Average Error: 34.9 → 10.1
Time: 18.9s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.680329042988888396603264581948851078331 \cdot 10^{148}:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\ \mathbf{elif}\;b \le 4.612990823111230552052602417245542305295 \cdot 10^{-104}:\\ \;\;\;\;\frac{\frac{1}{a}}{2} \cdot \left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -3.680329042988888396603264581948851078331 \cdot 10^{148}:\\
\;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\

\mathbf{elif}\;b \le 4.612990823111230552052602417245542305295 \cdot 10^{-104}:\\
\;\;\;\;\frac{\frac{1}{a}}{2} \cdot \left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right)\\

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

\end{array}
double f(double a, double b, double c) {
        double r4564439 = b;
        double r4564440 = -r4564439;
        double r4564441 = r4564439 * r4564439;
        double r4564442 = 4.0;
        double r4564443 = a;
        double r4564444 = c;
        double r4564445 = r4564443 * r4564444;
        double r4564446 = r4564442 * r4564445;
        double r4564447 = r4564441 - r4564446;
        double r4564448 = sqrt(r4564447);
        double r4564449 = r4564440 + r4564448;
        double r4564450 = 2.0;
        double r4564451 = r4564450 * r4564443;
        double r4564452 = r4564449 / r4564451;
        return r4564452;
}
↓
double f(double a, double b, double c) {
        double r4564453 = b;
        double r4564454 = -3.6803290429888884e+148;
        bool r4564455 = r4564453 <= r4564454;
        double r4564456 = c;
        double r4564457 = r4564456 / r4564453;
        double r4564458 = a;
        double r4564459 = r4564453 / r4564458;
        double r4564460 = r4564457 - r4564459;
        double r4564461 = 1.0;
        double r4564462 = r4564460 * r4564461;
        double r4564463 = 4.6129908231112306e-104;
        bool r4564464 = r4564453 <= r4564463;
        double r4564465 = 1.0;
        double r4564466 = r4564465 / r4564458;
        double r4564467 = 2.0;
        double r4564468 = r4564466 / r4564467;
        double r4564469 = r4564453 * r4564453;
        double r4564470 = 4.0;
        double r4564471 = r4564470 * r4564458;
        double r4564472 = r4564471 * r4564456;
        double r4564473 = r4564469 - r4564472;
        double r4564474 = sqrt(r4564473);
        double r4564475 = r4564474 - r4564453;
        double r4564476 = r4564468 * r4564475;
        double r4564477 = -1.0;
        double r4564478 = r4564477 * r4564457;
        double r4564479 = r4564464 ? r4564476 : r4564478;
        double r4564480 = r4564455 ? r4564462 : r4564479;
        return r4564480;
}

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

Original34.9
Target21.3
Herbie10.1
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -3.6803290429888884e+148

    1. Initial program 62.1

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

    2. Simplified62.1

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

    3. Taylor expanded around -inf 2.3

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

    4. Simplified2.3

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

    if -3.6803290429888884e+148 < b < 4.6129908231112306e-104

    1. Initial program 12.2

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

    2. Simplified12.2

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

    4. Applied *-un-lft-identity12.2

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

    5. Applied div-inv12.3

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

    6. Applied times-frac12.3

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

    7. Simplified12.3

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

    if 4.6129908231112306e-104 < b

    1. Initial program 52.7

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

    2. Simplified52.7

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

    3. Taylor expanded around inf 9.8

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

  4. Final simplification10.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.680329042988888396603264581948851078331 \cdot 10^{148}:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\ \mathbf{elif}\;b \le 4.612990823111230552052602417245542305295 \cdot 10^{-104}:\\ \;\;\;\;\frac{\frac{1}{a}}{2} \cdot \left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019170 
(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)))