Average Error: 33.9 → 6.7
Time: 18.0s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -5.674469085146396739103610609439188639717 \cdot 10^{110}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 7.061692521831335565675525372535211636164 \cdot 10^{-266}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}\\ \mathbf{elif}\;b \le 1.715181108188238274259588142060201574853 \cdot 10^{78}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot b}{2 \cdot a}\\ \end{array}\]
\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 -5.674469085146396739103610609439188639717 \cdot 10^{110}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 7.061692521831335565675525372535211636164 \cdot 10^{-266}:\\
\;\;\;\;\frac{2 \cdot c}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}\\

\mathbf{elif}\;b \le 1.715181108188238274259588142060201574853 \cdot 10^{78}:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;\frac{-2 \cdot b}{2 \cdot a}\\

\end{array}
double f(double a, double b, double c) {
        double r69702 = b;
        double r69703 = -r69702;
        double r69704 = r69702 * r69702;
        double r69705 = 4.0;
        double r69706 = a;
        double r69707 = c;
        double r69708 = r69706 * r69707;
        double r69709 = r69705 * r69708;
        double r69710 = r69704 - r69709;
        double r69711 = sqrt(r69710);
        double r69712 = r69703 - r69711;
        double r69713 = 2.0;
        double r69714 = r69713 * r69706;
        double r69715 = r69712 / r69714;
        return r69715;
}
↓
double f(double a, double b, double c) {
        double r69716 = b;
        double r69717 = -5.674469085146397e+110;
        bool r69718 = r69716 <= r69717;
        double r69719 = -1.0;
        double r69720 = c;
        double r69721 = r69720 / r69716;
        double r69722 = r69719 * r69721;
        double r69723 = 7.061692521831336e-266;
        bool r69724 = r69716 <= r69723;
        double r69725 = 2.0;
        double r69726 = r69725 * r69720;
        double r69727 = r69716 * r69716;
        double r69728 = 4.0;
        double r69729 = a;
        double r69730 = r69729 * r69720;
        double r69731 = r69728 * r69730;
        double r69732 = r69727 - r69731;
        double r69733 = sqrt(r69732);
        double r69734 = r69733 - r69716;
        double r69735 = r69726 / r69734;
        double r69736 = 1.7151811081882383e+78;
        bool r69737 = r69716 <= r69736;
        double r69738 = -r69716;
        double r69739 = r69738 - r69733;
        double r69740 = r69725 * r69729;
        double r69741 = r69739 / r69740;
        double r69742 = -2.0;
        double r69743 = r69742 * r69716;
        double r69744 = r69743 / r69740;
        double r69745 = r69737 ? r69741 : r69744;
        double r69746 = r69724 ? r69735 : r69745;
        double r69747 = r69718 ? r69722 : r69746;
        return r69747;
}

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

Original33.9
Target21.0
Herbie6.7
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -5.674469085146397e+110

    1. Initial program 59.7

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

    2. Taylor expanded around -inf 2.7

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

    if -5.674469085146397e+110 < b < 7.061692521831336e-266

    1. Initial program 31.8

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

    3. Applied flip--31.8

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

    4. Simplified16.1

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

    5. Simplified16.1

      \[\leadsto \frac{\frac{0 + \left(4 \cdot a\right) \cdot c}{\color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}}{2 \cdot a}\]
    6. Using strategy rm

    7. Applied div-inv16.1

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

    9. Applied associate-*l/15.2

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

    10. Simplified15.1

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

    11. Taylor expanded around 0 8.7

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

    if 7.061692521831336e-266 < b < 1.7151811081882383e+78

    1. Initial program 8.6

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

    if 1.7151811081882383e+78 < b

    1. Initial program 43.0

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

    3. Applied flip--62.6

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

    4. Simplified61.8

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

    5. Simplified61.8

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

    6. Taylor expanded around 0 4.8

      \[\leadsto \frac{\color{blue}{-2 \cdot b}}{2 \cdot a}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification6.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -5.674469085146396739103610609439188639717 \cdot 10^{110}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 7.061692521831335565675525372535211636164 \cdot 10^{-266}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}\\ \mathbf{elif}\;b \le 1.715181108188238274259588142060201574853 \cdot 10^{78}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot b}{2 \cdot a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 
(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)))