Average Error: 34.4 → 6.7
Time: 19.0s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.763315479739403460017265344144602342789 \cdot 10^{89}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -1.085000278636624341855070450537604684134 \cdot 10^{-297}:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) + \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}\\ \mathbf{elif}\;b \le 3.355858625783055094237525774982320834143 \cdot 10^{101}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\ \end{array}\]
\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 r4779535 = b;
        double r4779536 = -r4779535;
        double r4779537 = r4779535 * r4779535;
        double r4779538 = 4.0;
        double r4779539 = a;
        double r4779540 = c;
        double r4779541 = r4779539 * r4779540;
        double r4779542 = r4779538 * r4779541;
        double r4779543 = r4779537 - r4779542;
        double r4779544 = sqrt(r4779543);
        double r4779545 = r4779536 - r4779544;
        double r4779546 = 2.0;
        double r4779547 = r4779546 * r4779539;
        double r4779548 = r4779545 / r4779547;
        return r4779548;
}
↓
double f(double a, double b, double c) {
        double r4779549 = b;
        double r4779550 = -1.7633154797394035e+89;
        bool r4779551 = r4779549 <= r4779550;
        double r4779552 = -1.0;
        double r4779553 = c;
        double r4779554 = r4779553 / r4779549;
        double r4779555 = r4779552 * r4779554;
        double r4779556 = -1.0850002786366243e-297;
        bool r4779557 = r4779549 <= r4779556;
        double r4779558 = 2.0;
        double r4779559 = r4779553 * r4779558;
        double r4779560 = -r4779549;
        double r4779561 = r4779549 * r4779549;
        double r4779562 = a;
        double r4779563 = 4.0;
        double r4779564 = r4779562 * r4779563;
        double r4779565 = r4779553 * r4779564;
        double r4779566 = r4779561 - r4779565;
        double r4779567 = sqrt(r4779566);
        double r4779568 = r4779560 + r4779567;
        double r4779569 = r4779559 / r4779568;
        double r4779570 = 3.355858625783055e+101;
        bool r4779571 = r4779549 <= r4779570;
        double r4779572 = r4779560 - r4779567;
        double r4779573 = r4779562 * r4779558;
        double r4779574 = r4779572 / r4779573;
        double r4779575 = r4779549 / r4779562;
        double r4779576 = r4779554 - r4779575;
        double r4779577 = 1.0;
        double r4779578 = r4779576 * r4779577;
        double r4779579 = r4779571 ? r4779574 : r4779578;
        double r4779580 = r4779557 ? r4779569 : r4779579;
        double r4779581 = r4779551 ? r4779555 : r4779580;
        return r4779581;
}

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 "The quadratic formula (r2)"

  :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)))