Average Error: 34.5 → 10.5
Time: 5.5s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.279587145681289 \cdot 10^{-136}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 6.8526453862578789 \cdot 10^{139}:\\ \;\;\;\;1 \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\right)\\ \end{array}\]
\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 -1.279587145681289 \cdot 10^{-136}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 6.8526453862578789 \cdot 10^{139}:\\
\;\;\;\;1 \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;1 \cdot \left(1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\right)\\

\end{array}
double f(double a, double b, double c) {
        double r93647 = b;
        double r93648 = -r93647;
        double r93649 = r93647 * r93647;
        double r93650 = 4.0;
        double r93651 = a;
        double r93652 = c;
        double r93653 = r93651 * r93652;
        double r93654 = r93650 * r93653;
        double r93655 = r93649 - r93654;
        double r93656 = sqrt(r93655);
        double r93657 = r93648 - r93656;
        double r93658 = 2.0;
        double r93659 = r93658 * r93651;
        double r93660 = r93657 / r93659;
        return r93660;
}
↓
double f(double a, double b, double c) {
        double r93661 = b;
        double r93662 = -1.279587145681289e-136;
        bool r93663 = r93661 <= r93662;
        double r93664 = -1.0;
        double r93665 = c;
        double r93666 = r93665 / r93661;
        double r93667 = r93664 * r93666;
        double r93668 = 6.852645386257879e+139;
        bool r93669 = r93661 <= r93668;
        double r93670 = 1.0;
        double r93671 = -r93661;
        double r93672 = r93661 * r93661;
        double r93673 = 4.0;
        double r93674 = a;
        double r93675 = r93674 * r93665;
        double r93676 = r93673 * r93675;
        double r93677 = r93672 - r93676;
        double r93678 = sqrt(r93677);
        double r93679 = r93671 - r93678;
        double r93680 = 2.0;
        double r93681 = r93680 * r93674;
        double r93682 = r93679 / r93681;
        double r93683 = r93670 * r93682;
        double r93684 = 1.0;
        double r93685 = r93661 / r93674;
        double r93686 = r93666 - r93685;
        double r93687 = r93684 * r93686;
        double r93688 = r93670 * r93687;
        double r93689 = r93669 ? r93683 : r93688;
        double r93690 = r93663 ? r93667 : r93689;
        return r93690;
}

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

Original34.5
Target21.1
Herbie10.5
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -1.279587145681289e-136

    1. Initial program 51.1

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

    3. Applied div-inv51.1

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

    4. Taylor expanded around -inf 12.2

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

    if -1.279587145681289e-136 < b < 6.852645386257879e+139

    1. Initial program 11.1

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

    3. Applied *-un-lft-identity11.1

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

    if 6.852645386257879e+139 < b

    1. Initial program 57.7

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

    3. Applied *-un-lft-identity57.7

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

    4. Taylor expanded around inf 2.2

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

    5. Simplified2.2

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

  4. Final simplification10.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.279587145681289 \cdot 10^{-136}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 6.8526453862578789 \cdot 10^{139}:\\ \;\;\;\;1 \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020089 
(FPCore (a b c)
  :name "quadm (p42, negative)"
  :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)))