Average Error: 34.7 → 10.8
Time: 5.1s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -6.371698442415157100029538982618411822116 \cdot 10^{150}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 2.306544477380116301747543706493703838768 \cdot 10^{-129}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 r87491 = b;
        double r87492 = -r87491;
        double r87493 = r87491 * r87491;
        double r87494 = 4.0;
        double r87495 = a;
        double r87496 = c;
        double r87497 = r87495 * r87496;
        double r87498 = r87494 * r87497;
        double r87499 = r87493 - r87498;
        double r87500 = sqrt(r87499);
        double r87501 = r87492 + r87500;
        double r87502 = 2.0;
        double r87503 = r87502 * r87495;
        double r87504 = r87501 / r87503;
        return r87504;
}
↓
double f(double a, double b, double c) {
        double r87505 = b;
        double r87506 = -6.371698442415157e+150;
        bool r87507 = r87505 <= r87506;
        double r87508 = 1.0;
        double r87509 = c;
        double r87510 = r87509 / r87505;
        double r87511 = a;
        double r87512 = r87505 / r87511;
        double r87513 = r87510 - r87512;
        double r87514 = r87508 * r87513;
        double r87515 = 2.3065444773801163e-129;
        bool r87516 = r87505 <= r87515;
        double r87517 = -r87505;
        double r87518 = r87505 * r87505;
        double r87519 = 4.0;
        double r87520 = r87511 * r87509;
        double r87521 = r87519 * r87520;
        double r87522 = r87518 - r87521;
        double r87523 = sqrt(r87522);
        double r87524 = r87517 + r87523;
        double r87525 = 1.0;
        double r87526 = 2.0;
        double r87527 = r87526 * r87511;
        double r87528 = r87525 / r87527;
        double r87529 = r87524 * r87528;
        double r87530 = -1.0;
        double r87531 = r87530 * r87510;
        double r87532 = r87516 ? r87529 : r87531;
        double r87533 = r87507 ? r87514 : r87532;
        return r87533;
}

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 
(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)))