Average Error: 34.1 → 10.3
Time: 5.3s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -8.8665312552031243 \cdot 10^{65}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -1519208.93058992573:\\ \;\;\;\;\frac{0 + 1 \cdot \left(4 \cdot \left(a \cdot c\right)\right)}{\left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \left(2 \cdot a\right)}\\ \mathbf{elif}\;b \le -2.3901106171036543 \cdot 10^{-123}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 1.6414783348607511 \cdot 10^{125}:\\ \;\;\;\;1 \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\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 -8.8665312552031243 \cdot 10^{65}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le -1519208.93058992573:\\
\;\;\;\;\frac{0 + 1 \cdot \left(4 \cdot \left(a \cdot c\right)\right)}{\left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \left(2 \cdot a\right)}\\

\mathbf{elif}\;b \le -2.3901106171036543 \cdot 10^{-123}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 1.6414783348607511 \cdot 10^{125}:\\
\;\;\;\;1 \cdot \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 r51673 = b;
        double r51674 = -r51673;
        double r51675 = r51673 * r51673;
        double r51676 = 4.0;
        double r51677 = a;
        double r51678 = c;
        double r51679 = r51677 * r51678;
        double r51680 = r51676 * r51679;
        double r51681 = r51675 - r51680;
        double r51682 = sqrt(r51681);
        double r51683 = r51674 - r51682;
        double r51684 = 2.0;
        double r51685 = r51684 * r51677;
        double r51686 = r51683 / r51685;
        return r51686;
}
↓
double f(double a, double b, double c) {
        double r51687 = b;
        double r51688 = -8.866531255203124e+65;
        bool r51689 = r51687 <= r51688;
        double r51690 = -1.0;
        double r51691 = c;
        double r51692 = r51691 / r51687;
        double r51693 = r51690 * r51692;
        double r51694 = -1519208.9305899257;
        bool r51695 = r51687 <= r51694;
        double r51696 = 0.0;
        double r51697 = 1.0;
        double r51698 = 4.0;
        double r51699 = a;
        double r51700 = r51699 * r51691;
        double r51701 = r51698 * r51700;
        double r51702 = r51697 * r51701;
        double r51703 = r51696 + r51702;
        double r51704 = -r51687;
        double r51705 = r51687 * r51687;
        double r51706 = r51705 - r51701;
        double r51707 = sqrt(r51706);
        double r51708 = r51704 + r51707;
        double r51709 = 2.0;
        double r51710 = r51709 * r51699;
        double r51711 = r51708 * r51710;
        double r51712 = r51703 / r51711;
        double r51713 = -2.3901106171036543e-123;
        bool r51714 = r51687 <= r51713;
        double r51715 = 1.6414783348607511e+125;
        bool r51716 = r51687 <= r51715;
        double r51717 = r51704 - r51707;
        double r51718 = r51717 / r51710;
        double r51719 = r51697 * r51718;
        double r51720 = 1.0;
        double r51721 = r51687 / r51699;
        double r51722 = r51692 - r51721;
        double r51723 = r51720 * r51722;
        double r51724 = r51716 ? r51719 : r51723;
        double r51725 = r51714 ? r51693 : r51724;
        double r51726 = r51695 ? r51712 : r51725;
        double r51727 = r51689 ? r51693 : r51726;
        return r51727;
}

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

Original34.1
Target21.2
Herbie10.3
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -8.866531255203124e+65 or -1519208.9305899257 < b < -2.3901106171036543e-123

    1. Initial program 51.6

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

    2. Taylor expanded around -inf 11.3

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

    if -8.866531255203124e+65 < b < -1519208.9305899257

    1. Initial program 47.1

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

    3. Applied div-inv47.1

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

    5. Applied flip--47.1

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

    6. Applied frac-times49.4

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

    7. Simplified14.8

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

    if -2.3901106171036543e-123 < b < 1.6414783348607511e+125

    1. Initial program 11.4

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

    3. Applied div-inv11.5

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

    5. Applied *-un-lft-identity11.5

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

    6. Applied associate-*l*11.5

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

    7. Simplified11.4

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

    if 1.6414783348607511e+125 < b

    1. Initial program 53.5

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

    2. Taylor expanded around inf 2.6

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

    3. Simplified2.6

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

  4. Final simplification10.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -8.8665312552031243 \cdot 10^{65}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -1519208.93058992573:\\ \;\;\;\;\frac{0 + 1 \cdot \left(4 \cdot \left(a \cdot c\right)\right)}{\left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \left(2 \cdot a\right)}\\ \mathbf{elif}\;b \le -2.3901106171036543 \cdot 10^{-123}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 1.6414783348607511 \cdot 10^{125}:\\ \;\;\;\;1 \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020083 
(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)))