Average Error: 33.4 → 9.9
Time: 23.3s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.0027271082217074 \cdot 10^{+110}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 2.326372645943808 \cdot 10^{-74}:\\ \;\;\;\;\frac{\frac{1}{a} \cdot \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - \frac{b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]
\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 -1.0027271082217074 \cdot 10^{+110}:\\
\;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\

\mathbf{elif}\;b \le 2.326372645943808 \cdot 10^{-74}:\\
\;\;\;\;\frac{\frac{1}{a} \cdot \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - \frac{b}{a}}{2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r1611420 = b;
        double r1611421 = -r1611420;
        double r1611422 = r1611420 * r1611420;
        double r1611423 = 4.0;
        double r1611424 = a;
        double r1611425 = c;
        double r1611426 = r1611424 * r1611425;
        double r1611427 = r1611423 * r1611426;
        double r1611428 = r1611422 - r1611427;
        double r1611429 = sqrt(r1611428);
        double r1611430 = r1611421 + r1611429;
        double r1611431 = 2.0;
        double r1611432 = r1611431 * r1611424;
        double r1611433 = r1611430 / r1611432;
        return r1611433;
}
↓
double f(double a, double b, double c) {
        double r1611434 = b;
        double r1611435 = -1.0027271082217074e+110;
        bool r1611436 = r1611434 <= r1611435;
        double r1611437 = c;
        double r1611438 = r1611437 / r1611434;
        double r1611439 = a;
        double r1611440 = r1611434 / r1611439;
        double r1611441 = r1611438 - r1611440;
        double r1611442 = 2.0;
        double r1611443 = r1611441 * r1611442;
        double r1611444 = r1611443 / r1611442;
        double r1611445 = 2.326372645943808e-74;
        bool r1611446 = r1611434 <= r1611445;
        double r1611447 = 1.0;
        double r1611448 = r1611447 / r1611439;
        double r1611449 = r1611434 * r1611434;
        double r1611450 = 4.0;
        double r1611451 = r1611450 * r1611439;
        double r1611452 = r1611437 * r1611451;
        double r1611453 = r1611449 - r1611452;
        double r1611454 = sqrt(r1611453);
        double r1611455 = r1611448 * r1611454;
        double r1611456 = r1611455 - r1611440;
        double r1611457 = r1611456 / r1611442;
        double r1611458 = -2.0;
        double r1611459 = r1611458 * r1611438;
        double r1611460 = r1611459 / r1611442;
        double r1611461 = r1611446 ? r1611457 : r1611460;
        double r1611462 = r1611436 ? r1611444 : r1611461;
        return r1611462;
}

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

Original33.4
Target20.3
Herbie9.9
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -1.0027271082217074e+110

    1. Initial program 46.7

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

    2. Simplified46.7

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

    4. Applied div-sub46.7

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

    6. Applied *-un-lft-identity46.7

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

    7. Applied add-sqr-sqrt46.8

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

    8. Applied times-frac46.8

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

    9. Simplified46.8

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

    10. Taylor expanded around -inf 3.6

      \[\leadsto \frac{\color{blue}{2 \cdot \frac{c}{b} - 2 \cdot \frac{b}{a}}}{2}\]

    11. Simplified3.6

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

    if -1.0027271082217074e+110 < b < 2.326372645943808e-74

    1. Initial program 12.8

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

    2. Simplified12.8

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

    4. Applied div-sub12.8

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

    6. Applied div-inv12.9

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

    if 2.326372645943808e-74 < b

    1. Initial program 52.5

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

    2. Simplified52.5

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

    4. Applied div-sub53.2

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

    5. Taylor expanded around inf 8.8

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

  4. Final simplification9.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.0027271082217074 \cdot 10^{+110}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 2.326372645943808 \cdot 10^{-74}:\\ \;\;\;\;\frac{\frac{1}{a} \cdot \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - \frac{b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019151 
(FPCore (a b c)
  :name "quadp (p42, positive)"

  :herbie-target
  (if (< b 0) (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)) (/ c (* a (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))))

  (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))