Average Error: 34.9 → 10.1
Time: 14.8s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.680329042988888396603264581948851078331 \cdot 10^{148}:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\ \mathbf{elif}\;b \le 4.612990823111230552052602417245542305295 \cdot 10^{-104}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, \left(-c\right) \cdot \left(4 \cdot a\right)\right)} - b\right) \cdot \frac{1}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -1\\ \end{array}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \le -3.680329042988888396603264581948851078331 \cdot 10^{148}:\\
\;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\

\mathbf{elif}\;b \le 4.612990823111230552052602417245542305295 \cdot 10^{-104}:\\
\;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, \left(-c\right) \cdot \left(4 \cdot a\right)\right)} - b\right) \cdot \frac{1}{a \cdot 2}\\

\mathbf{else}:\\
\;\;\;\;\frac{c}{b} \cdot -1\\

\end{array}
double f(double a, double b, double c) {
        double r5616951 = b;
        double r5616952 = -r5616951;
        double r5616953 = r5616951 * r5616951;
        double r5616954 = 4.0;
        double r5616955 = a;
        double r5616956 = r5616954 * r5616955;
        double r5616957 = c;
        double r5616958 = r5616956 * r5616957;
        double r5616959 = r5616953 - r5616958;
        double r5616960 = sqrt(r5616959);
        double r5616961 = r5616952 + r5616960;
        double r5616962 = 2.0;
        double r5616963 = r5616962 * r5616955;
        double r5616964 = r5616961 / r5616963;
        return r5616964;
}
↓
double f(double a, double b, double c) {
        double r5616965 = b;
        double r5616966 = -3.6803290429888884e+148;
        bool r5616967 = r5616965 <= r5616966;
        double r5616968 = c;
        double r5616969 = r5616968 / r5616965;
        double r5616970 = a;
        double r5616971 = r5616965 / r5616970;
        double r5616972 = r5616969 - r5616971;
        double r5616973 = 1.0;
        double r5616974 = r5616972 * r5616973;
        double r5616975 = 4.6129908231112306e-104;
        bool r5616976 = r5616965 <= r5616975;
        double r5616977 = -r5616968;
        double r5616978 = 4.0;
        double r5616979 = r5616978 * r5616970;
        double r5616980 = r5616977 * r5616979;
        double r5616981 = fma(r5616965, r5616965, r5616980);
        double r5616982 = sqrt(r5616981);
        double r5616983 = r5616982 - r5616965;
        double r5616984 = 1.0;
        double r5616985 = 2.0;
        double r5616986 = r5616970 * r5616985;
        double r5616987 = r5616984 / r5616986;
        double r5616988 = r5616983 * r5616987;
        double r5616989 = -1.0;
        double r5616990 = r5616969 * r5616989;
        double r5616991 = r5616976 ? r5616988 : r5616990;
        double r5616992 = r5616967 ? r5616974 : r5616991;
        return r5616992;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original34.9
Target21.3
Herbie10.1
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -3.6803290429888884e+148

    1. Initial program 62.1

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

    2. Taylor expanded around -inf 2.3

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

    3. Simplified2.3

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

    if -3.6803290429888884e+148 < b < 4.6129908231112306e-104

    1. Initial program 12.2

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

    3. Applied clear-num12.3

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

    4. Simplified12.3

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

    6. Applied fma-neg12.3

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

    8. Applied associate-/r/12.3

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

    if 4.6129908231112306e-104 < b

    1. Initial program 52.7

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

    2. Taylor expanded around inf 9.8

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.680329042988888396603264581948851078331 \cdot 10^{148}:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\ \mathbf{elif}\;b \le 4.612990823111230552052602417245542305295 \cdot 10^{-104}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, \left(-c\right) \cdot \left(4 \cdot a\right)\right)} - b\right) \cdot \frac{1}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -1\\ \end{array}\]

Reproduce

herbie shell --seed 2019170 +o rules:numerics
(FPCore (a b c)
  :name "The quadratic formula (r1)"

  :herbie-target
  (if (< b 0.0) (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) (/ c (* a (/ (- (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))))

  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))