Average Error: 34.7 → 10.5
Time: 5.2s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -6.77190793437936228 \cdot 10^{-77}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -1.466065355378786 \cdot 10^{-87}:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(b, b, -\left(b \cdot b - 4 \cdot \left(a \cdot c\right)\right)\right)}{a}}{2}}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\ \mathbf{elif}\;b \le -6.27327853977469935 \cdot 10^{-139}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 1.57632464397167146 \cdot 10^{69}:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\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 -6.77190793437936228 \cdot 10^{-77}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le -1.466065355378786 \cdot 10^{-87}:\\
\;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(b, b, -\left(b \cdot b - 4 \cdot \left(a \cdot c\right)\right)\right)}{a}}{2}}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\

\mathbf{elif}\;b \le -6.27327853977469935 \cdot 10^{-139}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 1.57632464397167146 \cdot 10^{69}:\\
\;\;\;\;\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 r112577 = b;
        double r112578 = -r112577;
        double r112579 = r112577 * r112577;
        double r112580 = 4.0;
        double r112581 = a;
        double r112582 = c;
        double r112583 = r112581 * r112582;
        double r112584 = r112580 * r112583;
        double r112585 = r112579 - r112584;
        double r112586 = sqrt(r112585);
        double r112587 = r112578 - r112586;
        double r112588 = 2.0;
        double r112589 = r112588 * r112581;
        double r112590 = r112587 / r112589;
        return r112590;
}
↓
double f(double a, double b, double c) {
        double r112591 = b;
        double r112592 = -6.771907934379362e-77;
        bool r112593 = r112591 <= r112592;
        double r112594 = -1.0;
        double r112595 = c;
        double r112596 = r112595 / r112591;
        double r112597 = r112594 * r112596;
        double r112598 = -1.466065355378786e-87;
        bool r112599 = r112591 <= r112598;
        double r112600 = r112591 * r112591;
        double r112601 = 4.0;
        double r112602 = a;
        double r112603 = r112602 * r112595;
        double r112604 = r112601 * r112603;
        double r112605 = r112600 - r112604;
        double r112606 = -r112605;
        double r112607 = fma(r112591, r112591, r112606);
        double r112608 = r112607 / r112602;
        double r112609 = 2.0;
        double r112610 = r112608 / r112609;
        double r112611 = -r112591;
        double r112612 = sqrt(r112605);
        double r112613 = r112611 + r112612;
        double r112614 = r112610 / r112613;
        double r112615 = -6.273278539774699e-139;
        bool r112616 = r112591 <= r112615;
        double r112617 = 1.5763246439716715e+69;
        bool r112618 = r112591 <= r112617;
        double r112619 = r112611 - r112612;
        double r112620 = 1.0;
        double r112621 = r112609 * r112602;
        double r112622 = r112620 / r112621;
        double r112623 = r112619 * r112622;
        double r112624 = 1.0;
        double r112625 = r112591 / r112602;
        double r112626 = r112596 - r112625;
        double r112627 = r112624 * r112626;
        double r112628 = r112618 ? r112623 : r112627;
        double r112629 = r112616 ? r112597 : r112628;
        double r112630 = r112599 ? r112614 : r112629;
        double r112631 = r112593 ? r112597 : r112630;
        return r112631;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original34.7
Target21.3
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 4 regimes

  2. if b < -6.771907934379362e-77 or -1.466065355378786e-87 < b < -6.273278539774699e-139

    1. Initial program 51.1

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

    2. Taylor expanded around -inf 11.6

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

    if -6.771907934379362e-77 < b < -1.466065355378786e-87

    1. Initial program 28.6

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

    3. Applied div-inv28.6

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

    5. Applied flip--28.6

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

    6. Applied associate-*l/28.7

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

    7. Simplified28.8

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

    if -6.273278539774699e-139 < b < 1.5763246439716715e+69

    1. Initial program 11.6

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

    3. Applied div-inv11.7

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

    if 1.5763246439716715e+69 < b

    1. Initial program 42.5

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

    2. Taylor expanded around inf 4.8

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

    3. Simplified4.8

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

  4. Final simplification10.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -6.77190793437936228 \cdot 10^{-77}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -1.466065355378786 \cdot 10^{-87}:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(b, b, -\left(b \cdot b - 4 \cdot \left(a \cdot c\right)\right)\right)}{a}}{2}}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\ \mathbf{elif}\;b \le -6.27327853977469935 \cdot 10^{-139}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 1.57632464397167146 \cdot 10^{69}:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020020 +o rules:numerics
(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)))