Average Error: 33.2 → 10.0
Time: 24.6s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -7.397994825724217 \cdot 10^{+150}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 1.2158870426682226 \cdot 10^{-82}:\\ \;\;\;\;\frac{\frac{1}{\frac{a}{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}}} - \frac{b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]
\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 -7.397994825724217 \cdot 10^{+150}:\\
\;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\

\mathbf{elif}\;b \le 1.2158870426682226 \cdot 10^{-82}:\\
\;\;\;\;\frac{\frac{1}{\frac{a}{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}}} - \frac{b}{a}}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\

\end{array}
double f(double a, double b, double c) {
        double r4009682 = b;
        double r4009683 = -r4009682;
        double r4009684 = r4009682 * r4009682;
        double r4009685 = 4.0;
        double r4009686 = a;
        double r4009687 = r4009685 * r4009686;
        double r4009688 = c;
        double r4009689 = r4009687 * r4009688;
        double r4009690 = r4009684 - r4009689;
        double r4009691 = sqrt(r4009690);
        double r4009692 = r4009683 + r4009691;
        double r4009693 = 2.0;
        double r4009694 = r4009693 * r4009686;
        double r4009695 = r4009692 / r4009694;
        return r4009695;
}
↓
double f(double a, double b, double c) {
        double r4009696 = b;
        double r4009697 = -7.397994825724217e+150;
        bool r4009698 = r4009696 <= r4009697;
        double r4009699 = c;
        double r4009700 = r4009699 / r4009696;
        double r4009701 = a;
        double r4009702 = r4009696 / r4009701;
        double r4009703 = r4009700 - r4009702;
        double r4009704 = 2.0;
        double r4009705 = r4009703 * r4009704;
        double r4009706 = r4009705 / r4009704;
        double r4009707 = 1.2158870426682226e-82;
        bool r4009708 = r4009696 <= r4009707;
        double r4009709 = 1.0;
        double r4009710 = r4009696 * r4009696;
        double r4009711 = r4009701 * r4009699;
        double r4009712 = 4.0;
        double r4009713 = r4009711 * r4009712;
        double r4009714 = r4009710 - r4009713;
        double r4009715 = sqrt(r4009714);
        double r4009716 = r4009701 / r4009715;
        double r4009717 = r4009709 / r4009716;
        double r4009718 = r4009717 - r4009702;
        double r4009719 = r4009718 / r4009704;
        double r4009720 = -2.0;
        double r4009721 = r4009720 * r4009700;
        double r4009722 = r4009721 / r4009704;
        double r4009723 = r4009708 ? r4009719 : r4009722;
        double r4009724 = r4009698 ? r4009706 : r4009723;
        return r4009724;
}

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

Original33.2
Target20.6
Herbie10.0
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -7.397994825724217e+150

    1. Initial program 59.1

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

    2. Simplified59.1

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

    3. Taylor expanded around -inf 2.2

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

    4. Simplified2.2

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

    if -7.397994825724217e+150 < b < 1.2158870426682226e-82

    1. Initial program 11.8

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

    2. Simplified11.7

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

    4. Applied div-sub11.7

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

    6. Applied clear-num11.8

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

    if 1.2158870426682226e-82 < b

    1. Initial program 52.3

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

    2. Simplified52.3

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

    3. Taylor expanded around inf 9.9

      \[\leadsto \frac{\color{blue}{-2 \cdot \frac{c}{b}}}{2}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -7.397994825724217 \cdot 10^{+150}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 1.2158870426682226 \cdot 10^{-82}:\\ \;\;\;\;\frac{\frac{1}{\frac{a}{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}}} - \frac{b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019162 
(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)))