Average Error: 33.9 → 6.8
Time: 17.8s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.361733299857302083043096878302889042354 \cdot 10^{105}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -3.320360656741600748358420677927629618815 \cdot 10^{-289}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - a \cdot \left(4 \cdot c\right)}}{2 \cdot a}\\ \mathbf{elif}\;b \le 6.326287366549382745037046972324082366467 \cdot 10^{74}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -1.361733299857302083043096878302889042354 \cdot 10^{105}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le -3.320360656741600748358420677927629618815 \cdot 10^{-289}:\\
\;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - a \cdot \left(4 \cdot c\right)}}{2 \cdot a}\\

\mathbf{elif}\;b \le 6.326287366549382745037046972324082366467 \cdot 10^{74}:\\
\;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\

\mathbf{else}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\end{array}
double f(double a, double b, double c) {
        double r70970 = b;
        double r70971 = -r70970;
        double r70972 = r70970 * r70970;
        double r70973 = 4.0;
        double r70974 = a;
        double r70975 = c;
        double r70976 = r70974 * r70975;
        double r70977 = r70973 * r70976;
        double r70978 = r70972 - r70977;
        double r70979 = sqrt(r70978);
        double r70980 = r70971 + r70979;
        double r70981 = 2.0;
        double r70982 = r70981 * r70974;
        double r70983 = r70980 / r70982;
        return r70983;
}
↓
double f(double a, double b, double c) {
        double r70984 = b;
        double r70985 = -1.361733299857302e+105;
        bool r70986 = r70984 <= r70985;
        double r70987 = 1.0;
        double r70988 = c;
        double r70989 = r70988 / r70984;
        double r70990 = a;
        double r70991 = r70984 / r70990;
        double r70992 = r70989 - r70991;
        double r70993 = r70987 * r70992;
        double r70994 = -3.3203606567416007e-289;
        bool r70995 = r70984 <= r70994;
        double r70996 = -r70984;
        double r70997 = r70984 * r70984;
        double r70998 = 4.0;
        double r70999 = r70998 * r70988;
        double r71000 = r70990 * r70999;
        double r71001 = r70997 - r71000;
        double r71002 = sqrt(r71001);
        double r71003 = r70996 + r71002;
        double r71004 = 2.0;
        double r71005 = r71004 * r70990;
        double r71006 = r71003 / r71005;
        double r71007 = 6.326287366549383e+74;
        bool r71008 = r70984 <= r71007;
        double r71009 = r71004 * r70988;
        double r71010 = r70990 * r70988;
        double r71011 = r70998 * r71010;
        double r71012 = r70997 - r71011;
        double r71013 = sqrt(r71012);
        double r71014 = r70996 - r71013;
        double r71015 = r71009 / r71014;
        double r71016 = -1.0;
        double r71017 = r71016 * r70989;
        double r71018 = r71008 ? r71015 : r71017;
        double r71019 = r70995 ? r71006 : r71018;
        double r71020 = r70986 ? r70993 : r71019;
        return r71020;
}

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

Original33.9
Target21.1
Herbie6.8
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -1.361733299857302e+105

    1. Initial program 48.6

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

    2. Taylor expanded around -inf 3.6

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

    3. Simplified3.6

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

    if -1.361733299857302e+105 < b < -3.3203606567416007e-289

    1. Initial program 8.5

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

    2. Taylor expanded around 0 8.5

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

    3. Simplified8.5

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

    if -3.3203606567416007e-289 < b < 6.326287366549383e+74

    1. Initial program 29.9

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

    3. Applied flip-+29.9

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

    4. Simplified16.2

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

    6. Applied *-un-lft-identity16.2

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

    7. Applied *-un-lft-identity16.2

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

    8. Applied times-frac16.2

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

    9. Applied associate-/l*16.3

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

    10. Simplified15.8

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

    12. Applied associate-/r*15.6

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

    13. Simplified9.4

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

    14. 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 6.326287366549383e+74 < b

    1. Initial program 58.0

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

    2. Taylor expanded around inf 3.7

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

  4. Final simplification6.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.361733299857302083043096878302889042354 \cdot 10^{105}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -3.320360656741600748358420677927629618815 \cdot 10^{-289}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - a \cdot \left(4 \cdot c\right)}}{2 \cdot a}\\ \mathbf{elif}\;b \le 6.326287366549382745037046972324082366467 \cdot 10^{74}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019322 
(FPCore (a b c)
  :name "quadp (p42, positive)"
  :precision binary64

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

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