Average Error: 44.2 → 0.4
Time: 1.7m
Precision: 64
\[1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt a \lt 9007199254740992 \land 1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt b \lt 9007199254740992 \land 1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt c \lt 9007199254740992\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{\frac{c \cdot \left(4 \cdot a\right)}{\left(-b\right) - \sqrt{\frac{{b}^{4} - \left(\left(a \cdot c\right) \cdot 4\right) \cdot \left(\left(a \cdot c\right) \cdot 4\right)}{b \cdot b + \left(a \cdot c\right) \cdot 4}}}}{2 \cdot a}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\frac{\frac{c \cdot \left(4 \cdot a\right)}{\left(-b\right) - \sqrt{\frac{{b}^{4} - \left(\left(a \cdot c\right) \cdot 4\right) \cdot \left(\left(a \cdot c\right) \cdot 4\right)}{b \cdot b + \left(a \cdot c\right) \cdot 4}}}}{2 \cdot a}
double f(double a, double b, double c) {
        double r40792 = b;
        double r40793 = -r40792;
        double r40794 = r40792 * r40792;
        double r40795 = 4.0;
        double r40796 = a;
        double r40797 = r40795 * r40796;
        double r40798 = c;
        double r40799 = r40797 * r40798;
        double r40800 = r40794 - r40799;
        double r40801 = sqrt(r40800);
        double r40802 = r40793 + r40801;
        double r40803 = 2.0;
        double r40804 = r40803 * r40796;
        double r40805 = r40802 / r40804;
        return r40805;
}
↓
double f(double a, double b, double c) {
        double r40806 = c;
        double r40807 = 4.0;
        double r40808 = a;
        double r40809 = r40807 * r40808;
        double r40810 = r40806 * r40809;
        double r40811 = b;
        double r40812 = -r40811;
        double r40813 = 4.0;
        double r40814 = pow(r40811, r40813);
        double r40815 = r40808 * r40806;
        double r40816 = r40815 * r40807;
        double r40817 = r40816 * r40816;
        double r40818 = r40814 - r40817;
        double r40819 = r40811 * r40811;
        double r40820 = r40819 + r40816;
        double r40821 = r40818 / r40820;
        double r40822 = sqrt(r40821);
        double r40823 = r40812 - r40822;
        double r40824 = r40810 / r40823;
        double r40825 = 2.0;
        double r40826 = r40825 * r40808;
        double r40827 = r40824 / r40826;
        return r40827;
}

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 44.2

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

  3. Applied flip-+44.2

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

  4. Simplified0.4

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

  6. Applied flip--0.4

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

  7. Simplified0.4

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

  8. Simplified0.4

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

  9. Final simplification0.4

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

Reproduce

herbie shell --seed 2019202 
(FPCore (a b c)
  :name "Quadratic roots, medium range"
  :pre (and (< 1.11022e-16 a 9.0072e15) (< 1.11022e-16 b 9.0072e15) (< 1.11022e-16 c 9.0072e15))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))