Average Error: 34.2 → 6.5
Time: 5.6s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.948753525425808925296770585273107730084 \cdot 10^{142}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 2.477064787285889532736667463396647969795 \cdot 10^{-233}:\\ \;\;\;\;1 \cdot \frac{2 \cdot c}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}\\ \mathbf{elif}\;b \le 1.276518349713334879747784170879022589057 \cdot 10^{91}:\\ \;\;\;\;\frac{-b}{2 \cdot a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\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 -1.948753525425808925296770585273107730084 \cdot 10^{142}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 2.477064787285889532736667463396647969795 \cdot 10^{-233}:\\
\;\;\;\;1 \cdot \frac{2 \cdot c}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}\\

\mathbf{elif}\;b \le 1.276518349713334879747784170879022589057 \cdot 10^{91}:\\
\;\;\;\;\frac{-b}{2 \cdot a} - \frac{\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 r85413 = b;
        double r85414 = -r85413;
        double r85415 = r85413 * r85413;
        double r85416 = 4.0;
        double r85417 = a;
        double r85418 = c;
        double r85419 = r85417 * r85418;
        double r85420 = r85416 * r85419;
        double r85421 = r85415 - r85420;
        double r85422 = sqrt(r85421);
        double r85423 = r85414 - r85422;
        double r85424 = 2.0;
        double r85425 = r85424 * r85417;
        double r85426 = r85423 / r85425;
        return r85426;
}
↓
double f(double a, double b, double c) {
        double r85427 = b;
        double r85428 = -1.948753525425809e+142;
        bool r85429 = r85427 <= r85428;
        double r85430 = -1.0;
        double r85431 = c;
        double r85432 = r85431 / r85427;
        double r85433 = r85430 * r85432;
        double r85434 = 2.4770647872858895e-233;
        bool r85435 = r85427 <= r85434;
        double r85436 = 1.0;
        double r85437 = 2.0;
        double r85438 = r85437 * r85431;
        double r85439 = r85427 * r85427;
        double r85440 = 4.0;
        double r85441 = a;
        double r85442 = r85441 * r85431;
        double r85443 = r85440 * r85442;
        double r85444 = r85439 - r85443;
        double r85445 = sqrt(r85444);
        double r85446 = r85445 - r85427;
        double r85447 = r85438 / r85446;
        double r85448 = r85436 * r85447;
        double r85449 = 1.2765183497133349e+91;
        bool r85450 = r85427 <= r85449;
        double r85451 = -r85427;
        double r85452 = r85437 * r85441;
        double r85453 = r85451 / r85452;
        double r85454 = r85445 / r85452;
        double r85455 = r85453 - r85454;
        double r85456 = 1.0;
        double r85457 = r85427 / r85441;
        double r85458 = r85432 - r85457;
        double r85459 = r85456 * r85458;
        double r85460 = r85450 ? r85455 : r85459;
        double r85461 = r85435 ? r85448 : r85460;
        double r85462 = r85429 ? r85433 : r85461;
        return r85462;
}

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
Target21.1
Herbie6.5
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -1.948753525425809e+142

    1. Initial program 62.8

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

    2. Taylor expanded around -inf 1.4

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

    if -1.948753525425809e+142 < b < 2.4770647872858895e-233

    1. Initial program 32.3

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

    3. Applied div-inv32.3

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

    5. Applied flip--32.4

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

    6. Simplified17.0

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

    7. Simplified17.0

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

    9. Applied *-un-lft-identity17.0

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

    10. Applied *-un-lft-identity17.0

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

    11. Applied times-frac17.0

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

    12. Applied associate-*l*17.0

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

    13. Simplified15.6

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

    14. Taylor expanded around 0 9.3

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

    if 2.4770647872858895e-233 < b < 1.2765183497133349e+91

    1. Initial program 7.5

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

    3. Applied div-sub7.5

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

    if 1.2765183497133349e+91 < b

    1. Initial program 45.7

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

    2. Taylor expanded around inf 4.2

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

    3. Simplified4.2

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

  4. Final simplification6.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.948753525425808925296770585273107730084 \cdot 10^{142}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 2.477064787285889532736667463396647969795 \cdot 10^{-233}:\\ \;\;\;\;1 \cdot \frac{2 \cdot c}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}\\ \mathbf{elif}\;b \le 1.276518349713334879747784170879022589057 \cdot 10^{91}:\\ \;\;\;\;\frac{-b}{2 \cdot a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019354 
(FPCore (a b c)
  :name "The quadratic formula (r2)"
  :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)))