Average Error: 34.5 → 10.4
Time: 6.8s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -8.3645547041066157 \cdot 10^{-80}:\\ \;\;\;\;\frac{1}{1 \cdot \left(\frac{a}{b} - \frac{b}{c}\right)}\\ \mathbf{elif}\;b \le 4.1199128263687574 \cdot 10^{46}:\\ \;\;\;\;\frac{1}{\frac{2}{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a}}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\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 -8.3645547041066157 \cdot 10^{-80}:\\
\;\;\;\;\frac{1}{1 \cdot \left(\frac{a}{b} - \frac{b}{c}\right)}\\

\mathbf{elif}\;b \le 4.1199128263687574 \cdot 10^{46}:\\
\;\;\;\;\frac{1}{\frac{2}{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a}}}\\

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

\end{array}
double f(double a, double b, double c) {
        double r118994 = b;
        double r118995 = -r118994;
        double r118996 = r118994 * r118994;
        double r118997 = 4.0;
        double r118998 = a;
        double r118999 = c;
        double r119000 = r118998 * r118999;
        double r119001 = r118997 * r119000;
        double r119002 = r118996 - r119001;
        double r119003 = sqrt(r119002);
        double r119004 = r118995 - r119003;
        double r119005 = 2.0;
        double r119006 = r119005 * r118998;
        double r119007 = r119004 / r119006;
        return r119007;
}
↓
double f(double a, double b, double c) {
        double r119008 = b;
        double r119009 = -8.364554704106616e-80;
        bool r119010 = r119008 <= r119009;
        double r119011 = 1.0;
        double r119012 = 1.0;
        double r119013 = a;
        double r119014 = r119013 / r119008;
        double r119015 = c;
        double r119016 = r119008 / r119015;
        double r119017 = r119014 - r119016;
        double r119018 = r119012 * r119017;
        double r119019 = r119011 / r119018;
        double r119020 = 4.1199128263687574e+46;
        bool r119021 = r119008 <= r119020;
        double r119022 = 2.0;
        double r119023 = -r119008;
        double r119024 = r119008 * r119008;
        double r119025 = 4.0;
        double r119026 = r119013 * r119015;
        double r119027 = r119025 * r119026;
        double r119028 = r119024 - r119027;
        double r119029 = sqrt(r119028);
        double r119030 = r119023 - r119029;
        double r119031 = r119030 / r119013;
        double r119032 = r119022 / r119031;
        double r119033 = r119011 / r119032;
        double r119034 = r119015 / r119008;
        double r119035 = r119008 / r119013;
        double r119036 = r119034 - r119035;
        double r119037 = r119012 * r119036;
        double r119038 = r119021 ? r119033 : r119037;
        double r119039 = r119010 ? r119019 : r119038;
        return r119039;
}

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

Original34.5
Target21.1
Herbie10.4
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -8.364554704106616e-80

    1. Initial program 53.8

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

    3. Applied clear-num53.8

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

    5. Applied associate-/l*53.8

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

    6. Taylor expanded around -inf 9.6

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

    7. Simplified9.6

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

    if -8.364554704106616e-80 < b < 4.1199128263687574e+46

    1. Initial program 13.8

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

    3. Applied clear-num13.9

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

    5. Applied associate-/l*13.8

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

    if 4.1199128263687574e+46 < b

    1. Initial program 36.8

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

    2. Taylor expanded around inf 5.2

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

    3. Simplified5.2

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

  4. Final simplification10.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -8.3645547041066157 \cdot 10^{-80}:\\ \;\;\;\;\frac{1}{1 \cdot \left(\frac{a}{b} - \frac{b}{c}\right)}\\ \mathbf{elif}\;b \le 4.1199128263687574 \cdot 10^{46}:\\ \;\;\;\;\frac{1}{\frac{2}{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a}}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 +o rules:numerics
(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)))