Average Error: 32.6 → 9.7
Time: 27.6s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -6.90131991727783 \cdot 10^{-39}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 4.012768074517757 \cdot 10^{+87}:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4}\right) \cdot \frac{\frac{1}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array}\]
\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 -6.90131991727783 \cdot 10^{-39}:\\
\;\;\;\;-\frac{c}{b}\\

\mathbf{elif}\;b \le 4.012768074517757 \cdot 10^{+87}:\\
\;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4}\right) \cdot \frac{\frac{1}{2}}{a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r2602033 = b;
        double r2602034 = -r2602033;
        double r2602035 = r2602033 * r2602033;
        double r2602036 = 4.0;
        double r2602037 = a;
        double r2602038 = c;
        double r2602039 = r2602037 * r2602038;
        double r2602040 = r2602036 * r2602039;
        double r2602041 = r2602035 - r2602040;
        double r2602042 = sqrt(r2602041);
        double r2602043 = r2602034 - r2602042;
        double r2602044 = 2.0;
        double r2602045 = r2602044 * r2602037;
        double r2602046 = r2602043 / r2602045;
        return r2602046;
}
↓
double f(double a, double b, double c) {
        double r2602047 = b;
        double r2602048 = -6.90131991727783e-39;
        bool r2602049 = r2602047 <= r2602048;
        double r2602050 = c;
        double r2602051 = r2602050 / r2602047;
        double r2602052 = -r2602051;
        double r2602053 = 4.012768074517757e+87;
        bool r2602054 = r2602047 <= r2602053;
        double r2602055 = -r2602047;
        double r2602056 = r2602047 * r2602047;
        double r2602057 = a;
        double r2602058 = r2602050 * r2602057;
        double r2602059 = 4.0;
        double r2602060 = r2602058 * r2602059;
        double r2602061 = r2602056 - r2602060;
        double r2602062 = sqrt(r2602061);
        double r2602063 = r2602055 - r2602062;
        double r2602064 = 0.5;
        double r2602065 = r2602064 / r2602057;
        double r2602066 = r2602063 * r2602065;
        double r2602067 = r2602055 / r2602057;
        double r2602068 = r2602054 ? r2602066 : r2602067;
        double r2602069 = r2602049 ? r2602052 : r2602068;
        return r2602069;
}

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

Original32.6
Target20.0
Herbie9.7
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -6.90131991727783e-39

    1. Initial program 53.0

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

    3. Applied div-inv53.1

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

    4. Simplified53.1

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

    5. Taylor expanded around -inf 7.9

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

    6. Simplified7.9

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

    if -6.90131991727783e-39 < b < 4.012768074517757e+87

    1. Initial program 13.2

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

    3. Applied div-inv13.3

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

    4. Simplified13.3

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

    if 4.012768074517757e+87 < b

    1. Initial program 41.3

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

    3. Applied *-un-lft-identity41.3

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

    4. Applied *-un-lft-identity41.3

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

    5. Applied distribute-lft-out--41.3

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

    6. Applied associate-/l*41.3

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

    7. Simplified41.3

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

    8. Taylor expanded around 0 3.3

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

    9. Simplified3.3

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

  4. Final simplification9.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -6.90131991727783 \cdot 10^{-39}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 4.012768074517757 \cdot 10^{+87}:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4}\right) \cdot \frac{\frac{1}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019132 +o rules:numerics
(FPCore (a b c)
  :name "quadm (p42, negative)"

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