Average Error: 33.2 → 10.7
Time: 21.1s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.2415082771065304 \cdot 10^{-131}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \mathbf{elif}\;b \le 2.559678284282607 \cdot 10^{+69}:\\ \;\;\;\;\frac{\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(a \cdot -4, c, b \cdot b\right)}}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \end{array}\]
\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 -2.2415082771065304 \cdot 10^{-131}:\\
\;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\

\mathbf{elif}\;b \le 2.559678284282607 \cdot 10^{+69}:\\
\;\;\;\;\frac{\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(a \cdot -4, c, b \cdot b\right)}}{a}}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\

\end{array}
double f(double a, double b, double c) {
        double r2085600 = b;
        double r2085601 = -r2085600;
        double r2085602 = r2085600 * r2085600;
        double r2085603 = 4.0;
        double r2085604 = a;
        double r2085605 = c;
        double r2085606 = r2085604 * r2085605;
        double r2085607 = r2085603 * r2085606;
        double r2085608 = r2085602 - r2085607;
        double r2085609 = sqrt(r2085608);
        double r2085610 = r2085601 - r2085609;
        double r2085611 = 2.0;
        double r2085612 = r2085611 * r2085604;
        double r2085613 = r2085610 / r2085612;
        return r2085613;
}
↓
double f(double a, double b, double c) {
        double r2085614 = b;
        double r2085615 = -2.2415082771065304e-131;
        bool r2085616 = r2085614 <= r2085615;
        double r2085617 = -2.0;
        double r2085618 = c;
        double r2085619 = r2085618 / r2085614;
        double r2085620 = r2085617 * r2085619;
        double r2085621 = 2.0;
        double r2085622 = r2085620 / r2085621;
        double r2085623 = 2.559678284282607e+69;
        bool r2085624 = r2085614 <= r2085623;
        double r2085625 = -r2085614;
        double r2085626 = a;
        double r2085627 = -4.0;
        double r2085628 = r2085626 * r2085627;
        double r2085629 = r2085614 * r2085614;
        double r2085630 = fma(r2085628, r2085618, r2085629);
        double r2085631 = sqrt(r2085630);
        double r2085632 = r2085625 - r2085631;
        double r2085633 = r2085632 / r2085626;
        double r2085634 = r2085633 / r2085621;
        double r2085635 = r2085614 / r2085626;
        double r2085636 = r2085619 - r2085635;
        double r2085637 = r2085636 * r2085621;
        double r2085638 = r2085637 / r2085621;
        double r2085639 = r2085624 ? r2085634 : r2085638;
        double r2085640 = r2085616 ? r2085622 : r2085639;
        return r2085640;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.2
Target19.9
Herbie10.7
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -2.2415082771065304e-131

    1. Initial program 49.6

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

    2. Simplified49.7

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

    3. Taylor expanded around -inf 12.4

      \[\leadsto \frac{\color{blue}{-2 \cdot \frac{c}{b}}}{2}\]

    if -2.2415082771065304e-131 < b < 2.559678284282607e+69

    1. Initial program 11.4

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

    2. Simplified11.4

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

    4. Applied div-inv11.5

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

    6. Applied un-div-inv11.4

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

    if 2.559678284282607e+69 < b

    1. Initial program 38.9

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

    2. Simplified38.9

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

    3. Taylor expanded around inf 4.8

      \[\leadsto \frac{\color{blue}{2 \cdot \frac{c}{b} - 2 \cdot \frac{b}{a}}}{2}\]

    4. Simplified4.8

      \[\leadsto \frac{\color{blue}{2 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)}}{2}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.2415082771065304 \cdot 10^{-131}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \mathbf{elif}\;b \le 2.559678284282607 \cdot 10^{+69}:\\ \;\;\;\;\frac{\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(a \cdot -4, c, b \cdot b\right)}}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019152 +o rules:numerics
(FPCore (a b c)
  :name "quadm (p42, negative)"

  :herbie-target
  (if (< b 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)))