Average Error: 34.6 → 8.7
Time: 18.4s
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 r64033 = b;
        double r64034 = -r64033;
        double r64035 = r64033 * r64033;
        double r64036 = 4.0;
        double r64037 = a;
        double r64038 = c;
        double r64039 = r64037 * r64038;
        double r64040 = r64036 * r64039;
        double r64041 = r64035 - r64040;
        double r64042 = sqrt(r64041);
        double r64043 = r64034 - r64042;
        double r64044 = 2.0;
        double r64045 = r64044 * r64037;
        double r64046 = r64043 / r64045;
        return r64046;
}
↓
double f(double a, double b, double c) {
        double r64047 = b;
        double r64048 = -1.1509557557359616e+21;
        bool r64049 = r64047 <= r64048;
        double r64050 = -1.0;
        double r64051 = c;
        double r64052 = r64051 / r64047;
        double r64053 = r64050 * r64052;
        double r64054 = -3.1153949179978696e-213;
        bool r64055 = r64047 <= r64054;
        double r64056 = 4.0;
        double r64057 = a;
        double r64058 = r64057 * r64051;
        double r64059 = r64056 * r64058;
        double r64060 = -r64059;
        double r64061 = fma(r64047, r64047, r64060);
        double r64062 = sqrt(r64061);
        double r64063 = r64062 - r64047;
        double r64064 = r64059 / r64063;
        double r64065 = 1.0;
        double r64066 = 2.0;
        double r64067 = r64066 * r64057;
        double r64068 = r64065 / r64067;
        double r64069 = r64064 * r64068;
        double r64070 = 1.974261024048121e+145;
        bool r64071 = r64047 <= r64070;
        double r64072 = -r64047;
        double r64073 = r64047 * r64047;
        double r64074 = r64073 - r64059;
        double r64075 = sqrt(r64074);
        double r64076 = r64072 - r64075;
        double r64077 = r64076 / r64067;
        double r64078 = 1.0;
        double r64079 = r64047 / r64057;
        double r64080 = r64052 - r64079;
        double r64081 = r64078 * r64080;
        double r64082 = r64071 ? r64077 : r64081;
        double r64083 = r64055 ? r64069 : r64082;
        double r64084 = r64049 ? r64053 : r64083;
        return r64084;
}

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 "quadm (p42, negative)"
  :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)))