Average Error: 32.9 → 6.8
Time: 41.1s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.515406138267436 \cdot 10^{+130}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le -4.320492610336173 \cdot 10^{-222}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{b \cdot b + a \cdot \left(-4 \cdot c\right)} + \left(-b\right)}\\ \mathbf{elif}\;b \le 5.000815192005961 \cdot 10^{+134}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b + a \cdot \left(-4 \cdot c\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]
\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 -1.515406138267436 \cdot 10^{+130}:\\
\;\;\;\;-\frac{c}{b}\\

\mathbf{elif}\;b \le -4.320492610336173 \cdot 10^{-222}:\\
\;\;\;\;\frac{2 \cdot c}{\sqrt{b \cdot b + a \cdot \left(-4 \cdot c\right)} + \left(-b\right)}\\

\mathbf{elif}\;b \le 5.000815192005961 \cdot 10^{+134}:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b + a \cdot \left(-4 \cdot c\right)}}{a \cdot 2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r5831332 = b;
        double r5831333 = -r5831332;
        double r5831334 = r5831332 * r5831332;
        double r5831335 = 4.0;
        double r5831336 = a;
        double r5831337 = c;
        double r5831338 = r5831336 * r5831337;
        double r5831339 = r5831335 * r5831338;
        double r5831340 = r5831334 - r5831339;
        double r5831341 = sqrt(r5831340);
        double r5831342 = r5831333 - r5831341;
        double r5831343 = 2.0;
        double r5831344 = r5831343 * r5831336;
        double r5831345 = r5831342 / r5831344;
        return r5831345;
}
↓
double f(double a, double b, double c) {
        double r5831346 = b;
        double r5831347 = -1.515406138267436e+130;
        bool r5831348 = r5831346 <= r5831347;
        double r5831349 = c;
        double r5831350 = r5831349 / r5831346;
        double r5831351 = -r5831350;
        double r5831352 = -4.320492610336173e-222;
        bool r5831353 = r5831346 <= r5831352;
        double r5831354 = 2.0;
        double r5831355 = r5831354 * r5831349;
        double r5831356 = r5831346 * r5831346;
        double r5831357 = a;
        double r5831358 = -4.0;
        double r5831359 = r5831358 * r5831349;
        double r5831360 = r5831357 * r5831359;
        double r5831361 = r5831356 + r5831360;
        double r5831362 = sqrt(r5831361);
        double r5831363 = -r5831346;
        double r5831364 = r5831362 + r5831363;
        double r5831365 = r5831355 / r5831364;
        double r5831366 = 5.000815192005961e+134;
        bool r5831367 = r5831346 <= r5831366;
        double r5831368 = r5831363 - r5831362;
        double r5831369 = r5831357 * r5831354;
        double r5831370 = r5831368 / r5831369;
        double r5831371 = r5831346 / r5831357;
        double r5831372 = r5831350 - r5831371;
        double r5831373 = r5831367 ? r5831370 : r5831372;
        double r5831374 = r5831353 ? r5831365 : r5831373;
        double r5831375 = r5831348 ? r5831351 : r5831374;
        return r5831375;
}

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

Original32.9
Target20.6
Herbie6.8
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -1.515406138267436e+130

    1. Initial program 60.2

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

    2. Taylor expanded around -inf 2.3

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

    3. Simplified2.3

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

    if -1.515406138267436e+130 < b < -4.320492610336173e-222

    1. Initial program 36.2

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

    2. Taylor expanded around -inf 36.2

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

    3. Simplified36.2

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

    5. Applied div-inv36.2

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

    7. Applied flip--36.3

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

    8. Applied associate-*l/36.3

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

    9. Simplified14.6

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

    10. Taylor expanded around 0 7.2

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

    if -4.320492610336173e-222 < b < 5.000815192005961e+134

    1. Initial program 9.7

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

    2. Taylor expanded around -inf 9.7

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

    3. Simplified9.8

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

    if 5.000815192005961e+134 < b

    1. Initial program 53.5

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

    2. Taylor expanded around -inf 53.5

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

    3. Simplified53.5

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

    5. Applied div-inv53.5

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

    6. Taylor expanded around inf 3.5

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

  4. Final simplification6.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.515406138267436 \cdot 10^{+130}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le -4.320492610336173 \cdot 10^{-222}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{b \cdot b + a \cdot \left(-4 \cdot c\right)} + \left(-b\right)}\\ \mathbf{elif}\;b \le 5.000815192005961 \cdot 10^{+134}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b + a \cdot \left(-4 \cdot c\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019121 
(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)))