Average Error: 34.6 → 10.5
Time: 27.0s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.6581383089037873 \cdot 10^{81}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 2.45811587950602871 \cdot 10^{-136}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -1.6581383089037873 \cdot 10^{81}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 2.45811587950602871 \cdot 10^{-136}:\\
\;\;\;\;\frac{1}{\frac{2 \cdot a}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}\\

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

\end{array}
double f(double a, double b, double c) {
        double r61395 = b;
        double r61396 = -r61395;
        double r61397 = r61395 * r61395;
        double r61398 = 4.0;
        double r61399 = a;
        double r61400 = c;
        double r61401 = r61399 * r61400;
        double r61402 = r61398 * r61401;
        double r61403 = r61397 - r61402;
        double r61404 = sqrt(r61403);
        double r61405 = r61396 + r61404;
        double r61406 = 2.0;
        double r61407 = r61406 * r61399;
        double r61408 = r61405 / r61407;
        return r61408;
}
↓
double f(double a, double b, double c) {
        double r61409 = b;
        double r61410 = -1.6581383089037873e+81;
        bool r61411 = r61409 <= r61410;
        double r61412 = 1.0;
        double r61413 = c;
        double r61414 = r61413 / r61409;
        double r61415 = a;
        double r61416 = r61409 / r61415;
        double r61417 = r61414 - r61416;
        double r61418 = r61412 * r61417;
        double r61419 = 2.4581158795060287e-136;
        bool r61420 = r61409 <= r61419;
        double r61421 = 1.0;
        double r61422 = 2.0;
        double r61423 = r61422 * r61415;
        double r61424 = r61409 * r61409;
        double r61425 = 4.0;
        double r61426 = r61415 * r61413;
        double r61427 = r61425 * r61426;
        double r61428 = r61424 - r61427;
        double r61429 = sqrt(r61428);
        double r61430 = r61429 - r61409;
        double r61431 = r61423 / r61430;
        double r61432 = r61421 / r61431;
        double r61433 = -1.0;
        double r61434 = r61433 * r61414;
        double r61435 = r61420 ? r61432 : r61434;
        double r61436 = r61411 ? r61418 : r61435;
        return r61436;
}

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

Original34.6
Target21.3
Herbie10.5
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -1.6581383089037873e+81

    1. Initial program 43.9

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

    2. Simplified43.9

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

    4. Applied div-inv44.0

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

    6. Applied *-un-lft-identity44.0

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

    7. Applied associate-*l*44.0

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

    8. Simplified43.9

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

    9. Taylor expanded around -inf 3.6

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

    10. Simplified3.6

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

    if -1.6581383089037873e+81 < b < 2.4581158795060287e-136

    1. Initial program 11.7

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

    2. Simplified11.7

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

    4. Applied div-inv11.8

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

    6. Applied *-un-lft-identity11.8

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

    7. Applied associate-*l*11.8

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

    8. Simplified11.7

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

    10. Applied clear-num11.8

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

    if 2.4581158795060287e-136 < b

    1. Initial program 50.8

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

    2. Simplified50.8

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

    4. Applied div-inv50.8

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

    6. Applied *-un-lft-identity50.8

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

    7. Applied associate-*l*50.8

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

    8. Simplified50.8

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

    9. Taylor expanded around inf 12.0

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

  4. Final simplification10.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.6581383089037873 \cdot 10^{81}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 2.45811587950602871 \cdot 10^{-136}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019199 +o rules:numerics
(FPCore (a b c)
  :name "quadp (p42, positive)"

  :herbie-target
  (if (< b 0.0) (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)) (/ c (* a (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))))

  (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))