Average Error: 33.2 → 10.8
Time: 18.7s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.2415082771065304 \cdot 10^{-131}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \mathbf{elif}\;b \le 2.559678284282607 \cdot 10^{+69}:\\ \;\;\;\;\frac{\frac{1}{a} \cdot \left(\left(-b\right) - \sqrt{\mathsf{fma}\left(a \cdot -4, c, b \cdot b\right)}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \end{array}\]
\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{1}{a} \cdot \left(\left(-b\right) - \sqrt{\mathsf{fma}\left(a \cdot -4, c, b \cdot b\right)}\right)}{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 r1157341 = b;
        double r1157342 = -r1157341;
        double r1157343 = r1157341 * r1157341;
        double r1157344 = 4.0;
        double r1157345 = a;
        double r1157346 = c;
        double r1157347 = r1157345 * r1157346;
        double r1157348 = r1157344 * r1157347;
        double r1157349 = r1157343 - r1157348;
        double r1157350 = sqrt(r1157349);
        double r1157351 = r1157342 - r1157350;
        double r1157352 = 2.0;
        double r1157353 = r1157352 * r1157345;
        double r1157354 = r1157351 / r1157353;
        return r1157354;
}
↓
double f(double a, double b, double c) {
        double r1157355 = b;
        double r1157356 = -2.2415082771065304e-131;
        bool r1157357 = r1157355 <= r1157356;
        double r1157358 = -2.0;
        double r1157359 = c;
        double r1157360 = r1157359 / r1157355;
        double r1157361 = r1157358 * r1157360;
        double r1157362 = 2.0;
        double r1157363 = r1157361 / r1157362;
        double r1157364 = 2.559678284282607e+69;
        bool r1157365 = r1157355 <= r1157364;
        double r1157366 = 1.0;
        double r1157367 = a;
        double r1157368 = r1157366 / r1157367;
        double r1157369 = -r1157355;
        double r1157370 = -4.0;
        double r1157371 = r1157367 * r1157370;
        double r1157372 = r1157355 * r1157355;
        double r1157373 = fma(r1157371, r1157359, r1157372);
        double r1157374 = sqrt(r1157373);
        double r1157375 = r1157369 - r1157374;
        double r1157376 = r1157368 * r1157375;
        double r1157377 = r1157376 / r1157362;
        double r1157378 = r1157355 / r1157367;
        double r1157379 = r1157360 - r1157378;
        double r1157380 = r1157379 * r1157362;
        double r1157381 = r1157380 / r1157362;
        double r1157382 = r1157365 ? r1157377 : r1157381;
        double r1157383 = r1157357 ? r1157363 : r1157382;
        return r1157383;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.2
Target19.9
Herbie10.8
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

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

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

    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. Using strategy rm

    4. Applied div-inv39.0

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

    5. Taylor expanded around inf 4.8

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

    6. Simplified4.8

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

  4. Final simplification10.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.2415082771065304 \cdot 10^{-131}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \mathbf{elif}\;b \le 2.559678284282607 \cdot 10^{+69}:\\ \;\;\;\;\frac{\frac{1}{a} \cdot \left(\left(-b\right) - \sqrt{\mathsf{fma}\left(a \cdot -4, c, b \cdot b\right)}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \end{array}\]

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)))