Average Error: 34.0 → 6.8
Time: 5.2s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -8.7237121667270365 \cdot 10^{113}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -1.30082212909031857 \cdot 10^{-300}:\\ \;\;\;\;\frac{1}{\frac{0.5}{c} \cdot \left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}\\ \mathbf{elif}\;b \le 2.1545230570852197 \cdot 10^{80}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\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 -8.7237121667270365 \cdot 10^{113}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le -1.30082212909031857 \cdot 10^{-300}:\\
\;\;\;\;\frac{1}{\frac{0.5}{c} \cdot \left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}\\

\mathbf{elif}\;b \le 2.1545230570852197 \cdot 10^{80}:\\
\;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\\

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

\end{array}
double f(double a, double b, double c) {
        double r79967 = b;
        double r79968 = -r79967;
        double r79969 = r79967 * r79967;
        double r79970 = 4.0;
        double r79971 = a;
        double r79972 = c;
        double r79973 = r79971 * r79972;
        double r79974 = r79970 * r79973;
        double r79975 = r79969 - r79974;
        double r79976 = sqrt(r79975);
        double r79977 = r79968 - r79976;
        double r79978 = 2.0;
        double r79979 = r79978 * r79971;
        double r79980 = r79977 / r79979;
        return r79980;
}
↓
double f(double a, double b, double c) {
        double r79981 = b;
        double r79982 = -8.723712166727036e+113;
        bool r79983 = r79981 <= r79982;
        double r79984 = -1.0;
        double r79985 = c;
        double r79986 = r79985 / r79981;
        double r79987 = r79984 * r79986;
        double r79988 = -1.3008221290903186e-300;
        bool r79989 = r79981 <= r79988;
        double r79990 = 1.0;
        double r79991 = 0.5;
        double r79992 = r79991 / r79985;
        double r79993 = -r79981;
        double r79994 = r79981 * r79981;
        double r79995 = 4.0;
        double r79996 = a;
        double r79997 = r79996 * r79985;
        double r79998 = r79995 * r79997;
        double r79999 = r79994 - r79998;
        double r80000 = sqrt(r79999);
        double r80001 = r79993 + r80000;
        double r80002 = r79992 * r80001;
        double r80003 = r79990 / r80002;
        double r80004 = 2.1545230570852197e+80;
        bool r80005 = r79981 <= r80004;
        double r80006 = 2.0;
        double r80007 = r80006 * r79996;
        double r80008 = r79993 - r80000;
        double r80009 = r80007 / r80008;
        double r80010 = r79990 / r80009;
        double r80011 = 1.0;
        double r80012 = r79981 / r79996;
        double r80013 = r79986 - r80012;
        double r80014 = r80011 * r80013;
        double r80015 = r80005 ? r80010 : r80014;
        double r80016 = r79989 ? r80003 : r80015;
        double r80017 = r79983 ? r79987 : r80016;
        return r80017;
}

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

Original34.0
Target21.0
Herbie6.8
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -8.723712166727036e+113

    1. Initial program 60.6

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

    2. Taylor expanded around -inf 2.2

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

    if -8.723712166727036e+113 < b < -1.3008221290903186e-300

    1. Initial program 32.5

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

    3. Applied clear-num32.5

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

    5. Applied flip--32.5

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

    6. Applied associate-/r/32.6

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

    7. Simplified15.3

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

    8. Taylor expanded around 0 9.2

      \[\leadsto \frac{1}{\color{blue}{\frac{0.5}{c}} \cdot \left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}\]

    if -1.3008221290903186e-300 < b < 2.1545230570852197e+80

    1. Initial program 9.2

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

    3. Applied clear-num9.4

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

    if 2.1545230570852197e+80 < b

    1. Initial program 42.9

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

    2. Taylor expanded around inf 4.0

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

    3. Simplified4.0

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

  4. Final simplification6.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -8.7237121667270365 \cdot 10^{113}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -1.30082212909031857 \cdot 10^{-300}:\\ \;\;\;\;\frac{1}{\frac{0.5}{c} \cdot \left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}\\ \mathbf{elif}\;b \le 2.1545230570852197 \cdot 10^{80}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020021 
(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)))