Average Error: 33.7 → 10.9
Time: 6.4s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.98276540088900058 \cdot 10^{134}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 1.1860189201379418 \cdot 10^{-161}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -1.98276540088900058 \cdot 10^{134}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 1.1860189201379418 \cdot 10^{-161}:\\
\;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r96410 = b;
        double r96411 = -r96410;
        double r96412 = r96410 * r96410;
        double r96413 = 4.0;
        double r96414 = a;
        double r96415 = c;
        double r96416 = r96414 * r96415;
        double r96417 = r96413 * r96416;
        double r96418 = r96412 - r96417;
        double r96419 = sqrt(r96418);
        double r96420 = r96411 + r96419;
        double r96421 = 2.0;
        double r96422 = r96421 * r96414;
        double r96423 = r96420 / r96422;
        return r96423;
}
↓
double f(double a, double b, double c) {
        double r96424 = b;
        double r96425 = -1.9827654008890006e+134;
        bool r96426 = r96424 <= r96425;
        double r96427 = 1.0;
        double r96428 = c;
        double r96429 = r96428 / r96424;
        double r96430 = a;
        double r96431 = r96424 / r96430;
        double r96432 = r96429 - r96431;
        double r96433 = r96427 * r96432;
        double r96434 = 1.1860189201379418e-161;
        bool r96435 = r96424 <= r96434;
        double r96436 = -r96424;
        double r96437 = r96424 * r96424;
        double r96438 = 4.0;
        double r96439 = r96430 * r96428;
        double r96440 = r96438 * r96439;
        double r96441 = r96437 - r96440;
        double r96442 = sqrt(r96441);
        double r96443 = r96436 + r96442;
        double r96444 = 1.0;
        double r96445 = 2.0;
        double r96446 = r96445 * r96430;
        double r96447 = r96444 / r96446;
        double r96448 = r96443 * r96447;
        double r96449 = -1.0;
        double r96450 = r96449 * r96429;
        double r96451 = r96435 ? r96448 : r96450;
        double r96452 = r96426 ? r96433 : r96451;
        return r96452;
}

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

Original33.7
Target21.0
Herbie10.9
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -1.9827654008890006e+134

    1. Initial program 56.8

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

    2. Taylor expanded around -inf 3.1

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

    3. Simplified3.1

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

    if -1.9827654008890006e+134 < b < 1.1860189201379418e-161

    1. Initial program 10.3

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

    3. Applied div-inv10.5

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

    if 1.1860189201379418e-161 < b

    1. Initial program 49.6

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

    2. Taylor expanded around inf 13.7

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

  4. Final simplification10.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.98276540088900058 \cdot 10^{134}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 1.1860189201379418 \cdot 10^{-161}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (a b c)
  :name "quadp (p42, positive)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ (+ (- 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)))