Average Error: 32.9 → 10.1
Time: 16.2s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.7512236628315378 \cdot 10^{+131}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 1.489031291672483 \cdot 10^{-98}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - \left(4 \cdot c\right) \cdot a} - b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]
\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 -1.7512236628315378 \cdot 10^{+131}:\\
\;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\

\mathbf{elif}\;b \le 1.489031291672483 \cdot 10^{-98}:\\
\;\;\;\;\frac{\frac{\sqrt{b \cdot b - \left(4 \cdot c\right) \cdot a} - b}{a}}{2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r1638668 = b;
        double r1638669 = -r1638668;
        double r1638670 = r1638668 * r1638668;
        double r1638671 = 4.0;
        double r1638672 = a;
        double r1638673 = r1638671 * r1638672;
        double r1638674 = c;
        double r1638675 = r1638673 * r1638674;
        double r1638676 = r1638670 - r1638675;
        double r1638677 = sqrt(r1638676);
        double r1638678 = r1638669 + r1638677;
        double r1638679 = 2.0;
        double r1638680 = r1638679 * r1638672;
        double r1638681 = r1638678 / r1638680;
        return r1638681;
}
↓
double f(double a, double b, double c) {
        double r1638682 = b;
        double r1638683 = -1.7512236628315378e+131;
        bool r1638684 = r1638682 <= r1638683;
        double r1638685 = c;
        double r1638686 = r1638685 / r1638682;
        double r1638687 = a;
        double r1638688 = r1638682 / r1638687;
        double r1638689 = r1638686 - r1638688;
        double r1638690 = 2.0;
        double r1638691 = r1638689 * r1638690;
        double r1638692 = r1638691 / r1638690;
        double r1638693 = 1.489031291672483e-98;
        bool r1638694 = r1638682 <= r1638693;
        double r1638695 = r1638682 * r1638682;
        double r1638696 = 4.0;
        double r1638697 = r1638696 * r1638685;
        double r1638698 = r1638697 * r1638687;
        double r1638699 = r1638695 - r1638698;
        double r1638700 = sqrt(r1638699);
        double r1638701 = r1638700 - r1638682;
        double r1638702 = r1638701 / r1638687;
        double r1638703 = r1638702 / r1638690;
        double r1638704 = -2.0;
        double r1638705 = r1638704 * r1638686;
        double r1638706 = r1638705 / r1638690;
        double r1638707 = r1638694 ? r1638703 : r1638706;
        double r1638708 = r1638684 ? r1638692 : r1638707;
        return r1638708;
}

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

Original32.9
Target20.4
Herbie10.1
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -1.7512236628315378e+131

    1. Initial program 51.5

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

    2. Simplified51.5

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

    3. Taylor expanded around -inf 3.0

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

    4. Simplified3.0

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

    if -1.7512236628315378e+131 < b < 1.489031291672483e-98

    1. Initial program 11.5

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

    2. Simplified11.6

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

    4. Applied *-un-lft-identity11.6

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

    5. Applied associate-/r*11.6

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

    6. Simplified11.6

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

    if 1.489031291672483e-98 < b

    1. Initial program 51.5

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

    2. Simplified51.5

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

    3. Taylor expanded around inf 10.7

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

  4. Final simplification10.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.7512236628315378 \cdot 10^{+131}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 1.489031291672483 \cdot 10^{-98}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - \left(4 \cdot c\right) \cdot a} - b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019154 
(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)))