Average Error: 28.5 → 0.5
Time: 6.5s
Precision: 64
\[1.05367121277235087 \cdot 10^{-8} \lt a \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt b \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt c \lt 94906265.6242515594\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{\sqrt[3]{{\left(\frac{{b}^{4} - \left(4 \cdot \left(a \cdot c\right)\right) \cdot \left(4 \cdot \left(a \cdot c\right)\right)}{b \cdot b + \left(4 \cdot a\right) \cdot c}\right)}^{3}}}}}{2 \cdot a}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{\sqrt[3]{{\left(\frac{{b}^{4} - \left(4 \cdot \left(a \cdot c\right)\right) \cdot \left(4 \cdot \left(a \cdot c\right)\right)}{b \cdot b + \left(4 \cdot a\right) \cdot c}\right)}^{3}}}}}{2 \cdot a}
double f(double a, double b, double c) {
        double r35086 = b;
        double r35087 = -r35086;
        double r35088 = r35086 * r35086;
        double r35089 = 4.0;
        double r35090 = a;
        double r35091 = r35089 * r35090;
        double r35092 = c;
        double r35093 = r35091 * r35092;
        double r35094 = r35088 - r35093;
        double r35095 = sqrt(r35094);
        double r35096 = r35087 + r35095;
        double r35097 = 2.0;
        double r35098 = r35097 * r35090;
        double r35099 = r35096 / r35098;
        return r35099;
}
↓
double f(double a, double b, double c) {
        double r35100 = 0.0;
        double r35101 = 4.0;
        double r35102 = a;
        double r35103 = c;
        double r35104 = r35102 * r35103;
        double r35105 = r35101 * r35104;
        double r35106 = r35100 + r35105;
        double r35107 = b;
        double r35108 = -r35107;
        double r35109 = 4.0;
        double r35110 = pow(r35107, r35109);
        double r35111 = r35105 * r35105;
        double r35112 = r35110 - r35111;
        double r35113 = r35107 * r35107;
        double r35114 = r35101 * r35102;
        double r35115 = r35114 * r35103;
        double r35116 = r35113 + r35115;
        double r35117 = r35112 / r35116;
        double r35118 = 3.0;
        double r35119 = pow(r35117, r35118);
        double r35120 = cbrt(r35119);
        double r35121 = sqrt(r35120);
        double r35122 = r35108 - r35121;
        double r35123 = r35106 / r35122;
        double r35124 = 2.0;
        double r35125 = r35124 * r35102;
        double r35126 = r35123 / r35125;
        return r35126;
}

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 28.5

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

  3. Applied flip-+28.6

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

  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 flip--0.5

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

  7. Simplified0.5

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

  9. Applied add-cbrt-cube0.5

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

  10. Applied add-cbrt-cube0.5

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

  11. Applied cbrt-undiv0.5

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

  12. Simplified0.5

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

  13. Final simplification0.5

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

Reproduce

herbie shell --seed 2020021 
(FPCore (a b c)
  :name "Quadratic roots, narrow range"
  :precision binary64
  :pre (and (< 1.0536712127723509e-08 a 94906265.62425156) (< 1.0536712127723509e-08 b 94906265.62425156) (< 1.0536712127723509e-08 c 94906265.62425156))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))