Average Error: 33.8 → 6.5
Time: 15.3s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -9.155327002621879240971632151563348379742 \cdot 10^{117}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 3.622962172940872725615087320648120303101 \cdot 10^{-206}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\ \mathbf{elif}\;b \le 5.179201385701060896596521886473852187744 \cdot 10^{98}:\\ \;\;\;\;\frac{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \le -9.155327002621879240971632151563348379742 \cdot 10^{117}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 3.622962172940872725615087320648120303101 \cdot 10^{-206}:\\
\;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\

\mathbf{elif}\;b \le 5.179201385701060896596521886473852187744 \cdot 10^{98}:\\
\;\;\;\;\frac{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2}}{a}\\

\mathbf{else}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\end{array}
double f(double a, double b, double c) {
        double r78008 = b;
        double r78009 = -r78008;
        double r78010 = r78008 * r78008;
        double r78011 = 4.0;
        double r78012 = a;
        double r78013 = c;
        double r78014 = r78012 * r78013;
        double r78015 = r78011 * r78014;
        double r78016 = r78010 - r78015;
        double r78017 = sqrt(r78016);
        double r78018 = r78009 - r78017;
        double r78019 = 2.0;
        double r78020 = r78019 * r78012;
        double r78021 = r78018 / r78020;
        return r78021;
}
↓
double f(double a, double b, double c) {
        double r78022 = b;
        double r78023 = -9.155327002621879e+117;
        bool r78024 = r78022 <= r78023;
        double r78025 = -1.0;
        double r78026 = c;
        double r78027 = r78026 / r78022;
        double r78028 = r78025 * r78027;
        double r78029 = 3.622962172940873e-206;
        bool r78030 = r78022 <= r78029;
        double r78031 = 2.0;
        double r78032 = r78031 * r78026;
        double r78033 = -r78022;
        double r78034 = r78022 * r78022;
        double r78035 = 4.0;
        double r78036 = a;
        double r78037 = r78036 * r78026;
        double r78038 = r78035 * r78037;
        double r78039 = r78034 - r78038;
        double r78040 = sqrt(r78039);
        double r78041 = r78033 + r78040;
        double r78042 = r78032 / r78041;
        double r78043 = 5.179201385701061e+98;
        bool r78044 = r78022 <= r78043;
        double r78045 = r78033 - r78040;
        double r78046 = r78045 / r78031;
        double r78047 = r78046 / r78036;
        double r78048 = 1.0;
        double r78049 = r78022 / r78036;
        double r78050 = r78027 - r78049;
        double r78051 = r78048 * r78050;
        double r78052 = r78044 ? r78047 : r78051;
        double r78053 = r78030 ? r78042 : r78052;
        double r78054 = r78024 ? r78028 : r78053;
        return r78054;
}

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

Target

Original33.8
Target20.4
Herbie6.5
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -9.155327002621879e+117

    1. Initial program 60.8

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

    2. Taylor expanded around -inf 1.8

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}}\]

    if -9.155327002621879e+117 < b < 3.622962172940873e-206

    1. Initial program 30.1

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

    3. Applied associate-/r*30.1

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

    5. Applied *-un-lft-identity30.1

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

    6. Applied div-inv30.1

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

    7. Applied times-frac30.1

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

    8. Simplified30.1

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

    10. Applied flip--30.3

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

    11. Applied associate-*l/30.3

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

    12. Simplified14.7

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

    13. Taylor expanded around 0 9.4

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

    if 3.622962172940873e-206 < b < 5.179201385701061e+98

    1. Initial program 8.1

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

    3. Applied associate-/r*8.1

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

    if 5.179201385701061e+98 < b

    1. Initial program 44.9

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

    2. Taylor expanded around inf 3.5

      \[\leadsto \color{blue}{1 \cdot \frac{c}{b} - 1 \cdot \frac{b}{a}}\]

    3. Simplified3.5

      \[\leadsto \color{blue}{1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification6.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -9.155327002621879240971632151563348379742 \cdot 10^{117}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 3.622962172940872725615087320648120303101 \cdot 10^{-206}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\ \mathbf{elif}\;b \le 5.179201385701060896596521886473852187744 \cdot 10^{98}:\\ \;\;\;\;\frac{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019298 
(FPCore (a b c)
  :name "quadm (p42, negative)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ c (* a (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))) (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))

  (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))