
(FPCore (a b c)
  :name "quadm (p42, negative)"
  (/.p16 (-.p16 (neg.p16 b) (sqrt.p16 (-.p16 (*.p16 b b) (*.p16 (real->posit16 4) (*.p16 a c))))) (*.p16 (real->posit16 2) a)))