Average Error: 52.9 → 0.4
Time: 7.9s
Precision: 64
\[4.93038 \cdot 10^{-32} \lt a \lt 2.02824 \cdot 10^{31} \land 4.93038 \cdot 10^{-32} \lt b \lt 2.02824 \cdot 10^{31} \land 4.93038 \cdot 10^{-32} \lt c \lt 2.02824 \cdot 10^{31}\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b + \left(0 - 4 \cdot \left(a \cdot c\right)\right)}}}{2 \cdot a}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b + \left(0 - 4 \cdot \left(a \cdot c\right)\right)}}}{2 \cdot a}
double f(double a, double b, double c) {
        double r46734 = b;
        double r46735 = -r46734;
        double r46736 = r46734 * r46734;
        double r46737 = 4.0;
        double r46738 = a;
        double r46739 = r46737 * r46738;
        double r46740 = c;
        double r46741 = r46739 * r46740;
        double r46742 = r46736 - r46741;
        double r46743 = sqrt(r46742);
        double r46744 = r46735 + r46743;
        double r46745 = 2.0;
        double r46746 = r46745 * r46738;
        double r46747 = r46744 / r46746;
        return r46747;
}
↓
double f(double a, double b, double c) {
        double r46748 = 0.0;
        double r46749 = 4.0;
        double r46750 = a;
        double r46751 = c;
        double r46752 = r46750 * r46751;
        double r46753 = r46749 * r46752;
        double r46754 = r46748 + r46753;
        double r46755 = b;
        double r46756 = -r46755;
        double r46757 = r46755 * r46755;
        double r46758 = r46748 - r46753;
        double r46759 = r46757 + r46758;
        double r46760 = sqrt(r46759);
        double r46761 = r46756 - r46760;
        double r46762 = r46754 / r46761;
        double r46763 = 2.0;
        double r46764 = r46763 * r46750;
        double r46765 = r46762 / r46764;
        return r46765;
}

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 52.9

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

  3. Applied flip-+52.9

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

  4. Simplified0.4

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

  6. Applied sub-neg0.4

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

  7. Simplified0.4

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

  8. Final simplification0.4

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

Reproduce

herbie shell --seed 2020065 
(FPCore (a b c)
  :name "Quadratic roots, wide range"
  :precision binary64
  :pre (and (< 4.9303800000000003e-32 a 2.02824e+31) (< 4.9303800000000003e-32 b 2.02824e+31) (< 4.9303800000000003e-32 c 2.02824e+31))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))