Average Error: 43.7 → 0.4
Time: 6.7s
Precision: 64
\[1.11022 \cdot 10^{-16} \lt a \lt 9.0072 \cdot 10^{15} \land 1.11022 \cdot 10^{-16} \lt b \lt 9.0072 \cdot 10^{15} \land 1.11022 \cdot 10^{-16} \lt c \lt 9.0072 \cdot 10^{15}\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(2 \cdot a\right) \cdot \left(-b\right) + \left(2 \cdot a\right) \cdot \left(-\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(2 \cdot a\right) \cdot \left(-b\right) + \left(2 \cdot a\right) \cdot \left(-\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}
double f(double a, double b, double c) {
        double r33235 = b;
        double r33236 = -r33235;
        double r33237 = r33235 * r33235;
        double r33238 = 4.0;
        double r33239 = a;
        double r33240 = r33238 * r33239;
        double r33241 = c;
        double r33242 = r33240 * r33241;
        double r33243 = r33237 - r33242;
        double r33244 = sqrt(r33243);
        double r33245 = r33236 + r33244;
        double r33246 = 2.0;
        double r33247 = r33246 * r33239;
        double r33248 = r33245 / r33247;
        return r33248;
}
↓
double f(double a, double b, double c) {
        double r33249 = 0.0;
        double r33250 = 4.0;
        double r33251 = a;
        double r33252 = c;
        double r33253 = r33251 * r33252;
        double r33254 = r33250 * r33253;
        double r33255 = r33249 + r33254;
        double r33256 = 2.0;
        double r33257 = r33256 * r33251;
        double r33258 = b;
        double r33259 = -r33258;
        double r33260 = r33257 * r33259;
        double r33261 = r33258 * r33258;
        double r33262 = r33250 * r33251;
        double r33263 = r33262 * r33252;
        double r33264 = r33261 - r33263;
        double r33265 = sqrt(r33264);
        double r33266 = -r33265;
        double r33267 = r33257 * r33266;
        double r33268 = r33260 + r33267;
        double r33269 = r33255 / r33268;
        return r33269;
}

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 43.7

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

  3. Applied flip-+43.7

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

  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

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

  7. Applied associate-/l*0.5

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

  8. Simplified0.4

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

  10. Applied sub-neg0.4

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

  11. Applied distribute-lft-in0.4

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

  12. Final simplification0.4

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

Reproduce

herbie shell --seed 2020100 
(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)))