Average Error: 34.3 → 6.8
Time: 17.1s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.336444958885988609650625284454613720554 \cdot 10^{152}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 2.866557489300444694985602640640979475553 \cdot 10^{-301}:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\ \mathbf{elif}\;b \le 2.360908374051695590422690701366520544893 \cdot 10^{86}:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\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 -3.336444958885988609650625284454613720554 \cdot 10^{152}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 2.866557489300444694985602640640979475553 \cdot 10^{-301}:\\
\;\;\;\;\frac{c \cdot 2}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\

\mathbf{elif}\;b \le 2.360908374051695590422690701366520544893 \cdot 10^{86}:\\
\;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{a \cdot 2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r3440952 = b;
        double r3440953 = -r3440952;
        double r3440954 = r3440952 * r3440952;
        double r3440955 = 4.0;
        double r3440956 = a;
        double r3440957 = c;
        double r3440958 = r3440956 * r3440957;
        double r3440959 = r3440955 * r3440958;
        double r3440960 = r3440954 - r3440959;
        double r3440961 = sqrt(r3440960);
        double r3440962 = r3440953 - r3440961;
        double r3440963 = 2.0;
        double r3440964 = r3440963 * r3440956;
        double r3440965 = r3440962 / r3440964;
        return r3440965;
}
↓
double f(double a, double b, double c) {
        double r3440966 = b;
        double r3440967 = -3.3364449588859886e+152;
        bool r3440968 = r3440966 <= r3440967;
        double r3440969 = -1.0;
        double r3440970 = c;
        double r3440971 = r3440970 / r3440966;
        double r3440972 = r3440969 * r3440971;
        double r3440973 = 2.8665574893004447e-301;
        bool r3440974 = r3440966 <= r3440973;
        double r3440975 = 2.0;
        double r3440976 = r3440970 * r3440975;
        double r3440977 = -r3440966;
        double r3440978 = r3440966 * r3440966;
        double r3440979 = 4.0;
        double r3440980 = a;
        double r3440981 = r3440980 * r3440970;
        double r3440982 = r3440979 * r3440981;
        double r3440983 = r3440978 - r3440982;
        double r3440984 = sqrt(r3440983);
        double r3440985 = r3440977 + r3440984;
        double r3440986 = r3440976 / r3440985;
        double r3440987 = 2.3609083740516956e+86;
        bool r3440988 = r3440966 <= r3440987;
        double r3440989 = r3440977 - r3440984;
        double r3440990 = 1.0;
        double r3440991 = r3440980 * r3440975;
        double r3440992 = r3440990 / r3440991;
        double r3440993 = r3440989 * r3440992;
        double r3440994 = 1.0;
        double r3440995 = r3440966 / r3440980;
        double r3440996 = r3440971 - r3440995;
        double r3440997 = r3440994 * r3440996;
        double r3440998 = r3440988 ? r3440993 : r3440997;
        double r3440999 = r3440974 ? r3440986 : r3440998;
        double r3441000 = r3440968 ? r3440972 : r3440999;
        return r3441000;
}

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

Original34.3
Target21.3
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 < -3.3364449588859886e+152

    1. Initial program 63.9

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

    2. Taylor expanded around -inf 1.6

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

    if -3.3364449588859886e+152 < b < 2.8665574893004447e-301

    1. Initial program 35.0

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

    3. Applied div-inv35.0

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

    5. Applied flip--35.0

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

    6. Applied associate-*l/35.0

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

    7. Simplified15.0

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

    8. Taylor expanded around 0 8.9

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

    if 2.8665574893004447e-301 < b < 2.3609083740516956e+86

    1. Initial program 9.0

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

    3. Applied div-inv9.1

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

    if 2.3609083740516956e+86 < b

    1. Initial program 44.1

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

    2. Taylor expanded around inf 3.9

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

    3. Simplified3.9

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

  4. Final simplification6.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.336444958885988609650625284454613720554 \cdot 10^{152}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 2.866557489300444694985602640640979475553 \cdot 10^{-301}:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\ \mathbf{elif}\;b \le 2.360908374051695590422690701366520544893 \cdot 10^{86}:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019170 
(FPCore (a b c)
  :name "quadm (p42, negative)"

  :herbie-target
  (if (< b 0.0) (/ c (* a (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))) (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))

  (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))