Average Error: 34.7 → 6.9
Time: 10.7s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.277091394223363542557795301452975127355 \cdot 10^{71}:\\ \;\;\;\;\frac{1}{2} \cdot \left(-2 \cdot \frac{c}{b}\right)\\ \mathbf{elif}\;b \le \frac{1931930388468481}{2.19444962751747547330237450047488370803 \cdot 10^{304}}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{1}{\frac{\frac{1}{4}}{c} \cdot \left(\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b\right)}\\ \mathbf{elif}\;b \le 7.49320603738817106440277485916837905827 \cdot 10^{90}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{-2 \cdot b}{a}\\ \end{array}\]
\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 -1.277091394223363542557795301452975127355 \cdot 10^{71}:\\
\;\;\;\;\frac{1}{2} \cdot \left(-2 \cdot \frac{c}{b}\right)\\

\mathbf{elif}\;b \le \frac{1931930388468481}{2.19444962751747547330237450047488370803 \cdot 10^{304}}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{1}{\frac{\frac{1}{4}}{c} \cdot \left(\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b\right)}\\

\mathbf{elif}\;b \le 7.49320603738817106440277485916837905827 \cdot 10^{90}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{-2 \cdot b}{a}\\

\end{array}
double f(double a, double b, double c) {
        double r82591 = b;
        double r82592 = -r82591;
        double r82593 = r82591 * r82591;
        double r82594 = 4.0;
        double r82595 = a;
        double r82596 = c;
        double r82597 = r82595 * r82596;
        double r82598 = r82594 * r82597;
        double r82599 = r82593 - r82598;
        double r82600 = sqrt(r82599);
        double r82601 = r82592 - r82600;
        double r82602 = 2.0;
        double r82603 = r82602 * r82595;
        double r82604 = r82601 / r82603;
        return r82604;
}
↓
double f(double a, double b, double c) {
        double r82605 = b;
        double r82606 = -1.2770913942233635e+71;
        bool r82607 = r82605 <= r82606;
        double r82608 = 1.0;
        double r82609 = 2.0;
        double r82610 = r82608 / r82609;
        double r82611 = -2.0;
        double r82612 = c;
        double r82613 = r82612 / r82605;
        double r82614 = r82611 * r82613;
        double r82615 = r82610 * r82614;
        double r82616 = 1931930388468481.0;
        double r82617 = 2.1944496275174755e+304;
        double r82618 = r82616 / r82617;
        bool r82619 = r82605 <= r82618;
        double r82620 = 1.0;
        double r82621 = 4.0;
        double r82622 = r82620 / r82621;
        double r82623 = r82622 / r82612;
        double r82624 = r82605 * r82605;
        double r82625 = a;
        double r82626 = r82625 * r82612;
        double r82627 = r82621 * r82626;
        double r82628 = r82624 - r82627;
        double r82629 = sqrt(r82628);
        double r82630 = r82629 - r82605;
        double r82631 = r82623 * r82630;
        double r82632 = r82608 / r82631;
        double r82633 = r82610 * r82632;
        double r82634 = 7.493206037388171e+90;
        bool r82635 = r82605 <= r82634;
        double r82636 = -r82605;
        double r82637 = r82636 - r82629;
        double r82638 = r82637 / r82625;
        double r82639 = r82610 * r82638;
        double r82640 = r82611 * r82605;
        double r82641 = r82640 / r82625;
        double r82642 = r82610 * r82641;
        double r82643 = r82635 ? r82639 : r82642;
        double r82644 = r82619 ? r82633 : r82643;
        double r82645 = r82607 ? r82615 : r82644;
        return r82645;
}

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

Original34.7
Target21.6
Herbie6.9
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -1.2770913942233635e+71

    1. Initial program 58.1

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

    3. Applied *-un-lft-identity58.1

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

    4. Applied times-frac58.1

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

    5. Taylor expanded around -inf 3.4

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

    if -1.2770913942233635e+71 < b < 8.803712622257929e-290

    1. Initial program 30.4

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

    3. Applied *-un-lft-identity30.4

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

    4. Applied times-frac30.3

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

    6. Applied flip--30.4

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

    7. Simplified17.0

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

    8. Simplified17.0

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

    10. Applied clear-num17.1

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

    11. Simplified16.8

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

    12. Taylor expanded around 0 10.0

      \[\leadsto \frac{1}{2} \cdot \frac{1}{\color{blue}{\frac{0.25}{c}} \cdot \left(\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b\right)}\]

    13. Simplified10.0

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

    if 8.803712622257929e-290 < b < 7.493206037388171e+90

    1. Initial program 8.6

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

    3. Applied *-un-lft-identity8.6

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

    4. Applied times-frac8.7

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

    if 7.493206037388171e+90 < b

    1. Initial program 45.9

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

    3. Applied *-un-lft-identity45.9

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

    4. Applied times-frac45.9

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

    6. Applied flip--62.9

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

    7. Simplified62.0

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

    8. Simplified62.0

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

    9. Taylor expanded around 0 4.2

      \[\leadsto \frac{1}{2} \cdot \frac{\color{blue}{-2 \cdot b}}{a}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification6.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.277091394223363542557795301452975127355 \cdot 10^{71}:\\ \;\;\;\;\frac{1}{2} \cdot \left(-2 \cdot \frac{c}{b}\right)\\ \mathbf{elif}\;b \le \frac{1931930388468481}{2.19444962751747547330237450047488370803 \cdot 10^{304}}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{1}{\frac{\frac{1}{4}}{c} \cdot \left(\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b\right)}\\ \mathbf{elif}\;b \le 7.49320603738817106440277485916837905827 \cdot 10^{90}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{-2 \cdot b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 197574269 
(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)))