Average Error: 32.9 → 6.4
Time: 59.1s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.880394710329243 \cdot 10^{+120}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 5.818192251940127 \cdot 10^{-227}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b}{a \cdot 2}\\ \mathbf{elif}\;b \le 6.6006279600139335 \cdot 10^{+131}:\\ \;\;\;\;\left(c \cdot -2\right) \cdot \frac{1}{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} + b}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]
\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 -2.880394710329243 \cdot 10^{+120}:\\
\;\;\;\;\frac{c}{b} - \frac{b}{a}\\

\mathbf{elif}\;b \le 5.818192251940127 \cdot 10^{-227}:\\
\;\;\;\;\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b}{a \cdot 2}\\

\mathbf{elif}\;b \le 6.6006279600139335 \cdot 10^{+131}:\\
\;\;\;\;\left(c \cdot -2\right) \cdot \frac{1}{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} + b}\\

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

\end{array}
double f(double a, double b, double c) {
        double r9571944 = b;
        double r9571945 = -r9571944;
        double r9571946 = r9571944 * r9571944;
        double r9571947 = 4.0;
        double r9571948 = a;
        double r9571949 = r9571947 * r9571948;
        double r9571950 = c;
        double r9571951 = r9571949 * r9571950;
        double r9571952 = r9571946 - r9571951;
        double r9571953 = sqrt(r9571952);
        double r9571954 = r9571945 + r9571953;
        double r9571955 = 2.0;
        double r9571956 = r9571955 * r9571948;
        double r9571957 = r9571954 / r9571956;
        return r9571957;
}
↓
double f(double a, double b, double c) {
        double r9571958 = b;
        double r9571959 = -2.880394710329243e+120;
        bool r9571960 = r9571958 <= r9571959;
        double r9571961 = c;
        double r9571962 = r9571961 / r9571958;
        double r9571963 = a;
        double r9571964 = r9571958 / r9571963;
        double r9571965 = r9571962 - r9571964;
        double r9571966 = 5.818192251940127e-227;
        bool r9571967 = r9571958 <= r9571966;
        double r9571968 = r9571958 * r9571958;
        double r9571969 = r9571961 * r9571963;
        double r9571970 = 4.0;
        double r9571971 = r9571969 * r9571970;
        double r9571972 = r9571968 - r9571971;
        double r9571973 = sqrt(r9571972);
        double r9571974 = r9571973 - r9571958;
        double r9571975 = 2.0;
        double r9571976 = r9571963 * r9571975;
        double r9571977 = r9571974 / r9571976;
        double r9571978 = 6.6006279600139335e+131;
        bool r9571979 = r9571958 <= r9571978;
        double r9571980 = -2.0;
        double r9571981 = r9571961 * r9571980;
        double r9571982 = 1.0;
        double r9571983 = r9571973 + r9571958;
        double r9571984 = r9571982 / r9571983;
        double r9571985 = r9571981 * r9571984;
        double r9571986 = -r9571962;
        double r9571987 = r9571979 ? r9571985 : r9571986;
        double r9571988 = r9571967 ? r9571977 : r9571987;
        double r9571989 = r9571960 ? r9571965 : r9571988;
        return r9571989;
}

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

Original32.9
Target20.3
Herbie6.4
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -2.880394710329243e+120

    1. Initial program 49.1

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

    2. Simplified49.1

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

    3. Taylor expanded around -inf 2.7

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

    if -2.880394710329243e+120 < b < 5.818192251940127e-227

    1. Initial program 9.3

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

    2. Simplified9.3

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

    if 5.818192251940127e-227 < b < 6.6006279600139335e+131

    1. Initial program 35.9

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

    2. Simplified35.9

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

    4. Applied *-un-lft-identity35.9

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

    5. Applied *-un-lft-identity35.9

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

    6. Applied distribute-lft-out--35.9

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

    7. Applied associate-/l*35.9

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

    9. Applied flip--36.0

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

    10. Applied associate-/r/36.1

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

    11. Applied *-un-lft-identity36.1

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

    12. Applied times-frac36.1

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

    13. Simplified13.6

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

    14. Taylor expanded around inf 7.3

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

    if 6.6006279600139335e+131 < b

    1. Initial program 60.5

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

    2. Simplified60.5

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

    3. Taylor expanded around inf 2.2

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

    4. Simplified2.2

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

  4. Final simplification6.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.880394710329243 \cdot 10^{+120}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 5.818192251940127 \cdot 10^{-227}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b}{a \cdot 2}\\ \mathbf{elif}\;b \le 6.6006279600139335 \cdot 10^{+131}:\\ \;\;\;\;\left(c \cdot -2\right) \cdot \frac{1}{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} + b}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019124 
(FPCore (a b c)
  :name "The quadratic formula (r1)"

  :herbie-target
  (if (< b 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)))