Average Error: 34.9 → 9.1
Time: 16.5s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.773941363092559658836993715800348666263 \cdot 10^{46}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -1.052552064964455729256789034532554564117 \cdot 10^{-131}:\\ \;\;\;\;\frac{\frac{4 \cdot \left(a \cdot c\right)}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}{2 \cdot a}\\ \mathbf{elif}\;b \le 1.711794221765945466011689717230842352302 \cdot 10^{108}:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\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 -2.773941363092559658836993715800348666263 \cdot 10^{46}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le -1.052552064964455729256789034532554564117 \cdot 10^{-131}:\\
\;\;\;\;\frac{\frac{4 \cdot \left(a \cdot c\right)}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}{2 \cdot a}\\

\mathbf{elif}\;b \le 1.711794221765945466011689717230842352302 \cdot 10^{108}:\\
\;\;\;\;\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 \left(\frac{c}{b} - \frac{b}{a}\right)\\

\end{array}
double f(double a, double b, double c) {
        double r75633 = b;
        double r75634 = -r75633;
        double r75635 = r75633 * r75633;
        double r75636 = 4.0;
        double r75637 = a;
        double r75638 = c;
        double r75639 = r75637 * r75638;
        double r75640 = r75636 * r75639;
        double r75641 = r75635 - r75640;
        double r75642 = sqrt(r75641);
        double r75643 = r75634 - r75642;
        double r75644 = 2.0;
        double r75645 = r75644 * r75637;
        double r75646 = r75643 / r75645;
        return r75646;
}
↓
double f(double a, double b, double c) {
        double r75647 = b;
        double r75648 = -2.7739413630925597e+46;
        bool r75649 = r75647 <= r75648;
        double r75650 = -1.0;
        double r75651 = c;
        double r75652 = r75651 / r75647;
        double r75653 = r75650 * r75652;
        double r75654 = -1.0525520649644557e-131;
        bool r75655 = r75647 <= r75654;
        double r75656 = 4.0;
        double r75657 = a;
        double r75658 = r75657 * r75651;
        double r75659 = r75656 * r75658;
        double r75660 = r75647 * r75647;
        double r75661 = r75660 - r75659;
        double r75662 = sqrt(r75661);
        double r75663 = r75662 - r75647;
        double r75664 = r75659 / r75663;
        double r75665 = 2.0;
        double r75666 = r75665 * r75657;
        double r75667 = r75664 / r75666;
        double r75668 = 1.7117942217659455e+108;
        bool r75669 = r75647 <= r75668;
        double r75670 = -r75647;
        double r75671 = r75670 - r75662;
        double r75672 = 1.0;
        double r75673 = r75672 / r75666;
        double r75674 = r75671 * r75673;
        double r75675 = 1.0;
        double r75676 = r75647 / r75657;
        double r75677 = r75652 - r75676;
        double r75678 = r75675 * r75677;
        double r75679 = r75669 ? r75674 : r75678;
        double r75680 = r75655 ? r75667 : r75679;
        double r75681 = r75649 ? r75653 : r75680;
        return r75681;
}

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

Original34.9
Target21.4
Herbie9.1
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -2.7739413630925597e+46

    1. Initial program 58.0

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

    2. Taylor expanded around -inf 4.1

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

    if -2.7739413630925597e+46 < b < -1.0525520649644557e-131

    1. Initial program 38.9

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

    3. Applied flip--38.9

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

    4. Simplified17.4

      \[\leadsto \frac{\frac{\color{blue}{0 + \left(a \cdot c\right) \cdot 4}}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\]

    5. Simplified17.4

      \[\leadsto \frac{\frac{0 + \left(a \cdot c\right) \cdot 4}{\color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}}{2 \cdot a}\]

    if -1.0525520649644557e-131 < b < 1.7117942217659455e+108

    1. Initial program 11.7

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

    3. Applied div-inv11.9

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

    if 1.7117942217659455e+108 < b

    1. Initial program 49.7

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

    2. Taylor expanded around inf 3.0

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

    3. Simplified3.0

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

  4. Final simplification9.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.773941363092559658836993715800348666263 \cdot 10^{46}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -1.052552064964455729256789034532554564117 \cdot 10^{-131}:\\ \;\;\;\;\frac{\frac{4 \cdot \left(a \cdot c\right)}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}{2 \cdot a}\\ \mathbf{elif}\;b \le 1.711794221765945466011689717230842352302 \cdot 10^{108}:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019235 
(FPCore (a b c)
  :name "The quadratic formula (r2)"
  :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)))