Average Error: 34.4 → 6.7
Time: 18.8s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.763315479739403460017265344144602342789 \cdot 10^{89}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -1.085000278636624341855070450537604684134 \cdot 10^{-297}:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) + \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}\\ \mathbf{elif}\;b \le 3.355858625783055094237525774982320834143 \cdot 10^{101}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\ \end{array}\]
\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 -1.763315479739403460017265344144602342789 \cdot 10^{89}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le -1.085000278636624341855070450537604684134 \cdot 10^{-297}:\\
\;\;\;\;\frac{c \cdot 2}{\left(-b\right) + \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}\\

\mathbf{elif}\;b \le 3.355858625783055094237525774982320834143 \cdot 10^{101}:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}{a \cdot 2}\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\

\end{array}
double f(double a, double b, double c) {
        double r4324464 = b;
        double r4324465 = -r4324464;
        double r4324466 = r4324464 * r4324464;
        double r4324467 = 4.0;
        double r4324468 = a;
        double r4324469 = c;
        double r4324470 = r4324468 * r4324469;
        double r4324471 = r4324467 * r4324470;
        double r4324472 = r4324466 - r4324471;
        double r4324473 = sqrt(r4324472);
        double r4324474 = r4324465 - r4324473;
        double r4324475 = 2.0;
        double r4324476 = r4324475 * r4324468;
        double r4324477 = r4324474 / r4324476;
        return r4324477;
}
↓
double f(double a, double b, double c) {
        double r4324478 = b;
        double r4324479 = -1.7633154797394035e+89;
        bool r4324480 = r4324478 <= r4324479;
        double r4324481 = -1.0;
        double r4324482 = c;
        double r4324483 = r4324482 / r4324478;
        double r4324484 = r4324481 * r4324483;
        double r4324485 = -1.0850002786366243e-297;
        bool r4324486 = r4324478 <= r4324485;
        double r4324487 = 2.0;
        double r4324488 = r4324482 * r4324487;
        double r4324489 = -r4324478;
        double r4324490 = r4324478 * r4324478;
        double r4324491 = a;
        double r4324492 = 4.0;
        double r4324493 = r4324491 * r4324492;
        double r4324494 = r4324482 * r4324493;
        double r4324495 = r4324490 - r4324494;
        double r4324496 = sqrt(r4324495);
        double r4324497 = r4324489 + r4324496;
        double r4324498 = r4324488 / r4324497;
        double r4324499 = 3.355858625783055e+101;
        bool r4324500 = r4324478 <= r4324499;
        double r4324501 = r4324489 - r4324496;
        double r4324502 = r4324491 * r4324487;
        double r4324503 = r4324501 / r4324502;
        double r4324504 = r4324478 / r4324491;
        double r4324505 = r4324483 - r4324504;
        double r4324506 = 1.0;
        double r4324507 = r4324505 * r4324506;
        double r4324508 = r4324500 ? r4324503 : r4324507;
        double r4324509 = r4324486 ? r4324498 : r4324508;
        double r4324510 = r4324480 ? r4324484 : r4324509;
        return r4324510;
}

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

Original34.4
Target20.9
Herbie6.7
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -1.7633154797394035e+89

    1. Initial program 59.1

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

    2. Taylor expanded around -inf 2.7

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

    if -1.7633154797394035e+89 < b < -1.0850002786366243e-297

    1. Initial program 32.1

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

    2. Taylor expanded around 0 32.1

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

    3. Simplified32.1

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

    5. Applied div-inv32.2

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

    7. Applied flip--32.2

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

    8. Applied associate-*l/32.2

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

    9. Simplified15.8

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

    10. Taylor expanded around 0 8.4

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

    if -1.0850002786366243e-297 < b < 3.355858625783055e+101

    1. Initial program 9.5

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

    2. Taylor expanded around 0 9.5

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

    3. Simplified9.5

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

    if 3.355858625783055e+101 < b

    1. Initial program 46.8

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

    2. Taylor expanded around inf 4.4

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

    3. Simplified4.4

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

  4. Final simplification6.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.763315479739403460017265344144602342789 \cdot 10^{89}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -1.085000278636624341855070450537604684134 \cdot 10^{-297}:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) + \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}\\ \mathbf{elif}\;b \le 3.355858625783055094237525774982320834143 \cdot 10^{101}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 
(FPCore (a b c)
  :name "quadm (p42, negative)"

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

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