Average Error: 34.5 → 8.8
Time: 18.8s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.102895015780532348136946077262401346805 \cdot 10^{69}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -8.767251655423633534328588307438915014497 \cdot 10^{-253}:\\ \;\;\;\;\frac{\frac{4 \cdot \left(a \cdot c\right)}{2 \cdot a}}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}\\ \mathbf{elif}\;b \le 2.125563079851438727208684227808951636731 \cdot 10^{135}:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\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 -3.102895015780532348136946077262401346805 \cdot 10^{69}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le -8.767251655423633534328588307438915014497 \cdot 10^{-253}:\\
\;\;\;\;\frac{\frac{4 \cdot \left(a \cdot c\right)}{2 \cdot a}}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}\\

\mathbf{elif}\;b \le 2.125563079851438727208684227808951636731 \cdot 10^{135}:\\
\;\;\;\;\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 \left(\frac{c}{b} - \frac{b}{a}\right)\\

\end{array}
double f(double a, double b, double c) {
        double r63498 = b;
        double r63499 = -r63498;
        double r63500 = r63498 * r63498;
        double r63501 = 4.0;
        double r63502 = a;
        double r63503 = c;
        double r63504 = r63502 * r63503;
        double r63505 = r63501 * r63504;
        double r63506 = r63500 - r63505;
        double r63507 = sqrt(r63506);
        double r63508 = r63499 - r63507;
        double r63509 = 2.0;
        double r63510 = r63509 * r63502;
        double r63511 = r63508 / r63510;
        return r63511;
}
↓
double f(double a, double b, double c) {
        double r63512 = b;
        double r63513 = -3.1028950157805323e+69;
        bool r63514 = r63512 <= r63513;
        double r63515 = -1.0;
        double r63516 = c;
        double r63517 = r63516 / r63512;
        double r63518 = r63515 * r63517;
        double r63519 = -8.767251655423634e-253;
        bool r63520 = r63512 <= r63519;
        double r63521 = 4.0;
        double r63522 = a;
        double r63523 = r63522 * r63516;
        double r63524 = r63521 * r63523;
        double r63525 = 2.0;
        double r63526 = r63525 * r63522;
        double r63527 = r63524 / r63526;
        double r63528 = r63512 * r63512;
        double r63529 = r63528 - r63524;
        double r63530 = sqrt(r63529);
        double r63531 = r63530 - r63512;
        double r63532 = r63527 / r63531;
        double r63533 = 2.1255630798514387e+135;
        bool r63534 = r63512 <= r63533;
        double r63535 = -r63512;
        double r63536 = r63535 - r63530;
        double r63537 = 1.0;
        double r63538 = r63537 / r63526;
        double r63539 = r63536 * r63538;
        double r63540 = 1.0;
        double r63541 = r63512 / r63522;
        double r63542 = r63517 - r63541;
        double r63543 = r63540 * r63542;
        double r63544 = r63534 ? r63539 : r63543;
        double r63545 = r63520 ? r63532 : r63544;
        double r63546 = r63514 ? r63518 : r63545;
        return r63546;
}

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

Original34.5
Target21.3
Herbie8.8
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -3.1028950157805323e+69

    1. Initial program 58.6

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

    2. Taylor expanded around -inf 3.1

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

    if -3.1028950157805323e+69 < b < -8.767251655423634e-253

    1. Initial program 33.1

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

    3. Applied flip--33.1

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

    4. Simplified16.9

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

    5. Simplified16.9

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

    7. Applied div-inv17.0

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

    9. Applied associate-*l/16.4

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

    10. Simplified16.3

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

    if -8.767251655423634e-253 < b < 2.1255630798514387e+135

    1. Initial program 9.6

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

    3. Applied div-inv9.8

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

    if 2.1255630798514387e+135 < b

    1. Initial program 58.2

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

    2. Taylor expanded around inf 3.0

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

    3. Simplified3.0

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

  4. Final simplification8.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.102895015780532348136946077262401346805 \cdot 10^{69}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -8.767251655423633534328588307438915014497 \cdot 10^{-253}:\\ \;\;\;\;\frac{\frac{4 \cdot \left(a \cdot c\right)}{2 \cdot a}}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}\\ \mathbf{elif}\;b \le 2.125563079851438727208684227808951636731 \cdot 10^{135}:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019306 
(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)))