Average Error: 43.6 → 0.5
Time: 21.8s
Precision: 64
\[1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt a \lt 9007199254740992 \land 1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt b \lt 9007199254740992 \land 1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt c \lt 9007199254740992\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{\left(a \cdot c\right) \cdot 4}{2} \cdot \frac{\frac{1}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{a}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\frac{\left(a \cdot c\right) \cdot 4}{2} \cdot \frac{\frac{1}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{a}
double f(double a, double b, double c) {
        double r48154 = b;
        double r48155 = -r48154;
        double r48156 = r48154 * r48154;
        double r48157 = 4.0;
        double r48158 = a;
        double r48159 = r48157 * r48158;
        double r48160 = c;
        double r48161 = r48159 * r48160;
        double r48162 = r48156 - r48161;
        double r48163 = sqrt(r48162);
        double r48164 = r48155 + r48163;
        double r48165 = 2.0;
        double r48166 = r48165 * r48158;
        double r48167 = r48164 / r48166;
        return r48167;
}
↓
double f(double a, double b, double c) {
        double r48168 = a;
        double r48169 = c;
        double r48170 = r48168 * r48169;
        double r48171 = 4.0;
        double r48172 = r48170 * r48171;
        double r48173 = 2.0;
        double r48174 = r48172 / r48173;
        double r48175 = 1.0;
        double r48176 = b;
        double r48177 = -r48176;
        double r48178 = r48176 * r48176;
        double r48179 = r48171 * r48168;
        double r48180 = r48179 * r48169;
        double r48181 = r48178 - r48180;
        double r48182 = sqrt(r48181);
        double r48183 = r48177 - r48182;
        double r48184 = r48175 / r48183;
        double r48185 = r48184 / r48168;
        double r48186 = r48174 * r48185;
        return r48186;
}

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 43.6

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

  3. Applied flip-+43.6

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

  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 div-inv0.5

    \[\leadsto \frac{\color{blue}{\left(0 + \left(4 \cdot a\right) \cdot c\right) \cdot \frac{1}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}{2 \cdot a}\]

  7. Applied times-frac0.5

    \[\leadsto \color{blue}{\frac{0 + \left(4 \cdot a\right) \cdot c}{2} \cdot \frac{\frac{1}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{a}}\]

  8. Simplified0.5

    \[\leadsto \color{blue}{\frac{\left(a \cdot c\right) \cdot 4}{2}} \cdot \frac{\frac{1}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{a}\]

  9. Final simplification0.5

    \[\leadsto \frac{\left(a \cdot c\right) \cdot 4}{2} \cdot \frac{\frac{1}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{a}\]

Reproduce

herbie shell --seed 2019323 
(FPCore (a b c)
  :name "Quadratic roots, medium range"
  :precision binary64
  :pre (and (< 1.11022e-16 a 9.0072e+15) (< 1.11022e-16 b 9.0072e+15) (< 1.11022e-16 c 9.0072e+15))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))