Average Error: 33.8 → 10.0
Time: 16.4s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -8.301687926884188663878043402578250574713 \cdot 10^{98}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 7.029337360841496098479843453825374035485 \cdot 10^{-56}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -8.301687926884188663878043402578250574713 \cdot 10^{98}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 7.029337360841496098479843453825374035485 \cdot 10^{-56}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r218352 = b;
        double r218353 = -r218352;
        double r218354 = r218352 * r218352;
        double r218355 = 4.0;
        double r218356 = a;
        double r218357 = r218355 * r218356;
        double r218358 = c;
        double r218359 = r218357 * r218358;
        double r218360 = r218354 - r218359;
        double r218361 = sqrt(r218360);
        double r218362 = r218353 + r218361;
        double r218363 = 2.0;
        double r218364 = r218363 * r218356;
        double r218365 = r218362 / r218364;
        return r218365;
}
↓
double f(double a, double b, double c) {
        double r218366 = b;
        double r218367 = -8.301687926884189e+98;
        bool r218368 = r218366 <= r218367;
        double r218369 = 1.0;
        double r218370 = c;
        double r218371 = r218370 / r218366;
        double r218372 = a;
        double r218373 = r218366 / r218372;
        double r218374 = r218371 - r218373;
        double r218375 = r218369 * r218374;
        double r218376 = 7.029337360841496e-56;
        bool r218377 = r218366 <= r218376;
        double r218378 = 1.0;
        double r218379 = 2.0;
        double r218380 = r218378 / r218379;
        double r218381 = r218366 * r218366;
        double r218382 = 4.0;
        double r218383 = r218382 * r218372;
        double r218384 = r218383 * r218370;
        double r218385 = r218381 - r218384;
        double r218386 = sqrt(r218385);
        double r218387 = r218386 - r218366;
        double r218388 = r218387 / r218372;
        double r218389 = r218380 * r218388;
        double r218390 = -1.0;
        double r218391 = r218390 * r218371;
        double r218392 = r218377 ? r218389 : r218391;
        double r218393 = r218368 ? r218375 : r218392;
        return r218393;
}

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

Original33.8
Target20.8
Herbie10.0
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -8.301687926884189e+98

    1. Initial program 46.2

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

    2. Simplified46.2

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

    3. Taylor expanded around -inf 3.6

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

    4. Simplified3.6

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

    if -8.301687926884189e+98 < b < 7.029337360841496e-56

    1. Initial program 13.5

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

    2. Simplified13.5

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

    4. Applied *-un-lft-identity13.5

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

    5. Applied times-frac13.5

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

    if 7.029337360841496e-56 < b

    1. Initial program 53.8

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

    2. Simplified53.8

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

    3. Taylor expanded around inf 8.2

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

  4. Final simplification10.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -8.301687926884188663878043402578250574713 \cdot 10^{98}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 7.029337360841496098479843453825374035485 \cdot 10^{-56}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 +o rules:numerics
(FPCore (a b c)
  :name "The quadratic formula (r1)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ (+ (- 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)))