Average Error: 52.6 → 0.4
Time: 22.4s
Precision: 64
\[4.93038 \cdot 10^{-32} \lt a \lt 2.02824 \cdot 10^{31} \land 4.93038 \cdot 10^{-32} \lt b \lt 2.02824 \cdot 10^{31} \land 4.93038 \cdot 10^{-32} \lt c \lt 2.02824 \cdot 10^{31}\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{\frac{\left(4 \cdot a\right) \cdot c}{\left(-b\right) - \sqrt{\frac{\frac{{b}^{4} - \left(\left(a \cdot c\right) \cdot 4\right) \cdot \left(\left(a \cdot c\right) \cdot 4\right)}{\sqrt{b \cdot b + \left(a \cdot c\right) \cdot 4}}}{\sqrt{b \cdot b + \left(a \cdot c\right) \cdot 4}}}}}{2 \cdot a}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\frac{\frac{\left(4 \cdot a\right) \cdot c}{\left(-b\right) - \sqrt{\frac{\frac{{b}^{4} - \left(\left(a \cdot c\right) \cdot 4\right) \cdot \left(\left(a \cdot c\right) \cdot 4\right)}{\sqrt{b \cdot b + \left(a \cdot c\right) \cdot 4}}}{\sqrt{b \cdot b + \left(a \cdot c\right) \cdot 4}}}}}{2 \cdot a}
double f(double a, double b, double c) {
        double r38264 = b;
        double r38265 = -r38264;
        double r38266 = r38264 * r38264;
        double r38267 = 4.0;
        double r38268 = a;
        double r38269 = r38267 * r38268;
        double r38270 = c;
        double r38271 = r38269 * r38270;
        double r38272 = r38266 - r38271;
        double r38273 = sqrt(r38272);
        double r38274 = r38265 + r38273;
        double r38275 = 2.0;
        double r38276 = r38275 * r38268;
        double r38277 = r38274 / r38276;
        return r38277;
}
↓
double f(double a, double b, double c) {
        double r38278 = 4.0;
        double r38279 = a;
        double r38280 = r38278 * r38279;
        double r38281 = c;
        double r38282 = r38280 * r38281;
        double r38283 = b;
        double r38284 = -r38283;
        double r38285 = 4.0;
        double r38286 = pow(r38283, r38285);
        double r38287 = r38279 * r38281;
        double r38288 = r38287 * r38278;
        double r38289 = r38288 * r38288;
        double r38290 = r38286 - r38289;
        double r38291 = r38283 * r38283;
        double r38292 = r38291 + r38288;
        double r38293 = sqrt(r38292);
        double r38294 = r38290 / r38293;
        double r38295 = r38294 / r38293;
        double r38296 = sqrt(r38295);
        double r38297 = r38284 - r38296;
        double r38298 = r38282 / r38297;
        double r38299 = 2.0;
        double r38300 = r38299 * r38279;
        double r38301 = r38298 / r38300;
        return r38301;
}

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

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

  10. Applied add-sqr-sqrt0.4

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

  11. Applied associate-/r*0.4

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

  12. Final simplification0.4

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

Reproduce

herbie shell --seed 2019195 
(FPCore (a b c)
  :name "Quadratic roots, wide range"
  :pre (and (< 4.930380657631324e-32 a 2.028240960365167e+31) (< 4.930380657631324e-32 b 2.028240960365167e+31) (< 4.930380657631324e-32 c 2.028240960365167e+31))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))