Average Error: 34.2 → 6.5
Time: 5.4s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.948753525425808925296770585273107730084 \cdot 10^{142}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 2.477064787285889532736667463396647969795 \cdot 10^{-233}:\\ \;\;\;\;1 \cdot \frac{2 \cdot c}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}\\ \mathbf{elif}\;b \le 1.276518349713334879747784170879022589057 \cdot 10^{91}:\\ \;\;\;\;\frac{-b}{2 \cdot a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\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 -1.948753525425808925296770585273107730084 \cdot 10^{142}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 2.477064787285889532736667463396647969795 \cdot 10^{-233}:\\
\;\;\;\;1 \cdot \frac{2 \cdot c}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}\\

\mathbf{elif}\;b \le 1.276518349713334879747784170879022589057 \cdot 10^{91}:\\
\;\;\;\;\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 r145181 = b;
        double r145182 = -r145181;
        double r145183 = r145181 * r145181;
        double r145184 = 4.0;
        double r145185 = a;
        double r145186 = c;
        double r145187 = r145185 * r145186;
        double r145188 = r145184 * r145187;
        double r145189 = r145183 - r145188;
        double r145190 = sqrt(r145189);
        double r145191 = r145182 - r145190;
        double r145192 = 2.0;
        double r145193 = r145192 * r145185;
        double r145194 = r145191 / r145193;
        return r145194;
}
↓
double f(double a, double b, double c) {
        double r145195 = b;
        double r145196 = -1.948753525425809e+142;
        bool r145197 = r145195 <= r145196;
        double r145198 = -1.0;
        double r145199 = c;
        double r145200 = r145199 / r145195;
        double r145201 = r145198 * r145200;
        double r145202 = 2.4770647872858895e-233;
        bool r145203 = r145195 <= r145202;
        double r145204 = 1.0;
        double r145205 = 2.0;
        double r145206 = r145205 * r145199;
        double r145207 = r145195 * r145195;
        double r145208 = 4.0;
        double r145209 = a;
        double r145210 = r145209 * r145199;
        double r145211 = r145208 * r145210;
        double r145212 = r145207 - r145211;
        double r145213 = sqrt(r145212);
        double r145214 = r145213 - r145195;
        double r145215 = r145206 / r145214;
        double r145216 = r145204 * r145215;
        double r145217 = 1.2765183497133349e+91;
        bool r145218 = r145195 <= r145217;
        double r145219 = -r145195;
        double r145220 = r145205 * r145209;
        double r145221 = r145219 / r145220;
        double r145222 = r145213 / r145220;
        double r145223 = r145221 - r145222;
        double r145224 = 1.0;
        double r145225 = r145195 / r145209;
        double r145226 = r145200 - r145225;
        double r145227 = r145224 * r145226;
        double r145228 = r145218 ? r145223 : r145227;
        double r145229 = r145203 ? r145216 : r145228;
        double r145230 = r145197 ? r145201 : r145229;
        return r145230;
}

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
Target21.1
Herbie6.5
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -1.948753525425809e+142

    1. Initial program 62.8

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

    2. Taylor expanded around -inf 1.4

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

    if -1.948753525425809e+142 < b < 2.4770647872858895e-233

    1. Initial program 32.3

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

    3. Applied div-inv32.3

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

    5. Applied flip--32.4

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

    6. Simplified17.0

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

    7. Simplified17.0

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

    9. Applied *-un-lft-identity17.0

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

    10. Applied associate-*l*17.0

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

    11. Simplified15.6

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

    12. Taylor expanded around 0 9.3

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

    if 2.4770647872858895e-233 < b < 1.2765183497133349e+91

    1. Initial program 7.5

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

    3. Applied div-sub7.5

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

    if 1.2765183497133349e+91 < b

    1. Initial program 45.7

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

    2. Taylor expanded around inf 4.2

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

    3. Simplified4.2

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

  4. Final simplification6.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.948753525425808925296770585273107730084 \cdot 10^{142}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 2.477064787285889532736667463396647969795 \cdot 10^{-233}:\\ \;\;\;\;1 \cdot \frac{2 \cdot c}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}\\ \mathbf{elif}\;b \le 1.276518349713334879747784170879022589057 \cdot 10^{91}:\\ \;\;\;\;\frac{-b}{2 \cdot a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019354 
(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)))