Average Error: 33.1 → 10.3
Time: 21.9s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.7729369216517423 \cdot 10^{+64}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 9.831724396970673 \cdot 10^{-110}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]
\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 -1.7729369216517423 \cdot 10^{+64}:\\
\;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\

\mathbf{elif}\;b \le 9.831724396970673 \cdot 10^{-110}:\\
\;\;\;\;\frac{\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a}}{2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r2216005 = b;
        double r2216006 = -r2216005;
        double r2216007 = r2216005 * r2216005;
        double r2216008 = 4.0;
        double r2216009 = a;
        double r2216010 = r2216008 * r2216009;
        double r2216011 = c;
        double r2216012 = r2216010 * r2216011;
        double r2216013 = r2216007 - r2216012;
        double r2216014 = sqrt(r2216013);
        double r2216015 = r2216006 + r2216014;
        double r2216016 = 2.0;
        double r2216017 = r2216016 * r2216009;
        double r2216018 = r2216015 / r2216017;
        return r2216018;
}
↓
double f(double a, double b, double c) {
        double r2216019 = b;
        double r2216020 = -1.7729369216517423e+64;
        bool r2216021 = r2216019 <= r2216020;
        double r2216022 = c;
        double r2216023 = r2216022 / r2216019;
        double r2216024 = a;
        double r2216025 = r2216019 / r2216024;
        double r2216026 = r2216023 - r2216025;
        double r2216027 = 2.0;
        double r2216028 = r2216026 * r2216027;
        double r2216029 = r2216028 / r2216027;
        double r2216030 = 9.831724396970673e-110;
        bool r2216031 = r2216019 <= r2216030;
        double r2216032 = r2216019 * r2216019;
        double r2216033 = 4.0;
        double r2216034 = r2216024 * r2216022;
        double r2216035 = r2216033 * r2216034;
        double r2216036 = r2216032 - r2216035;
        double r2216037 = sqrt(r2216036);
        double r2216038 = r2216037 - r2216019;
        double r2216039 = r2216038 / r2216024;
        double r2216040 = r2216039 / r2216027;
        double r2216041 = -2.0;
        double r2216042 = r2216041 * r2216023;
        double r2216043 = r2216042 / r2216027;
        double r2216044 = r2216031 ? r2216040 : r2216043;
        double r2216045 = r2216021 ? r2216029 : r2216044;
        return r2216045;
}

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

Original33.1
Target20.2
Herbie10.3
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -1.7729369216517423e+64

    1. Initial program 37.7

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

    2. Simplified37.7

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

    3. Taylor expanded around -inf 5.2

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

    4. Simplified5.2

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

    if -1.7729369216517423e+64 < b < 9.831724396970673e-110

    1. Initial program 12.1

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

    2. Simplified12.1

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

    4. Applied div-inv12.2

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

    6. Applied un-div-inv12.1

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

    if 9.831724396970673e-110 < b

    1. Initial program 51.0

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

    2. Simplified51.0

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

    4. Applied div-inv51.0

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

    5. Taylor expanded around inf 10.8

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

  4. Final simplification10.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.7729369216517423 \cdot 10^{+64}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 9.831724396970673 \cdot 10^{-110}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019138 
(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)))