Average Error: 28.4 → 0.5
Time: 13.0s
Precision: 64
\[1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt a \lt 94906265.62425155937671661376953125 \land 1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt b \lt 94906265.62425155937671661376953125 \land 1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt c \lt 94906265.62425155937671661376953125\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{4 \cdot \left(a \cdot c\right)}{\left(2 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{\frac{{b}^{6} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}{\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c + {b}^{2}\right) + \left(b \cdot b\right) \cdot \left(b \cdot b\right)}}\right)}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\frac{4 \cdot \left(a \cdot c\right)}{\left(2 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{\frac{{b}^{6} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}{\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c + {b}^{2}\right) + \left(b \cdot b\right) \cdot \left(b \cdot b\right)}}\right)}
double f(double a, double b, double c) {
        double r45608 = b;
        double r45609 = -r45608;
        double r45610 = r45608 * r45608;
        double r45611 = 4.0;
        double r45612 = a;
        double r45613 = r45611 * r45612;
        double r45614 = c;
        double r45615 = r45613 * r45614;
        double r45616 = r45610 - r45615;
        double r45617 = sqrt(r45616);
        double r45618 = r45609 + r45617;
        double r45619 = 2.0;
        double r45620 = r45619 * r45612;
        double r45621 = r45618 / r45620;
        return r45621;
}
↓
double f(double a, double b, double c) {
        double r45622 = 4.0;
        double r45623 = a;
        double r45624 = c;
        double r45625 = r45623 * r45624;
        double r45626 = r45622 * r45625;
        double r45627 = 2.0;
        double r45628 = r45627 * r45623;
        double r45629 = b;
        double r45630 = -r45629;
        double r45631 = 6.0;
        double r45632 = pow(r45629, r45631);
        double r45633 = r45622 * r45623;
        double r45634 = r45633 * r45624;
        double r45635 = 3.0;
        double r45636 = pow(r45634, r45635);
        double r45637 = r45632 - r45636;
        double r45638 = 2.0;
        double r45639 = pow(r45629, r45638);
        double r45640 = r45634 + r45639;
        double r45641 = r45634 * r45640;
        double r45642 = r45629 * r45629;
        double r45643 = r45642 * r45642;
        double r45644 = r45641 + r45643;
        double r45645 = r45637 / r45644;
        double r45646 = sqrt(r45645);
        double r45647 = r45630 - r45646;
        double r45648 = r45628 * r45647;
        double r45649 = r45626 / r45648;
        return r45649;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 28.4

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

  3. Applied flip-+28.5

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

  4. Simplified0.5

    \[\leadsto \frac{\frac{\color{blue}{0 + 4 \cdot \left(a \cdot c\right)}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\]
  5. Using strategy rm

  6. Applied flip3--0.5

    \[\leadsto \frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{\color{blue}{\frac{{\left(b \cdot b\right)}^{3} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}}}{2 \cdot a}\]

  7. Simplified0.5

    \[\leadsto \frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{\frac{\color{blue}{{b}^{6} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}}{2 \cdot a}\]

  8. Simplified0.5

    \[\leadsto \frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{\frac{{b}^{6} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}{\color{blue}{\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c + {b}^{2}\right) + \left(b \cdot b\right) \cdot \left(b \cdot b\right)}}}}}{2 \cdot a}\]

  9. Final simplification0.5

    \[\leadsto \frac{4 \cdot \left(a \cdot c\right)}{\left(2 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{\frac{{b}^{6} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}{\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c + {b}^{2}\right) + \left(b \cdot b\right) \cdot \left(b \cdot b\right)}}\right)}\]

Reproduce

herbie shell --seed 2019308 
(FPCore (a b c)
  :name "Quadratic roots, narrow range"
  :precision binary64
  :pre (and (< 1.05367121277235087e-8 a 94906265.6242515594) (< 1.05367121277235087e-8 b 94906265.6242515594) (< 1.05367121277235087e-8 c 94906265.6242515594))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))