Average Error: 34.6 → 8.7
Time: 18.2s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1150955755735961567232:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -3.11539491799786956147131222652382589094 \cdot 10^{-213}:\\ \;\;\;\;\frac{4 \cdot \left(a \cdot c\right)}{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)} - b} \cdot \frac{1}{2 \cdot a}\\ \mathbf{elif}\;b \le 1.974261024048120880950549217298529943371 \cdot 10^{145}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\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 -1150955755735961567232:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le -3.11539491799786956147131222652382589094 \cdot 10^{-213}:\\
\;\;\;\;\frac{4 \cdot \left(a \cdot c\right)}{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)} - b} \cdot \frac{1}{2 \cdot a}\\

\mathbf{elif}\;b \le 1.974261024048120880950549217298529943371 \cdot 10^{145}:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\end{array}
double f(double a, double b, double c) {
        double r86980 = b;
        double r86981 = -r86980;
        double r86982 = r86980 * r86980;
        double r86983 = 4.0;
        double r86984 = a;
        double r86985 = c;
        double r86986 = r86984 * r86985;
        double r86987 = r86983 * r86986;
        double r86988 = r86982 - r86987;
        double r86989 = sqrt(r86988);
        double r86990 = r86981 - r86989;
        double r86991 = 2.0;
        double r86992 = r86991 * r86984;
        double r86993 = r86990 / r86992;
        return r86993;
}
↓
double f(double a, double b, double c) {
        double r86994 = b;
        double r86995 = -1.1509557557359616e+21;
        bool r86996 = r86994 <= r86995;
        double r86997 = -1.0;
        double r86998 = c;
        double r86999 = r86998 / r86994;
        double r87000 = r86997 * r86999;
        double r87001 = -3.1153949179978696e-213;
        bool r87002 = r86994 <= r87001;
        double r87003 = 4.0;
        double r87004 = a;
        double r87005 = r87004 * r86998;
        double r87006 = r87003 * r87005;
        double r87007 = -r87006;
        double r87008 = fma(r86994, r86994, r87007);
        double r87009 = sqrt(r87008);
        double r87010 = r87009 - r86994;
        double r87011 = r87006 / r87010;
        double r87012 = 1.0;
        double r87013 = 2.0;
        double r87014 = r87013 * r87004;
        double r87015 = r87012 / r87014;
        double r87016 = r87011 * r87015;
        double r87017 = 1.974261024048121e+145;
        bool r87018 = r86994 <= r87017;
        double r87019 = -r86994;
        double r87020 = r86994 * r86994;
        double r87021 = r87020 - r87006;
        double r87022 = sqrt(r87021);
        double r87023 = r87019 - r87022;
        double r87024 = r87023 / r87014;
        double r87025 = 1.0;
        double r87026 = r86994 / r87004;
        double r87027 = r86999 - r87026;
        double r87028 = r87025 * r87027;
        double r87029 = r87018 ? r87024 : r87028;
        double r87030 = r87002 ? r87016 : r87029;
        double r87031 = r86996 ? r87000 : r87030;
        return r87031;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original34.6
Target20.9
Herbie8.7
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -1.1509557557359616e+21

    1. Initial program 56.3

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

    2. Taylor expanded around -inf 4.5

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

    if -1.1509557557359616e+21 < b < -3.1153949179978696e-213

    1. Initial program 31.5

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

    3. Applied flip--31.5

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

    4. Simplified17.7

      \[\leadsto \frac{\frac{\color{blue}{0 + \left(a \cdot c\right) \cdot 4}}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\]

    5. Simplified17.7

      \[\leadsto \frac{\frac{0 + \left(a \cdot c\right) \cdot 4}{\color{blue}{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)} - b}}}{2 \cdot a}\]
    6. Using strategy rm

    7. Applied div-inv17.8

      \[\leadsto \color{blue}{\frac{0 + \left(a \cdot c\right) \cdot 4}{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)} - b} \cdot \frac{1}{2 \cdot a}}\]

    if -3.1153949179978696e-213 < b < 1.974261024048121e+145

    1. Initial program 9.9

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

    if 1.974261024048121e+145 < b

    1. Initial program 60.1

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

    2. Taylor expanded around inf 2.3

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

    3. Simplified2.3

      \[\leadsto \color{blue}{1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification8.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1150955755735961567232:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -3.11539491799786956147131222652382589094 \cdot 10^{-213}:\\ \;\;\;\;\frac{4 \cdot \left(a \cdot c\right)}{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)} - b} \cdot \frac{1}{2 \cdot a}\\ \mathbf{elif}\;b \le 1.974261024048120880950549217298529943371 \cdot 10^{145}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 +o rules:numerics
(FPCore (a b c)
  :name "The quadratic formula (r2)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ 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)))