Average Error: 32.8 → 8.9
Time: 33.6s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -9.877342284320474 \cdot 10^{+38}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le -4.726535681060057 \cdot 10^{-132}:\\ \;\;\;\;\frac{\frac{\left(a \cdot c\right) \cdot 2}{a}}{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} - b}\\ \mathbf{elif}\;b \le 9.19242293018462 \cdot 10^{+63}:\\ \;\;\;\;\frac{1}{\frac{a \cdot 2}{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]
\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 -9.877342284320474 \cdot 10^{+38}:\\
\;\;\;\;-\frac{c}{b}\\

\mathbf{elif}\;b \le -4.726535681060057 \cdot 10^{-132}:\\
\;\;\;\;\frac{\frac{\left(a \cdot c\right) \cdot 2}{a}}{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} - b}\\

\mathbf{elif}\;b \le 9.19242293018462 \cdot 10^{+63}:\\
\;\;\;\;\frac{1}{\frac{a \cdot 2}{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}}}\\

\mathbf{else}:\\
\;\;\;\;\frac{c}{b} - \frac{b}{a}\\

\end{array}
double f(double a, double b, double c) {
        double r4243175 = b;
        double r4243176 = -r4243175;
        double r4243177 = r4243175 * r4243175;
        double r4243178 = 4.0;
        double r4243179 = a;
        double r4243180 = c;
        double r4243181 = r4243179 * r4243180;
        double r4243182 = r4243178 * r4243181;
        double r4243183 = r4243177 - r4243182;
        double r4243184 = sqrt(r4243183);
        double r4243185 = r4243176 - r4243184;
        double r4243186 = 2.0;
        double r4243187 = r4243186 * r4243179;
        double r4243188 = r4243185 / r4243187;
        return r4243188;
}
↓
double f(double a, double b, double c) {
        double r4243189 = b;
        double r4243190 = -9.877342284320474e+38;
        bool r4243191 = r4243189 <= r4243190;
        double r4243192 = c;
        double r4243193 = r4243192 / r4243189;
        double r4243194 = -r4243193;
        double r4243195 = -4.726535681060057e-132;
        bool r4243196 = r4243189 <= r4243195;
        double r4243197 = a;
        double r4243198 = r4243197 * r4243192;
        double r4243199 = 2.0;
        double r4243200 = r4243198 * r4243199;
        double r4243201 = r4243200 / r4243197;
        double r4243202 = r4243189 * r4243189;
        double r4243203 = 4.0;
        double r4243204 = r4243198 * r4243203;
        double r4243205 = r4243202 - r4243204;
        double r4243206 = sqrt(r4243205);
        double r4243207 = r4243206 - r4243189;
        double r4243208 = r4243201 / r4243207;
        double r4243209 = 9.19242293018462e+63;
        bool r4243210 = r4243189 <= r4243209;
        double r4243211 = 1.0;
        double r4243212 = r4243197 * r4243199;
        double r4243213 = -r4243189;
        double r4243214 = r4243213 - r4243206;
        double r4243215 = r4243212 / r4243214;
        double r4243216 = r4243211 / r4243215;
        double r4243217 = r4243189 / r4243197;
        double r4243218 = r4243193 - r4243217;
        double r4243219 = r4243210 ? r4243216 : r4243218;
        double r4243220 = r4243196 ? r4243208 : r4243219;
        double r4243221 = r4243191 ? r4243194 : r4243220;
        return r4243221;
}

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

Original32.8
Target20.1
Herbie8.9
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -9.877342284320474e+38

    1. Initial program 55.4

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

    2. Taylor expanded around -inf 4.5

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

    3. Simplified4.5

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

    if -9.877342284320474e+38 < b < -4.726535681060057e-132

    1. Initial program 37.3

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

    3. Applied flip--37.4

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

    4. Simplified15.8

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

    5. Simplified15.8

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

    7. Applied div-inv15.9

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

    8. Simplified15.8

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

    10. Applied associate-*l/15.5

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

    11. Simplified15.4

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

    if -4.726535681060057e-132 < b < 9.19242293018462e+63

    1. Initial program 11.8

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

    3. Applied clear-num11.9

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

    if 9.19242293018462e+63 < b

    1. Initial program 38.2

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

    2. Taylor expanded around inf 4.6

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

  4. Final simplification8.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -9.877342284320474 \cdot 10^{+38}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le -4.726535681060057 \cdot 10^{-132}:\\ \;\;\;\;\frac{\frac{\left(a \cdot c\right) \cdot 2}{a}}{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} - b}\\ \mathbf{elif}\;b \le 9.19242293018462 \cdot 10^{+63}:\\ \;\;\;\;\frac{1}{\frac{a \cdot 2}{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019158 
(FPCore (a b c)
  :name "quadm (p42, negative)"

  :herbie-target
  (if (< b 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)))