Average Error: 34.0 → 9.3
Time: 11.6s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -14858297.0087451544:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -1.46153957938955092 \cdot 10^{-137}:\\ \;\;\;\;\frac{\frac{\frac{\left({b}^{2} - {b}^{2}\right) + 4 \cdot \left(a \cdot c\right)}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}{2}}{a}\\ \mathbf{elif}\;b \le 1.3845340503596435 \cdot 10^{70}:\\ \;\;\;\;\frac{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array}\]
\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 -14858297.0087451544:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le -1.46153957938955092 \cdot 10^{-137}:\\
\;\;\;\;\frac{\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}}{2}}{a}\\

\mathbf{elif}\;b \le 1.3845340503596435 \cdot 10^{70}:\\
\;\;\;\;\frac{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2}}{a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r363 = b;
        double r364 = -r363;
        double r365 = r363 * r363;
        double r366 = 4.0;
        double r367 = a;
        double r368 = c;
        double r369 = r367 * r368;
        double r370 = r366 * r369;
        double r371 = r365 - r370;
        double r372 = sqrt(r371);
        double r373 = r364 - r372;
        double r374 = 2.0;
        double r375 = r374 * r367;
        double r376 = r373 / r375;
        return r376;
}
↓
double f(double a, double b, double c) {
        double r377 = b;
        double r378 = -14858297.008745154;
        bool r379 = r377 <= r378;
        double r380 = -1.0;
        double r381 = c;
        double r382 = r381 / r377;
        double r383 = r380 * r382;
        double r384 = -1.461539579389551e-137;
        bool r385 = r377 <= r384;
        double r386 = 2.0;
        double r387 = pow(r377, r386);
        double r388 = r387 - r387;
        double r389 = 4.0;
        double r390 = a;
        double r391 = r390 * r381;
        double r392 = r389 * r391;
        double r393 = r388 + r392;
        double r394 = r377 * r377;
        double r395 = r394 - r392;
        double r396 = sqrt(r395);
        double r397 = r396 - r377;
        double r398 = r393 / r397;
        double r399 = 2.0;
        double r400 = r398 / r399;
        double r401 = r400 / r390;
        double r402 = 1.3845340503596435e+70;
        bool r403 = r377 <= r402;
        double r404 = -r377;
        double r405 = r404 - r396;
        double r406 = r405 / r399;
        double r407 = r406 / r390;
        double r408 = r377 / r390;
        double r409 = r380 * r408;
        double r410 = r403 ? r407 : r409;
        double r411 = r385 ? r401 : r410;
        double r412 = r379 ? r383 : r411;
        return r412;
}

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

Original34.0
Target20.8
Herbie9.3
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -14858297.008745154

    1. Initial program 55.2

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

    2. Taylor expanded around -inf 5.8

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

    if -14858297.008745154 < b < -1.461539579389551e-137

    1. Initial program 35.4

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

    3. Applied associate-/r*35.4

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

    5. Applied flip--35.5

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

    6. Simplified18.0

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

    7. Simplified18.0

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

    if -1.461539579389551e-137 < b < 1.3845340503596435e+70

    1. Initial program 11.6

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

    3. Applied associate-/r*11.6

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

    if 1.3845340503596435e+70 < b

    1. Initial program 41.5

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

    3. Applied associate-/r*41.5

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

    5. Applied clear-num41.6

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

    6. Taylor expanded around 0 5.7

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

  4. Final simplification9.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -14858297.0087451544:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -1.46153957938955092 \cdot 10^{-137}:\\ \;\;\;\;\frac{\frac{\frac{\left({b}^{2} - {b}^{2}\right) + 4 \cdot \left(a \cdot c\right)}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}{2}}{a}\\ \mathbf{elif}\;b \le 1.3845340503596435 \cdot 10^{70}:\\ \;\;\;\;\frac{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020025 
(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)))