Average Error: 34.7 → 10.8
Time: 5.7s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -6.371698442415157100029538982618411822116 \cdot 10^{150}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 2.306544477380116301747543706493703838768 \cdot 10^{-129}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -6.371698442415157100029538982618411822116 \cdot 10^{150}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 2.306544477380116301747543706493703838768 \cdot 10^{-129}:\\
\;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r69323 = b;
        double r69324 = -r69323;
        double r69325 = r69323 * r69323;
        double r69326 = 4.0;
        double r69327 = a;
        double r69328 = c;
        double r69329 = r69327 * r69328;
        double r69330 = r69326 * r69329;
        double r69331 = r69325 - r69330;
        double r69332 = sqrt(r69331);
        double r69333 = r69324 + r69332;
        double r69334 = 2.0;
        double r69335 = r69334 * r69327;
        double r69336 = r69333 / r69335;
        return r69336;
}
↓
double f(double a, double b, double c) {
        double r69337 = b;
        double r69338 = -6.371698442415157e+150;
        bool r69339 = r69337 <= r69338;
        double r69340 = 1.0;
        double r69341 = c;
        double r69342 = r69341 / r69337;
        double r69343 = a;
        double r69344 = r69337 / r69343;
        double r69345 = r69342 - r69344;
        double r69346 = r69340 * r69345;
        double r69347 = 2.3065444773801163e-129;
        bool r69348 = r69337 <= r69347;
        double r69349 = -r69337;
        double r69350 = r69337 * r69337;
        double r69351 = 4.0;
        double r69352 = r69343 * r69341;
        double r69353 = r69351 * r69352;
        double r69354 = r69350 - r69353;
        double r69355 = sqrt(r69354);
        double r69356 = r69349 + r69355;
        double r69357 = 1.0;
        double r69358 = 2.0;
        double r69359 = r69358 * r69343;
        double r69360 = r69357 / r69359;
        double r69361 = r69356 * r69360;
        double r69362 = -1.0;
        double r69363 = r69362 * r69342;
        double r69364 = r69348 ? r69361 : r69363;
        double r69365 = r69339 ? r69346 : r69364;
        return r69365;
}

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

Original34.7
Target21.4
Herbie10.8
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -6.371698442415157e+150

    1. Initial program 63.0

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

    2. Taylor expanded around -inf 2.5

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

    3. Simplified2.5

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

    if -6.371698442415157e+150 < b < 2.3065444773801163e-129

    1. Initial program 11.3

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

    3. Applied div-inv11.4

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

    if 2.3065444773801163e-129 < b

    1. Initial program 51.5

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

    2. Taylor expanded around inf 12.3

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

  4. Final simplification10.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -6.371698442415157100029538982618411822116 \cdot 10^{150}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 2.306544477380116301747543706493703838768 \cdot 10^{-129}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019344 +o rules:numerics
(FPCore (a b c)
  :name "quadp (p42, positive)"
  :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)))