Average Error: 34.0 → 10.4
Time: 21.4s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.541338025369698 \cdot 10^{+80}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 1.1094847447691107 \cdot 10^{-113}:\\ \;\;\;\;\frac{\frac{1}{\frac{a}{\sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)} - b}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \le -2.541338025369698 \cdot 10^{+80}:\\
\;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\

\mathbf{elif}\;b \le 1.1094847447691107 \cdot 10^{-113}:\\
\;\;\;\;\frac{\frac{1}{\frac{a}{\sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)} - b}}}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\

\end{array}
double f(double a, double b, double c) {
        double r1870333 = b;
        double r1870334 = -r1870333;
        double r1870335 = r1870333 * r1870333;
        double r1870336 = 4.0;
        double r1870337 = a;
        double r1870338 = r1870336 * r1870337;
        double r1870339 = c;
        double r1870340 = r1870338 * r1870339;
        double r1870341 = r1870335 - r1870340;
        double r1870342 = sqrt(r1870341);
        double r1870343 = r1870334 + r1870342;
        double r1870344 = 2.0;
        double r1870345 = r1870344 * r1870337;
        double r1870346 = r1870343 / r1870345;
        return r1870346;
}
↓
double f(double a, double b, double c) {
        double r1870347 = b;
        double r1870348 = -2.541338025369698e+80;
        bool r1870349 = r1870347 <= r1870348;
        double r1870350 = c;
        double r1870351 = r1870350 / r1870347;
        double r1870352 = a;
        double r1870353 = r1870347 / r1870352;
        double r1870354 = r1870351 - r1870353;
        double r1870355 = 2.0;
        double r1870356 = r1870354 * r1870355;
        double r1870357 = r1870356 / r1870355;
        double r1870358 = 1.1094847447691107e-113;
        bool r1870359 = r1870347 <= r1870358;
        double r1870360 = 1.0;
        double r1870361 = -4.0;
        double r1870362 = r1870361 * r1870352;
        double r1870363 = r1870362 * r1870350;
        double r1870364 = fma(r1870347, r1870347, r1870363);
        double r1870365 = sqrt(r1870364);
        double r1870366 = r1870365 - r1870347;
        double r1870367 = r1870352 / r1870366;
        double r1870368 = r1870360 / r1870367;
        double r1870369 = r1870368 / r1870355;
        double r1870370 = -2.0;
        double r1870371 = r1870370 * r1870351;
        double r1870372 = r1870371 / r1870355;
        double r1870373 = r1870359 ? r1870369 : r1870372;
        double r1870374 = r1870349 ? r1870357 : r1870373;
        return r1870374;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original34.0
Target20.9
Herbie10.4
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -2.541338025369698e+80

    1. Initial program 41.0

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

    2. Simplified41.0

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

    3. Taylor expanded around -inf 4.2

      \[\leadsto \frac{\color{blue}{2 \cdot \frac{c}{b} - 2 \cdot \frac{b}{a}}}{2}\]

    4. Simplified4.2

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

    if -2.541338025369698e+80 < b < 1.1094847447691107e-113

    1. Initial program 12.5

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

    2. Simplified12.5

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

    4. Applied clear-num12.7

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

    if 1.1094847447691107e-113 < b

    1. Initial program 51.6

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

    2. Simplified51.6

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

    3. Taylor expanded around inf 10.8

      \[\leadsto \frac{\color{blue}{-2 \cdot \frac{c}{b}}}{2}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.541338025369698 \cdot 10^{+80}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 1.1094847447691107 \cdot 10^{-113}:\\ \;\;\;\;\frac{\frac{1}{\frac{a}{\sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)} - b}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019141 +o rules:numerics
(FPCore (a b c)
  :name "The quadratic formula (r1)"

  :herbie-target
  (if (< b 0) (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)) (/ c (* a (/ (- (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))))

  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))