Average Error: 33.3 → 10.6
Time: 23.8s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -8.424937854855119 \cdot 10^{-129}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 3.912332224813067 \cdot 10^{+23}:\\ \;\;\;\;\frac{1}{\frac{2}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)}} \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]
\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 -8.424937854855119 \cdot 10^{-129}:\\
\;\;\;\;-\frac{c}{b}\\

\mathbf{elif}\;b \le 3.912332224813067 \cdot 10^{+23}:\\
\;\;\;\;\frac{1}{\frac{2}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)}} \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r1583328 = b;
        double r1583329 = -r1583328;
        double r1583330 = r1583328 * r1583328;
        double r1583331 = 4.0;
        double r1583332 = a;
        double r1583333 = c;
        double r1583334 = r1583332 * r1583333;
        double r1583335 = r1583331 * r1583334;
        double r1583336 = r1583330 - r1583335;
        double r1583337 = sqrt(r1583336);
        double r1583338 = r1583329 - r1583337;
        double r1583339 = 2.0;
        double r1583340 = r1583339 * r1583332;
        double r1583341 = r1583338 / r1583340;
        return r1583341;
}
↓
double f(double a, double b, double c) {
        double r1583342 = b;
        double r1583343 = -8.424937854855119e-129;
        bool r1583344 = r1583342 <= r1583343;
        double r1583345 = c;
        double r1583346 = r1583345 / r1583342;
        double r1583347 = -r1583346;
        double r1583348 = 3.912332224813067e+23;
        bool r1583349 = r1583342 <= r1583348;
        double r1583350 = 1.0;
        double r1583351 = 2.0;
        double r1583352 = -r1583342;
        double r1583353 = -4.0;
        double r1583354 = a;
        double r1583355 = r1583354 * r1583345;
        double r1583356 = r1583353 * r1583355;
        double r1583357 = fma(r1583342, r1583342, r1583356);
        double r1583358 = sqrt(r1583357);
        double r1583359 = r1583352 - r1583358;
        double r1583360 = r1583351 / r1583359;
        double r1583361 = r1583360 * r1583354;
        double r1583362 = r1583350 / r1583361;
        double r1583363 = r1583342 / r1583354;
        double r1583364 = r1583346 - r1583363;
        double r1583365 = r1583349 ? r1583362 : r1583364;
        double r1583366 = r1583344 ? r1583347 : r1583365;
        return r1583366;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.3
Target20.3
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 < -8.424937854855119e-129

    1. Initial program 50.5

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

    2. Taylor expanded around -inf 11.4

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

    3. Simplified11.4

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

    if -8.424937854855119e-129 < b < 3.912332224813067e+23

    1. Initial program 12.3

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

    3. Applied clear-num12.5

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

    4. Simplified12.5

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

    if 3.912332224813067e+23 < b

    1. Initial program 32.6

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

    2. Taylor expanded around inf 6.2

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

  4. Final simplification10.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -8.424937854855119 \cdot 10^{-129}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 3.912332224813067 \cdot 10^{+23}:\\ \;\;\;\;\frac{1}{\frac{2}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)}} \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019155 +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)))