Average Error: 33.5 → 10.6
Time: 3.4m
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -5.691277786452672 \cdot 10^{-38}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 1.8091015183831773 \cdot 10^{+43}:\\ \;\;\;\;\frac{\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(\left(c \cdot a\right), -4, \left(b \cdot b\right)\right)}}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]
\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 -5.691277786452672 \cdot 10^{-38}:\\
\;\;\;\;-\frac{c}{b}\\

\mathbf{elif}\;b \le 1.8091015183831773 \cdot 10^{+43}:\\
\;\;\;\;\frac{\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(\left(c \cdot a\right), -4, \left(b \cdot b\right)\right)}}{2}}{a}\\

\mathbf{else}:\\
\;\;\;\;\frac{c}{b} - \frac{b}{a}\\

\end{array}
double f(double a, double b, double c) {
        double r11113438 = b;
        double r11113439 = -r11113438;
        double r11113440 = r11113438 * r11113438;
        double r11113441 = 4.0;
        double r11113442 = a;
        double r11113443 = c;
        double r11113444 = r11113442 * r11113443;
        double r11113445 = r11113441 * r11113444;
        double r11113446 = r11113440 - r11113445;
        double r11113447 = sqrt(r11113446);
        double r11113448 = r11113439 - r11113447;
        double r11113449 = 2.0;
        double r11113450 = r11113449 * r11113442;
        double r11113451 = r11113448 / r11113450;
        return r11113451;
}
↓
double f(double a, double b, double c) {
        double r11113452 = b;
        double r11113453 = -5.691277786452672e-38;
        bool r11113454 = r11113452 <= r11113453;
        double r11113455 = c;
        double r11113456 = r11113455 / r11113452;
        double r11113457 = -r11113456;
        double r11113458 = 1.8091015183831773e+43;
        bool r11113459 = r11113452 <= r11113458;
        double r11113460 = -r11113452;
        double r11113461 = a;
        double r11113462 = r11113455 * r11113461;
        double r11113463 = -4.0;
        double r11113464 = r11113452 * r11113452;
        double r11113465 = fma(r11113462, r11113463, r11113464);
        double r11113466 = sqrt(r11113465);
        double r11113467 = r11113460 - r11113466;
        double r11113468 = 2.0;
        double r11113469 = r11113467 / r11113468;
        double r11113470 = r11113469 / r11113461;
        double r11113471 = r11113452 / r11113461;
        double r11113472 = r11113456 - r11113471;
        double r11113473 = r11113459 ? r11113470 : r11113472;
        double r11113474 = r11113454 ? r11113457 : r11113473;
        return r11113474;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.5
Target21.1
Herbie10.6
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -5.691277786452672e-38

    1. Initial program 54.0

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

    2. Simplified54.0

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

    3. Taylor expanded around -inf 54.0

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

    4. Simplified54.0

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

    5. Taylor expanded around -inf 7.9

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

    6. Simplified7.9

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

    if -5.691277786452672e-38 < b < 1.8091015183831773e+43

    1. Initial program 15.3

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

    2. Simplified15.3

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

    3. Taylor expanded around -inf 15.3

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

    4. Simplified15.3

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

    if 1.8091015183831773e+43 < b

    1. Initial program 36.3

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

    2. Simplified36.3

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

    3. Taylor expanded around -inf 36.3

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

    4. Simplified36.3

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

    5. Taylor expanded around inf 5.7

      \[\leadsto \color{blue}{\frac{c}{b} - \frac{b}{a}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -5.691277786452672 \cdot 10^{-38}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 1.8091015183831773 \cdot 10^{+43}:\\ \;\;\;\;\frac{\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(\left(c \cdot a\right), -4, \left(b \cdot b\right)\right)}}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019120 +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)))