Average Error: 28.1 → 0.5
Time: 20.2s
Precision: 64
\[1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt a \lt 94906265.62425155937671661376953125 \land 1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt b \lt 94906265.62425155937671661376953125 \land 1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt c \lt 94906265.62425155937671661376953125\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{\frac{1}{\frac{-1}{a}}}{\frac{\left(a \cdot 2\right) \cdot \frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} + b}{4}}{c}}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\frac{\frac{1}{\frac{-1}{a}}}{\frac{\left(a \cdot 2\right) \cdot \frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} + b}{4}}{c}}
double f(double a, double b, double c) {
        double r51928 = b;
        double r51929 = -r51928;
        double r51930 = r51928 * r51928;
        double r51931 = 4.0;
        double r51932 = a;
        double r51933 = r51931 * r51932;
        double r51934 = c;
        double r51935 = r51933 * r51934;
        double r51936 = r51930 - r51935;
        double r51937 = sqrt(r51936);
        double r51938 = r51929 + r51937;
        double r51939 = 2.0;
        double r51940 = r51939 * r51932;
        double r51941 = r51938 / r51940;
        return r51941;
}
↓
double f(double a, double b, double c) {
        double r51942 = 1.0;
        double r51943 = -1.0;
        double r51944 = a;
        double r51945 = r51943 / r51944;
        double r51946 = r51942 / r51945;
        double r51947 = 2.0;
        double r51948 = r51944 * r51947;
        double r51949 = b;
        double r51950 = r51949 * r51949;
        double r51951 = 4.0;
        double r51952 = r51944 * r51951;
        double r51953 = c;
        double r51954 = r51952 * r51953;
        double r51955 = r51950 - r51954;
        double r51956 = sqrt(r51955);
        double r51957 = r51956 + r51949;
        double r51958 = r51957 / r51951;
        double r51959 = r51948 * r51958;
        double r51960 = r51959 / r51953;
        double r51961 = r51946 / r51960;
        return r51961;
}

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 28.1

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

  3. Applied flip-+28.1

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

  4. Simplified0.5

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

  5. Simplified0.5

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

  7. Applied clear-num0.5

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

  8. Simplified0.5

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

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

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

  11. Applied neg-mul-10.5

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

  12. Applied times-frac0.6

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

  13. Applied times-frac0.6

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

  14. Applied add-cube-cbrt0.6

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

  15. Applied times-frac0.5

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

  16. Simplified0.5

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

  17. Simplified0.5

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

  19. Applied associate-/l*0.5

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

  20. Simplified0.5

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

  21. Final simplification0.5

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

Reproduce

herbie shell --seed 2019179 
(FPCore (a b c)
  :name "Quadratic roots, narrow range"
  :pre (and (< 1.0536712127723509e-08 a 94906265.62425156) (< 1.0536712127723509e-08 b 94906265.62425156) (< 1.0536712127723509e-08 c 94906265.62425156))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))