Average Error: 33.9 → 10.4
Time: 20.5s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.5961406266953245 \cdot 10^{-58}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 3.1115579814291686 \cdot 10^{+29}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array}\]
\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 -1.5961406266953245 \cdot 10^{-58}:\\
\;\;\;\;-\frac{c}{b}\\

\mathbf{elif}\;b \le 3.1115579814291686 \cdot 10^{+29}:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4}}{a \cdot 2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r3420717 = b;
        double r3420718 = -r3420717;
        double r3420719 = r3420717 * r3420717;
        double r3420720 = 4.0;
        double r3420721 = a;
        double r3420722 = c;
        double r3420723 = r3420721 * r3420722;
        double r3420724 = r3420720 * r3420723;
        double r3420725 = r3420719 - r3420724;
        double r3420726 = sqrt(r3420725);
        double r3420727 = r3420718 - r3420726;
        double r3420728 = 2.0;
        double r3420729 = r3420728 * r3420721;
        double r3420730 = r3420727 / r3420729;
        return r3420730;
}
↓
double f(double a, double b, double c) {
        double r3420731 = b;
        double r3420732 = -1.5961406266953245e-58;
        bool r3420733 = r3420731 <= r3420732;
        double r3420734 = c;
        double r3420735 = r3420734 / r3420731;
        double r3420736 = -r3420735;
        double r3420737 = 3.1115579814291686e+29;
        bool r3420738 = r3420731 <= r3420737;
        double r3420739 = -r3420731;
        double r3420740 = r3420731 * r3420731;
        double r3420741 = a;
        double r3420742 = r3420734 * r3420741;
        double r3420743 = 4.0;
        double r3420744 = r3420742 * r3420743;
        double r3420745 = r3420740 - r3420744;
        double r3420746 = sqrt(r3420745);
        double r3420747 = r3420739 - r3420746;
        double r3420748 = 2.0;
        double r3420749 = r3420741 * r3420748;
        double r3420750 = r3420747 / r3420749;
        double r3420751 = r3420739 / r3420741;
        double r3420752 = r3420738 ? r3420750 : r3420751;
        double r3420753 = r3420733 ? r3420736 : r3420752;
        return r3420753;
}

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
Target20.6
Herbie10.4
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -1.5961406266953245e-58

    1. Initial program 53.4

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

    2. Taylor expanded around -inf 8.1

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

    3. Simplified8.1

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

    if -1.5961406266953245e-58 < b < 3.1115579814291686e+29

    1. Initial program 14.7

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

    3. Applied *-un-lft-identity14.7

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

    4. Applied *-un-lft-identity14.7

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

    5. Applied distribute-lft-out--14.7

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

    6. Applied associate-/l*14.9

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

    8. Applied div-inv14.9

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

    9. Applied *-un-lft-identity14.9

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

    10. Applied times-frac14.9

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

    11. Simplified14.8

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

    13. Applied associate-*l/14.7

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

    14. Simplified14.7

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

    if 3.1115579814291686e+29 < b

    1. Initial program 34.4

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

    3. Applied *-un-lft-identity34.4

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

    4. Applied *-un-lft-identity34.4

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

    5. Applied distribute-lft-out--34.4

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

    6. Applied associate-/l*34.5

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

    8. Applied div-inv34.6

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

    9. Applied *-un-lft-identity34.6

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

    10. Applied times-frac34.6

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

    11. Simplified34.5

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

    12. Taylor expanded around 0 6.5

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

    13. Simplified6.5

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

  4. Final simplification10.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.5961406266953245 \cdot 10^{-58}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 3.1115579814291686 \cdot 10^{+29}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019134 
(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)))