Average Error: 34.2 → 6.6
Time: 5.1s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -4.4767676629755074 \cdot 10^{150}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -9.7864508816148999 \cdot 10^{-248}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{4 \cdot c}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}\\ \mathbf{elif}\;b \le 8.29347910705948456 \cdot 10^{98}:\\ \;\;\;\;\frac{-b}{2 \cdot a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\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 -4.4767676629755074 \cdot 10^{150}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le -9.7864508816148999 \cdot 10^{-248}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{4 \cdot c}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}\\

\mathbf{elif}\;b \le 8.29347910705948456 \cdot 10^{98}:\\
\;\;\;\;\frac{-b}{2 \cdot a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\end{array}
double f(double a, double b, double c) {
        double r101164 = b;
        double r101165 = -r101164;
        double r101166 = r101164 * r101164;
        double r101167 = 4.0;
        double r101168 = a;
        double r101169 = c;
        double r101170 = r101168 * r101169;
        double r101171 = r101167 * r101170;
        double r101172 = r101166 - r101171;
        double r101173 = sqrt(r101172);
        double r101174 = r101165 - r101173;
        double r101175 = 2.0;
        double r101176 = r101175 * r101168;
        double r101177 = r101174 / r101176;
        return r101177;
}
↓
double f(double a, double b, double c) {
        double r101178 = b;
        double r101179 = -4.4767676629755074e+150;
        bool r101180 = r101178 <= r101179;
        double r101181 = -1.0;
        double r101182 = c;
        double r101183 = r101182 / r101178;
        double r101184 = r101181 * r101183;
        double r101185 = -9.7864508816149e-248;
        bool r101186 = r101178 <= r101185;
        double r101187 = 1.0;
        double r101188 = 2.0;
        double r101189 = r101187 / r101188;
        double r101190 = 4.0;
        double r101191 = r101190 * r101182;
        double r101192 = r101178 * r101178;
        double r101193 = a;
        double r101194 = r101193 * r101182;
        double r101195 = r101190 * r101194;
        double r101196 = r101192 - r101195;
        double r101197 = sqrt(r101196);
        double r101198 = r101197 - r101178;
        double r101199 = r101191 / r101198;
        double r101200 = r101189 * r101199;
        double r101201 = 8.293479107059485e+98;
        bool r101202 = r101178 <= r101201;
        double r101203 = -r101178;
        double r101204 = r101188 * r101193;
        double r101205 = r101203 / r101204;
        double r101206 = r101197 / r101204;
        double r101207 = r101205 - r101206;
        double r101208 = 1.0;
        double r101209 = r101178 / r101193;
        double r101210 = r101183 - r101209;
        double r101211 = r101208 * r101210;
        double r101212 = r101202 ? r101207 : r101211;
        double r101213 = r101186 ? r101200 : r101212;
        double r101214 = r101180 ? r101184 : r101213;
        return r101214;
}

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

Original34.2
Target21.5
Herbie6.6
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -4.4767676629755074e+150

    1. Initial program 63.7

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

    2. Taylor expanded around -inf 1.4

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

    if -4.4767676629755074e+150 < b < -9.7864508816149e-248

    1. Initial program 36.5

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

    3. Applied flip--36.5

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

    4. Simplified16.2

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

    5. Simplified16.2

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

    7. Applied *-un-lft-identity16.2

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

    8. Applied *-un-lft-identity16.2

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

    9. Applied times-frac16.2

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

    10. Applied times-frac16.2

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

    11. Simplified16.2

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

    12. Simplified14.2

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

    14. Applied *-un-lft-identity14.2

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

    15. Applied times-frac14.2

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

    16. Simplified14.2

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

    17. Simplified7.3

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

    if -9.7864508816149e-248 < b < 8.293479107059485e+98

    1. Initial program 10.0

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

    3. Applied div-sub10.0

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

    if 8.293479107059485e+98 < b

    1. Initial program 46.8

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

    2. Taylor expanded around inf 3.8

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

    3. Simplified3.8

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

  4. Final simplification6.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.4767676629755074 \cdot 10^{150}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -9.7864508816148999 \cdot 10^{-248}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{4 \cdot c}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}\\ \mathbf{elif}\;b \le 8.29347910705948456 \cdot 10^{98}:\\ \;\;\;\;\frac{-b}{2 \cdot a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 
(FPCore (a b c)
  :name "quadm (p42, negative)"
  :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)))