Average Error: 33.4 → 9.9
Time: 22.1s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -5.148407540792454 \cdot 10^{+110}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 2.326372645943808 \cdot 10^{-74}:\\ \;\;\;\;\frac{\frac{1}{a} \cdot \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - \frac{b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]
\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 -5.148407540792454 \cdot 10^{+110}:\\
\;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\

\mathbf{elif}\;b \le 2.326372645943808 \cdot 10^{-74}:\\
\;\;\;\;\frac{\frac{1}{a} \cdot \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - \frac{b}{a}}{2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r1545033 = b;
        double r1545034 = -r1545033;
        double r1545035 = r1545033 * r1545033;
        double r1545036 = 4.0;
        double r1545037 = a;
        double r1545038 = r1545036 * r1545037;
        double r1545039 = c;
        double r1545040 = r1545038 * r1545039;
        double r1545041 = r1545035 - r1545040;
        double r1545042 = sqrt(r1545041);
        double r1545043 = r1545034 + r1545042;
        double r1545044 = 2.0;
        double r1545045 = r1545044 * r1545037;
        double r1545046 = r1545043 / r1545045;
        return r1545046;
}
↓
double f(double a, double b, double c) {
        double r1545047 = b;
        double r1545048 = -5.148407540792454e+110;
        bool r1545049 = r1545047 <= r1545048;
        double r1545050 = c;
        double r1545051 = r1545050 / r1545047;
        double r1545052 = a;
        double r1545053 = r1545047 / r1545052;
        double r1545054 = r1545051 - r1545053;
        double r1545055 = 2.0;
        double r1545056 = r1545054 * r1545055;
        double r1545057 = r1545056 / r1545055;
        double r1545058 = 2.326372645943808e-74;
        bool r1545059 = r1545047 <= r1545058;
        double r1545060 = 1.0;
        double r1545061 = r1545060 / r1545052;
        double r1545062 = r1545047 * r1545047;
        double r1545063 = 4.0;
        double r1545064 = r1545052 * r1545050;
        double r1545065 = r1545063 * r1545064;
        double r1545066 = r1545062 - r1545065;
        double r1545067 = sqrt(r1545066);
        double r1545068 = r1545061 * r1545067;
        double r1545069 = r1545068 - r1545053;
        double r1545070 = r1545069 / r1545055;
        double r1545071 = -2.0;
        double r1545072 = r1545071 * r1545051;
        double r1545073 = r1545072 / r1545055;
        double r1545074 = r1545059 ? r1545070 : r1545073;
        double r1545075 = r1545049 ? r1545057 : r1545074;
        return r1545075;
}

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 3 regimes

  2. if b < -5.148407540792454e+110

    1. Initial program 46.9

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

    2. Simplified46.9

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

    4. Applied div-sub46.9

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

    6. Applied div-inv47.0

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

    8. Applied add-sqr-sqrt47.0

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

    9. Applied associate-*l*47.0

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

    10. Simplified47.0

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

    11. Taylor expanded around -inf 3.6

      \[\leadsto \frac{\color{blue}{2 \cdot \frac{c}{b} - 2 \cdot \frac{b}{a}}}{2}\]

    12. Simplified3.6

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

    if -5.148407540792454e+110 < b < 2.326372645943808e-74

    1. Initial program 12.8

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

    2. Simplified12.7

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

    4. Applied div-sub12.7

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

    6. Applied div-inv12.8

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

    if 2.326372645943808e-74 < b

    1. Initial program 52.5

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

    2. Simplified52.5

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

    3. Taylor expanded around inf 8.8

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

  4. Final simplification9.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -5.148407540792454 \cdot 10^{+110}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 2.326372645943808 \cdot 10^{-74}:\\ \;\;\;\;\frac{\frac{1}{a} \cdot \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - \frac{b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019151 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))