Average Error: 52.6 → 0.1
Time: 6.1s
Precision: 64
\[4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt a \lt 20282409603651670423947251286016 \land 4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt b \lt 20282409603651670423947251286016 \land 4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt c \lt 20282409603651670423947251286016\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}
double f(double a, double b, double c) {
        double r32017 = b;
        double r32018 = -r32017;
        double r32019 = r32017 * r32017;
        double r32020 = 4.0;
        double r32021 = a;
        double r32022 = r32020 * r32021;
        double r32023 = c;
        double r32024 = r32022 * r32023;
        double r32025 = r32019 - r32024;
        double r32026 = sqrt(r32025);
        double r32027 = r32018 + r32026;
        double r32028 = 2.0;
        double r32029 = r32028 * r32021;
        double r32030 = r32027 / r32029;
        return r32030;
}
↓
double f(double a, double b, double c) {
        double r32031 = 2.0;
        double r32032 = c;
        double r32033 = r32031 * r32032;
        double r32034 = b;
        double r32035 = -r32034;
        double r32036 = r32034 * r32034;
        double r32037 = 4.0;
        double r32038 = a;
        double r32039 = r32037 * r32038;
        double r32040 = r32039 * r32032;
        double r32041 = r32036 - r32040;
        double r32042 = sqrt(r32041);
        double r32043 = r32035 - r32042;
        double r32044 = r32033 / r32043;
        return r32044;
}

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 52.6

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

  3. Applied flip-+52.6

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

  4. Simplified0.4

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

  6. Applied div-inv0.5

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

  8. Applied *-un-lft-identity0.5

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

  9. Applied associate-*l*0.5

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

  10. Simplified0.2

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

  11. Taylor expanded around 0 0.1

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

  12. Final simplification0.1

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

Reproduce

herbie shell --seed 2019347 
(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)))