Average Error: 33.6 → 10.0
Time: 22.8s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.2092322739463293 \cdot 10^{-86}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 2.891777552454845 \cdot 10^{+74}:\\ \;\;\;\;\frac{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, \left(\left(c \cdot a\right) \cdot -4\right)\right)}\right) \cdot \frac{1}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array}\]
\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 -3.2092322739463293 \cdot 10^{-86}:\\
\;\;\;\;-\frac{c}{b}\\

\mathbf{elif}\;b \le 2.891777552454845 \cdot 10^{+74}:\\
\;\;\;\;\frac{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, \left(\left(c \cdot a\right) \cdot -4\right)\right)}\right) \cdot \frac{1}{2}}{a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r2166290 = b;
        double r2166291 = -r2166290;
        double r2166292 = r2166290 * r2166290;
        double r2166293 = 4.0;
        double r2166294 = a;
        double r2166295 = c;
        double r2166296 = r2166294 * r2166295;
        double r2166297 = r2166293 * r2166296;
        double r2166298 = r2166292 - r2166297;
        double r2166299 = sqrt(r2166298);
        double r2166300 = r2166291 - r2166299;
        double r2166301 = 2.0;
        double r2166302 = r2166301 * r2166294;
        double r2166303 = r2166300 / r2166302;
        return r2166303;
}
↓
double f(double a, double b, double c) {
        double r2166304 = b;
        double r2166305 = -3.2092322739463293e-86;
        bool r2166306 = r2166304 <= r2166305;
        double r2166307 = c;
        double r2166308 = r2166307 / r2166304;
        double r2166309 = -r2166308;
        double r2166310 = 2.891777552454845e+74;
        bool r2166311 = r2166304 <= r2166310;
        double r2166312 = -r2166304;
        double r2166313 = a;
        double r2166314 = r2166307 * r2166313;
        double r2166315 = -4.0;
        double r2166316 = r2166314 * r2166315;
        double r2166317 = fma(r2166304, r2166304, r2166316);
        double r2166318 = sqrt(r2166317);
        double r2166319 = r2166312 - r2166318;
        double r2166320 = 0.5;
        double r2166321 = r2166319 * r2166320;
        double r2166322 = r2166321 / r2166313;
        double r2166323 = r2166312 / r2166313;
        double r2166324 = r2166311 ? r2166322 : r2166323;
        double r2166325 = r2166306 ? r2166309 : r2166324;
        return r2166325;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.6
Target20.7
Herbie10.0
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -3.2092322739463293e-86

    1. Initial program 52.3

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

    3. Applied div-inv52.3

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

    4. Simplified52.3

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

    6. Applied associate-*r/52.3

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

    7. Simplified52.3

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

    8. Taylor expanded around -inf 9.4

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

    9. Simplified9.4

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

    if -3.2092322739463293e-86 < b < 2.891777552454845e+74

    1. Initial program 13.1

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

    3. Applied div-inv13.2

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

    4. Simplified13.2

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

    6. Applied associate-*r/13.1

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

    7. Simplified13.1

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

    if 2.891777552454845e+74 < 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. Using strategy rm

    3. Applied *-un-lft-identity38.9

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

    4. Applied *-un-lft-identity38.9

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

    5. Applied distribute-rgt-neg-in38.9

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

    6. Applied distribute-lft-out--38.9

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

    7. Applied associate-/l*39.0

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

    8. Simplified39.0

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

    9. Taylor expanded around 0 4.4

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

    10. Simplified4.4

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

  4. Final simplification10.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.2092322739463293 \cdot 10^{-86}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 2.891777552454845 \cdot 10^{+74}:\\ \;\;\;\;\frac{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, \left(\left(c \cdot a\right) \cdot -4\right)\right)}\right) \cdot \frac{1}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019130 +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)))