Average Error: 34.2 → 6.6
Time: 18.6s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.513258824878011748257049801344805265531 \cdot 10^{152}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 2.677071638923404327650485520297885208573 \cdot 10^{-300}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}\\ \mathbf{elif}\;b \le 5.031608061939102936286074782173578716838 \cdot 10^{53}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\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 -3.513258824878011748257049801344805265531 \cdot 10^{152}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 2.677071638923404327650485520297885208573 \cdot 10^{-300}:\\
\;\;\;\;\frac{2 \cdot c}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}\\

\mathbf{elif}\;b \le 5.031608061939102936286074782173578716838 \cdot 10^{53}:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r55730 = b;
        double r55731 = -r55730;
        double r55732 = r55730 * r55730;
        double r55733 = 4.0;
        double r55734 = a;
        double r55735 = c;
        double r55736 = r55734 * r55735;
        double r55737 = r55733 * r55736;
        double r55738 = r55732 - r55737;
        double r55739 = sqrt(r55738);
        double r55740 = r55731 - r55739;
        double r55741 = 2.0;
        double r55742 = r55741 * r55734;
        double r55743 = r55740 / r55742;
        return r55743;
}
↓
double f(double a, double b, double c) {
        double r55744 = b;
        double r55745 = -3.5132588248780117e+152;
        bool r55746 = r55744 <= r55745;
        double r55747 = -1.0;
        double r55748 = c;
        double r55749 = r55748 / r55744;
        double r55750 = r55747 * r55749;
        double r55751 = 2.6770716389234043e-300;
        bool r55752 = r55744 <= r55751;
        double r55753 = 2.0;
        double r55754 = r55753 * r55748;
        double r55755 = r55744 * r55744;
        double r55756 = 4.0;
        double r55757 = a;
        double r55758 = r55757 * r55748;
        double r55759 = r55756 * r55758;
        double r55760 = r55755 - r55759;
        double r55761 = sqrt(r55760);
        double r55762 = r55761 - r55744;
        double r55763 = r55754 / r55762;
        double r55764 = 5.031608061939103e+53;
        bool r55765 = r55744 <= r55764;
        double r55766 = -r55744;
        double r55767 = r55766 - r55761;
        double r55768 = r55753 * r55757;
        double r55769 = r55767 / r55768;
        double r55770 = 1.0;
        double r55771 = r55744 / r55757;
        double r55772 = r55749 - r55771;
        double r55773 = r55770 * r55772;
        double r55774 = r55765 ? r55769 : r55773;
        double r55775 = r55752 ? r55763 : r55774;
        double r55776 = r55746 ? r55750 : r55775;
        return r55776;
}

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

Original34.2
Target21.3
Herbie6.6
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -3.5132588248780117e+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 1.4

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

    if -3.5132588248780117e+152 < b < 2.6770716389234043e-300

    1. Initial program 33.8

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

    3. Applied flip--33.9

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

    4. Simplified15.3

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

    5. Simplified15.3

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

    7. Applied div-inv15.3

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

    9. Applied associate-*l/13.9

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

    10. Simplified13.9

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

    11. Taylor expanded around 0 7.5

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

    if 2.6770716389234043e-300 < b < 5.031608061939103e+53

    1. Initial program 9.7

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

    if 5.031608061939103e+53 < b

    1. Initial program 39.6

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

    2. Taylor expanded around inf 5.7

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

    3. Simplified5.7

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

  4. Final simplification6.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.513258824878011748257049801344805265531 \cdot 10^{152}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 2.677071638923404327650485520297885208573 \cdot 10^{-300}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}\\ \mathbf{elif}\;b \le 5.031608061939102936286074782173578716838 \cdot 10^{53}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 
(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)))