Average Error: 34.3 → 8.7
Time: 5.2s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.8746290509448952 \cdot 10^{74}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -1.10841810137722214 \cdot 10^{-289}:\\ \;\;\;\;\frac{\frac{4 \cdot \left(a \cdot c\right)}{2 \cdot a}}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}\\ \mathbf{elif}\;b \le 1.1800329617120703 \cdot 10^{123}:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array}\]
\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 -1.8746290509448952 \cdot 10^{74}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le -1.10841810137722214 \cdot 10^{-289}:\\
\;\;\;\;\frac{\frac{4 \cdot \left(a \cdot c\right)}{2 \cdot a}}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}\\

\mathbf{elif}\;b \le 1.1800329617120703 \cdot 10^{123}:\\
\;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;-1 \cdot \frac{b}{a}\\

\end{array}
double f(double a, double b, double c) {
        double r72005 = b;
        double r72006 = -r72005;
        double r72007 = r72005 * r72005;
        double r72008 = 4.0;
        double r72009 = a;
        double r72010 = c;
        double r72011 = r72009 * r72010;
        double r72012 = r72008 * r72011;
        double r72013 = r72007 - r72012;
        double r72014 = sqrt(r72013);
        double r72015 = r72006 - r72014;
        double r72016 = 2.0;
        double r72017 = r72016 * r72009;
        double r72018 = r72015 / r72017;
        return r72018;
}
↓
double f(double a, double b, double c) {
        double r72019 = b;
        double r72020 = -1.8746290509448952e+74;
        bool r72021 = r72019 <= r72020;
        double r72022 = -1.0;
        double r72023 = c;
        double r72024 = r72023 / r72019;
        double r72025 = r72022 * r72024;
        double r72026 = -1.1084181013772221e-289;
        bool r72027 = r72019 <= r72026;
        double r72028 = 4.0;
        double r72029 = a;
        double r72030 = r72029 * r72023;
        double r72031 = r72028 * r72030;
        double r72032 = 2.0;
        double r72033 = r72032 * r72029;
        double r72034 = r72031 / r72033;
        double r72035 = r72019 * r72019;
        double r72036 = r72035 - r72031;
        double r72037 = sqrt(r72036);
        double r72038 = r72037 - r72019;
        double r72039 = r72034 / r72038;
        double r72040 = 1.1800329617120703e+123;
        bool r72041 = r72019 <= r72040;
        double r72042 = -r72019;
        double r72043 = r72042 - r72037;
        double r72044 = 1.0;
        double r72045 = r72044 / r72033;
        double r72046 = r72043 * r72045;
        double r72047 = r72019 / r72029;
        double r72048 = r72022 * r72047;
        double r72049 = r72041 ? r72046 : r72048;
        double r72050 = r72027 ? r72039 : r72049;
        double r72051 = r72021 ? r72025 : r72050;
        return r72051;
}

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

Original34.3
Target21.3
Herbie8.7
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -1.8746290509448952e+74

    1. Initial program 58.6

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

    2. Taylor expanded around -inf 3.3

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}}\]

    if -1.8746290509448952e+74 < b < -1.1084181013772221e-289

    1. Initial program 31.6

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

    3. Applied div-inv31.6

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

    5. Applied flip--31.6

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

    6. Simplified16.7

      \[\leadsto \frac{\color{blue}{0 + 1 \cdot \left(4 \cdot \left(a \cdot c\right)\right)}}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}} \cdot \frac{1}{2 \cdot a}\]

    7. Simplified16.7

      \[\leadsto \frac{0 + 1 \cdot \left(4 \cdot \left(a \cdot c\right)\right)}{\color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}} \cdot \frac{1}{2 \cdot a}\]
    8. Using strategy rm

    9. Applied associate-*l/16.0

      \[\leadsto \color{blue}{\frac{\left(0 + 1 \cdot \left(4 \cdot \left(a \cdot c\right)\right)\right) \cdot \frac{1}{2 \cdot a}}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}\]

    10. Simplified15.9

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

    if -1.1084181013772221e-289 < b < 1.1800329617120703e+123

    1. Initial program 9.1

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

    3. Applied div-inv9.2

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

    if 1.1800329617120703e+123 < b

    1. Initial program 53.2

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

    3. Applied clear-num53.3

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

    4. Taylor expanded around 0 3.2

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification8.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.8746290509448952 \cdot 10^{74}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -1.10841810137722214 \cdot 10^{-289}:\\ \;\;\;\;\frac{\frac{4 \cdot \left(a \cdot c\right)}{2 \cdot a}}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}\\ \mathbf{elif}\;b \le 1.1800329617120703 \cdot 10^{123}:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020036 
(FPCore (a b c)
  :name "The quadratic formula (r2)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ c (* a (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))) (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))

  (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))