Average Error: 33.8 → 6.7
Time: 1.2m
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -4.3671561050226844 \cdot 10^{+101}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 5.6646816643665726 \cdot 10^{-285}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}\right) \cdot \frac{1}{a \cdot 2}\\ \mathbf{elif}\;b \le 2.8546456093447043 \cdot 10^{+107}:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{2 \cdot \left(\frac{c}{b} \cdot a - b\right)}\\ \end{array}\]
double f(double a, double b, double c) {
        double r32163669 = b;
        double r32163670 = -r32163669;
        double r32163671 = r32163669 * r32163669;
        double r32163672 = 4.0;
        double r32163673 = a;
        double r32163674 = r32163672 * r32163673;
        double r32163675 = c;
        double r32163676 = r32163674 * r32163675;
        double r32163677 = r32163671 - r32163676;
        double r32163678 = sqrt(r32163677);
        double r32163679 = r32163670 + r32163678;
        double r32163680 = 2.0;
        double r32163681 = r32163680 * r32163673;
        double r32163682 = r32163679 / r32163681;
        return r32163682;
}
↓
double f(double a, double b, double c) {
        double r32163683 = b;
        double r32163684 = -4.3671561050226844e+101;
        bool r32163685 = r32163683 <= r32163684;
        double r32163686 = c;
        double r32163687 = r32163686 / r32163683;
        double r32163688 = a;
        double r32163689 = r32163683 / r32163688;
        double r32163690 = r32163687 - r32163689;
        double r32163691 = 5.6646816643665726e-285;
        bool r32163692 = r32163683 <= r32163691;
        double r32163693 = -r32163683;
        double r32163694 = r32163683 * r32163683;
        double r32163695 = 4.0;
        double r32163696 = r32163695 * r32163688;
        double r32163697 = r32163686 * r32163696;
        double r32163698 = r32163694 - r32163697;
        double r32163699 = sqrt(r32163698);
        double r32163700 = r32163693 + r32163699;
        double r32163701 = 1.0;
        double r32163702 = 2.0;
        double r32163703 = r32163688 * r32163702;
        double r32163704 = r32163701 / r32163703;
        double r32163705 = r32163700 * r32163704;
        double r32163706 = 2.8546456093447043e+107;
        bool r32163707 = r32163683 <= r32163706;
        double r32163708 = r32163686 * r32163702;
        double r32163709 = r32163693 - r32163699;
        double r32163710 = r32163708 / r32163709;
        double r32163711 = r32163687 * r32163688;
        double r32163712 = r32163711 - r32163683;
        double r32163713 = r32163702 * r32163712;
        double r32163714 = r32163708 / r32163713;
        double r32163715 = r32163707 ? r32163710 : r32163714;
        double r32163716 = r32163692 ? r32163705 : r32163715;
        double r32163717 = r32163685 ? r32163690 : r32163716;
        return r32163717;
}


\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \le -4.3671561050226844 \cdot 10^{+101}:\\
\;\;\;\;\frac{c}{b} - \frac{b}{a}\\

\mathbf{elif}\;b \le 5.6646816643665726 \cdot 10^{-285}:\\
\;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}\right) \cdot \frac{1}{a \cdot 2}\\

\mathbf{elif}\;b \le 2.8546456093447043 \cdot 10^{+107}:\\
\;\;\;\;\frac{c \cdot 2}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}\\

\mathbf{else}:\\
\;\;\;\;\frac{c \cdot 2}{2 \cdot \left(\frac{c}{b} \cdot a - b\right)}\\

\end{array}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.8
Target21.0
Herbie6.7
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -4.3671561050226844e+101

    1. Initial program 44.7

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

    3. Applied div-inv44.8

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

    4. Taylor expanded around -inf 3.8

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

    if -4.3671561050226844e+101 < b < 5.6646816643665726e-285

    1. Initial program 9.6

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

    3. Applied div-inv9.7

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

    if 5.6646816643665726e-285 < b < 2.8546456093447043e+107

    1. Initial program 33.4

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

    3. Applied div-inv33.5

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

    5. Applied flip-+33.6

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

    6. Applied associate-*l/33.6

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

    7. Simplified16.0

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

    8. Taylor expanded around -inf 8.7

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

    if 2.8546456093447043e+107 < b

    1. Initial program 59.5

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

    3. Applied div-inv59.5

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

    5. Applied flip-+59.6

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

    6. Applied associate-*l/59.6

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

    7. Simplified32.0

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

    8. Taylor expanded around -inf 31.4

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

    9. Taylor expanded around inf 6.5

      \[\leadsto \frac{--2 \cdot c}{\color{blue}{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}}\]

    10. Simplified2.1

      \[\leadsto \frac{--2 \cdot c}{\color{blue}{\left(\frac{c}{b} \cdot a - b\right) \cdot 2}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification6.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.3671561050226844 \cdot 10^{+101}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 5.6646816643665726 \cdot 10^{-285}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}\right) \cdot \frac{1}{a \cdot 2}\\ \mathbf{elif}\;b \le 2.8546456093447043 \cdot 10^{+107}:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{2 \cdot \left(\frac{c}{b} \cdot a - b\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019102 +o rules:numerics
(FPCore (a b c)
  :name "The quadratic formula (r1)"

  :herbie-target
  (if (< b 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)))