Average Error: 28.5 → 16.5
Time: 42.4s
Precision: 64
\[1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt a \lt 94906265.62425155937671661376953125 \land 1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt b \lt 94906265.62425155937671661376953125 \land 1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt c \lt 94906265.62425155937671661376953125\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 5152.464935290847279247827827930450439453:\\ \;\;\;\;\frac{\frac{\frac{\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - \left(b \cdot b\right) \cdot b}{\left(b \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} + \left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)\right) + b \cdot b}}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \le 5152.464935290847279247827827930450439453:\\
\;\;\;\;\frac{\frac{\frac{\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - \left(b \cdot b\right) \cdot b}{\left(b \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} + \left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)\right) + b \cdot b}}{a}}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\

\end{array}
double f(double a, double b, double c) {
        double r1862065 = b;
        double r1862066 = -r1862065;
        double r1862067 = r1862065 * r1862065;
        double r1862068 = 4.0;
        double r1862069 = a;
        double r1862070 = r1862068 * r1862069;
        double r1862071 = c;
        double r1862072 = r1862070 * r1862071;
        double r1862073 = r1862067 - r1862072;
        double r1862074 = sqrt(r1862073);
        double r1862075 = r1862066 + r1862074;
        double r1862076 = 2.0;
        double r1862077 = r1862076 * r1862069;
        double r1862078 = r1862075 / r1862077;
        return r1862078;
}
↓
double f(double a, double b, double c) {
        double r1862079 = b;
        double r1862080 = 5152.464935290847;
        bool r1862081 = r1862079 <= r1862080;
        double r1862082 = r1862079 * r1862079;
        double r1862083 = c;
        double r1862084 = a;
        double r1862085 = 4.0;
        double r1862086 = r1862084 * r1862085;
        double r1862087 = r1862083 * r1862086;
        double r1862088 = r1862082 - r1862087;
        double r1862089 = sqrt(r1862088);
        double r1862090 = r1862088 * r1862089;
        double r1862091 = r1862082 * r1862079;
        double r1862092 = r1862090 - r1862091;
        double r1862093 = r1862079 * r1862089;
        double r1862094 = r1862093 + r1862088;
        double r1862095 = r1862094 + r1862082;
        double r1862096 = r1862092 / r1862095;
        double r1862097 = r1862096 / r1862084;
        double r1862098 = 2.0;
        double r1862099 = r1862097 / r1862098;
        double r1862100 = -2.0;
        double r1862101 = r1862083 / r1862079;
        double r1862102 = r1862100 * r1862101;
        double r1862103 = r1862102 / r1862098;
        double r1862104 = r1862081 ? r1862099 : r1862103;
        return r1862104;
}

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. Split input into 2 regimes

  2. if b < 5152.464935290847

    1. Initial program 18.7

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

    2. Simplified18.7

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

    4. Applied flip3--18.9

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

    5. Simplified18.1

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

    6. Simplified18.1

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

    if 5152.464935290847 < b

    1. Initial program 37.9

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

    2. Simplified37.9

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

    3. Taylor expanded around inf 15.0

      \[\leadsto \frac{\color{blue}{-2 \cdot \frac{c}{b}}}{2}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification16.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 5152.464935290847279247827827930450439453:\\ \;\;\;\;\frac{\frac{\frac{\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - \left(b \cdot b\right) \cdot b}{\left(b \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} + \left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)\right) + b \cdot b}}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 
(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)))