Average Error: 52.5 → 0.2
Time: 13.6s
Precision: 64
\[4.93038 \cdot 10^{-32} \lt a \lt 2.02824 \cdot 10^{31} \land 4.93038 \cdot 10^{-32} \lt b \lt 2.02824 \cdot 10^{31} \land 4.93038 \cdot 10^{-32} \lt c \lt 2.02824 \cdot 10^{31}\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{\frac{4 \cdot \left(a \cdot c\right)}{a}}{\left(-b\right) - \sqrt{\frac{{b}^{6} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}{\left(4 \cdot \left(a \cdot c\right)\right) \cdot \left(b \cdot b + \left(4 \cdot a\right) \cdot c\right) + {b}^{4}}}} \cdot \frac{1}{2}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\frac{\frac{4 \cdot \left(a \cdot c\right)}{a}}{\left(-b\right) - \sqrt{\frac{{b}^{6} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}{\left(4 \cdot \left(a \cdot c\right)\right) \cdot \left(b \cdot b + \left(4 \cdot a\right) \cdot c\right) + {b}^{4}}}} \cdot \frac{1}{2}
double f(double a, double b, double c) {
        double r37211 = b;
        double r37212 = -r37211;
        double r37213 = r37211 * r37211;
        double r37214 = 4.0;
        double r37215 = a;
        double r37216 = r37214 * r37215;
        double r37217 = c;
        double r37218 = r37216 * r37217;
        double r37219 = r37213 - r37218;
        double r37220 = sqrt(r37219);
        double r37221 = r37212 + r37220;
        double r37222 = 2.0;
        double r37223 = r37222 * r37215;
        double r37224 = r37221 / r37223;
        return r37224;
}
↓
double f(double a, double b, double c) {
        double r37225 = 4.0;
        double r37226 = a;
        double r37227 = c;
        double r37228 = r37226 * r37227;
        double r37229 = r37225 * r37228;
        double r37230 = r37229 / r37226;
        double r37231 = b;
        double r37232 = -r37231;
        double r37233 = 6.0;
        double r37234 = pow(r37231, r37233);
        double r37235 = r37225 * r37226;
        double r37236 = r37235 * r37227;
        double r37237 = 3.0;
        double r37238 = pow(r37236, r37237);
        double r37239 = r37234 - r37238;
        double r37240 = r37231 * r37231;
        double r37241 = r37240 + r37236;
        double r37242 = r37229 * r37241;
        double r37243 = 4.0;
        double r37244 = pow(r37231, r37243);
        double r37245 = r37242 + r37244;
        double r37246 = r37239 / r37245;
        double r37247 = sqrt(r37246);
        double r37248 = r37232 - r37247;
        double r37249 = r37230 / r37248;
        double r37250 = 1.0;
        double r37251 = 2.0;
        double r37252 = r37250 / r37251;
        double r37253 = r37249 * r37252;
        return r37253;
}

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 52.5

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

  3. Applied flip-+52.5

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

  4. Simplified0.4

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

  6. Applied flip3--0.4

    \[\leadsto \frac{\frac{4 \cdot \left(a \cdot c\right) + b \cdot \left(b - b\right)}{\left(-b\right) - \sqrt{\color{blue}{\frac{{\left(b \cdot b\right)}^{3} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}}}{2 \cdot a}\]

  7. Simplified0.4

    \[\leadsto \frac{\frac{4 \cdot \left(a \cdot c\right) + b \cdot \left(b - b\right)}{\left(-b\right) - \sqrt{\frac{\color{blue}{{b}^{6} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}}{2 \cdot a}\]

  8. Simplified0.4

    \[\leadsto \frac{\frac{4 \cdot \left(a \cdot c\right) + b \cdot \left(b - b\right)}{\left(-b\right) - \sqrt{\frac{{b}^{6} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}{\color{blue}{\left(4 \cdot \left(a \cdot c\right)\right) \cdot \left(b \cdot b + \left(4 \cdot a\right) \cdot c\right) + {b}^{4}}}}}}{2 \cdot a}\]
  9. Using strategy rm

  10. Applied *-un-lft-identity0.4

    \[\leadsto \frac{\frac{4 \cdot \left(a \cdot c\right) + b \cdot \left(b - b\right)}{\color{blue}{1 \cdot \left(\left(-b\right) - \sqrt{\frac{{b}^{6} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}{\left(4 \cdot \left(a \cdot c\right)\right) \cdot \left(b \cdot b + \left(4 \cdot a\right) \cdot c\right) + {b}^{4}}}\right)}}}{2 \cdot a}\]

  11. Applied *-un-lft-identity0.4

    \[\leadsto \frac{\frac{\color{blue}{1 \cdot \left(4 \cdot \left(a \cdot c\right) + b \cdot \left(b - b\right)\right)}}{1 \cdot \left(\left(-b\right) - \sqrt{\frac{{b}^{6} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}{\left(4 \cdot \left(a \cdot c\right)\right) \cdot \left(b \cdot b + \left(4 \cdot a\right) \cdot c\right) + {b}^{4}}}\right)}}{2 \cdot a}\]

  12. Applied times-frac0.4

    \[\leadsto \frac{\color{blue}{\frac{1}{1} \cdot \frac{4 \cdot \left(a \cdot c\right) + b \cdot \left(b - b\right)}{\left(-b\right) - \sqrt{\frac{{b}^{6} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}{\left(4 \cdot \left(a \cdot c\right)\right) \cdot \left(b \cdot b + \left(4 \cdot a\right) \cdot c\right) + {b}^{4}}}}}}{2 \cdot a}\]

  13. Applied times-frac0.4

    \[\leadsto \color{blue}{\frac{\frac{1}{1}}{2} \cdot \frac{\frac{4 \cdot \left(a \cdot c\right) + b \cdot \left(b - b\right)}{\left(-b\right) - \sqrt{\frac{{b}^{6} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}{\left(4 \cdot \left(a \cdot c\right)\right) \cdot \left(b \cdot b + \left(4 \cdot a\right) \cdot c\right) + {b}^{4}}}}}{a}}\]

  14. Simplified0.4

    \[\leadsto \color{blue}{\frac{1}{2}} \cdot \frac{\frac{4 \cdot \left(a \cdot c\right) + b \cdot \left(b - b\right)}{\left(-b\right) - \sqrt{\frac{{b}^{6} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}{\left(4 \cdot \left(a \cdot c\right)\right) \cdot \left(b \cdot b + \left(4 \cdot a\right) \cdot c\right) + {b}^{4}}}}}{a}\]

  15. Simplified0.2

    \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{\frac{4 \cdot \left(a \cdot c\right) + 0}{a}}{\left(-b\right) - \sqrt{\frac{{b}^{6} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}{\left(4 \cdot \left(a \cdot c\right)\right) \cdot \left(b \cdot b + \left(4 \cdot a\right) \cdot c\right) + {b}^{4}}}}}\]

  16. Final simplification0.2

    \[\leadsto \frac{\frac{4 \cdot \left(a \cdot c\right)}{a}}{\left(-b\right) - \sqrt{\frac{{b}^{6} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}{\left(4 \cdot \left(a \cdot c\right)\right) \cdot \left(b \cdot b + \left(4 \cdot a\right) \cdot c\right) + {b}^{4}}}} \cdot \frac{1}{2}\]

Reproduce

herbie shell --seed 2020047 
(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)))